qu.1.topic=_Syntax_@

qu.1.1.question=<p>Enter&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>5.48</mn></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>12</mn></mrow></msup></mrow></mstyle></math>&nbsp;to&nbsp;2 significant figures.</p>
<ul>
    <li>Will give part marks for small numerical mistakes.&nbsp;</li>
</ul>@
qu.1.1.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,nSF=2,uUN,dUN=3,nUN=2,mUN=0.80)@
qu.1.1.allow2d=0@
qu.1.1.maple_answer=5.48e-12@
qu.1.1.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.1.1.type=maple@
qu.1.1.mode=Maple@
qu.1.1.name=Syntax - Numeric Scientific Notation - 2@
qu.1.1.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,nSF=2,uUN,dUN=3,nUN=2,mUN=0.80,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>
<p>&nbsp;</p>
<p>There are many ways to enter the correct answer.&nbsp; Try:</p>
<ul>
    <li>5.5e-12</li>
    <li>5.5E-12</li>
    <li>5.5*10^(-12)</li>
    <li>(5.5*10^(-12))</li>
</ul>
<p><em>Do not use:</em></p>
<p>&nbsp;</p>
<ul>
    <li>5.5e(-12)</li>
</ul>@
qu.1.1.editing=useHTML@
qu.1.1.solution=@
qu.1.1.algorithm=@
qu.1.1.uid=d439038b-69a4-434d-87b8-b350e3b47857@
qu.1.1.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Difficulty=Easy;
  Topic=Syntax;
  Features=Partial Grading;
@

qu.1.2.question=<p>Enter&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>1.046</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mn>5</mn></mrow></msup><mi>N</mi></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>&nbsp;to 3 significant figures.</p>
<ul>
    <li>Will give part marks for small numerical mistakes.&nbsp;</li>
    <li>Will give part marks for units.</li>
</ul>@
qu.1.2.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uUN,dUN=3,nUN=2,mUN=0.80)@
qu.1.2.allow2d=0@
qu.1.2.maple_answer=1.05e5*N@
qu.1.2.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.1.2.type=maple@
qu.1.2.mode=Maple@
qu.1.2.name=Syntax - Numeric Scientific Notation with Units@
qu.1.2.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uUN,dUN=3,nUN=2,mUN=0.80,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>
<p>&nbsp;</p>
<p>There are a number of ways to enter the correct answer.&nbsp; Try:</p>
<ul>
    <li>1.05*10^5*N</li>
    <li>1.05e5*N</li>
    <li>105*kN</li>
    <li>(1.05*10^(5))*N</li>
</ul>@
qu.1.2.editing=useHTML@
qu.1.2.solution=@
qu.1.2.algorithm=@
qu.1.2.uid=f01b3e93-f8ea-41cf-981d-39650f6cdcc9@
qu.1.2.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Difficulty=Easy;
  Features=Partial Grading;
  Topic=Syntax;
@

qu.1.3.question=<p>Enter&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>5.48</mn></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>12</mn></mrow></msup></mrow></mstyle></math>&nbsp;to 3 significant figures.</p>
<ul>
    <li>Will give part marks for small numerical mistakes.&nbsp;</li>
</ul>@
qu.1.3.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uUN,dUN=3,nUN=2,mUN=0.80)@
qu.1.3.allow2d=0@
qu.1.3.maple_answer=5.48e-12@
qu.1.3.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.1.3.type=maple@
qu.1.3.mode=Maple@
qu.1.3.name=Syntax - Numeric Scientific Notation@
qu.1.3.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uUN,dUN=3,nUN=2,mUN=0.80,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>
<p>&nbsp;</p>
<p>There are many ways to enter the correct answer. Try:</p>
<ul>
    <li>5.48e-12</li>
    <li>5.48E-12</li>
    <li>5.48*10^(-12)</li>
    <li>(5.48*10^(-12))</li>
</ul>
<p><em>Do not use:</em></p>
<p>&nbsp;</p>
<ul>
    <li>5.48e(-12)</li>
</ul>@
qu.1.3.editing=useHTML@
qu.1.3.solution=@
qu.1.3.algorithm=@
qu.1.3.uid=e4c8cb88-67f6-40cd-9281-1ad6b409c0a5@
qu.1.3.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Difficulty=Easy;
  Topic=Syntax;
  Features=Partial Grading;
@

qu.1.4.question=<p>Enter the answer <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mn>1.00</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mn>5</mn></mrow></msup><mi>m</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>2.00</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>m</mi><mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo></mrow><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.</p>
<ul>
    <li>It will give part marks if you, for example, mix up the&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math> components.</li>
    <li>You will receive marks for a correct component, even if the other components are incorrect.</li>
    <li>You will receive marks if you make small numerical errors, or unit errors.</li>
</ul>
<p>&nbsp;</p>
<p><em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>ihat</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>jhat</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>khat</mi></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.</p>@
qu.1.4.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uUN,dUN=3,nUN=2,mUN=0.80,uCT)@
qu.1.4.allow2d=0@
qu.1.4.maple_answer=1.00e5*m*ihat+2.00*m*khat@
qu.1.4.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.1.4.type=maple@
qu.1.4.mode=Maple@
qu.1.4.name=Syntax - Vector Numeric@
qu.1.4.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uUN,dUN=3,nUN=2,mUN=0.80,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>
<p align="left">&nbsp;</p>
<p align="left">There are many ways to enter the answer.&nbsp; Try:</p>
<ul>
    <li>
    <div align="left">1.00e5*m*ihat+2.00*m*khat</div>
    </li>
    <li>
    <div align="left">1.00*10^5*m*ihat+2.00*m*khat</div>
    </li>
    <li>
    <div align="left">2.00*m*khat+(1.00*10^5*m)*ihat</div>
    </li>
    <li>
    <div align="left">200*cm*khat+(1.00*10^2*km)*ihat</div>
    </li>
</ul>@
qu.1.4.editing=useHTML@
qu.1.4.solution=@
qu.1.4.algorithm=@
qu.1.4.uid=42fc9a05-1201-445d-b6b1-da2dd112c67c@
qu.1.4.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Difficulty=Easy;
  Topic=Syntax;
  Features=Partial Grading;
@

qu.1.5.question=<p>Enter the answer&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>451</mn></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>&nbsp;to 3 significant figures.</p>
<ul>
    <li>Will give part marks for small numerical mistakes.&nbsp;</li>
</ul>@
qu.1.5.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uUN,dUN=3,nUN=2,mUN=0.80)@
qu.1.5.allow2d=0@
qu.1.5.maple_answer=451@
qu.1.5.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.1.5.type=maple@
qu.1.5.mode=Maple@
qu.1.5.name=Syntax - Numeric@
qu.1.5.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uUN,dUN=3,nUN=2,mUN=0.80,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.1.5.editing=useHTML@
qu.1.5.solution=@
qu.1.5.algorithm=@
qu.1.5.uid=7d2b4628-65de-48f9-a9ee-a1559f5fe9c1@
qu.1.5.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Difficulty=Easy;
  Topic=Syntax;
  Features=Partial Grading;
@

qu.1.6.question=<p>Enter&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mi>a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>b</mi></mrow><mrow><mi>c</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>d</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mstyle></math>.</p>
<ul>
    <li>Will give part marks for extra/missing terms.</li>
</ul>@
qu.1.6.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=algebraicScalar)@
qu.1.6.allow2d=0@
qu.1.6.maple_answer=a*b/(c*d^2)@
qu.1.6.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.1.6.type=maple@
qu.1.6.mode=Maple@
qu.1.6.name=Syntax - Algebraic@
qu.1.6.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=algebraicScalar,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>
<p>&nbsp;</p>@
qu.1.6.editing=useHTML@
qu.1.6.solution=@
qu.1.6.algorithm=@
qu.1.6.uid=9186109c-16d1-414b-9bdb-71d41c39d702@
qu.1.6.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Difficulty=Easy;
  Features=Partial Grading;
  Topic=Syntax;
  Features=Algebraic;
@

qu.2.topic=Cross Product@

qu.2.1.question=<p>Given two vectors <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>A</mi><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$a1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$a2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>B</mi><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$b1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$b2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>&nbsp;calculate <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mover><mi>A</mi><mi>&rarr;</mi></mover></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><mrow><mover><mi>B</mi><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p><em>Entry Notes</em>:&nbsp;<br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ihat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>jhat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>&nbsp; and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>khat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp;<br />
Ensure that you explicitly enter the multiplication symbol * between terms.</p>@
qu.2.1.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT=false,uVS=false,uSF=false,uDM=false,uUN=false,uEM);@
qu.2.1.allow2d=0@
qu.2.1.maple_answer=with(Physics[Vectors]);
A:=($a1)*_i+($a2)*_j;
B:=($b1)*_i+($b2)*_j;
subs(_i=ihat,_j=jhat,_k=khat,A &x B);@
qu.2.1.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.2.1.type=maple@
qu.2.1.mode=Maple@
qu.2.1.name=Cross Product - 2D@
qu.2.1.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT=false,uVS=false,uSF=false,uDM=false,uUN=false,uEM,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.2.1.editing=useHTML@
qu.2.1.hint.1=Set up a determinant.&nbsp; The first row should be <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp; The second row should be the components of the first vector and the third row should be the components of the second vector.&nbsp; Evaluate the determinant.@
qu.2.1.solution=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mfenced open='[' close=']' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mover><mi>i</mi><mi>&#x005e;</mi></mover></mrow></mtd><mtd><mrow><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow></mtd><mtd><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mtd></mtr><mtr><mtd><mrow><mi>$a1</mi></mrow></mtd><mtd><mrow><mi>$a2</mi></mrow></mtd><mtd><mrow><mn>0</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>$b1</mi></mrow></mtd><mtd><mrow><mi>$b2</mi></mrow></mtd><mtd><mrow><mn>0</mn></mrow></mtd></mtr></mtable></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mrow></mfenced></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$a2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mn>0</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>0</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$b2</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$a1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mn>0</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>0</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$b1</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$a1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$b2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$a2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$b1</mi></mrow></mfenced></mrow></mstyle></math></p>@
qu.2.1.algorithm=$a1=switch(rint(0,2),rint(1,9),-rint(1,9));
$a2=switch(rint(0,2),rint(1,9),-rint(1,9));
$b1=switch(rint(0,2),rint(1,9),-rint(1,9));
$b2=switch(rint(0,2),rint(1,9),-rint(1,9));@
qu.2.1.uid=366aab2e-9cab-4c2e-93ee-6f5b5035fbf9@
qu.2.1.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Cross Product;
  Difficulty=Easy;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Partial Grading;
@

qu.2.2.question=<p>Calculate $aDisp<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow></mstyle></math>$bDisp.&nbsp;</p>
<p>&nbsp;</p>
<p><em>Entry Notes</em>:&nbsp;<br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ihat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>jhat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>&nbsp; and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>khat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp;<br />
Ensure that you explicitly enter the multiplication symbol * between terms.</p>@
qu.2.2.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF=false,uCT=false,uVS=false,uDM=false,uUN=false,uEM);@
qu.2.2.allow2d=0@
qu.2.2.maple_answer=with(Physics[Vectors]);
A:=($a1)*subs({i=_i,j=_j,k=_k},$adir);
B:=($b1)*subs({i=_i,j=_j,k=_k},$bdir);
subs(_i=ihat,_j=jhat,_k=khat,A &x B);@
qu.2.2.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.2.2.type=maple@
qu.2.2.mode=Maple@
qu.2.2.name=Cross Product - 1D@
qu.2.2.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT=false,uVS=false,uSF=false,uDM=false,uUN=false,giveComments,uEM);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.2.2.editing=useHTML@
qu.2.2.hint.1=Use the right-hand rule to find the direction, then multiply the coefficients.@
qu.2.2.hint.2=Alternatively, set up a determinant.&nbsp; The first row should be <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp; The second row should be the components of the first vector and the third row should be the components of the second vector.&nbsp; Evaluate the determinant.@
qu.2.2.solution=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mfenced open='[' close=']' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mover><mi>i</mi><mi>&#x005e;</mi></mover></mrow></mtd><mtd><mrow><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow></mtd><mtd><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mtd></mtr><mtr><mtd><mrow><mi>$a1</mi></mrow></mtd><mtd><mrow><mi>$a2</mi></mrow></mtd><mtd><mrow><mn>0</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>$b1</mi></mrow></mtd><mtd><mrow><mi>$b2</mi></mrow></mtd><mtd><mrow><mn>0</mn></mrow></mtd></mtr></mtable></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mrow></mfenced></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$a2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mn>0</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>0</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$b2</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$a1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mn>0</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>0</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$b1</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$a1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$b2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$a2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$b1</mi></mrow></mfenced></mrow></mstyle></math></p>@
qu.2.2.algorithm=$a1=switch(rint(0,2),rint(1,9),-rint(1,9));
$b1=switch(rint(0,2),rint(1,9),-rint(1,9));
$adir=switch(rint(0,3),'i','j','k');
$bdir=switch(rint(0,3),'i','j','k');
$aDisp='<math><mrow><mn>$a1</mn><mover><mi>$adir</mi><mo>&Hat;</mo></mover></mrow></math>';
$bDisp='<math><mrow><mn>$b1</mn><mover><mi>$bdir</mi><mo>&Hat;</mo></mover></mrow></math>';@
qu.2.2.uid=abd4b09c-28da-4640-979b-7c8a5a56ce07@
qu.2.2.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Cross Product;
  Difficulty=Easy;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Partial Grading;
@

qu.2.3.question=<p>A charge&nbsp;of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$q</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>C</mi></mrow></mstyle></math> is moving with velocity&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>v</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$v1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$v2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$v3</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mfenced></mrow></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>m</mi></mrow><mrow><mo lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&sol;</mo><mi>s</mi></mrow><mrow><mi></mi></mrow></mstyle></math>&nbsp;through<br />
a magnetic field&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>B</mi><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$b1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$b2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$b3</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mstyle></math>.<br />
&nbsp;<br />
<br />
What&nbsp;is the force acting on the charge due to the magnetic field?</p>
<p>&nbsp;</p>@
qu.2.3.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT=false,uVS=false,uUN);@
qu.2.3.allow2d=0@
qu.2.3.maple_answer=with(Physics[Vectors]);
with(SigFigs);
v:=($v1)*_i+($v2)*_j+($v3)*_k;
B:=($b1)*_i+($b2)*_j+($b3)*_k;
temp:=(($q)*(v &x B));
roundToSigFigs(Component(temp,1),3)*N*ihat+roundToSigFigs(Component(temp,2),3)*N*jhat+roundToSigFigs(Component(temp,3),3)*N*khat@
qu.2.3.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.2.3.type=maple@
qu.2.3.mode=Maple@
qu.2.3.name=Force on Charges in Magnetic Field - 3D@
qu.2.3.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT=false,uVS=false,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.2.3.editing=useHTML@
qu.2.3.hint.1=Set up a determinant.&nbsp; The first row should be <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp; The second row should be the components of the first vector and the third row should be the components of the second vector.&nbsp; Evaluate the determinant.@
qu.2.3.solution=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mover><mi>F</mi><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>q</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mover><mi>v</mi><mi>&rarr;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><mrow><mover><mi>B</mi><mi>&rarr;</mi></mover></mrow></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$q</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>C</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo></mrow><mi></mi></mover></mrow><mrow><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mfenced open='[' close=']' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mover><mi>i</mi><mi>&#x005e;</mi></mover></mrow></mtd><mtd><mrow><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow></mtd><mtd><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mi>$v1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mtd><mtd><mrow><mfrac><mrow><mi>$v2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mtd><mtd><mrow><mfrac><mrow><mi>$v3</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mtd></mtr><mtr><mtd><mrow><mi>$b1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mtd><mtd><mrow><mi>$b2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mtd><mtd><mrow><mi>$b3</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mtd></mtr></mtable></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mrow></mfenced></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$q</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>C</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mfenced open='(' close=')' separators=','><mrow><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$v2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$b3</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$v3</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$b2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$v1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$b3</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$v3</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$b1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$v1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$b2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$v2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$b1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mfenced></mrow></mfenced></mrow></mfenced></mrow></mrow><mrow></mrow></mstyle></math></p>@
qu.2.3.algorithm=$v1=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$v2=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$v3=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$b1=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$b2=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$b3=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$q=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));@
qu.2.3.uid=452bd2b4-3866-40e9-835a-d74fd6aada4d@
qu.2.3.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Force on Moving Charge in Magnetic Field;
  Difficulty=Hard;
  Features=Algorithmic;
  Features=Partial Grading;
@

qu.2.4.question=<p>Given two vectors&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>A</mi><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$b1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$b2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow><mrow><mi></mi></mrow></mstyle></math>&nbsp;and&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>B</mi><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$a1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$a2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow></mstyle></math>&nbsp;calculate <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><mrow><mover><mrow><mi>A</mi></mrow><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p><em>Entry Notes</em>:&nbsp;<br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ihat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>jhat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>&nbsp; and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>khat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp;<br />
Ensure that you explicitly enter the multiplication symbol * between terms.</p>@
qu.2.4.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT=false,uVS=false,uSF=false,uDM=false,uUN=false,uEM);@
qu.2.4.allow2d=0@
qu.2.4.maple_answer=with(Physics[Vectors]);
A:=($b1)*_i+($b2)*_j;
B:=($a1)*_i+($a2)*_j;
subs(_i=ihat,_j=jhat,_k=khat,B &x A);@
qu.2.4.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.2.4.type=maple@
qu.2.4.mode=Maple@
qu.2.4.name=Cross Product - 2D, Order Switch@
qu.2.4.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT=false,uVS=false,uSF=false,uDM=false,uUN=false,uEM,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.2.4.editing=useHTML@
qu.2.4.hint.1=Set up a determinant.&nbsp; The first row should be <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp; The second row should be the components of the first vector and the third row should be the components of the second vector.&nbsp; Evaluate the determinant.@
qu.2.4.solution=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mfenced open='[' close=']' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mover><mi>i</mi><mi>&#x005e;</mi></mover></mrow></mtd><mtd><mrow><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow></mtd><mtd><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mtd></mtr><mtr><mtd><mrow><mi>$a1</mi></mrow></mtd><mtd><mrow><mi>$a2</mi></mrow></mtd><mtd><mrow><mn>0</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>$b1</mi></mrow></mtd><mtd><mrow><mi>$b2</mi></mrow></mtd><mtd><mrow><mn>0</mn></mrow></mtd></mtr></mtable></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mrow></mfenced></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$a2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mn>0</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>0</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$b2</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$a1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mn>0</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>0</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$b1</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$a1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$b2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$a2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$b1</mi></mrow></mfenced></mrow></mstyle></math></p>@
qu.2.4.algorithm=$a1=switch(rint(0,2),rint(1,9),-rint(1,9));
$a2=switch(rint(0,2),rint(1,9),-rint(1,9));
$b1=switch(rint(0,2),rint(1,9),-rint(1,9));
$b2=switch(rint(0,2),rint(1,9),-rint(1,9));@
qu.2.4.uid=c75926b4-6f2c-4263-ae53-7f642a33ad84@
qu.2.4.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Cross Product;
  Difficulty=Easy;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Partial Grading;
@

qu.2.5.question=<p>A charge of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$q</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>C</mi></mrow></mstyle></math> is moving with velocity&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>v</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$v1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$v2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow></mfenced></mrow></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>m</mi></mrow><mrow><mo lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&sol;</mo><mi>s</mi></mrow></mstyle></math>&nbsp;through a magnetic field&nbsp;<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>B</mi><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$b1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>T</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math>.&nbsp; What is the force acting on the charge due to the magnetic field?</p>
<p>&nbsp;</p>
<p><em>Entry Notes</em>:&nbsp;<br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ihat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>jhat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>&nbsp; and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>khat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp;<br />
Ensure that you explicitly enter the multiplication symbol * between terms.</p>@
qu.2.5.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT=false,uVS=false,uUN);@
qu.2.5.allow2d=0@
qu.2.5.maple_answer=with(Physics[Vectors]);
with(SigFigs);
v:=($v1)*_i+($v2)*_j;
B:=($b1)*_k;
temp:=(($q)*(v &x B));
roundToSigFigs(Component(temp,1),3)*N*ihat+roundToSigFigs(Component(temp,2),3)*N*jhat@
qu.2.5.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.2.5.type=maple@
qu.2.5.mode=Maple@
qu.2.5.name=Force on Charges in Magnetic Field - 2D@
qu.2.5.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT=false,uVS=false,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.2.5.editing=useHTML@
qu.2.5.hint.1=Use the right-hand rule to find the direction, then multiply the coefficients.@
qu.2.5.hint.2=Set up a determinant.&nbsp; The first row should be <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp; The second row should be the components of the first vector and the third row should be the components of the second vector.&nbsp; Evaluate the determinant.@
qu.2.5.solution=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>F</mi><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>q</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mrow><mover><mi>v</mi><mi>&rarr;</mi></mover></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><mrow><mover><mi>B</mi><mi>&rarr;</mi></mover></mrow></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$q</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>C</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo></mrow><mrow><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mfenced open='[' close=']' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mover><mi>i</mi><mi>&#x005e;</mi></mover></mrow></mtd><mtd><mrow><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow></mtd><mtd><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mi>$v1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mtd><mtd><mrow><mfrac><mrow><mi>$v2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mtd><mtd><mrow><mn>0</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mi>$b1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mtd></mtr></mtable></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mrow></mfenced></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$q</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>C</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mfenced open='(' close=')' separators=','><mrow><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$v2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$b1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>0</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mn>0</mn></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$v1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$b1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>0</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mn>0</mn></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$v1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mn>0</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><mi>$v2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mn>0</mn></mrow></mfenced></mrow></mfenced></mrow></mrow></mstyle></math></p>@
qu.2.5.algorithm=$v1=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$v2=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$b1=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$b2=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$q=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));@
qu.2.5.uid=8908f121-1365-449c-bfea-f1a383730a62@
qu.2.5.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Force on Moving Charge in Magnetic Field;
  Difficulty=Medium;
  Features=Algorithmic;
  Features=Partial Grading;
@

qu.2.6.question=<p>Given two vectors&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>A</mi><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$a1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$a2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$a3</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo></mrow><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math>&nbsp;and&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>B</mi><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$b1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$b2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$b3</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo></mrow><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math>&nbsp;calculate <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mover><mi>A</mi><mi>&rarr;</mi></mover></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><mrow><mover><mi>B</mi><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p><em>Entry Notes</em>:&nbsp;<br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ihat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>jhat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>&nbsp; and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>khat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp;<br />
Ensure that you explicitly enter the multiplication symbol * between terms.</p>@
qu.2.6.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT=false,uVS=false,uSF=false,uDM=false,uUN=false,uEM);@
qu.2.6.allow2d=0@
qu.2.6.maple_answer=with(Physics[Vectors]);
A:=($a1)*_i+($a2)*_j+($a3)*_k;
B:=($b1)*_i+($b2)*_j+($b3)*_k;
subs(_i=ihat,_j=jhat,_k=khat,A &x B);@
qu.2.6.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.2.6.type=maple@
qu.2.6.mode=Maple@
qu.2.6.name=Cross Product - 3D@
qu.2.6.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT=false,uVS=false,uSF=false,uDM=false,uUN=false,uEM,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.2.6.editing=useHTML@
qu.2.6.hint.1=Set up a determinant.&nbsp; The first row should be <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp; The second row should be the components of the first vector and the third row should be the components of the second vector.&nbsp; Evaluate the determinant.@
qu.2.6.solution=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mfenced open='[' close=']' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mover><mi>i</mi><mi>&#x005e;</mi></mover></mrow></mtd><mtd><mrow><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow></mtd><mtd><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mtd></mtr><mtr><mtd><mrow><mi>$a1</mi></mrow></mtd><mtd><mrow><mi>$a2</mi></mrow></mtd><mtd><mrow><mi>$a3</mi></mrow></mtd></mtr><mtr><mtd><mrow><mi>$b1</mi></mrow></mtd><mtd><mrow><mi>$b2</mi></mrow></mtd><mtd><mrow><mi>$b3</mi></mrow></mtd></mtr></mtable></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mrow></mfenced></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$a2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$b3</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$a3</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$b2</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$a1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$b3</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$a3</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$b1</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$a1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$b2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$a2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$b1</mi></mrow></mfenced></mrow><mrow><mi></mi></mrow></mstyle></math></p>@
qu.2.6.algorithm=$a1=switch(rint(0,2),rint(1,9),-rint(1,9));
$a2=switch(rint(0,2),rint(1,9),-rint(1,9));
$a3=switch(rint(0,2),rint(1,9),-rint(1,9));
$b1=switch(rint(0,2),rint(1,9),-rint(1,9));
$b2=switch(rint(0,2),rint(1,9),-rint(1,9));
$b3=switch(rint(0,2),rint(1,9),-rint(1,9));@
qu.2.6.uid=9483a7c5-c144-4b23-8ebc-1b89d34abf0b@
qu.2.6.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Cross Product;
  Difficulty=Medium;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Partial Grading;
@

qu.2.7.question=<p>Given two vectors&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>A</mi><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$b1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$b2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$b3</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo></mrow><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math>&nbsp;and&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>B</mi><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$a1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$a2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$a3</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo></mrow><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math>&nbsp;calculate <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><mrow><mover><mrow><mi>A</mi></mrow><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p><em>Entry Notes</em>:&nbsp;<br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ihat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>jhat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>&nbsp; and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>khat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp;<br />
Ensure that you explicitly enter the multiplication symbol * between terms.</p>@
qu.2.7.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT=false,uVS=false,uSF=false,uDM=false,uUN=false,uEM);@
qu.2.7.allow2d=0@
qu.2.7.maple_answer=with(Physics[Vectors]);
A:=($b1)*_i+($b2)*_j+($b3)*_k;
B:=($a1)*_i+($a2)*_j+($a3)*_k;
subs(_i=ihat,_j=jhat,_k=khat,B &x A);@
qu.2.7.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.2.7.type=maple@
qu.2.7.mode=Maple@
qu.2.7.name=Cross Product - 3D, Order Switch@
qu.2.7.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT=false,uVS=false,uSF=false,uDM=false,uUN=false,uEM,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.2.7.editing=useHTML@
qu.2.7.hint.1=Set up a determinant.&nbsp; The first row should be <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp; The second row should be the components of the first vector and the third row should be the components of the second vector.&nbsp; Evaluate the determinant.@
qu.2.7.solution=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mfenced open='[' close=']' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mover><mi>i</mi><mi>&#x005e;</mi></mover></mrow></mtd><mtd><mrow><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow></mtd><mtd><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mtd></mtr><mtr><mtd><mrow><mi>$a1</mi></mrow></mtd><mtd><mrow><mi>$a2</mi></mrow></mtd><mtd><mrow><mi>$a3</mi></mrow></mtd></mtr><mtr><mtd><mrow><mi>$b1</mi></mrow></mtd><mtd><mrow><mi>$b2</mi></mrow></mtd><mtd><mrow><mi>$b3</mi></mrow></mtd></mtr></mtable></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mrow></mfenced></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$a2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$b3</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$a3</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$b2</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$a1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$b3</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$a3</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$b1</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$a1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$b2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$a2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$b1</mi></mrow></mfenced></mrow><mrow><mi></mi></mrow></mstyle></math></p>@
qu.2.7.algorithm=$a1=switch(rint(0,2),rint(1,9),-rint(1,9));
$a2=switch(rint(0,2),rint(1,9),-rint(1,9));
$a3=switch(rint(0,2),rint(1,9),-rint(1,9));
$b1=switch(rint(0,2),rint(1,9),-rint(1,9));
$b2=switch(rint(0,2),rint(1,9),-rint(1,9));
$b3=switch(rint(0,2),rint(1,9),-rint(1,9));@
qu.2.7.uid=80d0202a-8ae6-4316-ab80-f36649829b0e@
qu.2.7.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Cross Product;
  Difficulty=Medium;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Partial Grading;
@

qu.3.topic=Charge in E-Field@

qu.3.1.question=<p>A velocity selector is set up as in the diagram with uniform electric and magnetic fields. If the electric <br />
field is of magnitude&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$E</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$Eex</mi></mrow></msup><mfrac><mi>V</mi><mrow><mi>m</mi></mrow></mfrac></mrow></mstyle></math> and&nbsp;a charged particle of speed <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$v</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$vex</mi></mrow></msup><mfrac><mrow><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mstyle></math>&nbsp;will pass<br />
through undeflected, what is the magnitude and direction of the magnetic field?</p>
<p><img alt="" align="middle" width="600" height="171" src="__BASE_URI__img/MagneticFields/VelocitySelector/DiagramNoB1.png" /></p>
<p>&nbsp;</p>
<p>&nbsp;</p>@
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qu.3.1.allow2d=0@
qu.3.1.maple_answer=SigFigs[roundToSigFigs]($ans,3)*T*zhat@
qu.3.1.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.3.1.type=maple@
qu.3.1.mode=Maple@
qu.3.1.name=Velocity Selector - Find B - 1 ~ PGc@
qu.3.1.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.3.1.editing=useHTML@
qu.3.1.solution=<p>In a velocity selector, the forces due to a magnetic field and an electric field cancel eachother, for one velocity of charged particle:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>qE</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>qvB</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mi>v</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>E</mi><mrow><mi>B</mi></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>&nbsp;We are given the electric field and the velocity, so we can find the magnitude of the magnetic field.</p>
<p>Assume that the charge is positive and determine which direction the electric force will be - the magnetic force must be in the opposite direction.&nbsp; Still assuming the charge is positive, use <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>q</mi><mrow><mover><mrow><mi>v</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math>&nbsp;to determine the direction of the magnetic field.</p>
<p>&nbsp;</p>@
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$Eex=range(5,13);
$v=rand(1.00,9.99,3);
$vex=range(2,7);
$ans=($E*10^($Eex))/($v*10^($vex));@
qu.3.1.uid=8cdeaeb0-9e72-4b28-a95e-fcecdc21df06@
qu.3.1.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Motion of Charged Particles in Electromagnetic Fields;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.3.2.mode=Inline@
qu.3.2.name=Velocity Selector - Find Direction of E - 2@
qu.3.2.comment=@
qu.3.2.editing=useHTML@
qu.3.2.solution=<p>In a velocity selector, the forces due to a magnetic field and an electric field cancel eachother, for one velocity of charged particle:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>qE</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>qvB</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mi>v</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>E</mi><mrow><mi>B</mi></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>Assuming the charge is positive, use <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>q</mi><mrow><mover><mrow><mi>v</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math>&nbsp;to determine the direction of the magnetic force - the electric <em>force</em> must be in the opposite direction.&nbsp; Again, assume that the charge is positive and determine the direction of the electric <em>field.</em></p>
<p>&nbsp;</p>@
qu.3.2.algorithm=@
qu.3.2.uid=f67f93f7-ec2e-479b-9a66-dac7329e4675@
qu.3.2.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Motion of Charged Particles in Electromagnetic Fields;
  Difficulty=Easy;
  Features=Diagram;
@
qu.3.2.weighting=1@
qu.3.2.numbering=alpha@
qu.3.2.part.1.answer.6=- z@
qu.3.2.part.1.answer.5=+ z@
qu.3.2.part.1.answer.4=- y@
qu.3.2.part.1.editing=useHTML@
qu.3.2.part.1.answer.3=+ y@
qu.3.2.part.1.answer.2=- x@
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qu.3.2.part.1.question=(Unset)@
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qu.3.2.part.1.display=menu@
qu.3.2.part.1.name=sro_id_1@
qu.3.2.part.1.display.permute=false@
qu.3.2.question=<p>A velocity selector is set up as in the diagram with uniform electric and magnetic fields.&nbsp; If the magnetic&nbsp;<br />field is directed as shown, in what direction must the electric field point so that a charged particle of a&nbsp;<br />particular velocity can pass through undeflected?</p><p><img alt="" align="middle" width="600" height="178" src="__BASE_URI__img/MagneticFields/VelocitySelector/DiagramNoE2.png" /></p><p>&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.3.3.question=<p>A velocity selector is set up as in the diagram with uniform electric and magnetic fields. If the electric <br />
field is of magnitude&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$E</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$Eex</mi></mrow></msup><mfrac><mi>V</mi><mrow><mi>m</mi></mrow></mfrac></mrow></mstyle></math> and&nbsp;a charged particle of speed <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$v</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$vex</mi></mrow></msup><mfrac><mrow><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mstyle></math>&nbsp;will pass<br />
through undeflected, what is the magnitude and direction of the magnetic field?</p>
<p><img alt="" align="middle" width="600" height="171" src="__BASE_URI__img/MagneticFields/VelocitySelector/DiagramNoB2.png" /></p>
<p>&nbsp;</p>@
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qu.3.3.allow2d=0@
qu.3.3.maple_answer=SigFigs[roundToSigFigs]($ans,3)*T*(-zhat)@
qu.3.3.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.3.3.type=maple@
qu.3.3.mode=Maple@
qu.3.3.name=Velocity Selector - Find B - 2 ~ PGc@
qu.3.3.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.3.3.editing=useHTML@
qu.3.3.solution=<p>In a velocity selector, the forces due to a magnetic field and an electric field cancel eachother, for one velocity of charged particle:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>qE</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>qvB</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mi>v</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>E</mi><mrow><mi>B</mi></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>&nbsp;We are given the electric field and the velocity, so we can find the magnitude of the magnetic field.</p>
<p>Assume that the charge is positive and determine which direction the electric force will be - the magnetic force must be in the opposite direction.&nbsp; Still assuming the charge is positive, use <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>q</mi><mrow><mover><mrow><mi>v</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math>&nbsp;to determine the direction of the magnetic field.</p>
<p>&nbsp;</p>@
qu.3.3.algorithm=$E=rand(1.00,9.99,3);
$Eex=range(5,13);
$v=rand(1.00,9.99,3);
$vex=range(2,7);
$ans=($E*10^($Eex))/($v*10^($vex));@
qu.3.3.uid=1f45e28a-1bbc-47ad-a1f3-e5dfb9933a08@
qu.3.3.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Motion of Charged Particles in Electromagnetic Fields;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.3.4.question=<p>A charge of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$q</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>C</mi></mrow></mstyle></math> is moving with velocity&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>v</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$v1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$v2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>j</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$v3</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mfenced></mrow></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>m</mi></mrow><mrow><mo lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&sol;</mo><mi>s</mi></mrow><mrow><mi></mi></mrow></mstyle></math>&nbsp;through a magnetic&nbsp;<br />
field&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>B</mi><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$b1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mi>i</mi><mi>&#x005e;</mi></mover></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$b2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$b3</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mstyle></math>&nbsp;and an electric field <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$e1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mi>i</mi><mi>&#x005e;</mi></mover></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$e2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$e3</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mi>V</mi><mrow><mi>m</mi></mrow></mfrac></mrow></mfenced></mrow></mstyle></math>.&nbsp;<br />
What is the force acting on the charge due to the magnetic field?</p>
<p>&nbsp;</p>
<p><em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ihat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mi>i</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>jhat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>khat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.</p>@
qu.3.4.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT=false,uVS=false,uUN);@
qu.3.4.allow2d=0@
qu.3.4.maple_answer=with(Physics[Vectors]);
with(SigFigs);
v:=($v1)*_i+($v2)*_j+($v3)*_k;
B:=($b1)*_j+($b2)*_j+($b3)*_k;
E:=($e1)*_i+($e2)*_j+($e3)*_k;
temp:=(($q)*((v &x B)+E));
roundToSigFigs(Component(temp,1),3)*N*ihat+roundToSigFigs(Component(temp,2),3)*N*jhat+roundToSigFigs(Component(temp,3),3)*N*khat@
qu.3.4.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.3.4.type=maple@
qu.3.4.mode=Maple@
qu.3.4.name=Force on Charges in Electro-Magnetic Field - 3D@
qu.3.4.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT=false,uVS=false,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.3.4.editing=useHTML@
qu.3.4.hint.1=Use the right-hand rule to find the direction, then multiply the coefficients.@
qu.3.4.hint.2=Set up a determinant.&nbsp; The first row should be <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp; The second row should be the components of the first vector and the third row should be the components of the second vector.&nbsp; Evaluate the determinant.@
qu.3.4.solution=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>F</mi><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>q</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mover><mi>v</mi><mi>&rarr;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><mover><mi>B</mi><mi>&rarr;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$q</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>C</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo></mrow><mrow><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mfenced open='[' close=']' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mover><mi>i</mi><mi>&#x005e;</mi></mover></mrow></mtd><mtd><mrow><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow></mtd><mtd><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mi>$v1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mtd><mtd><mrow><mfrac><mrow><mi>$v2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mtd><mtd><mrow><mfrac><mrow><mi>$v3</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mtd></mtr><mtr><mtd><mrow><mi>$b1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mtd><mtd><mrow><mi>$b2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mtd><mtd><mrow><mi>$b3</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mtd></mtr></mtable></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mrow></mfenced></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$q</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>C</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$e1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mi>i</mi><mi>&#x005e;</mi></mover></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$e2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$e3</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mi>V</mi><mrow><mi>m</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$q</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>C</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$v2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$b3</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$v3</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$b2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$v1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$b3</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$v3</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$b1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$v1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$b2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$v2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$b1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mfenced></mrow></mfenced></mrow></mfenced></mrow><mrow><mi mathvariant='normal'></mi></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$q</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>C</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$e1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mi>i</mi><mi>&#x005e;</mi></mover></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$e2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$e3</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mi>V</mi><mrow><mi>m</mi></mrow></mfrac></mrow></mfenced></mrow></mstyle></math></p>@
qu.3.4.algorithm=$v1=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$v2=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$v3=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$b1=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$b2=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$b3=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$e1=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$e2=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$e3=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$q=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));@
qu.3.4.uid=30d65bf1-2b7d-4402-8723-63b2ff2c71b6@
qu.3.4.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Force on Moving Charge in Electro-Magnetic Field;
  Difficulty=Hard;
  Features=Algorithmic;
  Features=Partial Grading;
@

qu.3.5.question=<p>A velocity selector is set up as in the diagram with uniform electric and magnetic fields. If the electric <br />
field is of magnitude&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$E</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$Eex</mi></mrow></msup><mfrac><mi>V</mi><mrow><mi>m</mi></mrow></mfrac></mrow></mstyle></math> and the magnetic field is of magnitude <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$B</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$Bex</mi></mrow></msup><mi>T</mi></mrow></mstyle></math>, what <br />
speed of charged particle will pass through undeflected?</p>
<p><img alt="" align="middle" width="600" height="171" src="__BASE_URI__img/MagneticFields/VelocitySelector/Diagram2.png" /></p>
<p>&nbsp;</p>@
qu.3.5.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.3.5.allow2d=0@
qu.3.5.maple_answer=SigFigs[roundToSigFigs]($ans,3)*m/s@
qu.3.5.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.3.5.type=maple@
qu.3.5.mode=Maple@
qu.3.5.name=Velocity Selector - Find v - 2 ~ PGc@
qu.3.5.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.3.5.editing=useHTML@
qu.3.5.solution=<p>In a velocity selector, the forces due to a magnetic field and an electric field cancel eachother, for one velocity of charged particle:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>qE</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>qvB</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mi>v</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>E</mi><mrow><mi>B</mi></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>We are&nbsp;given the electric and magnetic fields.&nbsp;</p>@
qu.3.5.algorithm=$E=rand(1.00,9.99,3);
$Eex=range(5,13);
$v=range(10000,299790000);
$B=rand(1.00,9.99,3);
$Bex=int(log($v))-1;
$ans=($E*10^($Eex))/($B*10^($Bex));@
qu.3.5.uid=1d87bdc8-4a24-4df1-908d-977e61e451f4@
qu.3.5.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Motion of Charged Particles in Electromagnetic Fields;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.3.6.question=<p>A charged particle of mass&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$m</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$mex</mi></mrow></msup><mi>kg</mi></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>&nbsp;is suspended motionless in the air by an electric field.<br />
If the charge is&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$signOp$q</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$qex</mi></mrow></msup><mi>C</mi></mrow><mrow><mi></mi></mrow></mstyle></math>,&nbsp;what is the magnitude and sign of the&nbsp;electric field (take&nbsp;<br />
upwards to be positive and downwards negative)?</p>
<p>&nbsp;</p>@
qu.3.6.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.3.6.allow2d=0@
qu.3.6.maple_answer=SigFigs[roundToSigFigs]($ans,3)*N/C@
qu.3.6.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.3.6.type=maple@
qu.3.6.mode=Maple@
qu.3.6.name=Charge vs Gravity - Find Electric Field ~ PG(nc)@
qu.3.6.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.3.6.editing=useHTML@
qu.3.6.solution=<p>If a charged particle is held motionless by gravity and an electric field, then:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mg</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>qE</mi></mrow></mstyle></math>.</p>
<p>&nbsp;We are given <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>m</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>q</mi></mrow></mstyle></math>.&nbsp; The force of gravity will be downwards, so the direction of the electric field must combine with the sign of the charge to result in an electric&nbsp;force upwards.</p>@
qu.3.6.algorithm=$m=rand(1.0,9.8,3);
$mex=range(20,40);
$q=rand(1.0,9.8,3);
$qex=range(10,19);
$idx=range(0,1);
$signOp=switch($idx,"+","-");
$corDir=switch($idx,+1,-1);
$ans=$corDir*($m*10^(-$mex))*9.80/($q*10^(-$qex));@
qu.3.6.uid=9c4cc1ca-8fb0-4c7a-ba03-8d2e2f477886@
qu.3.6.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Forces on Charged Particles in an Electric Field;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
@

qu.3.7.question=<p>Find an algebraic expression for the electric force due to the two points charges on the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi></mrow></mstyle></math>-axis on&nbsp;a point charge <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$q3</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>q</mi></mrow></mstyle></math>&nbsp;at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math>, in terms of the given parameters.</p>
<p>&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="395" height="400">
<param name="image" value="__BASE_URI__img/ElectricFields/E-PtChr-2Chrg-2D-Sym/Diagram.png" />
<param name="size" value="10" />
<param name="label.1.x" value="20" />
<param name="label.1.y" value="105" />
<param name="label.1.text" value="q" />
<param name="label.2.x" value="10" />
<param name="label.2.y" value="105" />
<param name="label.2.text" value="$q1" />
<param name="label.3.x" value="20" />
<param name="label.3.y" value="317" />
<param name="label.3.text" value="q" />
<param name="label.4.x" value="10" />
<param name="label.4.y" value="317" />
<param name="label.4.text" value="$q2" />
<param name="label.5.x" value="50" />
<param name="label.5.y" value="155" />
<param name="label.5.text" value="a" />
<param name="label.6.x" value="40" />
<param name="label.6.y" value="155" />
<param name="label.6.text" value="$a1mag" />
<param name="label.7.x" value="50" />
<param name="label.7.y" value="265" />
<param name="label.7.text" value="a" />
<param name="label.8.x" value="40" />
<param name="label.8.y" value="265" />
<param name="label.8.text" value="$a2mag" />
<param name="label.9.x" value="160" />
<param name="label.9.y" value="200" />
<param name="label.9.text" value="x" />
<param name="label.10.x" value="150" />
<param name="label.10.y" value="200" />
<param name="label.10.text" value="$xmag" /></applet></p>
<p><br />
<em>Entry Notes</em>:&nbsp;&nbsp;<br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ihat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mi>i</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>jhat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>khat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp;&nbsp;<br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
If necessary, use the letter <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi></mrow></mstyle></math> to represent the Coulomb constant <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mi mathvariant='normal'>&pi;</mi><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>eg.&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mrow><mi>&pi;</mi><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mrow></mfrac><mfrac><mrow><msub><mi>q</mi><mrow><mn>1</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></msub><msub><mi>q</mi><mrow><mn>2</mn></mrow></msub></mrow><mrow><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mn>1</mn><mrow><mn>4</mn><mrow><mi>&pi;</mi></mrow><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mfrac><mfrac><mrow><msub><mi>q</mi><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></msub><msub><mi>q</mi><mrow><mn>3</mn></mrow></msub></mrow><mrow><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>&nbsp;could be entered as (k*q1*q2/(r^2))*ihat + (k*q2*q3/(r^2))*khat.</p>@
qu.3.7.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=vectorAlgebraicComponents);@
qu.3.7.allow2d=0@
qu.3.7.maple_answer=subs(_i=ihat,_j=jhat,_k=khat,collect(PhysFuncs[ElectricField](($q1)*q*($q3)*q,$a1mag*a*_j,$xmag*x*_i,constant=k)+PhysFuncs[ElectricField](($q2)*q*($q3)*q,-$a2mag*a*_j,$xmag*x*_i,constant=k),[_i,_j,_k]))@
qu.3.7.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.3.7.type=maple@
qu.3.7.mode=Maple@
qu.3.7.name=Electric Force Due to Point Charges - 2 Charges, 2D, Symmetric@
qu.3.7.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.3.7.editing=useHTML@
qu.3.7.hint.1=Remember that the superposition principle means that the electric force at a point is the sum of the electric forces produced by each of the nearby charges.&nbsp; Pretend that only one charge at a time&nbsp;is present and calculate the force due to that charge at the point.&nbsp; Finally, add all of the electric forces together to get the total.@
qu.3.7.hint.2=Use symmetry.@
qu.3.7.solution=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mi>top</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$q1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>q</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$q3</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>q</mi></mrow></mfenced></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>$a1mag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$xmag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow><mrow><mover><mrow><mi>r</mi></mrow><mi>&#x005e;</mi></mover></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math></p>
<p>The y-components cancel by symmetry.&nbsp; The total is thus just twice&nbsp;the x-component:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>F</mi><mrow><msub><mi>top</mi><mrow><mi>X</mi></mrow></msub></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><msub><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mrow><mi>top</mi></mrow></msub></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$xmag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow><mrow><msqrt><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>$a1mag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$xmag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mi>P</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><msub><mi>F</mi><mrow><msub><mi>top</mi><mrow><mi>X</mi></mrow></msub></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>F</mi><mrow><msub><mi>bottom</mi><mrow><mi>X</mi></mrow></msub></mrow></msub></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>F</mi><mrow><msub><mi>top</mi><mrow><mi>X</mi></mrow></msub></mrow></msub></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mi>i</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math></p>@
qu.3.7.algorithm=$a1mag=rint(2,5);
$a2mag=$a1mag;
$xmag=rint(2,5);
$q1=switch(rint(2),rint(1,10),-rint(1,10));
$q2=$q1;
$q3=switch(rint(2),rint(1,10),-rint(1,10));@
qu.3.7.uid=bd620c4d-9ad3-4944-92ba-064f6ef3243c@
qu.3.7.info=  Course=Introductory Electricity and Magnetism;
  Topic=Electric Force Due to Point Charges in 2D;
  Author=Aron Pasieka;
  Difficulty=Hard;
  Features=Algebraic;
  Features=Algorithmic;
  Features=Partial Grading;
  Features=Diagram;
@

qu.3.8.question=<p>A particle&nbsp;with charge&nbsp;of magnitude&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$chr</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$chrex</mi></mrow></msup><mi>C</mi></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>&nbsp;is suspended motionless in the air by an electric field.<br />
If the electric field is&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$E</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$Eex</mi></mrow></msup><mfrac><mrow><mi>N</mi></mrow><mrow><mi>C</mi></mrow></mfrac></mrow></mstyle></math>&nbsp;$EDir, what is the&nbsp;mass of the particle?</p>
<p>&nbsp;</p>@
qu.3.8.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.3.8.allow2d=0@
qu.3.8.maple_answer=SigFigs[roundToSigFigs]($ans,3)*kg@
qu.3.8.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.3.8.type=maple@
qu.3.8.mode=Maple@
qu.3.8.name=Charge vs Gravity - Find Mass ~ PG(nc)@
qu.3.8.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.3.8.editing=useHTML@
qu.3.8.solution=<p>If a charged particle is held motionless by gravity and an electric field, then:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mg</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>qE</mi></mrow></mstyle></math>.</p>
<p>&nbsp;We are given&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>q</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>E</mi></mrow></mstyle></math>.&nbsp;</p>@
qu.3.8.algorithm=$E=rand(1.0,9.8,3);
$Eex=range(10,20);
$idx=range(0,1);
$EDir=switch($idx,'upwards','downwards');
$corDir=switch($idx,+1,-1);
$chr=rand(1.0,9.8,3);
$chrex=range(10,15);
$ans=($E*10^(-$Eex))*($chr*10^(-$chrex))/9.80;@
qu.3.8.uid=3955b1ad-ed59-4d11-b049-abbbde85d2cd@
qu.3.8.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Forces on Charged Particles in an Electric Field;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
@

qu.3.9.question=<p>After rubbing a balloon against someone's hair, a balloon has accumulated a static electric charge.&nbsp; The<br />
air inside the balloon has the same density as the air outside of the balloon.&nbsp;&nbsp;If there are&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$n</mi></mrow></mstyle></math> free electrons<br />
on the surface of the balloon and&nbsp;an electric field of magnitude&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$E</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$eex</mi></mrow></msup><mfrac><mi>N</mi><mrow><mi>m</mi></mrow></mfrac></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>&nbsp;is required to keep the&nbsp;<br />
balloon floating at a constant height, what is the mass of the balloon?</p>@
qu.3.9.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.3.9.allow2d=0@
qu.3.9.maple_answer=SigFigs[roundToSigFigs]($m,3)*g@
qu.3.9.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.3.9.type=maple@
qu.3.9.mode=Maple@
qu.3.9.name=Charged Balloon vs Gravity - Find Mass ~ PGc@
qu.3.9.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.3.9.editing=useHTML@
qu.3.9.solution=<p>If&nbsp;the charged balloon is held motionless by gravity and an electric field, then:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mg</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>qE</mi></mrow></mstyle></math>.</p>
<p>&nbsp;We are given&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>q</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>n</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>e</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>E</mi></mrow></mstyle></math>.</p>@
qu.3.9.algorithm=$E=rand(0.5,9,3);
$eex=range(10,12);
$n=range(600000,900000);
$m=$n*1000*(1.60*10^(-19)*$E*10^($eex))/(9.80);@
qu.3.9.uid=dc457ee6-4c5a-40f8-a7d0-01301d080339@
qu.3.9.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Charges in an Electric Field;
  Difficulty=Hard;
  Features=Partial Grading;
  Features=Algorithmic;
@

qu.3.10.question=<p>After rubbing a balloon against someone's hair, a balloon has accumulated a static electric charge.&nbsp; The<br />
air inside the balloon has the same density as the air outside of the balloon, and the balloon itself&nbsp;<br />
weighs&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$m</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mN</mi></mrow></mstyle></math>.&nbsp;&nbsp;How many free electrons are there on the surface of the balloon if an electric field of<br />
magnitude&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$E</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$eex</mi></mrow></msup><mfrac><mi>N</mi><mrow><mi>C</mi></mrow></mfrac></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>&nbsp;is required to keep the balloon floating at a constant height?</p>
<p>&nbsp;</p>@
qu.3.10.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.3.10.allow2d=0@
qu.3.10.maple_answer=SigFigs[roundToSigFigs]($n,3)@
qu.3.10.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.3.10.type=maple@
qu.3.10.mode=Maple@
qu.3.10.name=Charged Balloon vs Gravity - Find Free Electrons ~ PGc@
qu.3.10.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.3.10.editing=useHTML@
qu.3.10.solution=<p>If&nbsp;the charged balloon is held motionless by gravity and an electric field, then:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mg</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>qE</mi></mrow></mstyle></math>.</p>
<p>&nbsp;We are given <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>m</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>E</mi></mrow></mstyle></math>&nbsp;which allows us to calculate <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>q</mi></mrow></mstyle></math>.&nbsp; We can then find the number of free electrons by using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>q</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>n</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>e</mi></mrow></mstyle></math>.</p>@
qu.3.10.algorithm=$m=rand(0.2,9.9,3);
$E=rand(0.5,9.9,3);
$eex=range(10,12);
$n=($m/1000)/(1.60*10^(-19)*$E*10^($eex));@
qu.3.10.uid=cb8edb26-bfe9-4872-8b62-e874002032cf@
qu.3.10.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Charges in an Electric Field;
  Difficulty=Hard;
  Features=Partial Grading;
  Features=Algorithmic;
@

qu.3.11.mode=Inline@
qu.3.11.name=Velocity Selector - Find Direction of B - 1@
qu.3.11.comment=@
qu.3.11.editing=useHTML@
qu.3.11.solution=<p>In a velocity selector, the forces due to a magnetic field and an electric field cancel eachother, for one velocity of charged particle:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>qE</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>qvB</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mi>v</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>E</mi><mrow><mi>B</mi></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>Assume that the charge is positive and determine which direction the electric force will be - the magnetic force must be in the opposite direction.&nbsp; Still assuming the charge is positive, use <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>q</mi><mrow><mover><mrow><mi>v</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math>&nbsp;to determine the direction of the magnetic field.</p>
<p>&nbsp;</p>@
qu.3.11.algorithm=@
qu.3.11.uid=21f1dbb0-5e96-4ca7-89a2-69e870d6a207@
qu.3.11.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Motion of Charged Particles in Electromagnetic Fields;
  Difficulty=Easy;
@
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qu.3.11.question=<p>A velocity selector is set up as in the diagram with uniform electric and magnetic fields.&nbsp; If the magnetic&nbsp;<br />field is directed as shown, in what direction must the electric field point so that a charged particle of a&nbsp;<br />particular velocity can pass through undeflected?</p><p><img alt="" align="middle" width="600" height="178" src="__BASE_URI__img/MagneticFields/VelocitySelector/DiagramNoB1.png" /></p><p>&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.3.12.mode=Inline@
qu.3.12.name=Velocity Selector - Find Direction of B - 2@
qu.3.12.comment=@
qu.3.12.editing=useHTML@
qu.3.12.solution=<p>In a velocity selector, the forces due to a magnetic field and an electric field cancel eachother, for one velocity of charged particle:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>qE</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>qvB</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mi>v</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>E</mi><mrow><mi>B</mi></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>Assume that the charge is positive and determine which direction the electric force will be - the magnetic force must be in the opposite direction.&nbsp; Still assuming the charge is positive, use <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>q</mi><mrow><mover><mrow><mi>v</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math>&nbsp;to determine the direction of the magnetic field.</p>
<p>&nbsp;</p>@
qu.3.12.algorithm=@
qu.3.12.uid=702b1a2b-8b58-452b-97fd-062f7ead199a@
qu.3.12.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Motion of Charged Particles in Electromagnetic Fields;
  Difficulty=Easy;
  Features=Diagram;
@
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qu.3.12.question=<p>A velocity selector is set up as in the diagram with uniform electric and magnetic fields.&nbsp; If the magnetic&nbsp;<br />field is directed as shown, in what direction must the electric field point so that a charged particle of a&nbsp;<br />particular velocity can pass through undeflected?</p><p><img alt="" align="middle" width="600" height="178" src="__BASE_URI__img/MagneticFields/VelocitySelector/DiagramNoB2.png" /></p><p>&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.3.13.question=<p align="left">A particle with charge q>0 and mass m is fired from a gun, with an initial speed v in the&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi></mi></mrow><mrow><mover><mrow><mi>x</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math>&nbsp;direction.&nbsp;&nbsp; The particle&nbsp;passes<br />
through a region between two plates where the electric field is (approximately)&nbsp;uniform&nbsp;and equal to E and in the positive&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>y</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math> direction. The particle<br />
then passes through a region with no fields, before finally hitting a&nbsp;screen.</p>
<p>&nbsp;</p>
<p>&nbsp;</p>
<div align="center">
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="600" height="266">
<param name="image" value="__BASE_URI__img/ElectricFields/ChargeGun-FindElectricField/Diagram.png" />
<param name="size" value="3" />
<param name="label.1.x" value="322" />
<param name="label.1.y" value="30" />
<param name="label.1.text" value="x1" />
<param name="label.2.x" value="487" />
<param name="label.2.y" value="30" />
<param name="label.2.text" value="x2" />
<param name="label.3.x" value="200" />
<param name="label.3.y" value="130" />
<param name="label.3.text" value="q" /></applet></p>
<p align="left">&nbsp;</p>
<p align="left">Relative to the point P, where does the particle hit the screen?&nbsp; Provide an algebraic answer in terms of&nbsp;<br />
the&nbsp;given parameters.&nbsp; Neglect gravity.&nbsp;</p>
</div>@
qu.3.13.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=algebraicScalar)@
qu.3.13.allow2d=0@
qu.3.13.maple_answer=(q*E*x1/(m*v^2))*((x1/2)+x2)@
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qu.3.13.editing=useHTML@
qu.3.13.solution=<p>Between the plates, the charge will have an electric force acting on it equal to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>qE</mi></mrow></mstyle></math>.&nbsp; The acceleration due to this force will be <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>qE</mi><mrow><mi>m</mi></mrow></mfrac></mrow></mstyle></math>.&nbsp; It will take the particle a time <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>t</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><msub><mi>x</mi><mrow><mn>1</mn></mrow></msub><mrow><mi>v</mi></mrow></mfrac></mrow></mstyle></math> to pass through the plates.&nbsp; As a result, the particle will have travelled a vertical distance of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&Delta;y</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac><mfrac><mi>qE</mi><mrow><mi>m</mi></mrow></mfrac><mfrac><msubsup><mi>x</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><msup><mi>v</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mstyle></math>&nbsp;and have a vertical speed of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>v</mi><mrow><mi>y</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><msub><mi>qEx</mi><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>mv</mi></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>After exiting the plate region, the particle will no longer be under acceleration.&nbsp; Since the horizontal speed has not changed, the time to reach the screen after exiting the plates will be<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>t</mi><mrow><mn>2</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><msub><mi>x</mi><mrow><mn>2</mn></mrow></msub><mrow><mi>v</mi></mrow></mfrac></mrow></mstyle></math>.&nbsp; During this time, it will travel&nbsp;a vertical distance of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&Delta;y</mi><mrow><mn>2</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>x</mi><mrow><mn>2</mn></mrow></msub><msub><mi>qEx</mi><mrow><mn>1</mn></mrow></msub></mrow><mrow><msup><mi>mv</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mstyle></math>.</p>
<p>Therefore, the particle will hit the screen at a distance of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&Delta;y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><msub><mi>qEx</mi><mrow><mn>1</mn></mrow></msub><mrow><msup><mi>mv</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mfenced open='(' close=')' separators=','><mrow><mfrac><msub><mi>x</mi><mrow><mn>1</mn></mrow></msub><mrow><mn>2</mn></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>x</mi><mrow><mn>2</mn></mrow></msub></mrow></mfenced></mrow></mstyle></math>&nbsp;from <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math>.</p>@
qu.3.13.algorithm=@
qu.3.13.uid=01b26182-bc81-4b6f-be62-9a23c886c5da@
qu.3.13.info=  Course=Introductory Electricity and Magnetism;
  Topic=Motion of Charges in Electric Fields;
  Author=Aron Pasieka;
  Difficulty=Hard;
  Features=Partial Grading;
  Features=Diagram;
@

qu.3.14.question=<p>A&nbsp;solid disc&nbsp;of radius&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$aMag$aLet</mi></mrow></mstyle></math>&nbsp;is centered at the origin in the&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mover><mi>z</mi><mi>&#x005e;</mi></mover></mrow></mstyle></math>&nbsp;plane.&nbsp;&nbsp;A&nbsp;charge<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$QSignLabel</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow></mstyle></math>&nbsp;is uniformly distributed on the disc.</p>
<p>Use symmetry to set up a single&nbsp;expression&nbsp;for the&nbsp;electric force&nbsp;due to the disc of charge on a point charge <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$qMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>q</mi></mrow></mstyle></math>&nbsp;at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math>.<br />
Use the fact&nbsp;that one can treat the disc as an infinite number of thin rings, and the fact that the electric force due to a&nbsp;single thin<br />
ring of radius <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>R</mi></mrow></mstyle></math>&nbsp;is:</p>
<p align="center"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi><mfrac><mrow><mi>dQ</mi><mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo></mrow><mi>x</mi></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>R</mi><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mrow><mfrac><mn>3</mn><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></mfrac><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.</p>
<p align="left">Provide the <em>integrand</em> of this integral as your answer, in terms of the given parameters.</p>
<p align="center">e.g. Integral: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathcolor='#800080'>a</mi></mrow><mrow><mi>a</mi></mrow></munderover><mfrac><mi>Q</mi><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow></mstyle></math>, Response: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>Q</mi><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>
<p align="left">In order to find the electric force, one needs to evaluate an integral that adds up the contributions from&nbsp;each infinitesimal ring that<br />
makes up the disc, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dr</mi></mrow></mstyle></math>, from <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>0</mn></mrow></mstyle></math> to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$aMag$aLet</mi></mrow></mstyle></math>.&nbsp;&nbsp;&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="585" height="500">
<param name="image" value="__BASE_URI__img/ElectricFields/DiscCharge/DiagramDetailed.png" />
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<p align="center">&nbsp;</p>
<p><em>Note: </em>Use the letter 'k' for the Coulomb constant, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mrow><mi>&pi;</mi></mrow><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mfrac></mrow></mstyle></math>.</p>@
qu.3.14.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSA2,subPackage=vectorAlgebraicComponents)@
qu.3.14.allow2d=0@
qu.3.14.maple_answer=with(Physics[Vectors]):
subs(_i=ihat,_j=jhat,_k=khat,simplify(PhysFuncs[ElectricField](($qMag)*q*($QSign)*(x*Q/($aMag*$aLet)),r*_j,x*_i,constant=k)._i)*_i);@
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qu.3.14.mode=Maple@
qu.3.14.name=Disc Charge - Find F - Set Up Integral@
qu.3.14.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSA2,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.3.14.editing=useHTML@
qu.3.14.solution=<p>Start by identifying a small ring of charge on the disc at a radius of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dr</mi></mrow></mstyle></math>.&nbsp; Then the small ring will produce&nbsp;an electric force:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$qMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>q</mi></mrow></mfenced><mi>x</mi></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mrow><mfrac><mn>3</mn><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></mfrac><mi>dQ</mi><mover><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math>.</p>
<p>Since the charge is uniformly distributed on the disc, the charge per unit area is constant, so:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mi>dQ</mi></mrow><mrow><mi>dr</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac></mrow></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mi>dQ</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac><mi>d</mi></mrow><mrow><mi mathvariant='normal'>r</mi></mrow></mstyle></math>.</p>
<p>Adding up the contributions from each unit of charge on the ring:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></munderover><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$qMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>q</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>r</mi></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mrow><mfrac><mn>3</mn><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></mfrac></mrow></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mover><mrow><mi>x</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math></p>@
qu.3.14.algorithm=$aMag=rint(2,10);
$aLet=switch(rint(8),'a','b','c','d','f','g','h','w');
$idx=rint(2);
$QSign=switch($idx,1,-1);
$QSignLabel=switch($idx,"+","-");
$QMag=rint(2,6);
$qMag=switch(rint(2),rint(2,6),-rint(2,6));@
qu.3.14.uid=e6c7c55b-7950-4e53-ba8b-9ece7da3470a@
qu.3.14.info=  Course=Introductory Electricity and Magnetism;
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  Topic=Force Due to A Line of Charge;
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  Features=Algorithmic;
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  Features=Diagram;
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@

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If the electric field is&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$E</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$Eex</mi></mrow></msup><mfrac><mrow><mi>N</mi></mrow><mrow><mi>C</mi></mrow></mfrac></mrow></mstyle></math>&nbsp;$EDir, what is the magnitude and sign of the charge of&nbsp;<br />
the particle?</p>
<p>&nbsp;</p>@
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qu.3.15.maple_answer=SigFigs[roundToSigFigs]($ans,3)*C@
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qu.3.15.name=Charge vs Gravity - Find Charge ~ PGc@
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qu.3.15.editing=useHTML@
qu.3.15.solution=<p>If a charged particle is held motionless by gravity and an electric field, then:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mg</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>qE</mi></mrow></mstyle></math>.</p>
<p>&nbsp;We are given <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>m</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>E</mi></mrow></mstyle></math>.&nbsp; The force of gravity will be downwards, so the direction of the electric field must combine with the sign of the charge to result in an electric&nbsp;force upwards.</p>@
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$mex=range(20,40);
$E=rand(1.0,9.8,3);
$Eex=range(15,25);
$idx=range(0,1);
$EDir=switch($idx,'upwards','downwards');
$corDir=switch($idx,+1,-1);
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qu.3.15.uid=7f8f9da0-2462-46a7-b9d5-7f2d03fcfb1b@
qu.3.15.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Forces on Charged Particles in an Electric Field;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
@

qu.3.16.mode=Inline@
qu.3.16.name=Velocity Selector - Find Direction of E - 1@
qu.3.16.comment=@
qu.3.16.editing=useHTML@
qu.3.16.solution=<p>In a velocity selector, the forces due to a magnetic field and an electric field cancel eachother, for one velocity of charged particle:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>qE</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>qvB</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mi>v</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>E</mi><mrow><mi>B</mi></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>Assuming the charge is positive, use <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>q</mi><mrow><mover><mrow><mi>v</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math>&nbsp;to determine the direction of the magnetic force - the electric <em>force</em> must be in the opposite direction.&nbsp; Again, assume that the charge is positive and determine the direction of the electric <em>field.</em></p>
<p>&nbsp;</p>@
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  Topic=Motion of Charged Particles in Electromagnetic Fields;
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  Features=Diagram;
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qu.3.16.question=<p>A velocity selector is set up as in the diagram with uniform electric and magnetic fields.&nbsp; If the magnetic&nbsp;<br />field is directed as shown, in what direction must the electric field point so that a charged particle of a&nbsp;<br />particular velocity can pass through undeflected?</p><p><img alt="" align="middle" width="600" height="178" src="__BASE_URI__img/MagneticFields/VelocitySelector/DiagramNoE1.png" /></p><p>&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.3.17.question=<p>A thin, uniform&nbsp;line of charge extends from&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$aMag$aLet</mi></mrow></mstyle></math>&nbsp;to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$aMag$aLet</mi></mrow></mstyle></math>&nbsp;along the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi></mrow></mstyle></math>-axis&nbsp;and&nbsp;contains a total charge <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$QSignLabel</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow></mstyle></math>.</p>
<p>Set up&nbsp;an integral for the&nbsp;electric force&nbsp;due to the line of charge on a charge <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$qMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>q</mi></mrow></mstyle></math>&nbsp;at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math>.&nbsp; Use symmetry to reduce the integral<br />
to a single expression.&nbsp; Provide the <em>integrand</em> of this integral as your answer, in terms of the given parameters.</p>
<p align="center">e.g. Integral: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathcolor='#800080'>a</mi></mrow><mrow><mi>a</mi></mrow></munderover><mfrac><mi>Q</mi><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow></mstyle></math>, Integrand: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>Q</mi><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>
<p>In order to find the electric force, one needs to evaluate an integral that adds up the contributions from each infinitesimal piece of<br />
the line, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dy</mi></mrow></mstyle></math>, from <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$aMag$aLet</mi></mrow></mstyle></math> to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$aMag$aLet</mi></mrow></mstyle></math>.&nbsp;</p>
<p align="center">&nbsp;</p>
<p align="center">&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="585" height="500">
<param name="image" value="__BASE_URI__img/ElectricFields/LineCharge/Diagram.png" />
<param name="size" value="4" />
<param name="label.1.x" value="70" />
<param name="label.1.y" value="250" />
<param name="label.1.text" value="$QSignLabel Q" />
<param name="label.2.x" value="70" />
<param name="label.2.y" value="175" />
<param name="label.2.text" value="$aMag$aLet" />
<param name="label.3.x" value="70" />
<param name="label.3.y" value="360" />
<param name="label.3.text" value="$aMag$aLet" />
<param name="label.4.x" value="270" />
<param name="label.4.y" value="250" />
<param name="label.4.text" value="x" /></applet></p>
<p align="center">&nbsp;</p>
<p><em>Note: </em>Use the letter 'k' for the Coulomb constant,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mrow><mi>&pi;</mi></mrow><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mfrac></mrow></mstyle></math>.</p>@
qu.3.17.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSA2,subPackage=vectorAlgebraicComponents)@
qu.3.17.allow2d=0@
qu.3.17.maple_answer=with(Physics[Vectors]):
subs(_i=ihat,_j=jhat,_k=khat,collect((PhysFuncs[ElectricField](($qMag)*q*($QSign)*(Q/(2*$aMag*$aLet)),y*_j,x*_i,constant=k)._i)*_i,[_i,_j,_k]))@
qu.3.17.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.3.17.type=maple@
qu.3.17.mode=Maple@
qu.3.17.name=Line Charge - Find F - Set Up Integral@
qu.3.17.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSA2,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.3.17.editing=useHTML@
qu.3.17.solution=<p>Start by identifying a small unit of charge on the line, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dq</mi></mrow></mstyle></math>.&nbsp; Then the small amount of electric force due to that unit of charge is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$qMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>q</mi></mrow></mfenced><mi>dq</mi></mrow><mrow><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mrow><mover><mi>r</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msqrt><mrow><msup><mi>y</mi><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>Since the charge is uniformly distributed on the line, the charge per unit length is constant, so:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>dq</mi><mrow><mi>dy</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac></mrow></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mi>dq</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac><mi>dy</mi></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>.</p>
<p>Finally, breaking into components:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>r</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mover><mrow><mi>r</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mi>r</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mrow><mi>x</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>y</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mrow><mi>y</mi></mrow><mi>&#x005e;</mi></mover></mrow><mrow><mi>r</mi></mrow></mfrac></mrow></mstyle></math></p>
<p>Recognizing that the y-components for units of charge above and below the x-axis will cancel, we can conclude that the y-component of the total force is zero.</p>
<p>Thus, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>dF</mi><mrow><mi>x</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$qMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>q</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mn>2</mn><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mi>dy</mi><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mrow><mfrac><mn>3</mn><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></mfrac></mrow></mfenced></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>.</p>
<p>Adding up the contributions from each unit of charge on the line:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$aMag$aLet</mi></mrow><mrow><mi>$aMag$aLet</mi></mrow></munderover><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$qMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>q</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mn>2</mn><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mrow><mfrac><mn>3</mn><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></mfrac></mrow></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mover><mrow><mi>x</mi></mrow><mi>&#x005e;</mi></mover></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math></p>@
qu.3.17.algorithm=$aMag=rint(2,10);
$aLet=switch(rint(8),'a','b','c','d','f','g','h','w');
$idx=rint(2);
$QSign=switch($idx,1,-1);
$QSignLabel=switch($idx,"+","-");
$QMag=rint(2,6);
$qMag=switch(rint(2),rint(2,6),-rint(2,6));@
qu.3.17.uid=46a6ddbd-e920-4923-b5d1-a2bcaedec71f@
qu.3.17.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Force Due to A Line of Charge;
  Difficulty=Hard;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Diagram;
  Features=Partial Grading;
@

qu.3.18.question=<p>Find an algebraic expression for the electric force due to the two points charges on the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi></mrow></mstyle></math>-axis on&nbsp;a point charge <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$q3</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>q</mi></mrow></mstyle></math>&nbsp;at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math>, in terms of the given parameters.</p>
<p>&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="395" height="400">
<param name="image" value="__BASE_URI__img/ElectricFields/E-PtChr-2Chrg-2D-Sym/Diagram.png" />
<param name="size" value="10" />
<param name="label.1.x" value="20" />
<param name="label.1.y" value="105" />
<param name="label.1.text" value="q" />
<param name="label.2.x" value="10" />
<param name="label.2.y" value="105" />
<param name="label.2.text" value="$q1" />
<param name="label.3.x" value="20" />
<param name="label.3.y" value="317" />
<param name="label.3.text" value="q" />
<param name="label.4.x" value="10" />
<param name="label.4.y" value="317" />
<param name="label.4.text" value="$q2" />
<param name="label.5.x" value="50" />
<param name="label.5.y" value="155" />
<param name="label.5.text" value="a" />
<param name="label.6.x" value="40" />
<param name="label.6.y" value="155" />
<param name="label.6.text" value="$a1mag" />
<param name="label.7.x" value="50" />
<param name="label.7.y" value="265" />
<param name="label.7.text" value="a" />
<param name="label.8.x" value="40" />
<param name="label.8.y" value="265" />
<param name="label.8.text" value="$a2mag" />
<param name="label.9.x" value="160" />
<param name="label.9.y" value="200" />
<param name="label.9.text" value="x" />
<param name="label.10.x" value="150" />
<param name="label.10.y" value="200" />
<param name="label.10.text" value="$xmag" /></applet></p>
<p><br />
<em>Entry Notes</em>:&nbsp;&nbsp;<br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ihat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mi>i</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>jhat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>khat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp;&nbsp;<br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
If necessary, use the letter <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi></mrow></mstyle></math> to represent the Coulomb constant <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mi mathvariant='normal'>&pi;</mi><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>eg.&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mrow><mi>&pi;</mi><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mrow></mfrac><mfrac><mrow><msub><mi>q</mi><mrow><mn>1</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></msub><msub><mi>q</mi><mrow><mn>2</mn></mrow></msub></mrow><mrow><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mn>1</mn><mrow><mn>4</mn><mrow><mi>&pi;</mi></mrow><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mfrac><mfrac><mrow><msub><mi>q</mi><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></msub><msub><mi>q</mi><mrow><mn>3</mn></mrow></msub></mrow><mrow><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>&nbsp;could be entered as (k*q1*q2/(r^2))*ihat + (k*q2*q3/(r^2))*khat.</p>@
qu.3.18.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=vectorAlgebraicComponents);@
qu.3.18.allow2d=0@
qu.3.18.maple_answer=subs(_i=ihat,_j=jhat,_k=khat,collect(PhysFuncs[ElectricField](($q1)*q*($q3)*q,$a1mag*a*_j,$xmag*x*_i,constant=k)+PhysFuncs[ElectricField](($q2)*q*($q3)*q,-$a2mag*a*_j,$xmag*x*_i,constant=k),[_i,_j,_k]))@
qu.3.18.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.3.18.type=maple@
qu.3.18.mode=Maple@
qu.3.18.name=Electric Force Due to Point Charges - 2 Charges, 2D, Different Charges, Different Distances@
qu.3.18.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.3.18.editing=useHTML@
qu.3.18.hint.1=Remember that the superposition principle means that the electric force at a point is the sum of the electric forces produced by each of the nearby charges.&nbsp; Pretend that only one charge at a time&nbsp;is present and calculate the force due to that charge at the point.&nbsp; Finally, add all of the electric forces together to get the total.@
qu.3.18.solution=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mi>top</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$q1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>q</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$q3</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>q</mi></mrow></mfenced></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>$a1mag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$xmag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow><mrow><mover><mrow><mi>r</mi></mrow><mi>&#x005e;</mi></mover></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math></p>
<p>The x-component is</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>F</mi><mrow><msub><mi>top</mi><mrow><mi>X</mi></mrow></msub></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><msub><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mrow><mi>top</mi></mrow></msub></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$xmag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow><mrow><msqrt><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>$a1mag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$xmag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac></mrow></mstyle></math></p>
<p>and the y-component is</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>F</mi><mrow><msub><mi>top</mi><mrow><mi>Y</mi></mrow></msub></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><msub><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mrow><mi>top</mi></mrow></msub></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow><mrow><mi mathvariant='normal'></mi></mrow></mstyle></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mi>sin</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$a1mag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow><mrow><msqrt><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>$a1mag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$xmag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac></mrow></mstyle></math></p>
<p>The components of the electric force due to the bottom charge can be similarly calculated.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mi>P</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><msub><mi>F</mi><mrow><msub><mi>top</mi><mrow><mi>X</mi></mrow></msub></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>F</mi><mrow><msub><mi>bottom</mi><mrow><mi>X</mi></mrow></msub></mrow></msub></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><msub><mi>F</mi><mrow><msub><mi>top</mi><mrow><mi>Y</mi></mrow></msub></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>F</mi><mrow><msub><mi>bottom</mi><mrow><mi>Y</mi></mrow></msub></mrow></msub></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math></p>@
qu.3.18.algorithm=$a1mag=rint(2,5);
$a2mag=rint(2,5);
$xmag=rint(2,5);
$q1=switch(rint(2),rint(1,10),-rint(1,10));
$q2=switch(rint(2),rint(1,10),-rint(1,10));
$q3=switch(rint(2),rint(1,10),-rint(1,10));@
qu.3.18.uid=94c70cb1-b573-488d-8eef-03f7ed889c71@
qu.3.18.info=  Course=Introductory Electricity and Magnetism;
  Topic=Electric Force Due to Point Charges in 2D;
  Author=Aron Pasieka;
  Difficulty=Hard;
  Features=Algebraic;
  Features=Algorithmic;
  Features=Partial Grading;
  Features=Diagram;
@

qu.3.19.question=<p>A charge of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$q</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>C</mi></mrow></mstyle></math> is moving with velocity&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>v</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$v1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$v2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow></mfenced></mrow></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>m</mi></mrow><mrow><mo lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&sol;</mo><mi>s</mi></mrow></mstyle></math>&nbsp;through a magnetic field<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>B</mi><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$b1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>T</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math>&nbsp;and an electric field <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$e</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>V</mi></mrow><mrow><mi>m</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp; What is the force acting on the charge due to&nbsp;<br />
the magnetic field?</p>
<p>&nbsp;</p>
<p><em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ihat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>jhat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>khat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.</p>@
qu.3.19.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT=false,uVS=false,uUN);@
qu.3.19.allow2d=0@
qu.3.19.maple_answer=with(Physics[Vectors]);
with(SigFigs);
v:=($v1)*_i+($v2)*_j;
B:=($b1)*_k;
E:=($e)*_k;
temp:=(($q)*((v &x B)+E));
roundToSigFigs(Component(temp,1),3)*N*ihat+roundToSigFigs(Component(temp,2),3)*N*jhat+roundToSigFigs(Component(temp,3),3)*N*khat@
qu.3.19.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.3.19.type=maple@
qu.3.19.mode=Maple@
qu.3.19.name=Force on Charges in Electro-Magnetic Field - Helix@
qu.3.19.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT=false,uVS=false,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.3.19.editing=useHTML@
qu.3.19.hint.1=Use the right-hand rule to find the direction, then multiply the coefficients.@
qu.3.19.hint.2=Set up a determinant.&nbsp; The first row should be <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp; The second row should be the components of the first vector and the third row should be the components of the second vector.&nbsp; Evaluate the determinant.@
qu.3.19.solution=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>F</mi><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>q</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mover><mi>v</mi><mi>&rarr;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><mover><mi>B</mi><mi>&rarr;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$q</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>C</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo></mrow><mrow><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mfenced open='[' close=']' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mover><mi>i</mi><mi>&#x005e;</mi></mover></mrow></mtd><mtd><mrow><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow></mtd><mtd><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mi>$v1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mtd><mtd><mrow><mfrac><mrow><mi>$v2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mtd><mtd><mrow><mn>0</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mi>$b1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mtd></mtr></mtable></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mrow></mfenced></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$q</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>C</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$e</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>V</mi></mrow><mrow><mi>m</mi></mrow></mfrac><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$q</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>C</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$v2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$b1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>0</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mn>0</mn></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$v1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$b1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>0</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mn>0</mn></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$v1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mn>0</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><mi>$v2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mn>0</mn></mrow></mfenced></mrow></mfenced></mrow><mrow></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$q</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>C</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$e</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>V</mi></mrow><mrow><mi>m</mi></mrow></mfrac><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'></mo></mrow></mstyle></math></p>@
qu.3.19.algorithm=$v1=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$v2=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$b1=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$b2=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$e=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$q=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));@
qu.3.19.uid=34044ffb-e074-4b22-b407-0e5a7c1075dc@
qu.3.19.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Force on Moving Charge in Electro-Magnetic Field;
  Difficulty=Medium;
  Features=Algorithmic;
  Features=Partial Grading;
@

qu.3.20.question=<p>Find an algebraic expression for the electric force due to the two points charges on the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi></mrow></mstyle></math>-axis on&nbsp;a point charge <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$q3</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>q</mi></mrow></mstyle></math>&nbsp;at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math>, in terms of the given parameters.</p>
<p>&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="395" height="400">
<param name="image" value="__BASE_URI__img/ElectricFields/E-PtChr-2Chrg-2D-Sym/Diagram.png" />
<param name="size" value="10" />
<param name="label.1.x" value="20" />
<param name="label.1.y" value="105" />
<param name="label.1.text" value="q" />
<param name="label.2.x" value="10" />
<param name="label.2.y" value="105" />
<param name="label.2.text" value="$q1" />
<param name="label.3.x" value="20" />
<param name="label.3.y" value="317" />
<param name="label.3.text" value="q" />
<param name="label.4.x" value="10" />
<param name="label.4.y" value="317" />
<param name="label.4.text" value="$q2" />
<param name="label.5.x" value="50" />
<param name="label.5.y" value="155" />
<param name="label.5.text" value="a" />
<param name="label.6.x" value="40" />
<param name="label.6.y" value="155" />
<param name="label.6.text" value="$a1mag" />
<param name="label.7.x" value="50" />
<param name="label.7.y" value="265" />
<param name="label.7.text" value="a" />
<param name="label.8.x" value="40" />
<param name="label.8.y" value="265" />
<param name="label.8.text" value="$a2mag" />
<param name="label.9.x" value="160" />
<param name="label.9.y" value="200" />
<param name="label.9.text" value="x" />
<param name="label.10.x" value="150" />
<param name="label.10.y" value="200" />
<param name="label.10.text" value="$xmag" /></applet></p>
<p><br />
<em>Entry Notes</em>:&nbsp;&nbsp;<br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ihat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mi>i</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>jhat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>khat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp;&nbsp;<br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
If necessary, use the letter <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi></mrow></mstyle></math> to represent the Coulomb constant <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mi mathvariant='normal'>&pi;</mi><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>eg.&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mrow><mi>&pi;</mi><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mrow></mfrac><mfrac><mrow><msub><mi>q</mi><mrow><mn>1</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></msub><msub><mi>q</mi><mrow><mn>2</mn></mrow></msub></mrow><mrow><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mn>1</mn><mrow><mn>4</mn><mrow><mi>&pi;</mi></mrow><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mfrac><mfrac><mrow><msub><mi>q</mi><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></msub><msub><mi>q</mi><mrow><mn>3</mn></mrow></msub></mrow><mrow><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>&nbsp;could be entered as (k*q1*q2/(r^2))*ihat + (k*q2*q3/(r^2))*khat.</p>@
qu.3.20.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=vectorAlgebraicComponents);@
qu.3.20.allow2d=0@
qu.3.20.maple_answer=subs(_i=ihat,_j=jhat,_k=khat,collect(PhysFuncs[ElectricField](($q1)*q*($q3)*q,$a1mag*a*_j,$xmag*x*_i,constant=k)+PhysFuncs[ElectricField](($q2)*q*($q3)*q,-$a2mag*a*_j,$xmag*x*_i,constant=k),[_i,_j,_k]))@
qu.3.20.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.3.20.type=maple@
qu.3.20.mode=Maple@
qu.3.20.name=Electric Force Due to Point Charges - 2 Charges, 2D, Different Charges@
qu.3.20.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.3.20.editing=useHTML@
qu.3.20.hint.1=Remember that the superposition principle means that the electric force at a point is the sum of the electric forces produced by each of the nearby charges.&nbsp; Pretend that only one charge at a time&nbsp;is present and calculate the force due to that charge at the point.&nbsp; Finally, add all of the electric forces together to get the total.@
qu.3.20.solution=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mi>top</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$q1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>q</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$q3</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>q</mi></mrow></mfenced></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>$a1mag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$xmag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow><mrow><mover><mrow><mi>r</mi></mrow><mi>&#x005e;</mi></mover></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math></p>
<p>The x-component is</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>F</mi><mrow><msub><mi>top</mi><mrow><mi>X</mi></mrow></msub></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><msub><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mrow><mi>top</mi></mrow></msub></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$xmag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow><mrow><msqrt><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>$a1mag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$xmag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac></mrow></mstyle></math></p>
<p>and the y-component is</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>F</mi><mrow><msub><mi>top</mi><mrow><mi>Y</mi></mrow></msub></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><msub><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mrow><mi>top</mi></mrow></msub></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow><mrow><mi mathvariant='normal'></mi></mrow></mstyle></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mi>sin</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$a1mag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow><mrow><msqrt><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>$a1mag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$xmag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac></mrow></mstyle></math></p>
<p>The components of the electric force due to the bottom charge can be similarly calculated.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mi>P</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><msub><mi>F</mi><mrow><msub><mi>top</mi><mrow><mi>X</mi></mrow></msub></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>F</mi><mrow><msub><mi>bottom</mi><mrow><mi>X</mi></mrow></msub></mrow></msub></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><msub><mi>F</mi><mrow><msub><mi>top</mi><mrow><mi>Y</mi></mrow></msub></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>F</mi><mrow><msub><mi>bottom</mi><mrow><mi>Y</mi></mrow></msub></mrow></msub></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math></p>@
qu.3.20.algorithm=$a1mag=rint(2,5);
$a2mag=$a1mag;
$xmag=rint(2,5);
$q1=switch(rint(2),rint(1,10),-rint(1,10));
$q2=switch(rint(2),rint(1,10),-rint(1,10));
$q3=switch(rint(2),rint(1,10),-rint(1,10));@
qu.3.20.uid=caafce01-8a3e-4aa4-93a2-e7774b63bce9@
qu.3.20.info=  Course=Introductory Electricity and Magnetism;
  Topic=Electric Force Due to Point Charges in 2D;
  Author=Aron Pasieka;
  Difficulty=Hard;
  Features=Algebraic;
  Features=Algorithmic;
  Features=Partial Grading;
  Features=Diagram;
@

qu.3.21.question=<p>A velocity selector is set up as in the diagram with uniform electric and magnetic fields. If the magnetic<br />
field is of magnitude&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$B</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$Bex</mi></mrow></msup><mi>T</mi></mrow></mstyle></math> and&nbsp;a charged particle of speed <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$v</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$vex</mi></mrow></msup><mfrac><mrow><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mstyle></math>&nbsp;will pass<br />
through undeflected, what is the magnitude and direction of the magnetic field?</p>
<p><img alt="" align="middle" width="600" height="171" src="__BASE_URI__img/MagneticFields/VelocitySelector/DiagramNoE2.png" /></p>
<p>&nbsp;</p>@
qu.3.21.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.3.21.allow2d=0@
qu.3.21.maple_answer=SigFigs[roundToSigFigs]($ans,3)*(V/m)*(yhat)@
qu.3.21.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.3.21.type=maple@
qu.3.21.mode=Maple@
qu.3.21.name=Velocity Selector - Find E - 2 ~ PGc@
qu.3.21.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.3.21.editing=useHTML@
qu.3.21.solution=<p>In a velocity selector, the forces due to a magnetic field and an electric field cancel eachother, for one velocity of charged particle:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>qE</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>qvB</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mi>v</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>E</mi><mrow><mi>B</mi></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>We are given the velocity and the magnetic field, which we can use to calculate the magnitude of the electric field.</p>
<p>Assuming the charge is positive, use <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>q</mi><mrow><mover><mrow><mi>v</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math>&nbsp;to determine the direction of the magnetic force - the electric <em>force</em> must be in the opposite direction.&nbsp; Again, assume that the charge is positive and determine the direction of the electric <em>field.</em></p>
<p>&nbsp;</p>@
qu.3.21.algorithm=$B=rand(1.00,9.99,3);
$Bex=range(2,10);
$v=rand(1.00,9.99,3);
$vex=range(2,7);
$ans=($B*10^($Bex))*($v*10^($vex));@
qu.3.21.uid=bfdb0318-c090-45db-9615-338dfa4f12dc@
qu.3.21.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Motion of Charged Particles in Electromagnetic Fields;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.3.22.question=<p>A velocity selector is set up as in the diagram with uniform electric and magnetic fields. If the magnetic<br />
field is of magnitude&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$B</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$Bex</mi></mrow></msup><mi>T</mi></mrow></mstyle></math> and&nbsp;a charged particle of speed <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$v</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$vex</mi></mrow></msup><mfrac><mrow><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mstyle></math>&nbsp;will pass<br />
through undeflected, what is the magnitude and direction of the magnetic field?</p>
<p><img alt="" align="middle" width="600" height="171" src="__BASE_URI__img/MagneticFields/VelocitySelector/DiagramNoE1.png" /></p>
<p>&nbsp;</p>@
qu.3.22.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.3.22.allow2d=0@
qu.3.22.maple_answer=SigFigs[roundToSigFigs]($ans,3)*(V/m)*(-yhat)@
qu.3.22.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.3.22.type=maple@
qu.3.22.mode=Maple@
qu.3.22.name=Velocity Selector - Find E - 1 ~ PGc@
qu.3.22.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.3.22.editing=useHTML@
qu.3.22.solution=<p>In a velocity selector, the forces due to a magnetic field and an electric field cancel eachother, for one velocity of charged particle:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>qE</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>qvB</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mi>v</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>E</mi><mrow><mi>B</mi></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>We are given the magnetic field and the velocity to calculate the magnitude of the electric field.</p>
<p>Assuming the charge is positive, use <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>q</mi><mrow><mover><mrow><mi>v</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math>&nbsp;to determine the direction of the magnetic force - the electric <em>force</em> must be in the opposite direction.&nbsp; Again, assume that the charge is positive and determine the direction of the electric <em>field.</em></p>
<p>&nbsp;</p>@
qu.3.22.algorithm=$B=rand(1.00,9.99,3);
$Bex=range(2,10);
$v=rand(1.00,9.99,3);
$vex=range(2,7);
$ans=($B*10^($Bex))*($v*10^($vex));@
qu.3.22.uid=a1360cb3-9d93-40c4-92a1-66f4fbf2e764@
qu.3.22.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Motion of Charged Particles in Electromagnetic Fields;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.3.23.question=<p>A thin ring of radius&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$aMag$aLet</mi></mrow></mstyle></math>&nbsp;is centered at the origin in the&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mover><mi>z</mi><mi>&#x005e;</mi></mover></mrow></mstyle></math>&nbsp;plane.&nbsp;&nbsp;A&nbsp;charge<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$QSignLabel</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow></mstyle></math>&nbsp;is uniformly distributed on the ring.&nbsp;</p>
<p>Use symmetry to set up a single&nbsp;expression&nbsp;for the&nbsp;electric force due to the ring of charge on a point charge <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$qMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>q</mi></mrow></mstyle></math>&nbsp;at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math>.<br />
Provide&nbsp;the <em>integrand</em> of this integral as your answer, in terms of the given parameters.</p>
<p align="center">e.g.&nbsp; Integral:&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathcolor='#800080'>a</mi></mrow><mrow><mi>a</mi></mrow></munderover><mfrac><mi>Q</mi><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow></mstyle></math>,&nbsp; Response:&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>Q</mi><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>
<p>In order to find the electric force, one needs to evaluate an integral that adds up the contributions from each infinitesimal piece<br />
of the line, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ds</mi></mrow></mstyle></math>, from <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>0</mn></mrow></mstyle></math> to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&pi;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mstyle></math>.</p>
<p align="center">&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="585" height="500">
<param name="image" value="__BASE_URI__img/ElectricFields/RingCharge/Diagram.png" />
<param name="size" value="3" />
<param name="label.1.x" value="25" />
<param name="label.1.y" value="300" />
<param name="label.1.text" value="$QSignLabel Q" />
<param name="label.2.x" value="220" />
<param name="label.2.y" value="22" />
<param name="label.2.text" value="$aMag$aLet" />
<param name="label.3.x" value="270" />
<param name="label.3.y" value="250" />
<param name="label.3.text" value="x" /></applet></p>
<p align="center">&nbsp;</p>
<p>&nbsp;</p>
<p><em>Note:&nbsp; </em>Use the letter 'k' for the Coulomb constant, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mrow><mi>&pi;</mi></mrow><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mfrac></mrow></mstyle></math>.</p>@
qu.3.23.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSA2,subPackage=vectorAlgebraicComponents)@
qu.3.23.allow2d=0@
qu.3.23.maple_answer=with(Physics[Vectors]):
subs(_i=ihat,_j=jhat,_k=khat,collect((PhysFuncs[ElectricField](($qMag)*q*($QSign)*(Q/(2*Pi*$aMag*$aLet)),$aMag*$aLet*_j,x*_i,constant=k)._i)*_i,[_i,_j,_k]))@
qu.3.23.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.3.23.type=maple@
qu.3.23.mode=Maple@
qu.3.23.name=Ring Charge - Find F - Set Up Integral@
qu.3.23.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSA2,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.3.23.editing=useHTML@
qu.3.23.solution=<p>Start by identifying a small unit of charge on the ring, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dq</mi></mrow></mstyle></math>.&nbsp; Then the small amount of electric force due to that unit of charge is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$qMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>q</mi></mrow></mfenced><mi>dq</mi></mrow><mrow><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mrow><mover><mi>r</mi><mi>&#x005e;</mi></mover></mrow></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msqrt><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>Since the charge is uniformly distributed on the ring, the charge per unit length is constant, so:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>dq</mi><mrow><mi>ds</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac></mrow></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mi>dq</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac><mi>ds</mi></mrow><mrow></mrow></mstyle></math>.</p>
<p>Finally, breaking into components:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>r</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mover><mrow><mi>r</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mi>r</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mrow><mi>x</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mrow><mi>y</mi></mrow><mi>&#x005e;</mi></mover></mrow><mrow><mi>r</mi></mrow></mfrac></mrow></mstyle></math></p>
<p>Recognizing that the y- and z-components for units of charge diametrically opposite will cancel, we can conclude that the y- and z-component of the total electric force is zero.</p>
<p>Thus, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>dF</mi><mrow><mi>x</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$qMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>q</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mn>2</mn><mi>&pi;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>ds</mi></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mrow><mfrac><mn>3</mn><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></mfrac></mrow></mfenced></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>.</p>
<p>Adding up the contributions from each unit of charge on the ring:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mover><mi>F</mi><mi>&rarr;</mi></mover></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi>s</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></munderover><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$qMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>q</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mrow><mfrac><mn>3</mn><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></mfrac></mrow></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mover><mrow><mi>x</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math></p>@
qu.3.23.algorithm=$aMag=rint(2,10);
$aLet=switch(rint(8),'a','b','c','d','f','g','h','w');
$idx=rint(2);
$QSign=switch($idx,1,-1);
$QSignLabel=switch($idx,"+","-");
$QMag=rint(2,6);
$qMag=switch(rint(2),rint(2,6),-rint(2,6));@
qu.3.23.uid=8f94a274-a913-4a98-9058-c53170ffe100@
qu.3.23.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Force Due to A Line of Charge;
  Difficulty=Hard;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Diagram;
  Features=Partial Grading;
@

qu.3.24.question=<p>A velocity selector is set up as in the diagram with uniform electric and magnetic fields. If the electric <br />
field is of magnitude&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$E</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$Eex</mi></mrow></msup><mfrac><mi>V</mi><mrow><mi>m</mi></mrow></mfrac></mrow></mstyle></math> and the magnetic field is of magnitude <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$B</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$Bex</mi></mrow></msup><mi>T</mi></mrow></mstyle></math>, what <br />
speed of charged particle will pass through undeflected?</p>
<p><img alt="" align="middle" width="600" height="171" src="__BASE_URI__img/MagneticFields/VelocitySelector/Diagram1.png" /></p>
<p>&nbsp;</p>
<p><span>&nbsp;</span></p>@
qu.3.24.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.3.24.allow2d=0@
qu.3.24.maple_answer=SigFigs[roundToSigFigs]($ans,3)*m/s@
qu.3.24.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.3.24.type=maple@
qu.3.24.mode=Maple@
qu.3.24.name=Velocity Selector - Find v - 1 ~ PGc@
qu.3.24.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.3.24.editing=useHTML@
qu.3.24.solution=<p>In a velocity selector, the forces due to a magnetic field and an electric field cancel eachother, for one velocity of charged particle:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>qE</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>qvB</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mi>v</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>E</mi><mrow><mi>B</mi></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>We are given the electric and magnetic fields.</p>@
qu.3.24.algorithm=$E=rand(1.00,9.99,3);
$Eex=range(5,13);
$v=range(10000,299790000);
$B=rand(1.00,9.99,3);
$Bex=int(log($v))-1;
$ans=($E*10^($Eex))/($B*10^($Bex));@
qu.3.24.uid=b8b96b46-23ce-41d7-a62a-d0745fd50c56@
qu.3.24.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Motion of Charged Particles in Electromagnetic Fields;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.4.topic=Electric Fields@

qu.4.1.question=<p>The magnitude of the potential difference between two infinite<font size="2"> parallel</font> plates separated by $x cm&nbsp;is $pot V.&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="400" height="360">
<param name="image" value="__BASE_URI__img/ElectricFields/InfiniteParallelPlates/Diagram.png" />
<param name="size" value="1" />
<param name="label.1.x" value="295" />
<param name="label.1.y" value="12" />
<param name="label.1.text" value="$x cm" /></applet>&nbsp;</p>
<p>&nbsp;</p>
<p>What is the magnitude of the electric field between the plates?</p>@
qu.4.1.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.4.1.allow2d=0@
qu.4.1.maple_answer=abs($E)*V/m@
qu.4.1.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.4.1.type=maple@
qu.4.1.mode=Maple@
qu.4.1.name=Infinite Parallel Plates - Find Field Magnitude ~ PG@
qu.4.1.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.4.1.editing=useHTML@
qu.4.1.hint.1=Recall that the electric field between infinite parallel plates is constant.&nbsp@
qu.4.1.hint.2=;The potential difference is related to the electric field and the separation of the plates.@
qu.4.1.hint.3=Ensure that the units work out.@
qu.4.1.solution=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&Delta;V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mi>&Delta;x</mi><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow></mstyle></math>and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>&Delta;x</mi></mrow><mi>&rarr;</mi></mover></mrow></mstyle></math>&nbsp;are parallel so the magnitude of the dot product is equal to the product of the two magnitudes.&nbsp; (ie. <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow></mstyle></math>)</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo></mrow><mrow><mrow><mover><mrow><mi>&Delta;x</mi></mrow><mi>&rarr;</mi></mover></mrow><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$x cm<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='|' close='|' separators=','><mrow><mi>V</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi></mi></mrow></mstyle></math>$pot V</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><mrow><mrow><mi mathvariant='normal'></mi></mrow><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow></mrow></mfenced></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>($pot V)/($x cm)<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow></mrow><mrow><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo></mrow><mrow><mrow><mover><mrow><mi>E</mi></mrow><mi>&rarr;</mi></mover></mrow><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>${sig(3,abs($E))} V/m</p>@
qu.4.1.algorithm=$potGen=rand(10,100,3);
$x=rand(2,15,3);
$EGen=sig(3,$potGen/($x/100));
$pot=sig(3,$EGen*($x/100));
$idx=rint(2);
$dir=switch($idx,1,-1);
$E=sig(3,$EGen*$dir);
$correctPlate=switch($idx,"A","B");
$wrongPlate=switch($idx,"B","A");@
qu.4.1.uid=6b488086-c97d-4b69-b50b-ac61fb771e11@
qu.4.1.info=  Course=Introductory Electricity and Magnetism;
  Topic=Electric Potential Between Infinite Parallel Plates;
  Author=Aron Pasieka;
  Difficulty=Easy;
  Features=Algorithmic;
  Features=Partial Grading;
  Features=Diagram;
@

qu.4.2.question=<p>Find an algebraic expression for the electric field at point P, in terms of the given parameters.</p>
<p>&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="395" height="400">
<param name="image" value="__BASE_URI__img/ElectricFields/E-PtChr-2Chrg-2D-Sym/Diagram.png" />
<param name="size" value="10" />
<param name="label.1.x" value="20" />
<param name="label.1.y" value="105" />
<param name="label.1.text" value="q" />
<param name="label.2.x" value="10" />
<param name="label.2.y" value="105" />
<param name="label.2.text" value="$q1" />
<param name="label.3.x" value="20" />
<param name="label.3.y" value="317" />
<param name="label.3.text" value="q" />
<param name="label.4.x" value="10" />
<param name="label.4.y" value="317" />
<param name="label.4.text" value="$q2" />
<param name="label.5.x" value="50" />
<param name="label.5.y" value="155" />
<param name="label.5.text" value="a" />
<param name="label.6.x" value="40" />
<param name="label.6.y" value="155" />
<param name="label.6.text" value="$a1mag" />
<param name="label.7.x" value="50" />
<param name="label.7.y" value="265" />
<param name="label.7.text" value="a" />
<param name="label.8.x" value="40" />
<param name="label.8.y" value="265" />
<param name="label.8.text" value="$a2mag" />
<param name="label.9.x" value="160" />
<param name="label.9.y" value="200" />
<param name="label.9.text" value="x" />
<param name="label.10.x" value="150" />
<param name="label.10.y" value="200" />
<param name="label.10.text" value="$xmag" /></applet></p>
<p><br />
<em>Entry Notes</em>:&nbsp;&nbsp;<br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ihat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mi>i</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>jhat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>khat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp;&nbsp;<br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
If necessary, use the letter <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi></mrow></mstyle></math> to represent the Coulomb constant <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mi mathvariant='normal'>&pi;</mi><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>eg.&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mrow><mi>&pi;</mi><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mrow></mfrac><mfrac><mrow><msub><mi>q</mi><mrow><mn>1</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></msub><msub><mi>q</mi><mrow><mn>2</mn></mrow></msub></mrow><mrow><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mn>1</mn><mrow><mn>4</mn><mrow><mi>&pi;</mi></mrow><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mfrac><mfrac><mrow><msub><mi>q</mi><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></msub><msub><mi>q</mi><mrow><mn>3</mn></mrow></msub></mrow><mrow><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>&nbsp;could be entered as (k*q1*q2/(r^2))*ihat + (k*q2*q3/(r^2))*khat.</p>@
qu.4.2.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=vectorAlgebraicComponents);@
qu.4.2.allow2d=0@
qu.4.2.maple_answer=subs(_i=ihat,_j=jhat,_k=khat,collect(PhysFuncs[ElectricField](($q1)*q,$a1mag*a*_j,$xmag*x*_i,constant=k)+PhysFuncs[ElectricField](($q2)*q,-$a2mag*a*_j,$xmag*x*_i,constant=k),[_i,_j,_k]))@
qu.4.2.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.4.2.type=maple@
qu.4.2.mode=Maple@
qu.4.2.name=Electric Field Due to Point Charges - 2 Charges, 2D, Different Charges, Different Distances@
qu.4.2.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.4.2.editing=useHTML@
qu.4.2.hint.1=Remember that the superposition principle means that the electric field at a point is the sum of the electric fields produced by each of the nearby charges.&nbsp; Pretend that only one charge at a time&nbsp;is present and calculate the field due to that charge at the point.&nbsp; Finally, add all of the electric fields together to get the total.@
qu.4.2.solution=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow><mrow><mi>top</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$q1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>q</mi></mrow></mfenced></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>$a1mag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$xmag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow><mrow><mover><mrow><mi>r</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math></p>
<p>The x-component is</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>E</mi><mrow><msub><mi>top</mi><mrow><mi>X</mi></mrow></msub></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><msub><mover><mi>E</mi><mi>&rarr;</mi></mover><mrow><mi>top</mi></mrow></msub></mfenced></mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mi>cos</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mrow></mstyle></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$xmag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow><mrow><msqrt><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>$a1mag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$xmag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac></mrow></mstyle></math></p>
<p>and the y-component is</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>E</mi><mrow><msub><mi>top</mi><mrow><mi>Y</mi></mrow></msub></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><msub><mover><mi>E</mi><mi>&rarr;</mi></mover><mrow><mi>top</mi></mrow></msub></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow><mrow></mrow></mstyle></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mi>sin</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$a1mag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow><mrow><msqrt><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>$a1mag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$xmag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac></mrow></mstyle></math></p>
<p>The components of the electric field due to the bottom charge can be similarly calculated.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow><mrow><mi>P</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><msub><mi>E</mi><mrow><msub><mi>top</mi><mrow><mi>X</mi></mrow></msub></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>E</mi><mrow><msub><mi>bottom</mi><mrow><mi>X</mi></mrow></msub></mrow></msub></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><msub><mi>E</mi><mrow><msub><mi>top</mi><mrow><mi>Y</mi></mrow></msub></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>E</mi><mrow><msub><mi>bottom</mi><mrow><mi>Y</mi></mrow></msub></mrow></msub></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math></p>@
qu.4.2.algorithm=$a1mag=rint(2,5);
$a2mag=rint(2,5);
$xmag=rint(2,5);
$q1=switch(rint(2),rint(1,10),-rint(1,10));
$q2=switch(rint(2),rint(1,10),-rint(1,10));@
qu.4.2.uid=e7aeed7b-e795-4cc0-b091-065be6e74a4d@
qu.4.2.info=  Course=Introductory Electricity and Magnetism;
  Topic=Electric Field Due to Point Charges in 2D;
  Author=Aron Pasieka;
  Difficulty=Hard;
  Features=Algebraic;
  Features=Algorithmic;
  Features=Partial Grading;
  Features=Diagram;
@

qu.4.3.question=<p>Two infinite<font size="2"> parallel</font> plates are separated by a distance $x cm . The electric field between the plates is&nbsp;<br />
measured to be $E V/m <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover></mrow></mstyle></math>.</p>
<p align="center">&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="400" height="360">
<param name="image" value="__BASE_URI__img/ElectricFields/InfiniteParallelPlates/Diagram.png" />
<param name="size" value="1" />
<param name="label.1.x" value="295" />
<param name="label.1.y" value="12" />
<param name="label.1.text" value="$x cm" /></applet>&nbsp;</p>
<p>&nbsp;</p>
<p>What is the magnitude of the potential difference between the plates?</p>@
qu.4.3.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.4.3.allow2d=0@
qu.4.3.maple_answer=($pot)*V@
qu.4.3.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.4.3.type=maple@
qu.4.3.mode=Maple@
qu.4.3.name=Infinite Parallel Plates - Magnitude of Potential Difference ~ PG@
qu.4.3.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.4.3.editing=useHTML@
qu.4.3.hint.1=Recall that the electric field between infinite parallel plates is constant.&nbsp@
qu.4.3.hint.2=;The potential difference is related to the electric field and the separation of the plates.@
qu.4.3.hint.3=Ensure that the units work out.@
qu.4.3.solution=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&Delta;V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mi>&Delta;x</mi><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow></mstyle></math>and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>&Delta;x</mi></mrow><mi>&rarr;</mi></mover></mrow></mstyle></math>&nbsp;are parallel so the magnitude of the dot product is equal to the product of the two magnitudes.&nbsp; (ie. <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow></mstyle></math>)</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo></mrow><mrow><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>${abs($E)} V/m<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='|' close='|' separators=','><mrow><mover><mi>&Delta;x</mi><mi>&rarr;</mi></mover></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$x cm</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='|' close='|' separators=','><mrow><mi>&Delta;V</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>(${abs($E)} V/m)($x cm)<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow></mrow><mrow><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo></mrow><mrow><mi>&Delta;V</mi><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$pot V</p>@
qu.4.3.algorithm=$potGen=rand(10,100,3);
$x=rand(2,15,3);
$EGen=sig(3,$potGen/($x/100));
$pot=sig(3,$EGen*($x/100));
$idx=rint(2);
$dir=switch($idx,1,-1);
$E=sig(3,$EGen*$dir);
$correctPlate=switch($idx,"A","B");
$wrongPlate=switch($idx,"B","A");@
qu.4.3.uid=77498cd7-2652-4afe-b893-68dd5fb3f703@
qu.4.3.info=  Course=Introductory Electricity and Magnetism;
  Topic=Electric Potential Between Infinite Parallel Plates;
  Author=Aron Pasieka;
  Difficulty=Easy;
  Features=Algorithmic;
  Features=Partial Grading;
  Features=Diagram;
@

qu.4.4.question=<p>Find an algebraic expression for the electric field at point P, in terms of the given parameters.</p>
<p>&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="395" height="400">
<param name="image" value="__BASE_URI__img/ElectricFields/E-PtChr-2Chrg-2D-Sym/Diagram.png" />
<param name="size" value="10" />
<param name="label.1.x" value="20" />
<param name="label.1.y" value="105" />
<param name="label.1.text" value="q" />
<param name="label.2.x" value="10" />
<param name="label.2.y" value="105" />
<param name="label.2.text" value="$q1" />
<param name="label.3.x" value="20" />
<param name="label.3.y" value="317" />
<param name="label.3.text" value="q" />
<param name="label.4.x" value="10" />
<param name="label.4.y" value="317" />
<param name="label.4.text" value="$q2" />
<param name="label.5.x" value="50" />
<param name="label.5.y" value="155" />
<param name="label.5.text" value="a" />
<param name="label.6.x" value="40" />
<param name="label.6.y" value="155" />
<param name="label.6.text" value="$amag" />
<param name="label.7.x" value="50" />
<param name="label.7.y" value="265" />
<param name="label.7.text" value="a" />
<param name="label.8.x" value="40" />
<param name="label.8.y" value="265" />
<param name="label.8.text" value="$amag" />
<param name="label.9.x" value="160" />
<param name="label.9.y" value="200" />
<param name="label.9.text" value="x" />
<param name="label.10.x" value="150" />
<param name="label.10.y" value="200" />
<param name="label.10.text" value="$xmag" /></applet></p>
<p><br />
<em>Entry Notes</em>:&nbsp;&nbsp;<br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ihat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mi>i</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>jhat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>khat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp;&nbsp;<br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
If necessary, use the letter <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi></mrow></mstyle></math> to represent the Coulomb constant <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mi mathvariant='normal'>&pi;</mi><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>eg.&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mrow><mi>&pi;</mi><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mrow></mfrac><mfrac><mrow><msub><mi>q</mi><mrow><mn>1</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></msub><msub><mi>q</mi><mrow><mn>2</mn></mrow></msub></mrow><mrow><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mn>1</mn><mrow><mn>4</mn><mrow><mi>&pi;</mi></mrow><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mfrac><mfrac><mrow><msub><mi>q</mi><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></msub><msub><mi>q</mi><mrow><mn>3</mn></mrow></msub></mrow><mrow><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>&nbsp;could be entered as (k*q1*q2/(r^2))*ihat + (k*q2*q3/(r^2))*khat.</p>@
qu.4.4.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=vectorAlgebraicComponents);@
qu.4.4.allow2d=0@
qu.4.4.maple_answer=subs(_i=ihat,_j=jhat,_k=khat,collect(PhysFuncs[ElectricField](($q1)*q,$amag*a*_j,$xmag*x*_i,constant=k)+PhysFuncs[ElectricField](($q2)*q,-$amag*a*_j,$xmag*x*_i,constant=k),[_i,_j,_k]))@
qu.4.4.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.4.4.type=maple@
qu.4.4.mode=Maple@
qu.4.4.name=Electric Field Due to Point Charges - 2 Charges, 2D, Different Charges@
qu.4.4.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.4.4.editing=useHTML@
qu.4.4.hint.1=Remember that the superposition principle means that the electric field at a point is the sum of the electric fields produced by each of the nearby charges.&nbsp; Pretend that only one charge at a time&nbsp;is present and calculate the field due to that charge at the point.&nbsp; Finally, add all of the electric fields together to get the total.@
qu.4.4.solution=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow><mrow><mi>top</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$q1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>q</mi></mrow></mfenced></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>$amag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$xmag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow><mrow><mover><mrow><mi>r</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math></p>
<p>The x-component is</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>E</mi><mrow><msub><mi>top</mi><mrow><mi>X</mi></mrow></msub></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><msub><mover><mi>E</mi><mi>&rarr;</mi></mover><mrow><mi>top</mi></mrow></msub></mfenced></mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mi>cos</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mrow></mstyle></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$xmag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow><mrow><msqrt><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>$amag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$xmag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac></mrow></mstyle></math></p>
<p>and the y-component is</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>E</mi><mrow><msub><mi>top</mi><mrow><mi>Y</mi></mrow></msub></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><msub><mover><mi>E</mi><mi>&rarr;</mi></mover><mrow><mi>top</mi></mrow></msub></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow><mrow></mrow></mstyle></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mi>sin</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$amag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow><mrow><msqrt><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>$amag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$xmag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac></mrow></mstyle></math></p>
<p>The components of the electric field due to the bottom charge can be similarly calculated.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow><mrow><mi>P</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><msub><mi>E</mi><mrow><msub><mi>top</mi><mrow><mi>X</mi></mrow></msub></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>E</mi><mrow><msub><mi>bottom</mi><mrow><mi>X</mi></mrow></msub></mrow></msub></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><msub><mi>E</mi><mrow><msub><mi>top</mi><mrow><mi>Y</mi></mrow></msub></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>E</mi><mrow><msub><mi>bottom</mi><mrow><mi>Y</mi></mrow></msub></mrow></msub></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math></p>@
qu.4.4.algorithm=$amag=rint(2,5);
$xmag=rint(2,5);
$q1=switch(rint(2),rint(1,10),-rint(1,10));
$q2=switch(rint(2),rint(1,10),-rint(1,10));@
qu.4.4.uid=0d7a7a9f-cd11-4818-98c3-3a846ef862a2@
qu.4.4.info=  Course=Introductory Electricity and Magnetism;
  Topic=Electric Field Due to Point Charges in 2D;
  Author=Aron Pasieka;
  Difficulty=Medium;
  Features=Algebraic;
  Features=Algorithmic;
  Features=Partial Grading;
  Features=Diagram;
@

qu.4.5.question=<p>The magnitude of the potential difference between two infinite<font size="2"> parallel</font> plates is $pot V. The electric field&nbsp;<br />
between the plates is measured to be $E V/m <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover></mrow></mstyle></math>.</p>
<p align="center">&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="400" height="360">
<param name="image" value="__BASE_URI__img/ElectricFields/InfiniteParallelPlates/Diagram.png" />
<param name="size" value="1" />
<param name="label.1.x" value="295" />
<param name="label.1.y" value="12" />
<param name="label.1.text" value="x" /></applet>&nbsp;</p>
<p>&nbsp;</p>
<p>What is the separation of the plates?</p>@
qu.4.5.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.4.5.allow2d=0@
qu.4.5.maple_answer=($x)*cm@
qu.4.5.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.4.5.type=maple@
qu.4.5.mode=Maple@
qu.4.5.name=Infinite Parallel Plates - Find Separation ~ PG@
qu.4.5.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.4.5.editing=useHTML@
qu.4.5.hint.1=Recall that the electric field between infinite parallel plates is constant.&nbsp@
qu.4.5.hint.2=;The potential difference is related to the electric field and the separation of the plates.@
qu.4.5.hint.3=Ensure that the units work out.@
qu.4.5.solution=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&Delta;V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mi>&Delta;x</mi><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow></mstyle></math>and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>&Delta;x</mi></mrow><mi>&rarr;</mi></mover></mrow></mstyle></math>&nbsp;are parallel so the magnitude of the dot product is equal to the product of the two magnitudes.&nbsp; (ie. <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow></mstyle></math>)</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo></mrow><mrow><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>${abs($E)} V/m<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='|' close='|' separators=','><mrow><mi>V</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi></mi></mrow></mstyle></math>$pot V</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='|' close='|' separators=','><mrow><mrow><mover><mi>&Delta;x</mi><mi>&rarr;</mi></mover></mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>($pot V)/(${abs($E)} V/m)<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'></mi></mrow><mrow><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo></mrow><mrow><mrow><mover><mi>&Delta;x</mi><mi>&rarr;</mi></mover></mrow><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$x cm</p>@
qu.4.5.algorithm=$potGen=rand(10,100,3);
$x=rand(2,15,3);
$EGen=sig(3,$potGen/($x/100));
$pot=sig(3,$EGen*($x/100));
$idx=rint(2);
$dir=switch($idx,1,-1);
$E=sig(3,$EGen*$dir);
$correctPlate=switch($idx,"A","B");
$wrongPlate=switch($idx,"B","A");@
qu.4.5.uid=9e7e4da9-e107-410b-9cbb-6884a5bdcaa1@
qu.4.5.info=  Course=Introductory Electricity and Magnetism;
  Topic=Electric Potential Between Infinite Parallel Plates;
  Author=Aron Pasieka;
  Difficulty=Easy;
  Features=Algorithmic;
  Features=Partial Grading;
  Features=Diagram;
@

qu.4.6.question=<p>A thin ring of radius&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$aMag$aLet</mi></mrow></mstyle></math>&nbsp;is centered at the origin in the&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mover><mi>z</mi><mi>&#x005e;</mi></mover></mrow></mstyle></math>&nbsp;plane.&nbsp;&nbsp;A&nbsp;charge<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$QSignLabel</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow></mstyle></math>&nbsp;is uniformly distributed on the ring.&nbsp;</p>
<p>Use symmetry to set up a single&nbsp;expression&nbsp;for the&nbsp;electric field at an&nbsp;arbitrary point along the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi></mrow></mstyle></math>-axis due to the ring of charge.<br />
Provide&nbsp;the <em>integrand</em> of this integral as your answer, in terms of the given parameters.</p>
<p align="center">e.g.&nbsp; Integral:&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathcolor='#800080'>a</mi></mrow><mrow><mi>a</mi></mrow></munderover><mfrac><mi>Q</mi><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow></mstyle></math>,&nbsp; Response:&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>Q</mi><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>
<p>In order to find the electric field, one needs to evaluate an integral that adds up the contributions from each infinitesimal piece<br />
of the line, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ds</mi></mrow></mstyle></math>, from <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>0</mn></mrow></mstyle></math> to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&pi;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mstyle></math>.</p>
<p align="center">&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="585" height="500">
<param name="image" value="__BASE_URI__img/ElectricFields/RingCharge/Diagram.png" />
<param name="size" value="3" />
<param name="label.1.x" value="25" />
<param name="label.1.y" value="300" />
<param name="label.1.text" value="$QSignLabel Q" />
<param name="label.2.x" value="220" />
<param name="label.2.y" value="22" />
<param name="label.2.text" value="$aMag$aLet" />
<param name="label.3.x" value="270" />
<param name="label.3.y" value="250" />
<param name="label.3.text" value="x" /></applet></p>
<p align="center">&nbsp;</p>
<p>&nbsp;</p>
<p><em>Note: </em>Use the letter 'k' for the Coulomb constant, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mrow><mi>&pi;</mi><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mrow></mfrac></mrow></mstyle></math>.</p>@
qu.4.6.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSA2,subPackage=vectorAlgebraicComponents)@
qu.4.6.allow2d=0@
qu.4.6.maple_answer=with(Physics[Vectors]):
subs(_i=ihat,_j=jhat,_k=khat,collect((PhysFuncs[ElectricField]($QSign*(Q/(2*Pi*$aMag*$aLet)),$aMag*$aLet*_j,x*_i,constant=k)._i)*_i,[_i,_j,_k]))@
qu.4.6.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.4.6.type=maple@
qu.4.6.mode=Maple@
qu.4.6.name=Ring Charge - Find E - Set Up Integral@
qu.4.6.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSA2,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.4.6.editing=useHTML@
qu.4.6.solution=<p>Start by identifying a small unit of charge on the ring, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dq</mi></mrow></mstyle></math>.&nbsp; Then the small amount of electric field due to that unit of charge is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi><mover><mi>E</mi><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi>kdq</mi><mrow><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mrow><mover><mi>r</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msqrt><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>Since the charge is uniformly distributed on the ring, the charge per unit length is constant, so:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>dq</mi><mrow><mi>ds</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac></mrow></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mi>dq</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac><mi>ds</mi></mrow><mrow></mrow></mstyle></math>.</p>
<p>Finally, breaking into components:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>r</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mover><mrow><mi>r</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mi>r</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mrow><mi>x</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mrow><mi>y</mi></mrow><mi>&#x005e;</mi></mover></mrow><mrow><mi>r</mi></mrow></mfrac></mrow></mstyle></math></p>
<p>Recognizing that the y- and z-components for units of charge&nbsp;diametrically opposite to eachother&nbsp;will cancel, we can conclude that the y- and z-component of the total electric field is zero.</p>
<p>Thus, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>dE</mi><mrow><mi>x</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>ds</mi></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mrow><mfrac><mn>3</mn><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></mfrac></mrow></mfenced></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>.</p>
<p>Adding up the contributions from each unit of charge on the ring:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi>s</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></munderover><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>s</mi></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mrow><mfrac><mn>3</mn><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></mfrac></mrow></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mover><mrow><mi>x</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math></p>@
qu.4.6.algorithm=$aMag=rint(2,10);
$aLet=switch(rint(8),'a','b','c','d','f','g','h','w');
$idx=rint(2);
$QSign=switch($idx,1,-1);
$QSignLabel=switch($idx,"+","-");
$QMag=rint(2,6);@
qu.4.6.uid=017e63c6-1d4f-4cb9-a36c-08d34fb73762@
qu.4.6.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Electric Field Due to A Line of Charge;
  Difficulty=Hard;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Diagram;
@

qu.4.7.question=<p>A&nbsp;solid disc&nbsp;of radius&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$aMag$aLet</mi></mrow></mstyle></math>&nbsp;is centered at the origin in the&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mover><mi>z</mi><mi>&#x005e;</mi></mover></mrow></mstyle></math>&nbsp;plane.&nbsp;&nbsp;A&nbsp;charge<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$QSignLabel</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow></mstyle></math>&nbsp;is uniformly distributed on the disc.</p>
<p>Use symmetry to set up a single&nbsp;expression&nbsp;for the&nbsp;electric field at an arbitrary point along the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi></mrow></mstyle></math>-axis due to the ring of charge.<br />
Use the fact&nbsp;that one can treat the disc as an infinite number of thin rings, and the fact that the electric field due to a&nbsp;single thin<br />
ring of radius <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>R</mi></mrow></mstyle></math>&nbsp;is:</p>
<p align="center"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi><mfrac><mrow><mi>dQ</mi><mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo></mrow><mi>x</mi></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>R</mi><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mrow><mfrac><mn>3</mn><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></mfrac><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.</p>
<p align="left">Provide the <em>integrand</em> of this integral as your answer, in terms of the given parameters.</p>
<p align="center">e.g. Integral: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathcolor='#800080'>a</mi></mrow><mrow><mi>a</mi></mrow></munderover><mfrac><mi>Q</mi><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow></mstyle></math>, Response: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>Q</mi><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>
<p align="left">In order to find the electric field, one needs to evaluate an integral that adds up the contributions from&nbsp;each infinitesimal ring that<br />
makes up the disc, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dr</mi></mrow></mstyle></math>, from <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>0</mn></mrow></mstyle></math> to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$aMag$aLet</mi></mrow></mstyle></math>.&nbsp;&nbsp;&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="585" height="500">
<param name="image" value="__BASE_URI__img/ElectricFields/DiscCharge/DiagramDetailed.png" />
<param name="size" value="3" />
<param name="label.1.x" value="25" />
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<param name="label.2.x" value="345" />
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<param name="label.2.text" value="$aMag$aLet" />
<param name="label.3.x" value="270" />
<param name="label.3.y" value="250" />
<param name="label.3.text" value="x" /></applet></p>
<p align="center">&nbsp;</p>
<p><em>Note: </em>Use the letter 'k' for the Coulomb constant, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mrow><mi>&pi;</mi></mrow><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mfrac></mrow></mstyle></math>.</p>@
qu.4.7.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSA2,subPackage=vectorAlgebraicComponents)@
qu.4.7.allow2d=0@
qu.4.7.maple_answer=with(Physics[Vectors]):
subs(_i=ihat,_j=jhat,_k=khat,simplify(PhysFuncs[ElectricField](($QSign)*(x*Q/($aMag*$aLet)),r*_j,x*_i,constant=k)._i)*_i);@
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qu.4.7.type=maple@
qu.4.7.mode=Maple@
qu.4.7.name=Disc Charge - Find E - Set Up Integral@
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qu.4.7.editing=useHTML@
qu.4.7.solution=<p>Start by identifying a small ring of charge on the disc at a radius of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dr</mi></mrow></mstyle></math>.&nbsp; Then the small ring will produce&nbsp;an electric field:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi><mover><mrow><mi>E</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mi>kx</mi></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mrow><mfrac><mn>3</mn><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></mfrac><mi>dQ</mi><mover><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math>.</p>
<p>Since the charge is uniformly distributed on the disc, the charge per unit area is constant, so:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mi>dQ</mi></mrow><mrow><mi>dr</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac></mrow></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mi>dQ</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac><mi>d</mi></mrow><mrow><mi mathvariant='normal'>r</mi></mrow></mstyle></math>.</p>
<p>Adding up the contributions from each unit of charge on the ring:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></munderover><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>r</mi></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mrow><mfrac><mn>3</mn><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></mfrac></mrow></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mover><mrow><mi>x</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math></p>@
qu.4.7.algorithm=$aMag=rint(2,10);
$aLet=switch(rint(8),'a','b','c','d','f','g','h','w');
$idx=rint(2);
$QSign=switch($idx,1,-1);
$QSignLabel=switch($idx,"+","-");
$QMag=rint(2,6);@
qu.4.7.uid=0ec66d60-1135-404f-961b-466732331b06@
qu.4.7.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Electric Field Due to A Line of Charge;
  Difficulty=Hard;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Diagram;
@

qu.4.8.question=<p>A thin, uniform&nbsp;line of charge extends from&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$aMag$aLet</mi></mrow></mstyle></math>&nbsp;to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$aMag$aLet</mi></mrow></mstyle></math>&nbsp;along the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi></mrow></mstyle></math>-axis&nbsp;and&nbsp;contains a total charge <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$QSignLabel</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow></mstyle></math>.</p>
<p>Set up&nbsp;an integral for the&nbsp;electric field at an arbitrary point along the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi></mrow></mstyle></math>-axis due to the line of charge.&nbsp; Use symmetry to reduce the integral<br />
to a single expression.&nbsp; Provide the <em>integrand</em> of this integral as your answer, in terms of the given parameters.</p>
<p align="center">e.g. Integral: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathcolor='#800080'>a</mi></mrow><mrow><mi>a</mi></mrow></munderover><mfrac><mi>Q</mi><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow></mstyle></math>, Integrand: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>Q</mi><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>
<p>In order to find the electric field, one needs to evaluate an integral that adds up the contributions from each infinitesimal piece of<br />
the line, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dy</mi></mrow></mstyle></math>, from <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$aMag$aLet</mi></mrow></mstyle></math> to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$aMag$aLet</mi></mrow></mstyle></math>.&nbsp;</p>
<p align="center">&nbsp;</p>
<p align="center">&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="585" height="500">
<param name="image" value="__BASE_URI__img/ElectricFields/LineCharge/Diagram.png" />
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<param name="label.4.text" value="x" /></applet></p>
<p align="center">&nbsp;</p>
<p><em>Note: </em>Use the letter 'k' for the Coulomb constant, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mrow><mi>&pi;</mi></mrow><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mfrac></mrow></mstyle></math>.</p>@
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qu.4.8.allow2d=0@
qu.4.8.maple_answer=with(Physics[Vectors]):
subs(_i=ihat,_j=jhat,_k=khat,collect((PhysFuncs[ElectricField](($QSign)*(Q/(2*$aMag*$aLet)),y*_j,x*_i,constant=k)._i)*_i,[_i,_j,_k]))@
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qu.4.8.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSA2,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.4.8.editing=useHTML@
qu.4.8.solution=<p>Start by identifying a small unit of charge on the line, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dq</mi></mrow></mstyle></math>.&nbsp; Then the small amount of electric field due to that unit of charge is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi><mover><mi>E</mi><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi>kdq</mi><mrow><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mrow><mover><mi>r</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msqrt><mrow><msup><mi>y</mi><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>Since the charge is uniformly distributed on the line, the charge per unit length is constant, so:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>dq</mi><mrow><mi>dy</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac></mrow></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mi>dq</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac><mi>dy</mi></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>.</p>
<p>Finally, breaking into components:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>r</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mover><mrow><mi>r</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mi>r</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mrow><mi>x</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>y</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mrow><mi>y</mi></mrow><mi>&#x005e;</mi></mover></mrow><mrow><mi>r</mi></mrow></mfrac></mrow></mstyle></math></p>
<p>Recognizing that the y-components for units of charge above and below the x-axis will cancel, we can conclude that the y-component of the total electric field is zero.</p>
<p>Thus, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>dE</mi><mrow><mi>x</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mn>2</mn><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mi>dy</mi><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mrow><mfrac><mn>3</mn><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></mfrac></mrow></mfenced></mrow></mstyle></math>.</p>
<p>Adding up the contributions from each unit of charge on the line:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$aMag$aLet</mi></mrow><mrow><mi>$aMag$aLet</mi></mrow></munderover><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mn>2</mn><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mrow><mfrac><mn>3</mn><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></mfrac></mrow></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mover><mrow><mi>x</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math></p>@
qu.4.8.algorithm=$aMag=rint(2,10);
$aLet=switch(rint(8),'a','b','c','d','f','g','h','w');
$idx=rint(2);
$QSign=switch($idx,1,-1);
$QSignLabel=switch($idx,"+","-");
$QMag=rint(2,6);@
qu.4.8.uid=6c17de1b-a818-45b3-94ec-659ebb27bf74@
qu.4.8.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Electric Field Due to A Line of Charge;
  Difficulty=Hard;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Diagram;
@

qu.4.9.question=<p align="left">Point charges $dir1Label q and $dir2Label q are positioned as shown.&nbsp;&nbsp;<br />
<br />
Find an algebraic expression for the electric field at point P, in terms of the given parameters.</p>
<p align="center">&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="395" height="400">
<param name="image" value="__BASE_URI__img/ElectricFields/E-PtChr-2Chrg-2D-Sym/Diagram.png" />
<param name="size" value="10" />
<param name="label.1.x" value="20" />
<param name="label.1.y" value="105" />
<param name="label.1.text" value="q" />
<param name="label.2.x" value="10" />
<param name="label.2.y" value="105" />
<param name="label.2.text" value="$dir1Label" />
<param name="label.3.x" value="20" />
<param name="label.3.y" value="317" />
<param name="label.3.text" value="q" />
<param name="label.4.x" value="10" />
<param name="label.4.y" value="317" />
<param name="label.4.text" value="$dir2Label" />
<param name="label.5.x" value="50" />
<param name="label.5.y" value="155" />
<param name="label.5.text" value="a" />
<param name="label.6.x" value="40" />
<param name="label.6.y" value="155" />
<param name="label.6.text" value="$amag" />
<param name="label.7.x" value="50" />
<param name="label.7.y" value="265" />
<param name="label.7.text" value="a" />
<param name="label.8.x" value="40" />
<param name="label.8.y" value="265" />
<param name="label.8.text" value="$amag" />
<param name="label.9.x" value="160" />
<param name="label.9.y" value="200" />
<param name="label.9.text" value="x" />
<param name="label.10.x" value="150" />
<param name="label.10.y" value="200" />
<param name="label.10.text" value="$xmag" /></applet></p>
<p align="center">&nbsp;</p>
<p align="left"><em>Entry Notes</em>: <br />
<br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
If necessary, use the letter&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi></mrow></mstyle></math> to represent the Coulomb constant <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mrow><mi>&pi;</mi></mrow><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mfrac></mrow></mrow></mstyle></math>.</p>@
qu.4.9.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=vectorAlgebraicComponents);@
qu.4.9.allow2d=0@
qu.4.9.maple_answer=subs(_i=ihat,_j=jhat,_k=khat,collect(PhysFuncs[ElectricField]($dir1*q,$amag*a*_j,$xmag*x*_i,constant=k)+PhysFuncs[ElectricField]($dir2*q,-$amag*a*_j,$xmag*x*_i,constant=k),[_i,_j,_k]))@
qu.4.9.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.4.9.type=maple@
qu.4.9.mode=Maple@
qu.4.9.name=Electric Field Due to Point Charges - 2 Charges, 2D, Symmetric@
qu.4.9.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.4.9.editing=useHTML@
qu.4.9.hint.1=Remember that the superposition principle means that the electric field at a point is the sum of the electric fields produced by each of the nearby charges.&nbsp; Pretend that only one charge at a time&nbsp;is present and calculate the field due to that charge at the point.&nbsp; Finally, add all of the electric fields together to get the total.@
qu.4.9.hint.2=You can use symmetry to save some calculation time here.@
qu.4.9.solution=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow><mrow><mi>top</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$dir1Label</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>q</mi></mrow></mfenced></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>$amag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$xmag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow><mrow><mover><mrow><mi>r</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math></p>
<p>The x-component is</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>E</mi><mrow><msub><mi>top</mi><mrow><mi>X</mi></mrow></msub></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><msub><mover><mi>E</mi><mi>&rarr;</mi></mover><mrow><mi>top</mi></mrow></msub></mfenced></mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mi>cos</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mrow></mstyle></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$xmag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow><mrow><msqrt><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>$amag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$xmag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac></mrow></mstyle></math></p>
<p>and the y-component is</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>E</mi><mrow><msub><mi>top</mi><mrow><mi>Y</mi></mrow></msub></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><msub><mover><mi>E</mi><mi>&rarr;</mi></mover><mrow><mi>top</mi></mrow></msub></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow><mrow></mrow></mstyle></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mi>sin</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$amag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow><mrow><msqrt><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>$amag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$xmag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac></mrow></mstyle></math></p>
<p>Since the two charges have the same magnitude and are the same distance away from P, the magnitudes of their electric fields at P will be the same.&nbsp; One component of the electric field at P will therefore cancel, while the other will be the sum of the contributions from the two charges..&nbsp; We can use this symmetry to save time in the calculation.&nbsp; Alternatively, you can calculate both components due to each charge and add them, which will yield the same result.</p>@
qu.4.9.algorithm=$amag=rint(2,5);
$xmag=rint(2,5);
$idx1=rint(2);
$dir1=switch($idx1,1,-1);
$dir1Label=switch($idx1,"+","-");
$idx2=rint(2);
$dir2=switch($idx2,1,-1);
$dir2Label=switch($idx2,"+","-");@
qu.4.9.uid=645c5efc-c47f-4c26-9541-08038e1905cd@
qu.4.9.info=  Course=Introductory Electricity and Magnetism;
  Topic=Electric Field Due to Point Charges in 2D;
  Author=Aron Pasieka;
  Difficulty=Medium;
  Features=Algebraic;
  Features=Algorithmic;
  Features=Partial Grading;
  Features=Diagram;
@

qu.5.topic=Integration@

qu.5.1.question=<p>A thin ring of radius&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$aMag$aLet</mi></mrow></mstyle></math>&nbsp;is centered at the origin in the&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mover><mi>z</mi><mi>&#x005e;</mi></mover></mrow></mstyle></math>&nbsp;plane.&nbsp;&nbsp;A&nbsp;charge<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$QSignLabel</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow></mstyle></math>&nbsp;is uniformly distributed on the ring.&nbsp;</p>
<p>Use symmetry to set up a single&nbsp;expression&nbsp;for the&nbsp;electric potential at an&nbsp;arbitrary point along the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi></mrow></mstyle></math>-axis due to the ring of charge.<br />
Provide&nbsp;the <em>integrand</em> of this integral as your answer, in terms of the given parameters.</p>
<p align="center">e.g.&nbsp; Integral:&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathcolor='#800080'>a</mi></mrow><mrow><mi>a</mi></mrow></munderover><mfrac><mi>Q</mi><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow></mstyle></math>,&nbsp; Response:&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>Q</mi><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mstyle></math></p>
<p>In order to find the electric potential, one needs to evaluate an integral that adds up the contributions from each infinitesimal piece<br />
of the line, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ds</mi></mrow></mstyle></math>, from <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>0</mn></mrow></mstyle></math> to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&pi;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mstyle></math>.</p>
<p align="center">&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="585" height="500">
<param name="image" value="__BASE_URI__img/ElectricFields/RingCharge/Diagram.png" />
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<param name="label.3.text" value="x" /></applet></p>
<p align="center">&nbsp;</p>
<p><em>Note: </em>Use the letter 'k' for the Coulomb constant, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mrow><mi>&pi;</mi></mrow><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mfrac></mrow></mstyle></math>.</p>@
qu.5.1.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSA2,subPackage=vectorAlgebraicComponents)@
qu.5.1.allow2d=0@
qu.5.1.maple_answer=k*($QSign)*(Q/(2*Pi*$aMag*$aLet))/sqrt(($aMag*$aLet)^2+x^2)@
qu.5.1.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.5.1.type=maple@
qu.5.1.mode=Maple@
qu.5.1.name=Ring Charge - Find V - Set Up Integral@
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qu.5.1.editing=useHTML@
qu.5.1.solution=<p>Start by identifying a small unit of charge on the ring, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dq</mi></mrow></mstyle></math>.&nbsp; Then the small amount of electric potential due to that unit of charge is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>kdq</mi><mrow><mi>r</mi></mrow></mfrac></mrow></mrow></mstyle></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msqrt><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>Since the charge is uniformly distributed on the ring, the charge per unit length is constant, so:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>dq</mi><mrow><mi>ds</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac></mrow></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mi>dq</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac><mi>ds</mi></mrow><mrow></mrow></mstyle></math>.</p>
<p>Adding up the contributions from each unit of charge on the ring:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi>s</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></munderover><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mi>&pi;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>s</mi></mrow><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac></mrow></mstyle></math></p>@
qu.5.1.algorithm=$aMag=rint(2,10);
$aLet=switch(rint(8),'a','b','c','d','f','g','h','w');
$idx=rint(2);
$QSign=switch($idx,1,-1);
$QSignLabel=switch($idx,"+","-");
$QMag=rint(2,6);@
qu.5.1.uid=b152c087-4167-4efb-9049-8e07a72ed227@
qu.5.1.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Electric Potential Due to A Line of Charge;
  Difficulty=Hard;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Diagram;
@

qu.5.2.question=<p>A thin, uniform&nbsp;line of charge extends from&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$aMag$aLet</mi></mrow></mstyle></math>&nbsp;to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$aMag$aLet</mi></mrow></mstyle></math>&nbsp;along the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi></mrow></mstyle></math>-axis&nbsp;and&nbsp;contains a total charge <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$QSignLabel</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow></mstyle></math>.</p>
<p>Set up&nbsp;an integral for the&nbsp;electric potential at an arbitrary point along the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi></mrow></mstyle></math>-axis due to the line of charge.&nbsp; Provide the <em>integrand<br />
</em>of this integral as your answer, in terms of the given parameters.</p>
<p align="center">e.g. Integral: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathcolor='#800080'>a</mi></mrow><mrow><mi>a</mi></mrow></munderover><mfrac><mi>Q</mi><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>, Integrand: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>Q</mi><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mstyle></math></p>
<p>In order to find the electric potential, one needs to evaluate an integral that adds up the contributions from each infinitesimal piece of<br />
the line, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dy</mi></mrow></mstyle></math>, from <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$aMag$aLet</mi></mrow></mstyle></math> to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$aMag$aLet</mi></mrow></mstyle></math>.&nbsp;</p>
<p align="center">&nbsp;</p>
<p align="center">&nbsp;</p>
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<param name="label.4.text" value="x" /></applet></p>
<p align="center">&nbsp;</p>
<p><em>Note: </em>Use the letter 'k' for the Coulomb constant, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mrow><mi>&pi;</mi></mrow><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mfrac></mrow></mstyle></math>.</p>@
qu.5.2.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSA2,subPackage=vectorAlgebraicComponents)@
qu.5.2.allow2d=0@
qu.5.2.maple_answer=k*($QSign)*(Q/(2*$aMag*$aLet))/sqrt(x^2+y^2);@
qu.5.2.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.5.2.type=maple@
qu.5.2.mode=Maple@
qu.5.2.name=Line Charge - Find V - Set Up Integral@
qu.5.2.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSA2,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.5.2.editing=useHTML@
qu.5.2.solution=<p>Start by identifying a small unit of charge on the line, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dq</mi></mrow></mstyle></math>.&nbsp; Then the small amount of potential&nbsp;due to that unit of charge is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>kdq</mi><mrow><mi>r</mi></mrow></mfrac></mrow></mrow></mstyle></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msqrt><mrow><msup><mi>y</mi><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>Since the charge is uniformly distributed on the line, the charge per unit length is constant, so:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>dq</mi><mrow><mi>dy</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac></mrow></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mi>dq</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac><mi>dy</mi></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>.</p>
<p>Adding up the contributions from each unit of charge on the line:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$aMag$aLet</mi></mrow><mrow><mi>$aMag$aLet</mi></mrow></munderover><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mrow><mi></mi></mrow></msup></mrow></mfrac></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mi></mi></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math></p>@
qu.5.2.algorithm=$aMag=rint(2,10);
$aLet=switch(rint(8),'a','b','c','d','f','g','h','w');
$idx=rint(2);
$QSign=switch($idx,1,-1);
$QSignLabel=switch($idx,"+","-");
$QMag=rint(2,6);@
qu.5.2.uid=b0985b64-3eac-40de-9bc1-c3770a52029d@
qu.5.2.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Electric Potential Due to A Line of Charge;
  Difficulty=Hard;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Diagram;
@

qu.5.3.question=<p>A thin ring of radius&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$aMag$aLet</mi></mrow></mstyle></math>&nbsp;is centered at the origin in the&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mover><mi>z</mi><mi>&#x005e;</mi></mover></mrow></mstyle></math>&nbsp;plane.&nbsp;&nbsp;A&nbsp;charge<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$QSignLabel</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow></mstyle></math>&nbsp;is uniformly distributed on the ring.&nbsp;</p>
<p>Use symmetry to set up a single&nbsp;expression&nbsp;for the&nbsp;electric field at an&nbsp;arbitrary point along the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi></mrow></mstyle></math>-axis due to the ring of charge.<br />
Provide&nbsp;the <em>integrand</em> of this integral as your answer, in terms of the given parameters.</p>
<p align="center">e.g.&nbsp; Integral:&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathcolor='#800080'>a</mi></mrow><mrow><mi>a</mi></mrow></munderover><mfrac><mi>Q</mi><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow></mstyle></math>,&nbsp; Response:&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>Q</mi><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>
<p>In order to find the electric field, one needs to evaluate an integral that adds up the contributions from each infinitesimal piece<br />
of the line, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ds</mi></mrow></mstyle></math>, from <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>0</mn></mrow></mstyle></math> to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&pi;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mstyle></math>.</p>
<p align="center">&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="585" height="500">
<param name="image" value="__BASE_URI__img/ElectricFields/RingCharge/Diagram.png" />
<param name="size" value="3" />
<param name="label.1.x" value="25" />
<param name="label.1.y" value="300" />
<param name="label.1.text" value="$QSignLabel Q" />
<param name="label.2.x" value="220" />
<param name="label.2.y" value="22" />
<param name="label.2.text" value="$aMag$aLet" />
<param name="label.3.x" value="270" />
<param name="label.3.y" value="250" />
<param name="label.3.text" value="x" /></applet></p>
<p align="center">&nbsp;</p>
<p>&nbsp;</p>
<p><em>Note: </em>Use the letter 'k' for the Coulomb constant, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mrow><mi>&pi;</mi><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mrow></mfrac></mrow></mstyle></math>.</p>@
qu.5.3.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSA2,subPackage=vectorAlgebraicComponents)@
qu.5.3.allow2d=0@
qu.5.3.maple_answer=with(Physics[Vectors]):
subs(_i=ihat,_j=jhat,_k=khat,collect((PhysFuncs[ElectricField]($QSign*(Q/(2*Pi*$aMag*$aLet)),$aMag*$aLet*_j,x*_i,constant=k)._i)*_i,[_i,_j,_k]))@
qu.5.3.libname=__BASE_URI__repositories/PartialGrading.lib@
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qu.5.3.mode=Maple@
qu.5.3.name=Ring Charge - Find E - Set Up Integral@
qu.5.3.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSA2,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.5.3.editing=useHTML@
qu.5.3.solution=<p>Start by identifying a small unit of charge on the ring, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dq</mi></mrow></mstyle></math>.&nbsp; Then the small amount of electric field due to that unit of charge is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi><mover><mi>E</mi><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi>kdq</mi><mrow><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mrow><mover><mi>r</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msqrt><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>Since the charge is uniformly distributed on the ring, the charge per unit length is constant, so:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>dq</mi><mrow><mi>ds</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac></mrow></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mi>dq</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac><mi>ds</mi></mrow><mrow></mrow></mstyle></math>.</p>
<p>Finally, breaking into components:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>r</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mover><mrow><mi>r</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mi>r</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mrow><mi>x</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mrow><mi>y</mi></mrow><mi>&#x005e;</mi></mover></mrow><mrow><mi>r</mi></mrow></mfrac></mrow></mstyle></math></p>
<p>Recognizing that the y- and z-components for units of charge&nbsp;diametrically opposite to eachother&nbsp;will cancel, we can conclude that the y- and z-component of the total electric field is zero.</p>
<p>Thus, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>dE</mi><mrow><mi>x</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>ds</mi></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mrow><mfrac><mn>3</mn><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></mfrac></mrow></mfenced></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>.</p>
<p>Adding up the contributions from each unit of charge on the ring:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi>s</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></munderover><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>s</mi></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mrow><mfrac><mn>3</mn><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></mfrac></mrow></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mover><mrow><mi>x</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math></p>@
qu.5.3.algorithm=$aMag=rint(2,10);
$aLet=switch(rint(8),'a','b','c','d','f','g','h','w');
$idx=rint(2);
$QSign=switch($idx,1,-1);
$QSignLabel=switch($idx,"+","-");
$QMag=rint(2,6);@
qu.5.3.uid=017e63c6-1d4f-4cb9-a36c-08d34fb73762@
qu.5.3.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Electric Field Due to A Line of Charge;
  Difficulty=Hard;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Diagram;
@

qu.5.4.question=<p>A thin ring of radius&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$aMag$aLet</mi></mrow></mstyle></math>&nbsp;is centered at the origin in the&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mover><mi>z</mi><mi>&#x005e;</mi></mover></mrow></mstyle></math>&nbsp;plane.&nbsp;&nbsp;A&nbsp;charge<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$QSignLabel</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow></mstyle></math>&nbsp;is uniformly distributed on the ring.&nbsp;</p>
<p>Use symmetry to set up a single&nbsp;expression&nbsp;for the&nbsp;electric force due to the ring of charge on a point charge <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$qMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>q</mi></mrow></mstyle></math>&nbsp;at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math>.<br />
Provide&nbsp;the <em>integrand</em> of this integral as your answer, in terms of the given parameters.</p>
<p align="center">e.g.&nbsp; Integral:&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathcolor='#800080'>a</mi></mrow><mrow><mi>a</mi></mrow></munderover><mfrac><mi>Q</mi><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow></mstyle></math>,&nbsp; Response:&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>Q</mi><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>
<p>In order to find the electric force, one needs to evaluate an integral that adds up the contributions from each infinitesimal piece<br />
of the line, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ds</mi></mrow></mstyle></math>, from <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>0</mn></mrow></mstyle></math> to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&pi;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mstyle></math>.</p>
<p align="center">&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="585" height="500">
<param name="image" value="__BASE_URI__img/ElectricFields/RingCharge/Diagram.png" />
<param name="size" value="3" />
<param name="label.1.x" value="25" />
<param name="label.1.y" value="300" />
<param name="label.1.text" value="$QSignLabel Q" />
<param name="label.2.x" value="220" />
<param name="label.2.y" value="22" />
<param name="label.2.text" value="$aMag$aLet" />
<param name="label.3.x" value="270" />
<param name="label.3.y" value="250" />
<param name="label.3.text" value="x" /></applet></p>
<p align="center">&nbsp;</p>
<p>&nbsp;</p>
<p><em>Note:&nbsp; </em>Use the letter 'k' for the Coulomb constant, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mrow><mi>&pi;</mi></mrow><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mfrac></mrow></mstyle></math>.</p>@
qu.5.4.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSA2,subPackage=vectorAlgebraicComponents)@
qu.5.4.allow2d=0@
qu.5.4.maple_answer=with(Physics[Vectors]):
subs(_i=ihat,_j=jhat,_k=khat,collect((PhysFuncs[ElectricField](($qMag)*q*($QSign)*(Q/(2*Pi*$aMag*$aLet)),$aMag*$aLet*_j,x*_i,constant=k)._i)*_i,[_i,_j,_k]))@
qu.5.4.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.5.4.type=maple@
qu.5.4.mode=Maple@
qu.5.4.name=Ring Charge - Find F - Set Up Integral@
qu.5.4.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSA2,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.5.4.editing=useHTML@
qu.5.4.solution=<p>Start by identifying a small unit of charge on the ring, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dq</mi></mrow></mstyle></math>.&nbsp; Then the small amount of electric force due to that unit of charge is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$qMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>q</mi></mrow></mfenced><mi>dq</mi></mrow><mrow><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mrow><mover><mi>r</mi><mi>&#x005e;</mi></mover></mrow></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msqrt><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>Since the charge is uniformly distributed on the ring, the charge per unit length is constant, so:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>dq</mi><mrow><mi>ds</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac></mrow></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mi>dq</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac><mi>ds</mi></mrow><mrow></mrow></mstyle></math>.</p>
<p>Finally, breaking into components:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>r</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mover><mrow><mi>r</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mi>r</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mrow><mi>x</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mrow><mi>y</mi></mrow><mi>&#x005e;</mi></mover></mrow><mrow><mi>r</mi></mrow></mfrac></mrow></mstyle></math></p>
<p>Recognizing that the y- and z-components for units of charge diametrically opposite will cancel, we can conclude that the y- and z-component of the total electric force is zero.</p>
<p>Thus, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>dF</mi><mrow><mi>x</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$qMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>q</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mn>2</mn><mi>&pi;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>ds</mi></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mrow><mfrac><mn>3</mn><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></mfrac></mrow></mfenced></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>.</p>
<p>Adding up the contributions from each unit of charge on the ring:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mover><mi>F</mi><mi>&rarr;</mi></mover></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi>s</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></munderover><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$qMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>q</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mrow><mfrac><mn>3</mn><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></mfrac></mrow></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mover><mrow><mi>x</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math></p>@
qu.5.4.algorithm=$aMag=rint(2,10);
$aLet=switch(rint(8),'a','b','c','d','f','g','h','w');
$idx=rint(2);
$QSign=switch($idx,1,-1);
$QSignLabel=switch($idx,"+","-");
$QMag=rint(2,6);
$qMag=switch(rint(2),rint(2,6),-rint(2,6));@
qu.5.4.uid=8f94a274-a913-4a98-9058-c53170ffe100@
qu.5.4.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Force Due to A Line of Charge;
  Difficulty=Hard;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Diagram;
  Features=Partial Grading;
@

qu.5.5.question=<p>A&nbsp;solid disc&nbsp;of radius&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$aMag$aLet</mi></mrow></mstyle></math>&nbsp;is centered at the origin in the&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mover><mi>z</mi><mi>&#x005e;</mi></mover></mrow></mstyle></math>&nbsp;plane.&nbsp;&nbsp;A&nbsp;charge<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$QSignLabel</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow></mstyle></math>&nbsp;is uniformly distributed on the disc.</p>
<p>Use symmetry to set up a single&nbsp;expression&nbsp;for the&nbsp;electric force&nbsp;due to the disc of charge on a point charge <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$qMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>q</mi></mrow></mstyle></math>&nbsp;at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math>.<br />
Use the fact&nbsp;that one can treat the disc as an infinite number of thin rings, and the fact that the electric force due to a&nbsp;single thin<br />
ring of radius <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>R</mi></mrow></mstyle></math>&nbsp;is:</p>
<p align="center"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi><mfrac><mrow><mi>dQ</mi><mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo></mrow><mi>x</mi></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>R</mi><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mrow><mfrac><mn>3</mn><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></mfrac><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.</p>
<p align="left">Provide the <em>integrand</em> of this integral as your answer, in terms of the given parameters.</p>
<p align="center">e.g. Integral: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathcolor='#800080'>a</mi></mrow><mrow><mi>a</mi></mrow></munderover><mfrac><mi>Q</mi><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow></mstyle></math>, Response: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>Q</mi><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>
<p align="left">In order to find the electric force, one needs to evaluate an integral that adds up the contributions from&nbsp;each infinitesimal ring that<br />
makes up the disc, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dr</mi></mrow></mstyle></math>, from <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>0</mn></mrow></mstyle></math> to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$aMag$aLet</mi></mrow></mstyle></math>.&nbsp;&nbsp;&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="585" height="500">
<param name="image" value="__BASE_URI__img/ElectricFields/DiscCharge/DiagramDetailed.png" />
<param name="size" value="3" />
<param name="label.1.x" value="25" />
<param name="label.1.y" value="300" />
<param name="label.1.text" value="$QSignLabel Q" />
<param name="label.2.x" value="345" />
<param name="label.2.y" value="80" />
<param name="label.2.text" value="$aMag$aLet" />
<param name="label.3.x" value="270" />
<param name="label.3.y" value="250" />
<param name="label.3.text" value="x" /></applet></p>
<p align="center">&nbsp;</p>
<p><em>Note: </em>Use the letter 'k' for the Coulomb constant, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mrow><mi>&pi;</mi></mrow><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mfrac></mrow></mstyle></math>.</p>@
qu.5.5.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSA2,subPackage=vectorAlgebraicComponents)@
qu.5.5.allow2d=0@
qu.5.5.maple_answer=with(Physics[Vectors]):
subs(_i=ihat,_j=jhat,_k=khat,simplify(PhysFuncs[ElectricField](($qMag)*q*($QSign)*(x*Q/($aMag*$aLet)),r*_j,x*_i,constant=k)._i)*_i);@
qu.5.5.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.5.5.type=maple@
qu.5.5.mode=Maple@
qu.5.5.name=Disc Charge - Find F - Set Up Integral@
qu.5.5.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSA2,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.5.5.editing=useHTML@
qu.5.5.solution=<p>Start by identifying a small ring of charge on the disc at a radius of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dr</mi></mrow></mstyle></math>.&nbsp; Then the small ring will produce&nbsp;an electric force:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$qMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>q</mi></mrow></mfenced><mi>x</mi></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mrow><mfrac><mn>3</mn><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></mfrac><mi>dQ</mi><mover><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math>.</p>
<p>Since the charge is uniformly distributed on the disc, the charge per unit area is constant, so:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mi>dQ</mi></mrow><mrow><mi>dr</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac></mrow></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mi>dQ</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac><mi>d</mi></mrow><mrow><mi mathvariant='normal'>r</mi></mrow></mstyle></math>.</p>
<p>Adding up the contributions from each unit of charge on the ring:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></munderover><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$qMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>q</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>r</mi></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mrow><mfrac><mn>3</mn><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></mfrac></mrow></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mover><mrow><mi>x</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math></p>@
qu.5.5.algorithm=$aMag=rint(2,10);
$aLet=switch(rint(8),'a','b','c','d','f','g','h','w');
$idx=rint(2);
$QSign=switch($idx,1,-1);
$QSignLabel=switch($idx,"+","-");
$QMag=rint(2,6);
$qMag=switch(rint(2),rint(2,6),-rint(2,6));@
qu.5.5.uid=e6c7c55b-7950-4e53-ba8b-9ece7da3470a@
qu.5.5.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Force Due to A Line of Charge;
  Difficulty=Hard;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Diagram;
  Features=Partial Grading;
@

qu.5.6.question=<p>A&nbsp;solid disc&nbsp;of radius&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$aMag$aLet</mi></mrow></mstyle></math>&nbsp;is centered at the origin in the&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mover><mi>z</mi><mi>&#x005e;</mi></mover></mrow></mstyle></math>&nbsp;plane.&nbsp;&nbsp;A&nbsp;charge<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$QSignLabel</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow></mstyle></math>&nbsp;is uniformly distributed on the disc.</p>
<p>Use symmetry to set up a single&nbsp;expression&nbsp;for the&nbsp;electric field at an arbitrary point along the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi></mrow></mstyle></math>-axis due to the ring of charge.<br />
Use the fact&nbsp;that one can treat the disc as an infinite number of thin rings, and the fact that the electric field due to a&nbsp;single thin<br />
ring of radius <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>R</mi></mrow></mstyle></math>&nbsp;is:</p>
<p align="center"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi><mfrac><mrow><mi>dQ</mi><mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo></mrow><mi>x</mi></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>R</mi><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mrow><mfrac><mn>3</mn><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></mfrac><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.</p>
<p align="left">Provide the <em>integrand</em> of this integral as your answer, in terms of the given parameters.</p>
<p align="center">e.g. Integral: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathcolor='#800080'>a</mi></mrow><mrow><mi>a</mi></mrow></munderover><mfrac><mi>Q</mi><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow></mstyle></math>, Response: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>Q</mi><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>
<p align="left">In order to find the electric field, one needs to evaluate an integral that adds up the contributions from&nbsp;each infinitesimal ring that<br />
makes up the disc, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dr</mi></mrow></mstyle></math>, from <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>0</mn></mrow></mstyle></math> to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$aMag$aLet</mi></mrow></mstyle></math>.&nbsp;&nbsp;&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="585" height="500">
<param name="image" value="__BASE_URI__img/ElectricFields/DiscCharge/DiagramDetailed.png" />
<param name="size" value="3" />
<param name="label.1.x" value="25" />
<param name="label.1.y" value="300" />
<param name="label.1.text" value="$QSignLabel Q" />
<param name="label.2.x" value="345" />
<param name="label.2.y" value="80" />
<param name="label.2.text" value="$aMag$aLet" />
<param name="label.3.x" value="270" />
<param name="label.3.y" value="250" />
<param name="label.3.text" value="x" /></applet></p>
<p align="center">&nbsp;</p>
<p><em>Note: </em>Use the letter 'k' for the Coulomb constant, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mrow><mi>&pi;</mi></mrow><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mfrac></mrow></mstyle></math>.</p>@
qu.5.6.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSA2,subPackage=vectorAlgebraicComponents)@
qu.5.6.allow2d=0@
qu.5.6.maple_answer=with(Physics[Vectors]):
subs(_i=ihat,_j=jhat,_k=khat,simplify(PhysFuncs[ElectricField](($QSign)*(x*Q/($aMag*$aLet)),r*_j,x*_i,constant=k)._i)*_i);@
qu.5.6.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.5.6.type=maple@
qu.5.6.mode=Maple@
qu.5.6.name=Disc Charge - Find E - Set Up Integral@
qu.5.6.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSA2,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.5.6.editing=useHTML@
qu.5.6.solution=<p>Start by identifying a small ring of charge on the disc at a radius of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dr</mi></mrow></mstyle></math>.&nbsp; Then the small ring will produce&nbsp;an electric field:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi><mover><mrow><mi>E</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mi>kx</mi></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mrow><mfrac><mn>3</mn><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></mfrac><mi>dQ</mi><mover><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math>.</p>
<p>Since the charge is uniformly distributed on the disc, the charge per unit area is constant, so:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mi>dQ</mi></mrow><mrow><mi>dr</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac></mrow></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mi>dQ</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac><mi>d</mi></mrow><mrow><mi mathvariant='normal'>r</mi></mrow></mstyle></math>.</p>
<p>Adding up the contributions from each unit of charge on the ring:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></munderover><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>r</mi></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mrow><mfrac><mn>3</mn><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></mfrac></mrow></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mover><mrow><mi>x</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math></p>@
qu.5.6.algorithm=$aMag=rint(2,10);
$aLet=switch(rint(8),'a','b','c','d','f','g','h','w');
$idx=rint(2);
$QSign=switch($idx,1,-1);
$QSignLabel=switch($idx,"+","-");
$QMag=rint(2,6);@
qu.5.6.uid=0ec66d60-1135-404f-961b-466732331b06@
qu.5.6.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Electric Field Due to A Line of Charge;
  Difficulty=Hard;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Diagram;
@

qu.5.7.question=<p>A thin, uniform&nbsp;line of charge extends from&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$aMag$aLet</mi></mrow></mstyle></math>&nbsp;to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$aMag$aLet</mi></mrow></mstyle></math>&nbsp;along the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi></mrow></mstyle></math>-axis&nbsp;and&nbsp;contains a total charge <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$QSignLabel</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow></mstyle></math>.</p>
<p>Set up&nbsp;an integral for the&nbsp;electric field at an arbitrary point along the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi></mrow></mstyle></math>-axis due to the line of charge.&nbsp; Use symmetry to reduce the integral<br />
to a single expression.&nbsp; Provide the <em>integrand</em> of this integral as your answer, in terms of the given parameters.</p>
<p align="center">e.g. Integral: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathcolor='#800080'>a</mi></mrow><mrow><mi>a</mi></mrow></munderover><mfrac><mi>Q</mi><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow></mstyle></math>, Integrand: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>Q</mi><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>
<p>In order to find the electric field, one needs to evaluate an integral that adds up the contributions from each infinitesimal piece of<br />
the line, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dy</mi></mrow></mstyle></math>, from <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$aMag$aLet</mi></mrow></mstyle></math> to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$aMag$aLet</mi></mrow></mstyle></math>.&nbsp;</p>
<p align="center">&nbsp;</p>
<p align="center">&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="585" height="500">
<param name="image" value="__BASE_URI__img/ElectricFields/LineCharge/Diagram.png" />
<param name="size" value="4" />
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<param name="label.3.y" value="360" />
<param name="label.3.text" value="$aMag$aLet" />
<param name="label.4.x" value="270" />
<param name="label.4.y" value="250" />
<param name="label.4.text" value="x" /></applet></p>
<p align="center">&nbsp;</p>
<p><em>Note: </em>Use the letter 'k' for the Coulomb constant, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mrow><mi>&pi;</mi></mrow><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mfrac></mrow></mstyle></math>.</p>@
qu.5.7.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSA2,subPackage=vectorAlgebraicComponents)@
qu.5.7.allow2d=0@
qu.5.7.maple_answer=with(Physics[Vectors]):
subs(_i=ihat,_j=jhat,_k=khat,collect((PhysFuncs[ElectricField](($QSign)*(Q/(2*$aMag*$aLet)),y*_j,x*_i,constant=k)._i)*_i,[_i,_j,_k]))@
qu.5.7.libname=__BASE_URI__repositories/PartialGrading.lib@
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qu.5.7.mode=Maple@
qu.5.7.name=Line Charge - Find E - Set Up Integral@
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qu.5.7.editing=useHTML@
qu.5.7.solution=<p>Start by identifying a small unit of charge on the line, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dq</mi></mrow></mstyle></math>.&nbsp; Then the small amount of electric field due to that unit of charge is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi><mover><mi>E</mi><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi>kdq</mi><mrow><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mrow><mover><mi>r</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msqrt><mrow><msup><mi>y</mi><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>Since the charge is uniformly distributed on the line, the charge per unit length is constant, so:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>dq</mi><mrow><mi>dy</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac></mrow></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mi>dq</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac><mi>dy</mi></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>.</p>
<p>Finally, breaking into components:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>r</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mover><mrow><mi>r</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mi>r</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mrow><mi>x</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>y</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mrow><mi>y</mi></mrow><mi>&#x005e;</mi></mover></mrow><mrow><mi>r</mi></mrow></mfrac></mrow></mstyle></math></p>
<p>Recognizing that the y-components for units of charge above and below the x-axis will cancel, we can conclude that the y-component of the total electric field is zero.</p>
<p>Thus, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>dE</mi><mrow><mi>x</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mn>2</mn><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mi>dy</mi><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mrow><mfrac><mn>3</mn><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></mfrac></mrow></mfenced></mrow></mstyle></math>.</p>
<p>Adding up the contributions from each unit of charge on the line:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$aMag$aLet</mi></mrow><mrow><mi>$aMag$aLet</mi></mrow></munderover><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mn>2</mn><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mrow><mfrac><mn>3</mn><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></mfrac></mrow></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mover><mrow><mi>x</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math></p>@
qu.5.7.algorithm=$aMag=rint(2,10);
$aLet=switch(rint(8),'a','b','c','d','f','g','h','w');
$idx=rint(2);
$QSign=switch($idx,1,-1);
$QSignLabel=switch($idx,"+","-");
$QMag=rint(2,6);@
qu.5.7.uid=6c17de1b-a818-45b3-94ec-659ebb27bf74@
qu.5.7.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Electric Field Due to A Line of Charge;
  Difficulty=Hard;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Diagram;
@

qu.5.8.question=<p>A&nbsp;solid disc&nbsp;of radius&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$aMag$aLet</mi></mrow></mstyle></math>&nbsp;is centered at the origin in the&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mover><mi>z</mi><mi>&#x005e;</mi></mover></mrow></mstyle></math>&nbsp;plane.&nbsp;&nbsp;A&nbsp;charge<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$QSignLabel</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow></mstyle></math>&nbsp;is uniformly distributed on the disc.</p>
<p>Use symmetry to set up a single&nbsp;expression&nbsp;for the&nbsp;electric potential at an arbitrary point along the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi></mrow></mstyle></math>-axis due to the ring of charge.<br />
Use the fact&nbsp;that one can treat the disc as an infinite number of thin rings, and the fact that the electric field due to a&nbsp;single thin<br />
ring of radius <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>R</mi></mrow></mstyle></math>&nbsp;is:</p>
<p align="center"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi><mrow><mfrac><mi>dQ</mi><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>R</mi><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p align="left">Provide the <em>integrand</em> of this integral as your answer, in terms of the given parameters.</p>
<p align="center">e.g. Integral: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathcolor='#800080'>a</mi></mrow><mrow><mi>a</mi></mrow></munderover><mfrac><mi>Q</mi><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow></mstyle></math>, Response: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>Q</mi><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mstyle></math></p>
<p align="left">In order to find the electric potential, one needs to evaluate an integral that adds up the contributions from&nbsp;each infinitesimal ring that<br />
makes up the disc, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dr</mi></mrow></mstyle></math>, from <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>0</mn></mrow></mstyle></math> to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$aMag$aLet</mi></mrow></mstyle></math>.&nbsp;&nbsp;&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="585" height="500">
<param name="image" value="__BASE_URI__img/ElectricFields/DiscCharge/DiagramDetailed.png" />
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<param name="label.3.text" value="x" /></applet></p>
<p align="center">&nbsp;</p>
<p>&nbsp;</p>
<p><em>Note: </em>Use the letter 'k' for the Coulomb constant, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mrow><mi>&pi;</mi></mrow><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mfrac></mrow></mstyle></math>.</p>@
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qu.5.8.allow2d=0@
qu.5.8.maple_answer=k*($QSign)*(Q/($aMag*$aLet))/sqrt(r^2+x^2)@
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qu.5.8.name=Disc Charge - Find V - Set Up Integral@
qu.5.8.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSA2,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.5.8.editing=useHTML@
qu.5.8.solution=<p>Start by identifying a small ring of charge on the disc at a radius of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dr</mi></mrow></mstyle></math>.&nbsp; Then the small ring will produce&nbsp;an electric potential:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dV</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mi>k</mi></mrow><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac><mi>dQ</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>.</p>
<p>Since the charge is uniformly distributed on the disc, the charge per unit area is constant, so:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mi>dQ</mi></mrow><mrow><mi>dr</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac></mrow></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mi>dQ</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac><mi>d</mi></mrow><mrow><mi mathvariant='normal'>r</mi></mrow></mstyle></math>.</p>
<p>Adding up the contributions from each unit of charge on the ring:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></munderover><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow></mfenced></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>r</mi></mrow><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mi></mi></mrow></mstyle></math></p>@
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$aLet=switch(rint(8),'a','b','c','d','f','g','h','w');
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qu.5.8.uid=35505a14-ff76-4dc4-80a9-b275dbf68831@
qu.5.8.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Electric Potential Due to A Line of Charge;
  Difficulty=Hard;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Diagram;
@

qu.5.9.question=<p>A thin, uniform&nbsp;line of charge extends from&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$aMag$aLet</mi></mrow></mstyle></math>&nbsp;to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$aMag$aLet</mi></mrow></mstyle></math>&nbsp;along the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi></mrow></mstyle></math>-axis&nbsp;and&nbsp;contains a total charge <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$QSignLabel</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow></mstyle></math>.</p>
<p>Set up&nbsp;an integral for the&nbsp;electric force&nbsp;due to the line of charge on a charge <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$qMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>q</mi></mrow></mstyle></math>&nbsp;at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math>.&nbsp; Use symmetry to reduce the integral<br />
to a single expression.&nbsp; Provide the <em>integrand</em> of this integral as your answer, in terms of the given parameters.</p>
<p align="center">e.g. Integral: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathcolor='#800080'>a</mi></mrow><mrow><mi>a</mi></mrow></munderover><mfrac><mi>Q</mi><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow></mstyle></math>, Integrand: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>Q</mi><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>
<p>In order to find the electric force, one needs to evaluate an integral that adds up the contributions from each infinitesimal piece of<br />
the line, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dy</mi></mrow></mstyle></math>, from <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$aMag$aLet</mi></mrow></mstyle></math> to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$aMag$aLet</mi></mrow></mstyle></math>.&nbsp;</p>
<p align="center">&nbsp;</p>
<p align="center">&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="585" height="500">
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<p align="center">&nbsp;</p>
<p><em>Note: </em>Use the letter 'k' for the Coulomb constant,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mrow><mi>&pi;</mi></mrow><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mfrac></mrow></mstyle></math>.</p>@
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qu.5.9.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSA2,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.5.9.editing=useHTML@
qu.5.9.solution=<p>Start by identifying a small unit of charge on the line, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dq</mi></mrow></mstyle></math>.&nbsp; Then the small amount of electric force due to that unit of charge is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$qMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>q</mi></mrow></mfenced><mi>dq</mi></mrow><mrow><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mrow><mover><mi>r</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msqrt><mrow><msup><mi>y</mi><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>Since the charge is uniformly distributed on the line, the charge per unit length is constant, so:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>dq</mi><mrow><mi>dy</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac></mrow></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mi>dq</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac><mi>dy</mi></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>.</p>
<p>Finally, breaking into components:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>r</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mover><mrow><mi>r</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mi>r</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mrow><mi>x</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>y</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mrow><mi>y</mi></mrow><mi>&#x005e;</mi></mover></mrow><mrow><mi>r</mi></mrow></mfrac></mrow></mstyle></math></p>
<p>Recognizing that the y-components for units of charge above and below the x-axis will cancel, we can conclude that the y-component of the total force is zero.</p>
<p>Thus, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>dF</mi><mrow><mi>x</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$qMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>q</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mn>2</mn><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mi>dy</mi><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mrow><mfrac><mn>3</mn><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></mfrac></mrow></mfenced></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>.</p>
<p>Adding up the contributions from each unit of charge on the line:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$aMag$aLet</mi></mrow><mrow><mi>$aMag$aLet</mi></mrow></munderover><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$qMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>q</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mn>2</mn><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mrow><mfrac><mn>3</mn><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></mfrac></mrow></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mover><mrow><mi>x</mi></mrow><mi>&#x005e;</mi></mover></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math></p>@
qu.5.9.algorithm=$aMag=rint(2,10);
$aLet=switch(rint(8),'a','b','c','d','f','g','h','w');
$idx=rint(2);
$QSign=switch($idx,1,-1);
$QSignLabel=switch($idx,"+","-");
$QMag=rint(2,6);
$qMag=switch(rint(2),rint(2,6),-rint(2,6));@
qu.5.9.uid=46a6ddbd-e920-4923-b5d1-a2bcaedec71f@
qu.5.9.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Force Due to A Line of Charge;
  Difficulty=Hard;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Diagram;
  Features=Partial Grading;
@

qu.6.topic=Point Charges@

qu.6.1.question=<p>Find an algebraic expression for the electric potential at point P, in terms of the given parameters.</p>
<p>&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="395" height="400">
<param name="image" value="__BASE_URI__img/ElectricFields/E-PtChr-2Chrg-2D-Sym/Diagram.png" />
<param name="size" value="10" />
<param name="label.1.x" value="20" />
<param name="label.1.y" value="105" />
<param name="label.1.text" value="q" />
<param name="label.2.x" value="10" />
<param name="label.2.y" value="105" />
<param name="label.2.text" value="$q1" />
<param name="label.3.x" value="20" />
<param name="label.3.y" value="317" />
<param name="label.3.text" value="q" />
<param name="label.4.x" value="10" />
<param name="label.4.y" value="317" />
<param name="label.4.text" value="$q2" />
<param name="label.5.x" value="50" />
<param name="label.5.y" value="155" />
<param name="label.5.text" value="a" />
<param name="label.6.x" value="40" />
<param name="label.6.y" value="155" />
<param name="label.6.text" value="$a1mag" />
<param name="label.7.x" value="50" />
<param name="label.7.y" value="265" />
<param name="label.7.text" value="a" />
<param name="label.8.x" value="40" />
<param name="label.8.y" value="265" />
<param name="label.8.text" value="$a2mag" />
<param name="label.9.x" value="160" />
<param name="label.9.y" value="200" />
<param name="label.9.text" value="x" />
<param name="label.10.x" value="150" />
<param name="label.10.y" value="200" />
<param name="label.10.text" value="$xmag" /></applet></p>
<p><br />
<em>Entry Notes</em>:&nbsp;&nbsp;<br />
<br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
If necessary, use the letter <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi></mrow></mstyle></math> to represent the Coulomb constant <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mi mathvariant='normal'>&pi;</mi><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mfrac></mrow></mrow></mstyle></math>.</p>@
qu.6.1.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=vectorAlgebraicComponents);@
qu.6.1.allow2d=0@
qu.6.1.maple_answer=k*($q1)*q/sqrt(($a1mag*a)^2+($xmag*x)^2)
+k*($q2)*q/sqrt((-$a2mag*a)^2+($xmag*x)^2)@
qu.6.1.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.6.1.type=maple@
qu.6.1.mode=Maple@
qu.6.1.name=Electric Potential Due to Point Charges - 2 Charges, 2D, Different Charges@
qu.6.1.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.6.1.editing=useHTML@
qu.6.1.hint.1=Calculate the potential due to each charge separately, then add them together.@
qu.6.1.solution=<p>The electric potential due to a point charge is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>kq</mi><mrow><mi>r</mi></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>We can find the potential due to each charge separately, then add them together to find the total.</p>
<p>&nbsp;</p>
<p align="left">From both charges to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math>, we have <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$a1mag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow></mstyle></math>.</p>
<p align="left">Thus,</p>
<p align="left"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi>kq</mi><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$a1mag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$q1</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$q2</mi></mrow></mfenced></mrow></mfenced></mrow></mstyle></math>.</p>@
qu.6.1.algorithm=$a1mag=rint(2,5);
$a2mag=$a1mag;
$xmag=rint(2,5);
$q1=switch(rint(2),rint(1,10),-rint(1,10));
$q2=switch(rint(2),rint(1,10),-rint(1,10));@
qu.6.1.uid=128f2d6f-f073-4865-91be-0bb1b6ce6b2f@
qu.6.1.info=  Course=Introductory Electricity and Magnetism;
  Topic=Electric Potential Due to Point Charges in 2D;
  Author=Aron Pasieka;
  Difficulty=Hard;
  Features=Algebraic;
  Features=Algorithmic;
  Features=Partial Grading;
  Features=Diagram;
@

qu.6.2.question=<p>Find an algebraic expression for the electric force due to the two points charges on the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi></mrow></mstyle></math>-axis on&nbsp;a point charge <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$q3</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>q</mi></mrow></mstyle></math>&nbsp;at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math>, in terms of the given parameters.</p>
<p>&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="395" height="400">
<param name="image" value="__BASE_URI__img/ElectricFields/E-PtChr-2Chrg-2D-Sym/Diagram.png" />
<param name="size" value="10" />
<param name="label.1.x" value="20" />
<param name="label.1.y" value="105" />
<param name="label.1.text" value="q" />
<param name="label.2.x" value="10" />
<param name="label.2.y" value="105" />
<param name="label.2.text" value="$q1" />
<param name="label.3.x" value="20" />
<param name="label.3.y" value="317" />
<param name="label.3.text" value="q" />
<param name="label.4.x" value="10" />
<param name="label.4.y" value="317" />
<param name="label.4.text" value="$q2" />
<param name="label.5.x" value="50" />
<param name="label.5.y" value="155" />
<param name="label.5.text" value="a" />
<param name="label.6.x" value="40" />
<param name="label.6.y" value="155" />
<param name="label.6.text" value="$a1mag" />
<param name="label.7.x" value="50" />
<param name="label.7.y" value="265" />
<param name="label.7.text" value="a" />
<param name="label.8.x" value="40" />
<param name="label.8.y" value="265" />
<param name="label.8.text" value="$a2mag" />
<param name="label.9.x" value="160" />
<param name="label.9.y" value="200" />
<param name="label.9.text" value="x" />
<param name="label.10.x" value="150" />
<param name="label.10.y" value="200" />
<param name="label.10.text" value="$xmag" /></applet></p>
<p><br />
<em>Entry Notes</em>:&nbsp;&nbsp;<br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ihat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mi>i</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>jhat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>khat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp;&nbsp;<br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
If necessary, use the letter <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi></mrow></mstyle></math> to represent the Coulomb constant <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mi mathvariant='normal'>&pi;</mi><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>eg.&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mrow><mi>&pi;</mi><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mrow></mfrac><mfrac><mrow><msub><mi>q</mi><mrow><mn>1</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></msub><msub><mi>q</mi><mrow><mn>2</mn></mrow></msub></mrow><mrow><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mn>1</mn><mrow><mn>4</mn><mrow><mi>&pi;</mi></mrow><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mfrac><mfrac><mrow><msub><mi>q</mi><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></msub><msub><mi>q</mi><mrow><mn>3</mn></mrow></msub></mrow><mrow><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>&nbsp;could be entered as (k*q1*q2/(r^2))*ihat + (k*q2*q3/(r^2))*khat.</p>@
qu.6.2.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=vectorAlgebraicComponents);@
qu.6.2.allow2d=0@
qu.6.2.maple_answer=subs(_i=ihat,_j=jhat,_k=khat,collect(PhysFuncs[ElectricField](($q1)*q*($q3)*q,$a1mag*a*_j,$xmag*x*_i,constant=k)+PhysFuncs[ElectricField](($q2)*q*($q3)*q,-$a2mag*a*_j,$xmag*x*_i,constant=k),[_i,_j,_k]))@
qu.6.2.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.6.2.type=maple@
qu.6.2.mode=Maple@
qu.6.2.name=Electric Force Due to Point Charges - 2 Charges, 2D, Different Charges, Different Distances@
qu.6.2.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.6.2.editing=useHTML@
qu.6.2.hint.1=Remember that the superposition principle means that the electric force at a point is the sum of the electric forces produced by each of the nearby charges.&nbsp; Pretend that only one charge at a time&nbsp;is present and calculate the force due to that charge at the point.&nbsp; Finally, add all of the electric forces together to get the total.@
qu.6.2.solution=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mi>top</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$q1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>q</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$q3</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>q</mi></mrow></mfenced></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>$a1mag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$xmag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow><mrow><mover><mrow><mi>r</mi></mrow><mi>&#x005e;</mi></mover></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math></p>
<p>The x-component is</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>F</mi><mrow><msub><mi>top</mi><mrow><mi>X</mi></mrow></msub></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><msub><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mrow><mi>top</mi></mrow></msub></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$xmag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow><mrow><msqrt><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>$a1mag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$xmag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac></mrow></mstyle></math></p>
<p>and the y-component is</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>F</mi><mrow><msub><mi>top</mi><mrow><mi>Y</mi></mrow></msub></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><msub><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mrow><mi>top</mi></mrow></msub></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow><mrow><mi mathvariant='normal'></mi></mrow></mstyle></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mi>sin</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$a1mag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow><mrow><msqrt><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>$a1mag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$xmag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac></mrow></mstyle></math></p>
<p>The components of the electric force due to the bottom charge can be similarly calculated.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mi>P</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><msub><mi>F</mi><mrow><msub><mi>top</mi><mrow><mi>X</mi></mrow></msub></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>F</mi><mrow><msub><mi>bottom</mi><mrow><mi>X</mi></mrow></msub></mrow></msub></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><msub><mi>F</mi><mrow><msub><mi>top</mi><mrow><mi>Y</mi></mrow></msub></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>F</mi><mrow><msub><mi>bottom</mi><mrow><mi>Y</mi></mrow></msub></mrow></msub></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math></p>@
qu.6.2.algorithm=$a1mag=rint(2,5);
$a2mag=rint(2,5);
$xmag=rint(2,5);
$q1=switch(rint(2),rint(1,10),-rint(1,10));
$q2=switch(rint(2),rint(1,10),-rint(1,10));
$q3=switch(rint(2),rint(1,10),-rint(1,10));@
qu.6.2.uid=94c70cb1-b573-488d-8eef-03f7ed889c71@
qu.6.2.info=  Course=Introductory Electricity and Magnetism;
  Topic=Electric Force Due to Point Charges in 2D;
  Author=Aron Pasieka;
  Difficulty=Hard;
  Features=Algebraic;
  Features=Algorithmic;
  Features=Partial Grading;
  Features=Diagram;
@

qu.6.3.question=<p>Find an algebraic expression for the electric field at point P, in terms of the given parameters.</p>
<p>&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="395" height="400">
<param name="image" value="__BASE_URI__img/ElectricFields/E-PtChr-2Chrg-2D-Sym/Diagram.png" />
<param name="size" value="10" />
<param name="label.1.x" value="20" />
<param name="label.1.y" value="105" />
<param name="label.1.text" value="q" />
<param name="label.2.x" value="10" />
<param name="label.2.y" value="105" />
<param name="label.2.text" value="$q1" />
<param name="label.3.x" value="20" />
<param name="label.3.y" value="317" />
<param name="label.3.text" value="q" />
<param name="label.4.x" value="10" />
<param name="label.4.y" value="317" />
<param name="label.4.text" value="$q2" />
<param name="label.5.x" value="50" />
<param name="label.5.y" value="155" />
<param name="label.5.text" value="a" />
<param name="label.6.x" value="40" />
<param name="label.6.y" value="155" />
<param name="label.6.text" value="$a1mag" />
<param name="label.7.x" value="50" />
<param name="label.7.y" value="265" />
<param name="label.7.text" value="a" />
<param name="label.8.x" value="40" />
<param name="label.8.y" value="265" />
<param name="label.8.text" value="$a2mag" />
<param name="label.9.x" value="160" />
<param name="label.9.y" value="200" />
<param name="label.9.text" value="x" />
<param name="label.10.x" value="150" />
<param name="label.10.y" value="200" />
<param name="label.10.text" value="$xmag" /></applet></p>
<p><br />
<em>Entry Notes</em>:&nbsp;&nbsp;<br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ihat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mi>i</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>jhat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>khat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp;&nbsp;<br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
If necessary, use the letter <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi></mrow></mstyle></math> to represent the Coulomb constant <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mi mathvariant='normal'>&pi;</mi><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>eg.&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mrow><mi>&pi;</mi><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mrow></mfrac><mfrac><mrow><msub><mi>q</mi><mrow><mn>1</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></msub><msub><mi>q</mi><mrow><mn>2</mn></mrow></msub></mrow><mrow><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mn>1</mn><mrow><mn>4</mn><mrow><mi>&pi;</mi></mrow><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mfrac><mfrac><mrow><msub><mi>q</mi><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></msub><msub><mi>q</mi><mrow><mn>3</mn></mrow></msub></mrow><mrow><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>&nbsp;could be entered as (k*q1*q2/(r^2))*ihat + (k*q2*q3/(r^2))*khat.</p>@
qu.6.3.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=vectorAlgebraicComponents);@
qu.6.3.allow2d=0@
qu.6.3.maple_answer=subs(_i=ihat,_j=jhat,_k=khat,collect(PhysFuncs[ElectricField](($q1)*q,$a1mag*a*_j,$xmag*x*_i,constant=k)+PhysFuncs[ElectricField](($q2)*q,-$a2mag*a*_j,$xmag*x*_i,constant=k),[_i,_j,_k]))@
qu.6.3.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.6.3.type=maple@
qu.6.3.mode=Maple@
qu.6.3.name=Electric Field Due to Point Charges - 2 Charges, 2D, Different Charges, Different Distances@
qu.6.3.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.6.3.editing=useHTML@
qu.6.3.hint.1=Remember that the superposition principle means that the electric field at a point is the sum of the electric fields produced by each of the nearby charges.&nbsp; Pretend that only one charge at a time&nbsp;is present and calculate the field due to that charge at the point.&nbsp; Finally, add all of the electric fields together to get the total.@
qu.6.3.solution=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow><mrow><mi>top</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$q1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>q</mi></mrow></mfenced></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>$a1mag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$xmag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow><mrow><mover><mrow><mi>r</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math></p>
<p>The x-component is</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>E</mi><mrow><msub><mi>top</mi><mrow><mi>X</mi></mrow></msub></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><msub><mover><mi>E</mi><mi>&rarr;</mi></mover><mrow><mi>top</mi></mrow></msub></mfenced></mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mi>cos</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mrow></mstyle></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$xmag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow><mrow><msqrt><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>$a1mag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$xmag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac></mrow></mstyle></math></p>
<p>and the y-component is</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>E</mi><mrow><msub><mi>top</mi><mrow><mi>Y</mi></mrow></msub></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><msub><mover><mi>E</mi><mi>&rarr;</mi></mover><mrow><mi>top</mi></mrow></msub></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow><mrow></mrow></mstyle></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mi>sin</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$a1mag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow><mrow><msqrt><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>$a1mag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$xmag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac></mrow></mstyle></math></p>
<p>The components of the electric field due to the bottom charge can be similarly calculated.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow><mrow><mi>P</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><msub><mi>E</mi><mrow><msub><mi>top</mi><mrow><mi>X</mi></mrow></msub></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>E</mi><mrow><msub><mi>bottom</mi><mrow><mi>X</mi></mrow></msub></mrow></msub></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><msub><mi>E</mi><mrow><msub><mi>top</mi><mrow><mi>Y</mi></mrow></msub></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>E</mi><mrow><msub><mi>bottom</mi><mrow><mi>Y</mi></mrow></msub></mrow></msub></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math></p>@
qu.6.3.algorithm=$a1mag=rint(2,5);
$a2mag=rint(2,5);
$xmag=rint(2,5);
$q1=switch(rint(2),rint(1,10),-rint(1,10));
$q2=switch(rint(2),rint(1,10),-rint(1,10));@
qu.6.3.uid=e7aeed7b-e795-4cc0-b091-065be6e74a4d@
qu.6.3.info=  Course=Introductory Electricity and Magnetism;
  Topic=Electric Field Due to Point Charges in 2D;
  Author=Aron Pasieka;
  Difficulty=Hard;
  Features=Algebraic;
  Features=Algorithmic;
  Features=Partial Grading;
  Features=Diagram;
@

qu.6.4.question=<p>Find an algebraic expression for the electric force due to the two points charges on the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi></mrow></mstyle></math>-axis on&nbsp;a point charge <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$q3</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>q</mi></mrow></mstyle></math>&nbsp;at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math>, in terms of the given parameters.</p>
<p>&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="395" height="400">
<param name="image" value="__BASE_URI__img/ElectricFields/E-PtChr-2Chrg-2D-Sym/Diagram.png" />
<param name="size" value="10" />
<param name="label.1.x" value="20" />
<param name="label.1.y" value="105" />
<param name="label.1.text" value="q" />
<param name="label.2.x" value="10" />
<param name="label.2.y" value="105" />
<param name="label.2.text" value="$q1" />
<param name="label.3.x" value="20" />
<param name="label.3.y" value="317" />
<param name="label.3.text" value="q" />
<param name="label.4.x" value="10" />
<param name="label.4.y" value="317" />
<param name="label.4.text" value="$q2" />
<param name="label.5.x" value="50" />
<param name="label.5.y" value="155" />
<param name="label.5.text" value="a" />
<param name="label.6.x" value="40" />
<param name="label.6.y" value="155" />
<param name="label.6.text" value="$a1mag" />
<param name="label.7.x" value="50" />
<param name="label.7.y" value="265" />
<param name="label.7.text" value="a" />
<param name="label.8.x" value="40" />
<param name="label.8.y" value="265" />
<param name="label.8.text" value="$a2mag" />
<param name="label.9.x" value="160" />
<param name="label.9.y" value="200" />
<param name="label.9.text" value="x" />
<param name="label.10.x" value="150" />
<param name="label.10.y" value="200" />
<param name="label.10.text" value="$xmag" /></applet></p>
<p><br />
<em>Entry Notes</em>:&nbsp;&nbsp;<br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ihat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mi>i</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>jhat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>khat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp;&nbsp;<br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
If necessary, use the letter <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi></mrow></mstyle></math> to represent the Coulomb constant <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mi mathvariant='normal'>&pi;</mi><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>eg.&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mrow><mi>&pi;</mi><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mrow></mfrac><mfrac><mrow><msub><mi>q</mi><mrow><mn>1</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></msub><msub><mi>q</mi><mrow><mn>2</mn></mrow></msub></mrow><mrow><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mn>1</mn><mrow><mn>4</mn><mrow><mi>&pi;</mi></mrow><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mfrac><mfrac><mrow><msub><mi>q</mi><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></msub><msub><mi>q</mi><mrow><mn>3</mn></mrow></msub></mrow><mrow><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>&nbsp;could be entered as (k*q1*q2/(r^2))*ihat + (k*q2*q3/(r^2))*khat.</p>@
qu.6.4.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=vectorAlgebraicComponents);@
qu.6.4.allow2d=0@
qu.6.4.maple_answer=subs(_i=ihat,_j=jhat,_k=khat,collect(PhysFuncs[ElectricField](($q1)*q*($q3)*q,$a1mag*a*_j,$xmag*x*_i,constant=k)+PhysFuncs[ElectricField](($q2)*q*($q3)*q,-$a2mag*a*_j,$xmag*x*_i,constant=k),[_i,_j,_k]))@
qu.6.4.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.6.4.type=maple@
qu.6.4.mode=Maple@
qu.6.4.name=Electric Force Due to Point Charges - 2 Charges, 2D, Symmetric@
qu.6.4.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.6.4.editing=useHTML@
qu.6.4.hint.1=Remember that the superposition principle means that the electric force at a point is the sum of the electric forces produced by each of the nearby charges.&nbsp; Pretend that only one charge at a time&nbsp;is present and calculate the force due to that charge at the point.&nbsp; Finally, add all of the electric forces together to get the total.@
qu.6.4.hint.2=Use symmetry.@
qu.6.4.solution=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mi>top</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$q1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>q</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$q3</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>q</mi></mrow></mfenced></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>$a1mag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$xmag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow><mrow><mover><mrow><mi>r</mi></mrow><mi>&#x005e;</mi></mover></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math></p>
<p>The y-components cancel by symmetry.&nbsp; The total is thus just twice&nbsp;the x-component:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>F</mi><mrow><msub><mi>top</mi><mrow><mi>X</mi></mrow></msub></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><msub><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mrow><mi>top</mi></mrow></msub></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$xmag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow><mrow><msqrt><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>$a1mag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$xmag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mi>P</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><msub><mi>F</mi><mrow><msub><mi>top</mi><mrow><mi>X</mi></mrow></msub></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>F</mi><mrow><msub><mi>bottom</mi><mrow><mi>X</mi></mrow></msub></mrow></msub></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>F</mi><mrow><msub><mi>top</mi><mrow><mi>X</mi></mrow></msub></mrow></msub></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mi>i</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math></p>@
qu.6.4.algorithm=$a1mag=rint(2,5);
$a2mag=$a1mag;
$xmag=rint(2,5);
$q1=switch(rint(2),rint(1,10),-rint(1,10));
$q2=$q1;
$q3=switch(rint(2),rint(1,10),-rint(1,10));@
qu.6.4.uid=bd620c4d-9ad3-4944-92ba-064f6ef3243c@
qu.6.4.info=  Course=Introductory Electricity and Magnetism;
  Topic=Electric Force Due to Point Charges in 2D;
  Author=Aron Pasieka;
  Difficulty=Hard;
  Features=Algebraic;
  Features=Algorithmic;
  Features=Partial Grading;
  Features=Diagram;
@

qu.6.5.question=<p>Find an algebraic expression for the electric force due to the two points charges on the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi></mrow></mstyle></math>-axis on&nbsp;a point charge <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$q3</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>q</mi></mrow></mstyle></math>&nbsp;at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math>, in terms of the given parameters.</p>
<p>&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="395" height="400">
<param name="image" value="__BASE_URI__img/ElectricFields/E-PtChr-2Chrg-2D-Sym/Diagram.png" />
<param name="size" value="10" />
<param name="label.1.x" value="20" />
<param name="label.1.y" value="105" />
<param name="label.1.text" value="q" />
<param name="label.2.x" value="10" />
<param name="label.2.y" value="105" />
<param name="label.2.text" value="$q1" />
<param name="label.3.x" value="20" />
<param name="label.3.y" value="317" />
<param name="label.3.text" value="q" />
<param name="label.4.x" value="10" />
<param name="label.4.y" value="317" />
<param name="label.4.text" value="$q2" />
<param name="label.5.x" value="50" />
<param name="label.5.y" value="155" />
<param name="label.5.text" value="a" />
<param name="label.6.x" value="40" />
<param name="label.6.y" value="155" />
<param name="label.6.text" value="$a1mag" />
<param name="label.7.x" value="50" />
<param name="label.7.y" value="265" />
<param name="label.7.text" value="a" />
<param name="label.8.x" value="40" />
<param name="label.8.y" value="265" />
<param name="label.8.text" value="$a2mag" />
<param name="label.9.x" value="160" />
<param name="label.9.y" value="200" />
<param name="label.9.text" value="x" />
<param name="label.10.x" value="150" />
<param name="label.10.y" value="200" />
<param name="label.10.text" value="$xmag" /></applet></p>
<p><br />
<em>Entry Notes</em>:&nbsp;&nbsp;<br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ihat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mi>i</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>jhat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>khat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp;&nbsp;<br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
If necessary, use the letter <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi></mrow></mstyle></math> to represent the Coulomb constant <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mi mathvariant='normal'>&pi;</mi><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>eg.&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mrow><mi>&pi;</mi><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mrow></mfrac><mfrac><mrow><msub><mi>q</mi><mrow><mn>1</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></msub><msub><mi>q</mi><mrow><mn>2</mn></mrow></msub></mrow><mrow><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mn>1</mn><mrow><mn>4</mn><mrow><mi>&pi;</mi></mrow><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mfrac><mfrac><mrow><msub><mi>q</mi><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></msub><msub><mi>q</mi><mrow><mn>3</mn></mrow></msub></mrow><mrow><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>&nbsp;could be entered as (k*q1*q2/(r^2))*ihat + (k*q2*q3/(r^2))*khat.</p>@
qu.6.5.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=vectorAlgebraicComponents);@
qu.6.5.allow2d=0@
qu.6.5.maple_answer=subs(_i=ihat,_j=jhat,_k=khat,collect(PhysFuncs[ElectricField](($q1)*q*($q3)*q,$a1mag*a*_j,$xmag*x*_i,constant=k)+PhysFuncs[ElectricField](($q2)*q*($q3)*q,-$a2mag*a*_j,$xmag*x*_i,constant=k),[_i,_j,_k]))@
qu.6.5.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.6.5.type=maple@
qu.6.5.mode=Maple@
qu.6.5.name=Electric Force Due to Point Charges - 2 Charges, 2D, Different Charges@
qu.6.5.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.6.5.editing=useHTML@
qu.6.5.hint.1=Remember that the superposition principle means that the electric force at a point is the sum of the electric forces produced by each of the nearby charges.&nbsp; Pretend that only one charge at a time&nbsp;is present and calculate the force due to that charge at the point.&nbsp; Finally, add all of the electric forces together to get the total.@
qu.6.5.solution=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mi>top</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$q1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>q</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$q3</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>q</mi></mrow></mfenced></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>$a1mag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$xmag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow><mrow><mover><mrow><mi>r</mi></mrow><mi>&#x005e;</mi></mover></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math></p>
<p>The x-component is</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>F</mi><mrow><msub><mi>top</mi><mrow><mi>X</mi></mrow></msub></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><msub><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mrow><mi>top</mi></mrow></msub></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$xmag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow><mrow><msqrt><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>$a1mag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$xmag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac></mrow></mstyle></math></p>
<p>and the y-component is</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>F</mi><mrow><msub><mi>top</mi><mrow><mi>Y</mi></mrow></msub></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><msub><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mrow><mi>top</mi></mrow></msub></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow><mrow><mi mathvariant='normal'></mi></mrow></mstyle></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mi>sin</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$a1mag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow><mrow><msqrt><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>$a1mag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$xmag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac></mrow></mstyle></math></p>
<p>The components of the electric force due to the bottom charge can be similarly calculated.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mi>P</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><msub><mi>F</mi><mrow><msub><mi>top</mi><mrow><mi>X</mi></mrow></msub></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>F</mi><mrow><msub><mi>bottom</mi><mrow><mi>X</mi></mrow></msub></mrow></msub></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><msub><mi>F</mi><mrow><msub><mi>top</mi><mrow><mi>Y</mi></mrow></msub></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>F</mi><mrow><msub><mi>bottom</mi><mrow><mi>Y</mi></mrow></msub></mrow></msub></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math></p>@
qu.6.5.algorithm=$a1mag=rint(2,5);
$a2mag=$a1mag;
$xmag=rint(2,5);
$q1=switch(rint(2),rint(1,10),-rint(1,10));
$q2=switch(rint(2),rint(1,10),-rint(1,10));
$q3=switch(rint(2),rint(1,10),-rint(1,10));@
qu.6.5.uid=caafce01-8a3e-4aa4-93a2-e7774b63bce9@
qu.6.5.info=  Course=Introductory Electricity and Magnetism;
  Topic=Electric Force Due to Point Charges in 2D;
  Author=Aron Pasieka;
  Difficulty=Hard;
  Features=Algebraic;
  Features=Algorithmic;
  Features=Partial Grading;
  Features=Diagram;
@

qu.6.6.question=<p>Find an algebraic expression for the electric field at point P, in terms of the given parameters.</p>
<p>&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="395" height="400">
<param name="image" value="__BASE_URI__img/ElectricFields/E-PtChr-2Chrg-2D-Sym/Diagram.png" />
<param name="size" value="10" />
<param name="label.1.x" value="20" />
<param name="label.1.y" value="105" />
<param name="label.1.text" value="q" />
<param name="label.2.x" value="10" />
<param name="label.2.y" value="105" />
<param name="label.2.text" value="$q1" />
<param name="label.3.x" value="20" />
<param name="label.3.y" value="317" />
<param name="label.3.text" value="q" />
<param name="label.4.x" value="10" />
<param name="label.4.y" value="317" />
<param name="label.4.text" value="$q2" />
<param name="label.5.x" value="50" />
<param name="label.5.y" value="155" />
<param name="label.5.text" value="a" />
<param name="label.6.x" value="40" />
<param name="label.6.y" value="155" />
<param name="label.6.text" value="$amag" />
<param name="label.7.x" value="50" />
<param name="label.7.y" value="265" />
<param name="label.7.text" value="a" />
<param name="label.8.x" value="40" />
<param name="label.8.y" value="265" />
<param name="label.8.text" value="$amag" />
<param name="label.9.x" value="160" />
<param name="label.9.y" value="200" />
<param name="label.9.text" value="x" />
<param name="label.10.x" value="150" />
<param name="label.10.y" value="200" />
<param name="label.10.text" value="$xmag" /></applet></p>
<p><br />
<em>Entry Notes</em>:&nbsp;&nbsp;<br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ihat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mi>i</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>jhat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>khat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp;&nbsp;<br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
If necessary, use the letter <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi></mrow></mstyle></math> to represent the Coulomb constant <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mi mathvariant='normal'>&pi;</mi><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>eg.&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mrow><mi>&pi;</mi><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mrow></mfrac><mfrac><mrow><msub><mi>q</mi><mrow><mn>1</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></msub><msub><mi>q</mi><mrow><mn>2</mn></mrow></msub></mrow><mrow><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mn>1</mn><mrow><mn>4</mn><mrow><mi>&pi;</mi></mrow><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mfrac><mfrac><mrow><msub><mi>q</mi><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></msub><msub><mi>q</mi><mrow><mn>3</mn></mrow></msub></mrow><mrow><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>&nbsp;could be entered as (k*q1*q2/(r^2))*ihat + (k*q2*q3/(r^2))*khat.</p>@
qu.6.6.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=vectorAlgebraicComponents);@
qu.6.6.allow2d=0@
qu.6.6.maple_answer=subs(_i=ihat,_j=jhat,_k=khat,collect(PhysFuncs[ElectricField](($q1)*q,$amag*a*_j,$xmag*x*_i,constant=k)+PhysFuncs[ElectricField](($q2)*q,-$amag*a*_j,$xmag*x*_i,constant=k),[_i,_j,_k]))@
qu.6.6.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.6.6.type=maple@
qu.6.6.mode=Maple@
qu.6.6.name=Electric Field Due to Point Charges - 2 Charges, 2D, Different Charges@
qu.6.6.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.6.6.editing=useHTML@
qu.6.6.hint.1=Remember that the superposition principle means that the electric field at a point is the sum of the electric fields produced by each of the nearby charges.&nbsp; Pretend that only one charge at a time&nbsp;is present and calculate the field due to that charge at the point.&nbsp; Finally, add all of the electric fields together to get the total.@
qu.6.6.solution=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow><mrow><mi>top</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$q1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>q</mi></mrow></mfenced></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>$amag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$xmag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow><mrow><mover><mrow><mi>r</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math></p>
<p>The x-component is</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>E</mi><mrow><msub><mi>top</mi><mrow><mi>X</mi></mrow></msub></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><msub><mover><mi>E</mi><mi>&rarr;</mi></mover><mrow><mi>top</mi></mrow></msub></mfenced></mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mi>cos</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mrow></mstyle></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$xmag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow><mrow><msqrt><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>$amag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$xmag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac></mrow></mstyle></math></p>
<p>and the y-component is</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>E</mi><mrow><msub><mi>top</mi><mrow><mi>Y</mi></mrow></msub></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><msub><mover><mi>E</mi><mi>&rarr;</mi></mover><mrow><mi>top</mi></mrow></msub></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow><mrow></mrow></mstyle></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mi>sin</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$amag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow><mrow><msqrt><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>$amag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$xmag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac></mrow></mstyle></math></p>
<p>The components of the electric field due to the bottom charge can be similarly calculated.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow><mrow><mi>P</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><msub><mi>E</mi><mrow><msub><mi>top</mi><mrow><mi>X</mi></mrow></msub></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>E</mi><mrow><msub><mi>bottom</mi><mrow><mi>X</mi></mrow></msub></mrow></msub></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><msub><mi>E</mi><mrow><msub><mi>top</mi><mrow><mi>Y</mi></mrow></msub></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>E</mi><mrow><msub><mi>bottom</mi><mrow><mi>Y</mi></mrow></msub></mrow></msub></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math></p>@
qu.6.6.algorithm=$amag=rint(2,5);
$xmag=rint(2,5);
$q1=switch(rint(2),rint(1,10),-rint(1,10));
$q2=switch(rint(2),rint(1,10),-rint(1,10));@
qu.6.6.uid=0d7a7a9f-cd11-4818-98c3-3a846ef862a2@
qu.6.6.info=  Course=Introductory Electricity and Magnetism;
  Topic=Electric Field Due to Point Charges in 2D;
  Author=Aron Pasieka;
  Difficulty=Medium;
  Features=Algebraic;
  Features=Algorithmic;
  Features=Partial Grading;
  Features=Diagram;
@

qu.6.7.question=<p>Find an algebraic expression for the electric potential at point P, in terms of the given parameters.</p>
<p>&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="395" height="400">
<param name="image" value="__BASE_URI__img/ElectricFields/E-PtChr-2Chrg-2D-Sym/Diagram.png" />
<param name="size" value="10" />
<param name="label.1.x" value="20" />
<param name="label.1.y" value="105" />
<param name="label.1.text" value="q" />
<param name="label.2.x" value="10" />
<param name="label.2.y" value="105" />
<param name="label.2.text" value="$q1" />
<param name="label.3.x" value="20" />
<param name="label.3.y" value="317" />
<param name="label.3.text" value="q" />
<param name="label.4.x" value="10" />
<param name="label.4.y" value="317" />
<param name="label.4.text" value="$q2" />
<param name="label.5.x" value="50" />
<param name="label.5.y" value="155" />
<param name="label.5.text" value="a" />
<param name="label.6.x" value="40" />
<param name="label.6.y" value="155" />
<param name="label.6.text" value="$a1mag" />
<param name="label.7.x" value="50" />
<param name="label.7.y" value="265" />
<param name="label.7.text" value="a" />
<param name="label.8.x" value="40" />
<param name="label.8.y" value="265" />
<param name="label.8.text" value="$a2mag" />
<param name="label.9.x" value="160" />
<param name="label.9.y" value="200" />
<param name="label.9.text" value="x" />
<param name="label.10.x" value="150" />
<param name="label.10.y" value="200" />
<param name="label.10.text" value="$xmag" /></applet></p>
<p><br />
<em>Entry Notes</em>:&nbsp;&nbsp;<br />
<br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
If necessary, use the letter <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi></mrow></mstyle></math> to represent the Coulomb constant <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mi mathvariant='normal'>&pi;</mi><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mfrac></mrow></mrow></mstyle></math>.</p>@
qu.6.7.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=vectorAlgebraicComponents);@
qu.6.7.allow2d=0@
qu.6.7.maple_answer=k*($q1)*q/sqrt(($a1mag*a)^2+($xmag*x)^2)
+k*($q2)*q/sqrt((-$a2mag*a)^2+($xmag*x)^2)@
qu.6.7.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.6.7.type=maple@
qu.6.7.mode=Maple@
qu.6.7.name=Electric Potential Due to Point Charges - 2 Charges, 2D, Different Charges, Different Distances@
qu.6.7.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.6.7.editing=useHTML@
qu.6.7.hint.1=Calculate the potential due to each charge separately, then add them together.@
qu.6.7.solution=<p>The electric potential due to a point charge is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>kq</mi><mrow><mi>r</mi></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>We can find the potential due to each charge separately, then add them together to find the total.</p>
<p>&nbsp;</p>
<p align="left">From the top charge to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math>, we have <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$a1mag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow></mstyle></math>.</p>
<p align="left">From the bottom, we have <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$a2mag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow></mstyle></math>.</p>
<p align="left">Thus,</p>
<p align="left"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$q1</mi></mrow></mfenced><mi>q</mi></mrow><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$a1mag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$q2</mi></mrow></mfenced><mi>q</mi></mrow><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$a2mag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac></mrow></mstyle></math>.</p>@
qu.6.7.algorithm=$a1mag=rint(2,5);
$a2mag=rint(2,5);
$xmag=rint(2,5);
$q1=switch(rint(2),rint(1,10),-rint(1,10));
$q2=switch(rint(2),rint(1,10),-rint(1,10));@
qu.6.7.uid=a099f30d-cd91-44ab-ac25-dffc42512c41@
qu.6.7.info=  Course=Introductory Electricity and Magnetism;
  Topic=Electric Potential Due to Point Charges in 2D;
  Author=Aron Pasieka;
  Difficulty=Hard;
  Features=Algebraic;
  Features=Algorithmic;
  Features=Partial Grading;
  Features=Diagram;
@

qu.6.8.question=<p>Find an algebraic expression for the electric potential at point P, in terms of the given parameters.</p>
<p>&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="395" height="400">
<param name="image" value="__BASE_URI__img/ElectricFields/E-PtChr-2Chrg-2D-Sym/Diagram.png" />
<param name="size" value="10" />
<param name="label.1.x" value="20" />
<param name="label.1.y" value="105" />
<param name="label.1.text" value="q" />
<param name="label.2.x" value="10" />
<param name="label.2.y" value="105" />
<param name="label.2.text" value="$q1" />
<param name="label.3.x" value="20" />
<param name="label.3.y" value="317" />
<param name="label.3.text" value="q" />
<param name="label.4.x" value="10" />
<param name="label.4.y" value="317" />
<param name="label.4.text" value="$q2" />
<param name="label.5.x" value="50" />
<param name="label.5.y" value="155" />
<param name="label.5.text" value="a" />
<param name="label.6.x" value="40" />
<param name="label.6.y" value="155" />
<param name="label.6.text" value="$a1mag" />
<param name="label.7.x" value="50" />
<param name="label.7.y" value="265" />
<param name="label.7.text" value="a" />
<param name="label.8.x" value="40" />
<param name="label.8.y" value="265" />
<param name="label.8.text" value="$a2mag" />
<param name="label.9.x" value="160" />
<param name="label.9.y" value="200" />
<param name="label.9.text" value="x" />
<param name="label.10.x" value="150" />
<param name="label.10.y" value="200" />
<param name="label.10.text" value="$xmag" /></applet></p>
<p><br />
<em>Entry Notes</em>:&nbsp;&nbsp;<br />
<br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
If necessary, use the letter <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi></mrow></mstyle></math> to represent the Coulomb constant <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mi mathvariant='normal'>&pi;</mi><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mfrac></mrow></mrow></mstyle></math>.</p>@
qu.6.8.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=vectorAlgebraicComponents);@
qu.6.8.allow2d=0@
qu.6.8.maple_answer=k*($q1)*q/sqrt(($a1mag*a)^2+($xmag*x)^2)
+k*($q2)*q/sqrt((-$a2mag*a)^2+($xmag*x)^2)@
qu.6.8.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.6.8.type=maple@
qu.6.8.mode=Maple@
qu.6.8.name=Electric Potential Due to Point Charges - 2 Charges, 2D, Different Charges, Symmetric@
qu.6.8.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.6.8.editing=useHTML@
qu.6.8.hint.1=Calculate the potential due to each charge separately, then add them together.@
qu.6.8.solution=<p>The electric potential due to a point charge is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>kq</mi><mrow><mi>r</mi></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>We can find the potential due to each charge separately, then add them together to find the total.</p>
<p>&nbsp;</p>
<p align="left">From both charges to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math>, we have <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$a1mag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow></mstyle></math>.</p>
<p align="left">Thus,</p>
<p align="left"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$q1</mi></mrow></mfenced><mi>q</mi></mrow><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$a1mag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac></mrow></mstyle></math>.</p>@
qu.6.8.algorithm=$a1mag=rint(2,5);
$a2mag=$a1mag;
$xmag=rint(2,5);
$q1=switch(rint(2),rint(1,10),-rint(1,10));
$q2=$q1;@
qu.6.8.uid=f8b51ab6-e145-4316-b347-ad895f1cf696@
qu.6.8.info=  Course=Introductory Electricity and Magnetism;
  Topic=Electric Potential Due to Point Charges in 2D;
  Author=Aron Pasieka;
  Difficulty=Hard;
  Features=Algebraic;
  Features=Algorithmic;
  Features=Partial Grading;
  Features=Diagram;
@

qu.6.9.question=<p align="left">Point charges $dir1Label q and $dir2Label q are positioned as shown.&nbsp;&nbsp;<br />
<br />
Find an algebraic expression for the electric field at point P, in terms of the given parameters.</p>
<p align="center">&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="395" height="400">
<param name="image" value="__BASE_URI__img/ElectricFields/E-PtChr-2Chrg-2D-Sym/Diagram.png" />
<param name="size" value="10" />
<param name="label.1.x" value="20" />
<param name="label.1.y" value="105" />
<param name="label.1.text" value="q" />
<param name="label.2.x" value="10" />
<param name="label.2.y" value="105" />
<param name="label.2.text" value="$dir1Label" />
<param name="label.3.x" value="20" />
<param name="label.3.y" value="317" />
<param name="label.3.text" value="q" />
<param name="label.4.x" value="10" />
<param name="label.4.y" value="317" />
<param name="label.4.text" value="$dir2Label" />
<param name="label.5.x" value="50" />
<param name="label.5.y" value="155" />
<param name="label.5.text" value="a" />
<param name="label.6.x" value="40" />
<param name="label.6.y" value="155" />
<param name="label.6.text" value="$amag" />
<param name="label.7.x" value="50" />
<param name="label.7.y" value="265" />
<param name="label.7.text" value="a" />
<param name="label.8.x" value="40" />
<param name="label.8.y" value="265" />
<param name="label.8.text" value="$amag" />
<param name="label.9.x" value="160" />
<param name="label.9.y" value="200" />
<param name="label.9.text" value="x" />
<param name="label.10.x" value="150" />
<param name="label.10.y" value="200" />
<param name="label.10.text" value="$xmag" /></applet></p>
<p align="center">&nbsp;</p>
<p align="left"><em>Entry Notes</em>: <br />
<br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
If necessary, use the letter&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi></mrow></mstyle></math> to represent the Coulomb constant <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mrow><mi>&pi;</mi></mrow><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mfrac></mrow></mrow></mstyle></math>.</p>@
qu.6.9.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=vectorAlgebraicComponents);@
qu.6.9.allow2d=0@
qu.6.9.maple_answer=subs(_i=ihat,_j=jhat,_k=khat,collect(PhysFuncs[ElectricField]($dir1*q,$amag*a*_j,$xmag*x*_i,constant=k)+PhysFuncs[ElectricField]($dir2*q,-$amag*a*_j,$xmag*x*_i,constant=k),[_i,_j,_k]))@
qu.6.9.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.6.9.type=maple@
qu.6.9.mode=Maple@
qu.6.9.name=Electric Field Due to Point Charges - 2 Charges, 2D, Symmetric@
qu.6.9.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.6.9.editing=useHTML@
qu.6.9.hint.1=Remember that the superposition principle means that the electric field at a point is the sum of the electric fields produced by each of the nearby charges.&nbsp; Pretend that only one charge at a time&nbsp;is present and calculate the field due to that charge at the point.&nbsp; Finally, add all of the electric fields together to get the total.@
qu.6.9.hint.2=You can use symmetry to save some calculation time here.@
qu.6.9.solution=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow><mrow><mi>top</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$dir1Label</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>q</mi></mrow></mfenced></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>$amag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$xmag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow><mrow><mover><mrow><mi>r</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math></p>
<p>The x-component is</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>E</mi><mrow><msub><mi>top</mi><mrow><mi>X</mi></mrow></msub></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><msub><mover><mi>E</mi><mi>&rarr;</mi></mover><mrow><mi>top</mi></mrow></msub></mfenced></mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mi>cos</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mrow></mstyle></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$xmag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow><mrow><msqrt><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>$amag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$xmag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac></mrow></mstyle></math></p>
<p>and the y-component is</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>E</mi><mrow><msub><mi>top</mi><mrow><mi>Y</mi></mrow></msub></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><msub><mover><mi>E</mi><mi>&rarr;</mi></mover><mrow><mi>top</mi></mrow></msub></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow><mrow></mrow></mstyle></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mi>sin</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$amag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow><mrow><msqrt><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>$amag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$xmag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac></mrow></mstyle></math></p>
<p>Since the two charges have the same magnitude and are the same distance away from P, the magnitudes of their electric fields at P will be the same.&nbsp; One component of the electric field at P will therefore cancel, while the other will be the sum of the contributions from the two charges..&nbsp; We can use this symmetry to save time in the calculation.&nbsp; Alternatively, you can calculate both components due to each charge and add them, which will yield the same result.</p>@
qu.6.9.algorithm=$amag=rint(2,5);
$xmag=rint(2,5);
$idx1=rint(2);
$dir1=switch($idx1,1,-1);
$dir1Label=switch($idx1,"+","-");
$idx2=rint(2);
$dir2=switch($idx2,1,-1);
$dir2Label=switch($idx2,"+","-");@
qu.6.9.uid=645c5efc-c47f-4c26-9541-08038e1905cd@
qu.6.9.info=  Course=Introductory Electricity and Magnetism;
  Topic=Electric Field Due to Point Charges in 2D;
  Author=Aron Pasieka;
  Difficulty=Medium;
  Features=Algebraic;
  Features=Algorithmic;
  Features=Partial Grading;
  Features=Diagram;
@

qu.7.topic=Potential@

qu.7.1.question=<p>Two infinite<font size="2"> parallel</font> plates are separated by a distance $x cm . The electric field between the plates is&nbsp;<br />
measured to be $E V/m <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover></mrow></mstyle></math>.</p>
<p align="center">&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="400" height="360">
<param name="image" value="__BASE_URI__img/ElectricFields/InfiniteParallelPlates/Diagram.png" />
<param name="size" value="1" />
<param name="label.1.x" value="295" />
<param name="label.1.y" value="12" />
<param name="label.1.text" value="$x cm" /></applet>&nbsp;</p>
<p>&nbsp;</p>
<p>What is the magnitude of the potential difference between the plates?</p>@
qu.7.1.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.7.1.allow2d=0@
qu.7.1.maple_answer=($pot)*V@
qu.7.1.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.7.1.type=maple@
qu.7.1.mode=Maple@
qu.7.1.name=Infinite Parallel Plates - Magnitude of Potential Difference ~ PG@
qu.7.1.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.7.1.editing=useHTML@
qu.7.1.hint.1=Recall that the electric field between infinite parallel plates is constant.&nbsp@
qu.7.1.hint.2=;The potential difference is related to the electric field and the separation of the plates.@
qu.7.1.hint.3=Ensure that the units work out.@
qu.7.1.solution=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&Delta;V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mi>&Delta;x</mi><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow></mstyle></math>and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>&Delta;x</mi></mrow><mi>&rarr;</mi></mover></mrow></mstyle></math>&nbsp;are parallel so the magnitude of the dot product is equal to the product of the two magnitudes.&nbsp; (ie. <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow></mstyle></math>)</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo></mrow><mrow><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>${abs($E)} V/m<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='|' close='|' separators=','><mrow><mover><mi>&Delta;x</mi><mi>&rarr;</mi></mover></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$x cm</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='|' close='|' separators=','><mrow><mi>&Delta;V</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>(${abs($E)} V/m)($x cm)<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow></mrow><mrow><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo></mrow><mrow><mi>&Delta;V</mi><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$pot V</p>@
qu.7.1.algorithm=$potGen=rand(10,100,3);
$x=rand(2,15,3);
$EGen=sig(3,$potGen/($x/100));
$pot=sig(3,$EGen*($x/100));
$idx=rint(2);
$dir=switch($idx,1,-1);
$E=sig(3,$EGen*$dir);
$correctPlate=switch($idx,"A","B");
$wrongPlate=switch($idx,"B","A");@
qu.7.1.uid=77498cd7-2652-4afe-b893-68dd5fb3f703@
qu.7.1.info=  Course=Introductory Electricity and Magnetism;
  Topic=Electric Potential Between Infinite Parallel Plates;
  Author=Aron Pasieka;
  Difficulty=Easy;
  Features=Algorithmic;
  Features=Partial Grading;
  Features=Diagram;
@

qu.7.2.question=<p>The magnitude of the potential difference between two infinite<font size="2"> parallel</font> plates is $pot V. The electric field&nbsp;<br />
between the plates is measured to be $E V/m <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover></mrow></mstyle></math>.</p>
<p align="center">&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="400" height="360">
<param name="image" value="__BASE_URI__img/ElectricFields/InfiniteParallelPlates/Diagram.png" />
<param name="size" value="1" />
<param name="label.1.x" value="295" />
<param name="label.1.y" value="12" />
<param name="label.1.text" value="x" /></applet>&nbsp;</p>
<p>&nbsp;</p>
<p>What is the separation of the plates?</p>@
qu.7.2.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.7.2.allow2d=0@
qu.7.2.maple_answer=($x)*cm@
qu.7.2.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.7.2.type=maple@
qu.7.2.mode=Maple@
qu.7.2.name=Infinite Parallel Plates - Find Separation ~ PG@
qu.7.2.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.7.2.editing=useHTML@
qu.7.2.hint.1=Recall that the electric field between infinite parallel plates is constant.&nbsp@
qu.7.2.hint.2=;The potential difference is related to the electric field and the separation of the plates.@
qu.7.2.hint.3=Ensure that the units work out.@
qu.7.2.solution=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&Delta;V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mi>&Delta;x</mi><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow></mstyle></math>and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>&Delta;x</mi></mrow><mi>&rarr;</mi></mover></mrow></mstyle></math>&nbsp;are parallel so the magnitude of the dot product is equal to the product of the two magnitudes.&nbsp; (ie. <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow></mstyle></math>)</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo></mrow><mrow><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>${abs($E)} V/m<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='|' close='|' separators=','><mrow><mi>V</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi></mi></mrow></mstyle></math>$pot V</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='|' close='|' separators=','><mrow><mrow><mover><mi>&Delta;x</mi><mi>&rarr;</mi></mover></mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>($pot V)/(${abs($E)} V/m)<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'></mi></mrow><mrow><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo></mrow><mrow><mrow><mover><mi>&Delta;x</mi><mi>&rarr;</mi></mover></mrow><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$x cm</p>@
qu.7.2.algorithm=$potGen=rand(10,100,3);
$x=rand(2,15,3);
$EGen=sig(3,$potGen/($x/100));
$pot=sig(3,$EGen*($x/100));
$idx=rint(2);
$dir=switch($idx,1,-1);
$E=sig(3,$EGen*$dir);
$correctPlate=switch($idx,"A","B");
$wrongPlate=switch($idx,"B","A");@
qu.7.2.uid=9e7e4da9-e107-410b-9cbb-6884a5bdcaa1@
qu.7.2.info=  Course=Introductory Electricity and Magnetism;
  Topic=Electric Potential Between Infinite Parallel Plates;
  Author=Aron Pasieka;
  Difficulty=Easy;
  Features=Algorithmic;
  Features=Partial Grading;
  Features=Diagram;
@

qu.7.3.question=<p>Find an algebraic expression for the electric potential at point P, in terms of the given parameters.</p>
<p>&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="395" height="400">
<param name="image" value="__BASE_URI__img/ElectricFields/E-PtChr-2Chrg-2D-Sym/Diagram.png" />
<param name="size" value="10" />
<param name="label.1.x" value="20" />
<param name="label.1.y" value="105" />
<param name="label.1.text" value="q" />
<param name="label.2.x" value="10" />
<param name="label.2.y" value="105" />
<param name="label.2.text" value="$q1" />
<param name="label.3.x" value="20" />
<param name="label.3.y" value="317" />
<param name="label.3.text" value="q" />
<param name="label.4.x" value="10" />
<param name="label.4.y" value="317" />
<param name="label.4.text" value="$q2" />
<param name="label.5.x" value="50" />
<param name="label.5.y" value="155" />
<param name="label.5.text" value="a" />
<param name="label.6.x" value="40" />
<param name="label.6.y" value="155" />
<param name="label.6.text" value="$a1mag" />
<param name="label.7.x" value="50" />
<param name="label.7.y" value="265" />
<param name="label.7.text" value="a" />
<param name="label.8.x" value="40" />
<param name="label.8.y" value="265" />
<param name="label.8.text" value="$a2mag" />
<param name="label.9.x" value="160" />
<param name="label.9.y" value="200" />
<param name="label.9.text" value="x" />
<param name="label.10.x" value="150" />
<param name="label.10.y" value="200" />
<param name="label.10.text" value="$xmag" /></applet></p>
<p><br />
<em>Entry Notes</em>:&nbsp;&nbsp;<br />
<br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
If necessary, use the letter <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi></mrow></mstyle></math> to represent the Coulomb constant <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mi mathvariant='normal'>&pi;</mi><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mfrac></mrow></mrow></mstyle></math>.</p>@
qu.7.3.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=vectorAlgebraicComponents);@
qu.7.3.allow2d=0@
qu.7.3.maple_answer=k*($q1)*q/sqrt(($a1mag*a)^2+($xmag*x)^2)
+k*($q2)*q/sqrt((-$a2mag*a)^2+($xmag*x)^2)@
qu.7.3.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.7.3.type=maple@
qu.7.3.mode=Maple@
qu.7.3.name=Electric Potential Due to Point Charges - 2 Charges, 2D, Different Charges, Different Distances@
qu.7.3.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.7.3.editing=useHTML@
qu.7.3.hint.1=Calculate the potential due to each charge separately, then add them together.@
qu.7.3.solution=<p>The electric potential due to a point charge is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>kq</mi><mrow><mi>r</mi></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>We can find the potential due to each charge separately, then add them together to find the total.</p>
<p>&nbsp;</p>
<p align="left">From the top charge to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math>, we have <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$a1mag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow></mstyle></math>.</p>
<p align="left">From the bottom, we have <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$a2mag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow></mstyle></math>.</p>
<p align="left">Thus,</p>
<p align="left"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$q1</mi></mrow></mfenced><mi>q</mi></mrow><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$a1mag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$q2</mi></mrow></mfenced><mi>q</mi></mrow><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$a2mag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac></mrow></mstyle></math>.</p>@
qu.7.3.algorithm=$a1mag=rint(2,5);
$a2mag=rint(2,5);
$xmag=rint(2,5);
$q1=switch(rint(2),rint(1,10),-rint(1,10));
$q2=switch(rint(2),rint(1,10),-rint(1,10));@
qu.7.3.uid=a099f30d-cd91-44ab-ac25-dffc42512c41@
qu.7.3.info=  Course=Introductory Electricity and Magnetism;
  Topic=Electric Potential Due to Point Charges in 2D;
  Author=Aron Pasieka;
  Difficulty=Hard;
  Features=Algebraic;
  Features=Algorithmic;
  Features=Partial Grading;
  Features=Diagram;
@

qu.7.4.question=<p>Find an algebraic expression for the electric potential at point P, in terms of the given parameters.</p>
<p>&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="395" height="400">
<param name="image" value="__BASE_URI__img/ElectricFields/E-PtChr-2Chrg-2D-Sym/Diagram.png" />
<param name="size" value="10" />
<param name="label.1.x" value="20" />
<param name="label.1.y" value="105" />
<param name="label.1.text" value="q" />
<param name="label.2.x" value="10" />
<param name="label.2.y" value="105" />
<param name="label.2.text" value="$q1" />
<param name="label.3.x" value="20" />
<param name="label.3.y" value="317" />
<param name="label.3.text" value="q" />
<param name="label.4.x" value="10" />
<param name="label.4.y" value="317" />
<param name="label.4.text" value="$q2" />
<param name="label.5.x" value="50" />
<param name="label.5.y" value="155" />
<param name="label.5.text" value="a" />
<param name="label.6.x" value="40" />
<param name="label.6.y" value="155" />
<param name="label.6.text" value="$a1mag" />
<param name="label.7.x" value="50" />
<param name="label.7.y" value="265" />
<param name="label.7.text" value="a" />
<param name="label.8.x" value="40" />
<param name="label.8.y" value="265" />
<param name="label.8.text" value="$a2mag" />
<param name="label.9.x" value="160" />
<param name="label.9.y" value="200" />
<param name="label.9.text" value="x" />
<param name="label.10.x" value="150" />
<param name="label.10.y" value="200" />
<param name="label.10.text" value="$xmag" /></applet></p>
<p><br />
<em>Entry Notes</em>:&nbsp;&nbsp;<br />
<br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
If necessary, use the letter <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi></mrow></mstyle></math> to represent the Coulomb constant <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mi mathvariant='normal'>&pi;</mi><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mfrac></mrow></mrow></mstyle></math>.</p>@
qu.7.4.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=vectorAlgebraicComponents);@
qu.7.4.allow2d=0@
qu.7.4.maple_answer=k*($q1)*q/sqrt(($a1mag*a)^2+($xmag*x)^2)
+k*($q2)*q/sqrt((-$a2mag*a)^2+($xmag*x)^2)@
qu.7.4.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.7.4.type=maple@
qu.7.4.mode=Maple@
qu.7.4.name=Electric Potential Due to Point Charges - 2 Charges, 2D, Different Charges, Symmetric@
qu.7.4.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.7.4.editing=useHTML@
qu.7.4.hint.1=Calculate the potential due to each charge separately, then add them together.@
qu.7.4.solution=<p>The electric potential due to a point charge is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>kq</mi><mrow><mi>r</mi></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>We can find the potential due to each charge separately, then add them together to find the total.</p>
<p>&nbsp;</p>
<p align="left">From both charges to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math>, we have <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$a1mag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow></mstyle></math>.</p>
<p align="left">Thus,</p>
<p align="left"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$q1</mi></mrow></mfenced><mi>q</mi></mrow><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$a1mag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac></mrow></mstyle></math>.</p>@
qu.7.4.algorithm=$a1mag=rint(2,5);
$a2mag=$a1mag;
$xmag=rint(2,5);
$q1=switch(rint(2),rint(1,10),-rint(1,10));
$q2=$q1;@
qu.7.4.uid=f8b51ab6-e145-4316-b347-ad895f1cf696@
qu.7.4.info=  Course=Introductory Electricity and Magnetism;
  Topic=Electric Potential Due to Point Charges in 2D;
  Author=Aron Pasieka;
  Difficulty=Hard;
  Features=Algebraic;
  Features=Algorithmic;
  Features=Partial Grading;
  Features=Diagram;
@

qu.7.5.question=<p>A thin, uniform&nbsp;line of charge extends from&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$aMag$aLet</mi></mrow></mstyle></math>&nbsp;to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$aMag$aLet</mi></mrow></mstyle></math>&nbsp;along the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi></mrow></mstyle></math>-axis&nbsp;and&nbsp;contains a total charge <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$QSignLabel</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow></mstyle></math>.</p>
<p>Set up&nbsp;an integral for the&nbsp;electric potential at an arbitrary point along the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi></mrow></mstyle></math>-axis due to the line of charge.&nbsp; Provide the <em>integrand<br />
</em>of this integral as your answer, in terms of the given parameters.</p>
<p align="center">e.g. Integral: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathcolor='#800080'>a</mi></mrow><mrow><mi>a</mi></mrow></munderover><mfrac><mi>Q</mi><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>, Integrand: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>Q</mi><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mstyle></math></p>
<p>In order to find the electric potential, one needs to evaluate an integral that adds up the contributions from each infinitesimal piece of<br />
the line, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dy</mi></mrow></mstyle></math>, from <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$aMag$aLet</mi></mrow></mstyle></math> to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$aMag$aLet</mi></mrow></mstyle></math>.&nbsp;</p>
<p align="center">&nbsp;</p>
<p align="center">&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="585" height="500">
<param name="image" value="__BASE_URI__img/ElectricFields/LineCharge/Diagram.png" />
<param name="size" value="4" />
<param name="label.1.x" value="70" />
<param name="label.1.y" value="250" />
<param name="label.1.text" value="$QSignLabel Q" />
<param name="label.2.x" value="70" />
<param name="label.2.y" value="175" />
<param name="label.2.text" value="$aMag$aLet" />
<param name="label.3.x" value="70" />
<param name="label.3.y" value="360" />
<param name="label.3.text" value="$aMag$aLet" />
<param name="label.4.x" value="270" />
<param name="label.4.y" value="250" />
<param name="label.4.text" value="x" /></applet></p>
<p align="center">&nbsp;</p>
<p><em>Note: </em>Use the letter 'k' for the Coulomb constant, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mrow><mi>&pi;</mi></mrow><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mfrac></mrow></mstyle></math>.</p>@
qu.7.5.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSA2,subPackage=vectorAlgebraicComponents)@
qu.7.5.allow2d=0@
qu.7.5.maple_answer=k*($QSign)*(Q/(2*$aMag*$aLet))/sqrt(x^2+y^2);@
qu.7.5.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.7.5.type=maple@
qu.7.5.mode=Maple@
qu.7.5.name=Line Charge - Find V - Set Up Integral@
qu.7.5.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSA2,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.7.5.editing=useHTML@
qu.7.5.solution=<p>Start by identifying a small unit of charge on the line, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dq</mi></mrow></mstyle></math>.&nbsp; Then the small amount of potential&nbsp;due to that unit of charge is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>kdq</mi><mrow><mi>r</mi></mrow></mfrac></mrow></mrow></mstyle></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msqrt><mrow><msup><mi>y</mi><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>Since the charge is uniformly distributed on the line, the charge per unit length is constant, so:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>dq</mi><mrow><mi>dy</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac></mrow></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mi>dq</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac><mi>dy</mi></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>.</p>
<p>Adding up the contributions from each unit of charge on the line:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$aMag$aLet</mi></mrow><mrow><mi>$aMag$aLet</mi></mrow></munderover><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mrow><mi></mi></mrow></msup></mrow></mfrac></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mi></mi></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math></p>@
qu.7.5.algorithm=$aMag=rint(2,10);
$aLet=switch(rint(8),'a','b','c','d','f','g','h','w');
$idx=rint(2);
$QSign=switch($idx,1,-1);
$QSignLabel=switch($idx,"+","-");
$QMag=rint(2,6);@
qu.7.5.uid=b0985b64-3eac-40de-9bc1-c3770a52029d@
qu.7.5.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Electric Potential Due to A Line of Charge;
  Difficulty=Hard;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Diagram;
@

qu.7.6.mode=Inline@
qu.7.6.name=Equipotential Lines 1@
qu.7.6.comment=@
qu.7.6.editing=useHTML@
qu.7.6.solution=<p>Assuming that the lines are equally spaced, if the electric field is constant then the change in potential between each pair of lines should also be constant.</p>
<p>&nbsp;</p>
<p>The electric field lines are always perpendicular to the equipotential lines.</p>
<p>&nbsp;</p>
<p>The electric field will always point from higher to lower potential.</p>@
qu.7.6.algorithm=$idx=rint(2);
$corMag=switch($idx,'Constant','Changing');
$incMag=switch($idx,'Changing','Constant');
$V1=range(-500,500);
$d1=range(-100,100);
$corDirN=($d1/abs($d1));
$V2=$V1+$d1;
$d2=switch($idx,$d1,$corDirN*range(1,100));
$V3=$V2+$d2;
$d3=switch($idx,$d1,$corDirN*range(1,100));
$V4=$V3+$d3;
$corDir=switch(-$corDirN+1,'-',0,'+');
$incDir=switch(-$corDirN+1,'+',0,'-');@
qu.7.6.uid=67aa8df7-c6eb-4b54-83b8-293821bcc69e@
qu.7.6.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Equipotential Lines;
  Difficulty=Medium;
  Features=Algorithmic;
  Features=Diagram;
@
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qu.7.6.numbering=alpha@
qu.7.6.part.1.grader=exact@
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qu.7.6.part.1.editing=useHTML@
qu.7.6.part.1.display.permute=true@
qu.7.6.part.1.answer.3=Impossible to Tell@
qu.7.6.part.1.question=(Unset)@
qu.7.6.part.1.answer.2=$incMag@
qu.7.6.part.1.answer.1=$corMag@
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qu.7.6.part.2.answer.3=Impossible to Tell@
qu.7.6.part.2.question=(Unset)@
qu.7.6.part.2.answer.2=Changing@
qu.7.6.part.2.answer.1=Constant@
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qu.7.6.part.3.answer.6=Between - x and + y@
qu.7.6.part.3.answer.5=Between - x and - y@
qu.7.6.part.3.answer.4=- y@
qu.7.6.part.3.editing=useHTML@
qu.7.6.part.3.answer.3=+ y@
qu.7.6.part.3.answer.2=$incDir x@
qu.7.6.part.3.answer.1=$corDir x@
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qu.7.6.part.3.answer.8=Between + x and - y@
qu.7.6.part.3.name=sro_id_3@
qu.7.6.part.3.display.permute=true@
qu.7.6.part.3.answer.7=Between + x and + y@
qu.7.6.question=<p><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="625" height="567"><param name="image" value="__BASE_URI__img/EquipotentialLines/EquipotentialLines1/Diagram.png" /><param name="size" value="4" /><param name="label.1.x" value="235" /><param name="label.1.y" value="95" /><param name="label.1.text" value="$V1 V" /><param name="label.2.x" value="360" /><param name="label.2.y" value="95" /><param name="label.2.text" value="$V2 V" /><param name="label.3.x" value="480" /><param name="label.3.y" value="95" /><param name="label.3.text" value="$V3 V" /><param name="label.4.x" value="600" /><param name="label.4.y" value="95" /><param name="label.4.text" value="$V4 V" /></applet></p><p>&nbsp;</p><p>Please select the most appropriate description of the electric field associated with the equipotential lines in the diagram.&nbsp;</p><p>&nbsp;</p><p><span>The magnitude of the electric field is:&nbsp; </span><span>&nbsp;</span><1><span>&nbsp;<br /><br /></span><span><span>The direction of the electric field is:&nbsp;&nbsp;<span>&nbsp;</span><2><span>&nbsp;</span><br /><br />The direction of the electric field at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math>&nbsp;is in&nbsp;the direction: &nbsp; </span><3><span>&nbsp;</span></span></p>@

qu.7.7.mode=Inline@
qu.7.7.name=Equipotential Lines 2@
qu.7.7.comment=@
qu.7.7.editing=useHTML@
qu.7.7.solution=<p>Assuming that the lines are equally spaced, if the electric field is constant then the change in potential between each pair of lines should also be constant.</p>
<p>&nbsp;</p>
<p>The electric field lines are always perpendicular to the equipotential lines.</p>
<p>&nbsp;</p>
<p>The electric field will always point from higher to lower potential.</p>@
qu.7.7.algorithm=$idx=rint(2);
$corMag=switch($idx,'Constant','Changing');
$incMag=switch($idx,'Changing','Constant');
$V1=range(-500,500);
$d1=range(-100,100);
$corDirN=($d1/abs($d1));
$V2=$V1+$d1;
$d2=switch($idx,$d1,$corDirN*range(1,100));
$V3=$V2+$d2;
$d3=switch($idx,$d1,$corDirN*range(1,100));
$V4=$V3+$d3;
$corDir=switch(-$corDirN+1,'+',0,'-');
$incDir=switch(-$corDirN+1,'-',0,'+');@
qu.7.7.uid=be0319c5-e78e-4b78-8187-31b436a90cd9@
qu.7.7.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Equipotential Lines;
  Difficulty=Medium;
  Features=Algorithmic;
  Features=Diagram;
@
qu.7.7.weighting=1,1,1@
qu.7.7.numbering=alpha@
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qu.7.7.part.1.answer.3=Impossible to Tell@
qu.7.7.part.1.question=(Unset)@
qu.7.7.part.1.answer.2=$incMag@
qu.7.7.part.1.answer.1=$corMag@
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qu.7.7.part.2.editing=useHTML@
qu.7.7.part.2.display.permute=true@
qu.7.7.part.2.answer.3=Impossible to Tell@
qu.7.7.part.2.question=(Unset)@
qu.7.7.part.2.answer.2=Changing@
qu.7.7.part.2.answer.1=Constant@
qu.7.7.part.2.mode=List@
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qu.7.7.part.2.credit.2=1.0@
qu.7.7.part.2.credit.1=0.0@
qu.7.7.part.3.answer.6=- (+ x - y)@
qu.7.7.part.3.answer.5=+ (+ x - y)@
qu.7.7.part.3.answer.4=- y@
qu.7.7.part.3.editing=useHTML@
qu.7.7.part.3.answer.3=+ y@
qu.7.7.part.3.answer.2=- x@
qu.7.7.part.3.answer.1=+ x@
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qu.7.7.part.3.mode=List@
qu.7.7.part.3.credit.3=0.0@
qu.7.7.part.3.credit.2=0.0@
qu.7.7.part.3.credit.1=0.0@
qu.7.7.part.3.grader=exact@
qu.7.7.part.3.display=menu@
qu.7.7.part.3.answer.8=$incDir (+ x + y)@
qu.7.7.part.3.name=sro_id_3@
qu.7.7.part.3.display.permute=true@
qu.7.7.part.3.answer.7=$corDir (+ x + y)@
qu.7.7.question=<p><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="600" height="399"><param name="image" value="__BASE_URI__img/EquipotentialLines/EquipotentialLines2/Diagram.png" /><param name="size" value="4" /><param name="label.1.x" value="235" /><param name="label.1.y" value="95" /><param name="label.1.text" value="$V1 V" /><param name="label.2.x" value="285" /><param name="label.2.y" value="130" /><param name="label.2.text" value="$V2 V" /><param name="label.3.x" value="335" /><param name="label.3.y" value="165" /><param name="label.3.text" value="$V3 V" /><param name="label.4.x" value="385" /><param name="label.4.y" value="200" /><param name="label.4.text" value="$V4 V" /></applet></p><p>&nbsp;</p><p>Please select the most appropriate description of the electric field associated with the equipotential lines in the diagram.&nbsp;</p><p>&nbsp;</p><p><span>The magnitude of the electric field is:&nbsp; </span><span>&nbsp;</span><1><span>&nbsp;<br /><br /></span><span><span>The direction of the electric field is:&nbsp;&nbsp;<span>&nbsp;</span><2><span>&nbsp;</span><br /><br />The direction of the electric field at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math>&nbsp;is in&nbsp;the direction: &nbsp; </span><3><span>&nbsp;</span></span></p>@

qu.7.8.question=<p>A&nbsp;solid disc&nbsp;of radius&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$aMag$aLet</mi></mrow></mstyle></math>&nbsp;is centered at the origin in the&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mover><mi>z</mi><mi>&#x005e;</mi></mover></mrow></mstyle></math>&nbsp;plane.&nbsp;&nbsp;A&nbsp;charge<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$QSignLabel</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow></mstyle></math>&nbsp;is uniformly distributed on the disc.</p>
<p>Use symmetry to set up a single&nbsp;expression&nbsp;for the&nbsp;electric potential at an arbitrary point along the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi></mrow></mstyle></math>-axis due to the ring of charge.<br />
Use the fact&nbsp;that one can treat the disc as an infinite number of thin rings, and the fact that the electric field due to a&nbsp;single thin<br />
ring of radius <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>R</mi></mrow></mstyle></math>&nbsp;is:</p>
<p align="center"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi><mrow><mfrac><mi>dQ</mi><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>R</mi><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p align="left">Provide the <em>integrand</em> of this integral as your answer, in terms of the given parameters.</p>
<p align="center">e.g. Integral: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathcolor='#800080'>a</mi></mrow><mrow><mi>a</mi></mrow></munderover><mfrac><mi>Q</mi><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow></mstyle></math>, Response: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>Q</mi><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mstyle></math></p>
<p align="left">In order to find the electric potential, one needs to evaluate an integral that adds up the contributions from&nbsp;each infinitesimal ring that<br />
makes up the disc, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dr</mi></mrow></mstyle></math>, from <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>0</mn></mrow></mstyle></math> to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$aMag$aLet</mi></mrow></mstyle></math>.&nbsp;&nbsp;&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="585" height="500">
<param name="image" value="__BASE_URI__img/ElectricFields/DiscCharge/DiagramDetailed.png" />
<param name="size" value="3" />
<param name="label.1.x" value="25" />
<param name="label.1.y" value="300" />
<param name="label.1.text" value="$QSignLabel Q" />
<param name="label.2.x" value="345" />
<param name="label.2.y" value="80" />
<param name="label.2.text" value="$aMag$aLet" />
<param name="label.3.x" value="270" />
<param name="label.3.y" value="250" />
<param name="label.3.text" value="x" /></applet></p>
<p align="center">&nbsp;</p>
<p>&nbsp;</p>
<p><em>Note: </em>Use the letter 'k' for the Coulomb constant, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mrow><mi>&pi;</mi></mrow><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mfrac></mrow></mstyle></math>.</p>@
qu.7.8.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSA2,subPackage=vectorAlgebraicComponents)@
qu.7.8.allow2d=0@
qu.7.8.maple_answer=k*($QSign)*(Q/($aMag*$aLet))/sqrt(r^2+x^2)@
qu.7.8.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.7.8.type=maple@
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qu.7.8.name=Disc Charge - Find V - Set Up Integral@
qu.7.8.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSA2,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.7.8.editing=useHTML@
qu.7.8.solution=<p>Start by identifying a small ring of charge on the disc at a radius of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dr</mi></mrow></mstyle></math>.&nbsp; Then the small ring will produce&nbsp;an electric potential:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dV</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mi>k</mi></mrow><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac><mi>dQ</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>.</p>
<p>Since the charge is uniformly distributed on the disc, the charge per unit area is constant, so:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mi>dQ</mi></mrow><mrow><mi>dr</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac></mrow></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mi>dQ</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac><mi>d</mi></mrow><mrow><mi mathvariant='normal'>r</mi></mrow></mstyle></math>.</p>
<p>Adding up the contributions from each unit of charge on the ring:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></munderover><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow></mfenced></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>r</mi></mrow><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mi></mi></mrow></mstyle></math></p>@
qu.7.8.algorithm=$aMag=rint(2,10);
$aLet=switch(rint(8),'a','b','c','d','f','g','h','w');
$idx=rint(2);
$QSign=switch($idx,1,-1);
$QSignLabel=switch($idx,"+","-");
$QMag=rint(2,6);@
qu.7.8.uid=35505a14-ff76-4dc4-80a9-b275dbf68831@
qu.7.8.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Electric Potential Due to A Line of Charge;
  Difficulty=Hard;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Diagram;
@

qu.7.9.question=<p>Find an algebraic expression for the electric potential at point P, in terms of the given parameters.</p>
<p>&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="395" height="400">
<param name="image" value="__BASE_URI__img/ElectricFields/E-PtChr-2Chrg-2D-Sym/Diagram.png" />
<param name="size" value="10" />
<param name="label.1.x" value="20" />
<param name="label.1.y" value="105" />
<param name="label.1.text" value="q" />
<param name="label.2.x" value="10" />
<param name="label.2.y" value="105" />
<param name="label.2.text" value="$q1" />
<param name="label.3.x" value="20" />
<param name="label.3.y" value="317" />
<param name="label.3.text" value="q" />
<param name="label.4.x" value="10" />
<param name="label.4.y" value="317" />
<param name="label.4.text" value="$q2" />
<param name="label.5.x" value="50" />
<param name="label.5.y" value="155" />
<param name="label.5.text" value="a" />
<param name="label.6.x" value="40" />
<param name="label.6.y" value="155" />
<param name="label.6.text" value="$a1mag" />
<param name="label.7.x" value="50" />
<param name="label.7.y" value="265" />
<param name="label.7.text" value="a" />
<param name="label.8.x" value="40" />
<param name="label.8.y" value="265" />
<param name="label.8.text" value="$a2mag" />
<param name="label.9.x" value="160" />
<param name="label.9.y" value="200" />
<param name="label.9.text" value="x" />
<param name="label.10.x" value="150" />
<param name="label.10.y" value="200" />
<param name="label.10.text" value="$xmag" /></applet></p>
<p><br />
<em>Entry Notes</em>:&nbsp;&nbsp;<br />
<br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
If necessary, use the letter <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi></mrow></mstyle></math> to represent the Coulomb constant <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mi mathvariant='normal'>&pi;</mi><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mfrac></mrow></mrow></mstyle></math>.</p>@
qu.7.9.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=vectorAlgebraicComponents);@
qu.7.9.allow2d=0@
qu.7.9.maple_answer=k*($q1)*q/sqrt(($a1mag*a)^2+($xmag*x)^2)
+k*($q2)*q/sqrt((-$a2mag*a)^2+($xmag*x)^2)@
qu.7.9.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.7.9.type=maple@
qu.7.9.mode=Maple@
qu.7.9.name=Electric Potential Due to Point Charges - 2 Charges, 2D, Different Charges@
qu.7.9.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.7.9.editing=useHTML@
qu.7.9.hint.1=Calculate the potential due to each charge separately, then add them together.@
qu.7.9.solution=<p>The electric potential due to a point charge is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>kq</mi><mrow><mi>r</mi></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>We can find the potential due to each charge separately, then add them together to find the total.</p>
<p>&nbsp;</p>
<p align="left">From both charges to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math>, we have <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$a1mag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow></mstyle></math>.</p>
<p align="left">Thus,</p>
<p align="left"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi>kq</mi><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$a1mag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$q1</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$q2</mi></mrow></mfenced></mrow></mfenced></mrow></mstyle></math>.</p>@
qu.7.9.algorithm=$a1mag=rint(2,5);
$a2mag=$a1mag;
$xmag=rint(2,5);
$q1=switch(rint(2),rint(1,10),-rint(1,10));
$q2=switch(rint(2),rint(1,10),-rint(1,10));@
qu.7.9.uid=128f2d6f-f073-4865-91be-0bb1b6ce6b2f@
qu.7.9.info=  Course=Introductory Electricity and Magnetism;
  Topic=Electric Potential Due to Point Charges in 2D;
  Author=Aron Pasieka;
  Difficulty=Hard;
  Features=Algebraic;
  Features=Algorithmic;
  Features=Partial Grading;
  Features=Diagram;
@

qu.7.10.question=<p>The magnitude of the potential difference between two infinite<font size="2"> parallel</font> plates separated by $x cm&nbsp;is $pot V.&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="400" height="360">
<param name="image" value="__BASE_URI__img/ElectricFields/InfiniteParallelPlates/Diagram.png" />
<param name="size" value="1" />
<param name="label.1.x" value="295" />
<param name="label.1.y" value="12" />
<param name="label.1.text" value="$x cm" /></applet>&nbsp;</p>
<p>&nbsp;</p>
<p>What is the magnitude of the electric field between the plates?</p>@
qu.7.10.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.7.10.allow2d=0@
qu.7.10.maple_answer=abs($E)*V/m@
qu.7.10.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.7.10.type=maple@
qu.7.10.mode=Maple@
qu.7.10.name=Infinite Parallel Plates - Find Field Magnitude ~ PG@
qu.7.10.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.7.10.editing=useHTML@
qu.7.10.hint.1=Recall that the electric field between infinite parallel plates is constant.&nbsp@
qu.7.10.hint.2=;The potential difference is related to the electric field and the separation of the plates.@
qu.7.10.hint.3=Ensure that the units work out.@
qu.7.10.solution=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&Delta;V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mi>&Delta;x</mi><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow></mstyle></math>and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>&Delta;x</mi></mrow><mi>&rarr;</mi></mover></mrow></mstyle></math>&nbsp;are parallel so the magnitude of the dot product is equal to the product of the two magnitudes.&nbsp; (ie. <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow></mstyle></math>)</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo></mrow><mrow><mrow><mover><mrow><mi>&Delta;x</mi></mrow><mi>&rarr;</mi></mover></mrow><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$x cm<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='|' close='|' separators=','><mrow><mi>V</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi></mi></mrow></mstyle></math>$pot V</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><mrow><mrow><mi mathvariant='normal'></mi></mrow><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow></mrow></mfenced></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>($pot V)/($x cm)<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow></mrow><mrow><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo></mrow><mrow><mrow><mover><mrow><mi>E</mi></mrow><mi>&rarr;</mi></mover></mrow><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>${sig(3,abs($E))} V/m</p>@
qu.7.10.algorithm=$potGen=rand(10,100,3);
$x=rand(2,15,3);
$EGen=sig(3,$potGen/($x/100));
$pot=sig(3,$EGen*($x/100));
$idx=rint(2);
$dir=switch($idx,1,-1);
$E=sig(3,$EGen*$dir);
$correctPlate=switch($idx,"A","B");
$wrongPlate=switch($idx,"B","A");@
qu.7.10.uid=6b488086-c97d-4b69-b50b-ac61fb771e11@
qu.7.10.info=  Course=Introductory Electricity and Magnetism;
  Topic=Electric Potential Between Infinite Parallel Plates;
  Author=Aron Pasieka;
  Difficulty=Easy;
  Features=Algorithmic;
  Features=Partial Grading;
  Features=Diagram;
@

qu.7.11.mode=Inline@
qu.7.11.name=Equipotential Lines 3@
qu.7.11.comment=@
qu.7.11.editing=useHTML@
qu.7.11.solution=<p>Assuming that the lines are equally spaced, if the electric field is constant then the change in potential between each pair of lines should also be constant.</p>
<p>&nbsp;</p>
<p>The electric field lines are always perpendicular to the equipotential lines.</p>
<p>&nbsp;</p>
<p>The electric field will always point from higher to lower potential.</p>@
qu.7.11.algorithm=$idx=rint(2);
$corMag=switch($idx,'Constant','Changing');
$incMag=switch($idx,'Changing','Constant');
$V1=range(-500,500);
$d1=range(-100,100);
$corDirN=($d1/abs($d1));
$V2=$V1+$d1;
$d2=switch($idx,$d1,$corDirN*range(1,100));
$V3=$V2+$d2;
$d3=switch($idx,$d1,$corDirN*range(1,100));
$V4=$V3+$d3;
$corDir=switch(-$corDirN+1,'-',0,'+');
$incDir=switch(-$corDirN+1,'+',0,'-');@
qu.7.11.uid=3e261377-ade5-4e7e-b0b3-decae3fb0a27@
qu.7.11.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Equipotential Lines;
  Difficulty=Medium;
  Features=Algorithmic;
  Features=Diagram;
@
qu.7.11.weighting=1,1,1@
qu.7.11.numbering=alpha@
qu.7.11.part.1.grader=exact@
qu.7.11.part.1.name=sro_id_1@
qu.7.11.part.1.editing=useHTML@
qu.7.11.part.1.display.permute=true@
qu.7.11.part.1.answer.3=Impossible to Tell@
qu.7.11.part.1.question=(Unset)@
qu.7.11.part.1.answer.2=$incMag@
qu.7.11.part.1.answer.1=$corMag@
qu.7.11.part.1.mode=List@
qu.7.11.part.1.display=menu@
qu.7.11.part.1.credit.3=0.0@
qu.7.11.part.1.credit.2=0.0@
qu.7.11.part.1.credit.1=1.0@
qu.7.11.part.2.grader=exact@
qu.7.11.part.2.name=sro_id_2@
qu.7.11.part.2.editing=useHTML@
qu.7.11.part.2.display.permute=true@
qu.7.11.part.2.answer.3=Impossible to Tell@
qu.7.11.part.2.question=(Unset)@
qu.7.11.part.2.answer.2=Changing@
qu.7.11.part.2.answer.1=Constant@
qu.7.11.part.2.mode=List@
qu.7.11.part.2.display=menu@
qu.7.11.part.2.credit.3=0.0@
qu.7.11.part.2.credit.2=1.0@
qu.7.11.part.2.credit.1=0.0@
qu.7.11.part.3.answer.6=- (+ x - y)@
qu.7.11.part.3.answer.5=+ (+ x - y)@
qu.7.11.part.3.answer.4=- y@
qu.7.11.part.3.editing=useHTML@
qu.7.11.part.3.answer.3=+ y@
qu.7.11.part.3.answer.2=- x@
qu.7.11.part.3.answer.1=+ x@
qu.7.11.part.3.credit.8=0.0@
qu.7.11.part.3.credit.7=1.0@
qu.7.11.part.3.credit.6=0.0@
qu.7.11.part.3.credit.5=0.0@
qu.7.11.part.3.question=(Unset)@
qu.7.11.part.3.credit.4=0.0@
qu.7.11.part.3.mode=List@
qu.7.11.part.3.credit.3=0.0@
qu.7.11.part.3.credit.2=0.0@
qu.7.11.part.3.credit.1=0.0@
qu.7.11.part.3.grader=exact@
qu.7.11.part.3.display=menu@
qu.7.11.part.3.answer.8=$incDir (+ x + y)@
qu.7.11.part.3.name=sro_id_3@
qu.7.11.part.3.display.permute=true@
qu.7.11.part.3.answer.7=$corDir (+ x + y)@
qu.7.11.question=<p><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="600" height="349"><param name="image" value="__BASE_URI__img/EquipotentialLines/EquipotentialLines3/Diagram.png" /><param name="size" value="4" /><param name="label.1.x" value="300" /><param name="label.1.y" value="95" /><param name="label.1.text" value="$V1 V" /><param name="label.2.x" value="380" /><param name="label.2.y" value="95" /><param name="label.2.text" value="$V2 V" /><param name="label.3.x" value="465" /><param name="label.3.y" value="95" /><param name="label.3.text" value="$V3 V" /><param name="label.4.x" value="550" /><param name="label.4.y" value="95" /><param name="label.4.text" value="$V4 V" /></applet></p><p>&nbsp;</p><p>Please select the most appropriate description of the electric field associated with the equipotential lines in the diagram.&nbsp;</p><p>&nbsp;</p><p><span>The magnitude of the electric field is:&nbsp; </span><span>&nbsp;</span><1><span>&nbsp;<br /><br /></span><span><span>The direction of the electric field is:&nbsp;&nbsp;<span>&nbsp;</span><2><span>&nbsp;</span><br /><br />The direction of the electric field at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math>&nbsp;is in&nbsp;the direction: &nbsp; </span><3><span>&nbsp;</span></span></p>@

qu.7.12.question=<p>A thin ring of radius&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$aMag$aLet</mi></mrow></mstyle></math>&nbsp;is centered at the origin in the&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mover><mi>z</mi><mi>&#x005e;</mi></mover></mrow></mstyle></math>&nbsp;plane.&nbsp;&nbsp;A&nbsp;charge<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$QSignLabel</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow></mstyle></math>&nbsp;is uniformly distributed on the ring.&nbsp;</p>
<p>Use symmetry to set up a single&nbsp;expression&nbsp;for the&nbsp;electric potential at an&nbsp;arbitrary point along the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi></mrow></mstyle></math>-axis due to the ring of charge.<br />
Provide&nbsp;the <em>integrand</em> of this integral as your answer, in terms of the given parameters.</p>
<p align="center">e.g.&nbsp; Integral:&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathcolor='#800080'>a</mi></mrow><mrow><mi>a</mi></mrow></munderover><mfrac><mi>Q</mi><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow></mstyle></math>,&nbsp; Response:&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>Q</mi><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mstyle></math></p>
<p>In order to find the electric potential, one needs to evaluate an integral that adds up the contributions from each infinitesimal piece<br />
of the line, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ds</mi></mrow></mstyle></math>, from <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>0</mn></mrow></mstyle></math> to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&pi;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mstyle></math>.</p>
<p align="center">&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="585" height="500">
<param name="image" value="__BASE_URI__img/ElectricFields/RingCharge/Diagram.png" />
<param name="size" value="3" />
<param name="label.1.x" value="25" />
<param name="label.1.y" value="300" />
<param name="label.1.text" value="$QSignLabel Q" />
<param name="label.2.x" value="220" />
<param name="label.2.y" value="22" />
<param name="label.2.text" value="$aMag$aLet" />
<param name="label.3.x" value="270" />
<param name="label.3.y" value="250" />
<param name="label.3.text" value="x" /></applet></p>
<p align="center">&nbsp;</p>
<p><em>Note: </em>Use the letter 'k' for the Coulomb constant, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mrow><mi>&pi;</mi></mrow><mrow><msub><mi>&varepsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mfrac></mrow></mstyle></math>.</p>@
qu.7.12.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSA2,subPackage=vectorAlgebraicComponents)@
qu.7.12.allow2d=0@
qu.7.12.maple_answer=k*($QSign)*(Q/(2*Pi*$aMag*$aLet))/sqrt(($aMag*$aLet)^2+x^2)@
qu.7.12.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.7.12.type=maple@
qu.7.12.mode=Maple@
qu.7.12.name=Ring Charge - Find V - Set Up Integral@
qu.7.12.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSA2,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.7.12.editing=useHTML@
qu.7.12.solution=<p>Start by identifying a small unit of charge on the ring, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dq</mi></mrow></mstyle></math>.&nbsp; Then the small amount of electric potential due to that unit of charge is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>kdq</mi><mrow><mi>r</mi></mrow></mfrac></mrow></mrow></mstyle></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msqrt><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>Since the charge is uniformly distributed on the ring, the charge per unit length is constant, so:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>dq</mi><mrow><mi>ds</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac></mrow></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mi>dq</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac><mi>ds</mi></mrow><mrow></mrow></mstyle></math>.</p>
<p>Adding up the contributions from each unit of charge on the ring:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi>s</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></munderover><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>$QSignLabel$QMag</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Q</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mi>&pi;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>s</mi></mrow><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$aMag$aLet</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac></mrow></mstyle></math></p>@
qu.7.12.algorithm=$aMag=rint(2,10);
$aLet=switch(rint(8),'a','b','c','d','f','g','h','w');
$idx=rint(2);
$QSign=switch($idx,1,-1);
$QSignLabel=switch($idx,"+","-");
$QMag=rint(2,6);@
qu.7.12.uid=b152c087-4167-4efb-9049-8e07a72ed227@
qu.7.12.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Electric Potential Due to A Line of Charge;
  Difficulty=Hard;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Diagram;
@

qu.8.topic=Capacitance@

qu.8.1.question=<p>A cylindrical capacitor has inner and outer radii as labelled in following diagram.</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="400" height="420">
<param name="image" value="__BASE_URI__img/Capacitance/Capacitance-Coaxial/Diagram.png" />
<param name="size" value="4" />
<param name="label.1.x" value="83" />
<param name="label.1.y" value="24" />
<param name="label.1.text" value="$rout mm" />
<param name="label.2.x" value="315" />
<param name="label.2.y" value="42" />
<param name="label.2.text" value="$rin mm" />
<param name="label.3.x" value="85" />
<param name="label.3.y" value="150" />
<param name="label.3.text" value="+D" />
<param name="label.4.x" value="310" />
<param name="label.4.y" value="150" />
<param name="label.4.text" value="-D" /></applet></p>
<p>Assume that the surface charge densities are&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>D</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>D</mi></mrow></mstyle></math>.&nbsp; Find the length of capacitor required&nbsp;<br />
to acheive&nbsp;a capacitance of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$C</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$Cex</mi></mrow></msup><mi>F</mi></mrow></mstyle></math>.</p>@
qu.8.1.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.8.1.allow2d=0@
qu.8.1.maple_answer=SigFigs[roundToSigFigs](evalf(($C*10^(-$Cex))/(2*Pi*8.854187817*10^(-12)/ln($rout/$rin))),3)*m;@
qu.8.1.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.8.1.type=maple@
qu.8.1.mode=Maple@
qu.8.1.name=Capacitance - Coaxial - Find L ~ PG@
qu.8.1.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.8.1.editing=useHTML@
qu.8.1.solution=<p>The capacitance per unit length for a cylindrical capacitor is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mfrac><mi>C</mi><mrow><mi>L</mi></mrow></mfrac></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mrow><msub><mi>&epsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow><mrow><mi mathvariant='normal'>ln</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><msub><mi>r</mi><mrow><mi>out</mi></mrow></msub><mrow><msub><mi>r</mi><mrow><mi>inner</mi></mrow></msub></mrow></mfrac></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>.</p>
<p>Therefore:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>L</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>C</mi><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mrow><msub><mi>&epsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mfrac></mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mi mathvariant='normal'>ln</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><msub><mi>r</mi><mrow><mi>out</mi></mrow></msub></mrow><mrow><msub><mi>r</mi><mrow><mo mathvariant='bold' fontweight='bold' lspace='0.0em' rspace='0.0em'>in</mo></mrow></msub></mrow></mfrac></mrow></mfenced></mrow></mrow><mrow><mi></mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>L</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$C</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$Cex</mi></mrow></msup><mi>F</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mrow><msub><mi>&epsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mfrac></mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mi mathvariant='normal'>ln</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$rout</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow><mrow><mi>$rin</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mfrac></mrow></mfenced></mrow></mrow><mrow><mi mathvariant='normal'></mi></mrow></mstyle></math></p>@
qu.8.1.algorithm=$rin=rand(0.2,15,3);
$rout=rand($rin,25,3);
$C=rand(1.00,9.99,3);
$Cex=range(4,7);@
qu.8.1.uid=57d5827b-e701-45e7-a584-82e52bbc36e9@
qu.8.1.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Capacitance;
  Difficulty=Medium;
  Features=Algorithmic;
  Features=Diagram;
  Features=Partial Grading;
@

qu.8.2.question=<p>How far apart must the plates of the capacitor in the diagram be to give it a capacitance of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$C</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nF</mi></mrow></mstyle></math>?</p>
<p align="center">&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="500" height="560">
<param name="image" value="__BASE_URI__img/Capacitance/Capacitance-ParallelPlate/Diagram.png" />
<param name="size" value="3" />
<param name="label.1.x" value="280" />
<param name="label.1.y" value="20" />
<param name="label.1.text" value="d" />
<param name="label.2.x" value="130" />
<param name="label.2.y" value="385" />
<param name="label.2.text" value="$x cm" />
<param name="label.3.x" value="305" />
<param name="label.3.y" value="480" />
<param name="label.3.text" value="$y cm" /></applet></p>
<p align="center">&nbsp;</p>
<p align="center">&nbsp;</p>
<p>&nbsp;</p>
<p>&nbsp;</p>@
qu.8.2.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.8.2.allow2d=0@
qu.8.2.maple_answer=SigFigs[roundToSigFigs]($ans,3)*mm@
qu.8.2.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.8.2.type=maple@
qu.8.2.mode=Maple@
qu.8.2.name=Capacitor Geometry - Parallel Plate - Find Separation ~ PGc@
qu.8.2.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.8.2.editing=useHTML@
qu.8.2.solution=<p>The capacitance of a parallel-plate capacitor is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>&epsilon;</mi><mrow><mn>0</mn></mrow></msub><mfrac><mi>A</mi><mrow><mi>d</mi></mrow></mfrac></mrow></mstyle></math>.</p>
<p>Therefore:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>&epsilon;</mi><mrow><mn>0</mn></mrow></msub><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>cm</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$y</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>cm</mi></mrow></mfenced></mrow><mrow><mi>$C</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nF</mi></mrow></mfrac></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>.</p>
<p>Now, we can calculate <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>@
qu.8.2.algorithm=$x=rand(10.0,99.9,3);
$y=rand(10.0,99.9,3);
$C=rand(1.00,9.99,3);
$ans=1000*($x*10^(-2))*($y*10^(-2))*(8.85*10^(-12))/($C*10^(-9));@
qu.8.2.uid=65f3da19-f1a4-4252-ae56-7ac028e5ea2f@
qu.8.2.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Geometry of a Capacitor;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.8.3.question=<p align="center"><img alt="" align="middle" width="438" height="400" src="__BASE_URI__img/Capacitance/CapacitorsSeriesParallel2/Diagram.png" /></p>
<p>&nbsp;</p>
<p>If <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C1</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$C1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$C1ex</mi></mrow></msup><mi>F</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C2</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$C2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$C2ex</mi></mrow></msup><mi>F</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C3</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$C3</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$C3ex</mi></mrow></msup><mi>F</mi></mrow></mstyle></math><br />
and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C4</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$C4</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$C4ex</mi></mrow></msup><mi>F</mi></mrow></mstyle></math>, what is the total capacitance of the circuit between <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>?</p>@
qu.8.3.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN)@
qu.8.3.allow2d=0@
qu.8.3.maple_answer=SigFigs[roundToSigFigs](((1/($C1*10^(-$C1ex))+1/($C2*10^(-$C2ex)))^(-1)+($C3*10^(-$C3ex))+($C4*10^(-$C4ex))),3)*F;@
qu.8.3.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.8.3.type=maple@
qu.8.3.mode=Maple@
qu.8.3.name=Capacitors in Series and Parallel 2 - Numeric ~ PG@
qu.8.3.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.8.3.editing=useHTML@
qu.8.3.solution=<p>For the bottom arm, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>C</mi><mrow><mi>eq</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><msub><mi>C</mi><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mn>1</mn><mrow><msub><mi>C</mi><mrow><mn>2</mn></mrow></msub></mrow></mfrac></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup></mrow></mstyle></math>.</p>
<p>The total is then:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>C</mi><mrow><mi>AB</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>C</mi><mrow><mi>eq</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>C</mi><mrow><mn>3</mn></mrow></msub></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>C</mi><mrow><mn>4</mn></mrow></msub></mrow></mstyle></math>.</p>@
qu.8.3.algorithm=$C1=rand(1.00,9.99,3);
$C2=rand(1.00,9.99,3);
$C3=rand(1.00,9.99,3);
$C4=rand(1.00,9.99,3);
$C1ex=range(5,7);
$C2ex=range(5,7);
$C3ex=range(5,7);
$C4ex=range(5,7);@
qu.8.3.uid=3455e9a3-80bc-4d05-b452-f2963576da0e@
qu.8.3.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Capacitors in Series and Parallel;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Diagram;
  Features=Algorithmic;
@

qu.8.4.question=<p><span>A cylindrical capacitor of capacitance&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$C</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&mu;F</mi></mrow></mstyle></math> has accumulated&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$nex</mi></mrow></msup></mrow></mstyle></math>&nbsp;electrons on the negative<br />
&nbsp;surface.&nbsp; What is the potential difference between the plates?</span></p>
<p>&nbsp;</p>
<p>&nbsp;</p>@
qu.8.4.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.8.4.allow2d=0@
qu.8.4.maple_answer=SigFigs[roundToSigFigs]($ans,3)*V@
qu.8.4.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.8.4.type=maple@
qu.8.4.mode=Maple@
qu.8.4.name=Simple Capacitance - Find V ~ PGc@
qu.8.4.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.8.4.editing=useHTML@
qu.8.4.solution=<p>The relationship between charge, potential difference and capacitance is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>Q</mi><mrow><mi>V</mi></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>We are given&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Q</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$nex</mi></mrow></msup></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>e</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>@
qu.8.4.algorithm=$C=rand(1.00,9.99,3);
$n=rand(1.00,9.99,3);
$nex=range(13,16);
$ans=(($n*10^($nex))*(1.60*10^(-19)))/($C*10^(-6));@
qu.8.4.uid=ffc9c9e8-4ef9-4121-9d8b-1e529aa99bb8@
qu.8.4.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Capacitance;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Algorithmic;
@

qu.8.5.question=<p>What is the capacitance of the following capacitor?&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="500" height="560">
<param name="image" value="__BASE_URI__img/Capacitance/Capacitance-ParallelPlate/Diagram.png" />
<param name="size" value="3" />
<param name="label.1.x" value="280" />
<param name="label.1.y" value="20" />
<param name="label.1.text" value="$d mm" />
<param name="label.2.x" value="130" />
<param name="label.2.y" value="385" />
<param name="label.2.text" value="$x cm" />
<param name="label.3.x" value="305" />
<param name="label.3.y" value="480" />
<param name="label.3.text" value="$y cm" /></applet></p>
<p align="center">&nbsp;</p>
<p align="center">&nbsp;</p>
<p>&nbsp;</p>
<p><span>&nbsp;</span></p>@
qu.8.5.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
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qu.8.5.maple_answer=SigFigs[roundToSigFigs]($ans,3)*F@
qu.8.5.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.8.5.type=maple@
qu.8.5.mode=Maple@
qu.8.5.name=Capacitor Geometry - Parallel Plate - Find Capacitance ~ PGc@
qu.8.5.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.8.5.editing=useHTML@
qu.8.5.solution=<p>The capacitance of a parallel-plate capacitor is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>&epsilon;</mi><mrow><mn>0</mn></mrow></msub><mfrac><mi>A</mi><mrow><mi>d</mi></mrow></mfrac></mrow></mstyle></math>.</p>
<p>Therefore:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>&epsilon;</mi><mrow><mn>0</mn></mrow></msub><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>cm</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$y</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>cm</mi></mrow></mfenced></mrow><mrow><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mfrac></mrow></mstyle></math></p>@
qu.8.5.algorithm=$x=rand(10.0,99.9,3);
$y=rand(10.0,99.9,3);
$d=rand(.10,3.0,3);
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qu.8.5.uid=bc669094-4948-4f64-9876-741d8c50dc4d@
qu.8.5.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Geometry of a Capacitor;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.8.6.question=<p><span>A cylindrical capacitor of capacitance&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$C</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&mu;F</mi></mrow></mstyle></math> has developed a potential difference of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$V</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>V</mi></mrow></mstyle></math> between<br />
the positive and negative surfaces. How many electrons are there on the negative surface?</span></p>@
qu.8.6.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.8.6.allow2d=0@
qu.8.6.maple_answer=SigFigs[roundToSigFigs]($ans,3)@
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qu.8.6.type=maple@
qu.8.6.mode=Maple@
qu.8.6.name=Simple Capacitance - Find Electrons ~ PGc@
qu.8.6.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.8.6.editing=useHTML@
qu.8.6.solution=<p>The relationship between charge, potential difference and capacitance is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>Q</mi><mrow><mi>V</mi></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>We are given <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>.&nbsp; We can then use <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Q</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>n</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>e</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>@
qu.8.6.algorithm=$C=rand(1.00,9.99,3);
$V=rand(1.00,9.99,3);
$ans=($C*10^(-6))*$V/(1.60*10^(-19));@
qu.8.6.uid=57167ccb-d19a-423c-9e03-562845986f9c@
qu.8.6.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Capacitance;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Algorithmic;
@

qu.8.7.question=<p>A cylindrical capacitor has inner and outer radii as labelled in following diagram.</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="400" height="420">
<param name="image" value="__BASE_URI__img/Capacitance/Capacitance-Coaxial/Diagram.png" />
<param name="size" value="4" />
<param name="label.1.x" value="83" />
<param name="label.1.y" value="24" />
<param name="label.1.text" value="$rout mm" />
<param name="label.2.x" value="315" />
<param name="label.2.y" value="42" />
<param name="label.2.text" value="$rin mm" />
<param name="label.3.x" value="85" />
<param name="label.3.y" value="150" />
<param name="label.3.text" value="+D" />
<param name="label.4.x" value="310" />
<param name="label.4.y" value="150" />
<param name="label.4.text" value="-D" /></applet></p>
<p>Assuming that the surface charge densities are&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>D</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>D</mi></mrow></mstyle></math>, calculate<br />
the capacitance for a section of&nbsp;length&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$L</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>cm</mi></mrow></mstyle></math>of the capacitor.</p>@
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qu.8.7.allow2d=0@
qu.8.7.maple_answer=SigFigs[roundToSigFigs](evalf($L*10^(-2)*2*Pi*8.854187817*10^(-12)/ln($rout/$rin)),3)*F;@
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qu.8.7.type=maple@
qu.8.7.mode=Maple@
qu.8.7.name=Capacitance - Coaxial - Find C ~ PG@
qu.8.7.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.8.7.editing=useHTML@
qu.8.7.solution=<p>The capacitance per unit length for a cylindrical capacitor is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>C</mi><mrow><mi>L</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mrow><msub><mi>&epsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow><mrow><mi mathvariant='normal'>ln</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><msub><mi>r</mi><mrow><mi>out</mi></mrow></msub><mrow><msub><mi>r</mi><mrow><mi>inner</mi></mrow></msub></mrow></mfrac></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>.</p>
<p>Therefore:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>L</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfrac><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mrow><msub><mi>&epsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow><mrow><mi mathvariant='normal'>ln</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><msub><mi>r</mi><mrow><mi>out</mi></mrow></msub></mrow><mrow><msub><mi>r</mi><mrow><mo mathvariant='bold' fontweight='bold' lspace='0.0em' rspace='0.0em'>in</mo></mrow></msub></mrow></mfrac></mrow></mfenced></mrow></mfrac></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$L</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>cm</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfrac><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mrow><msub><mi>&epsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow><mrow><mi mathvariant='normal'>ln</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$rout</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow><mrow><mi>$rin</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mfrac></mrow></mfenced></mrow></mfrac></mrow></mstyle></math></p>@
qu.8.7.algorithm=$rin=rand(0.2,15,3);
$rout=rand($rin,25,3);
$L=rand(1.00,9.99,3);@
qu.8.7.uid=e1cb1310-d5f2-4997-8596-cae8da279f99@
qu.8.7.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Capacitance;
  Difficulty=Medium;
  Features=Algorithmic;
  Features=Diagram;
  Features=Partial Grading;
@

qu.8.8.question=<p><span>A cylindrical capacitor has accumulated&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$nex</mi></mrow></msup></mrow></mstyle></math>&nbsp;electrons on the negative&nbsp;surface, causing a&nbsp;<br />
potential difference of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$V</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>V</mi></mrow></mstyle></math>&nbsp;between the positive and negative surface.&nbsp; What is the capacitance of the&nbsp;<br />
capacitor?</span></p>
<p>&nbsp;</p>
<p>&nbsp;</p>@
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qu.8.8.maple_answer=SigFigs[roundToSigFigs]($ans,3)*F@
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qu.8.8.type=maple@
qu.8.8.mode=Maple@
qu.8.8.name=Simple Capacitance - Find C ~ PGc@
qu.8.8.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.8.8.editing=useHTML@
qu.8.8.solution=<p>The relationship between charge, potential difference and capacitance is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>Q</mi><mrow><mi>V</mi></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>We are given <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Q</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$nex</mi></mrow></msup></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>e</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>@
qu.8.8.algorithm=$V=rand(1.00,9.99,3);
$n=rand(1.00,9.99,3);
$nex=range(13,16);
$ans=(($n*10^($nex))*(1.60*10^(-19)))/($V);@
qu.8.8.uid=8dc5d35e-9f8e-4c85-b626-6d3cbcccc70f@
qu.8.8.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Capacitance;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Algorithmic;
@

qu.8.9.question=<p>A spherical capacitor has <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$U</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>J</mi></mrow></mstyle></math>&nbsp;of energy stored inside when <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$nex</mi></mrow></msup></mrow></mstyle></math>&nbsp;electrons are on the<br />
&nbsp;$wPlate plate.&nbsp; Calculate the capacitance.</p>
<p>&nbsp;</p>
<p>&nbsp;</p>@
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qu.8.9.allow2d=0@
qu.8.9.maple_answer=SigFigs[roundToSigFigs]($ans,3)*F@
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qu.8.9.type=maple@
qu.8.9.mode=Maple@
qu.8.9.name=Energy Stored in Capacitor - Find Capacitance ~ PGc@
qu.8.9.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.8.9.editing=useHTML@
qu.8.9.solution=<p>The energy stored in a capacitor can be expressed as:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>U</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><msup><mi>Q</mi><mrow><mn>2</mn></mrow></msup><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>C</mi></mrow></mfrac></mrow></mstyle></math>.</p>
<p>We are given <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>U</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$U</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>J</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Q</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$nex</mi></mrow></msup></mrow></mfenced><mi>e</mi></mrow></mstyle></math>.</p>@
qu.8.9.algorithm=$U=rand(1.00,9.99,3);
$n=rand(1.00,9.99,3);
$nex=range(13,15);
$ans=((($n*10^$nex)*(1.60*10^(-19)))^2)/(2*$U);
$wPlate=switch(rint(2),'positive','negative');@
qu.8.9.uid=d330f38f-cbb5-4f02-b5b4-60fb5c7891ed@
qu.8.9.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Energy Stored in a Capacitor;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Algorithmic;
@

qu.8.10.question=<p>If <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$nex</mi></mrow></msup></mrow></mstyle></math>&nbsp;electrons are accumulated on the negative plate of the capacitor in the diagram, what is<br />
the potential difference between the plates?&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="500" height="560">
<param name="image" value="__BASE_URI__img/Capacitance/Capacitance-ParallelPlate/Diagram.png" />
<param name="size" value="3" />
<param name="label.1.x" value="280" />
<param name="label.1.y" value="20" />
<param name="label.1.text" value="$d mm" />
<param name="label.2.x" value="130" />
<param name="label.2.y" value="385" />
<param name="label.2.text" value="$x cm" />
<param name="label.3.x" value="305" />
<param name="label.3.y" value="480" />
<param name="label.3.text" value="$y cm" /></applet></p>
<p align="center">&nbsp;</p>
<p align="center">&nbsp;</p>
<p>&nbsp;</p>
<p><span>&nbsp;</span></p>@
qu.8.10.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.8.10.allow2d=0@
qu.8.10.maple_answer=SigFigs[roundToSigFigs]($ans,3)*V@
qu.8.10.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.8.10.type=maple@
qu.8.10.mode=Maple@
qu.8.10.name=Capacitor Geometry - Parallel Plate - Find Potential ~ PGc@
qu.8.10.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.8.10.editing=useHTML@
qu.8.10.solution=<p>The relationship between charge, capacitance and voltage for a capacitor is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>Q</mi><mrow><mi>V</mi></mrow></mfrac></mrow></mrow></mstyle></math>.&nbsp; We know <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Q</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$nex</mi></mrow></msup></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>e</mi></mrow></mstyle></math>, so we need to find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;in order to calculate <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>The capacitance of a parallel-plate capacitor is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>&epsilon;</mi><mrow><mn>0</mn></mrow></msub><mfrac><mi>A</mi><mrow><mi>d</mi></mrow></mfrac></mrow></mstyle></math>.</p>
<p>Therefore:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>&epsilon;</mi><mrow><mn>0</mn></mrow></msub><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>cm</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$y</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>cm</mi></mrow></mfenced></mrow><mrow><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mfrac></mrow></mstyle></math>.</p>
<p>Now, we can calculate <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>@
qu.8.10.algorithm=$x=rand(10.0,99.9,3);
$y=rand(10.0,99.9,3);
$d=rand(.10,3.0,3);
$n=rand(1,9.99,3);
$nex=range(7,10);
$Q=(1.60*10^(-19))*($n*10^($nex));
$C=($x*10^(-2))*($y*10^(-2))*(8.85*10^(-12))/($d*10^(-3));
$ans=$Q/$C;@
qu.8.10.uid=957b7972-71ef-4620-ac19-aec324597cbb@
qu.8.10.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Geometry of a Capacitor;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.8.11.question=<p><img alt="" align="middle" width="438" height="400" src="__BASE_URI__img/Capacitance/CapacitorsSeriesParallel1/Diagram.png" /></p>
<p>&nbsp;</p>
<p>If <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C1</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$C1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$C1ex</mi></mrow></msup><mi>F</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C2</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$C2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$C2ex</mi></mrow></msup><mi>F</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C3</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$C3</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$C3ex</mi></mrow></msup><mi>F</mi></mrow></mstyle></math>,<br />
what is the total capacitance of the circuit between <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>?</p>@
qu.8.11.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN)@
qu.8.11.allow2d=0@
qu.8.11.maple_answer=SigFigs[roundToSigFigs](((1/($C1*10^(-$C1ex))+1/($C2*10^(-$C2ex)))^(-1)+($C3*10^(-$C3ex))),3)*F;@
qu.8.11.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.8.11.type=maple@
qu.8.11.mode=Maple@
qu.8.11.name=Capacitors in Series and Parallel 1 - Numeric ~ PG@
qu.8.11.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.8.11.editing=useHTML@
qu.8.11.solution=<p>For the bottom arm, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>C</mi><mrow><mi>eq</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><msub><mi>C</mi><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mn>1</mn><mrow><msub><mi>C</mi><mrow><mn>2</mn></mrow></msub></mrow></mfrac></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup></mrow></mstyle></math>.</p>
<p>The total is then:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>C</mi><mrow><mi>AB</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>C</mi><mrow><mi>eq</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>C</mi><mrow><mn>3</mn></mrow></msub></mrow></mstyle></math>.</p>@
qu.8.11.algorithm=$C1=rand(1.00,9.99,3);
$C2=rand(1.00,9.99,3);
$C3=rand(1.00,9.99,3);
$C1ex=range(5,7);
$C2ex=range(5,7);
$C3ex=range(5,7);@
qu.8.11.uid=b122aa02-0f9d-44ef-b4bd-e9d7ca81be1c@
qu.8.11.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Capacitors in Series and Parallel;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Diagram;
  Features=Algorithmic;
@

qu.8.12.question=<p><img alt="" align="middle" width="438" height="400" src="__BASE_URI__img/Capacitance/CapacitorsSeriesParallel1/Diagram.png" /></p>
<p>&nbsp;</p>
<p>What is the total capacitance of the circuit between <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>?</p>@
qu.8.12.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=algebraicScalar)@
qu.8.12.allow2d=0@
qu.8.12.maple_answer=(1/C1+1/C2)^(-1)+C3@
qu.8.12.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.8.12.type=maple@
qu.8.12.mode=Maple@
qu.8.12.name=Capacitors in Series and Parallel 1 ~ PG@
qu.8.12.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=algebraicScalar,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.8.12.editing=useHTML@
qu.8.12.solution=<p>For the bottom arm, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>C</mi><mrow><mi>eq</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><msub><mi>C</mi><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mn>1</mn><mrow><msub><mi>C</mi><mrow><mn>2</mn></mrow></msub></mrow></mfrac></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup></mrow></mstyle></math>.</p>
<p>The total is then:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>C</mi><mrow><mi>AB</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>C</mi><mrow><mi>eq</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>C</mi><mrow><mn>3</mn></mrow></msub></mrow></mstyle></math>.</p>@
qu.8.12.algorithm=@
qu.8.12.uid=fedaa5f7-74d0-43ee-a3a7-d98a0b855805@
qu.8.12.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Capacitors in Series and Parallel;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Diagram;
  Features=Algorithmic;
  Features=Algebraic;
@

qu.8.13.question=<p>A spherical capacitor has&nbsp;a capacitance of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$C</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$Cex</mi></mrow></msup><mi>F</mi></mrow></mstyle></math>.&nbsp; When <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$nex</mi></mrow></msup></mrow></mstyle></math>&nbsp;electrons are on&nbsp;<br />
the&nbsp;$wPlate plate,&nbsp;how much energy is stored in the capacitor?</p>
<p>&nbsp;</p>
<p>&nbsp;</p>@
qu.8.13.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.8.13.allow2d=0@
qu.8.13.maple_answer=SigFigs[roundToSigFigs]($ans,3)*J@
qu.8.13.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.8.13.type=maple@
qu.8.13.mode=Maple@
qu.8.13.name=Energy Stored in Capacitor - Find Energy ~ PGc@
qu.8.13.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.8.13.editing=useHTML@
qu.8.13.solution=<p>The energy stored in a capacitor can be expressed as:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>U</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><msup><mi>Q</mi><mrow><mn>2</mn></mrow></msup><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>C</mi></mrow></mfrac></mrow></mstyle></math>.</p>
<p>We are given <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Q</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$nex</mi></mrow></msup></mrow></mfenced><mi>e</mi></mrow></mstyle></math>.</p>@
qu.8.13.algorithm=$C=rand(1.00,9.99,3);
$Cex=range(5,9);
$n=rand(1.00,9.99,3);
$nex=range(13,15);
$ans=((($n*10^$nex)*(1.60*10^(-19)))^2)/(2*($C*10^(-$Cex)));
$wPlate=switch(rint(2),'positive','negative');@
qu.8.13.uid=979fba97-462b-4dd5-b597-c5ba3bb0a58d@
qu.8.13.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Energy Stored in a Capacitor;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Algorithmic;
@

qu.8.14.question=<p><img alt="" align="middle" width="438" height="400" src="__BASE_URI__img/Capacitance/CapacitorsSeriesParallel2/Diagram.png" /></p>
<p>&nbsp;</p>
<p>What is the total capacitance of the circuit between <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>?</p>@
qu.8.14.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=algebraicScalar)@
qu.8.14.allow2d=0@
qu.8.14.maple_answer=(1/C1+1/C2)^(-1)+C3+C4@
qu.8.14.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.8.14.type=maple@
qu.8.14.mode=Maple@
qu.8.14.name=Capacitors in Series and Parallel 2 ~ PG@
qu.8.14.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=algebraicScalar,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.8.14.editing=useHTML@
qu.8.14.solution=<p>For the bottom arm, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>C</mi><mrow><mi>eq</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><msub><mi>C</mi><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mn>1</mn><mrow><msub><mi>C</mi><mrow><mn>2</mn></mrow></msub></mrow></mfrac></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup></mrow></mstyle></math>.</p>
<p>The total is then:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>C</mi><mrow><mi>AB</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>C</mi><mrow><mi>eq</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>C</mi><mrow><mn>3</mn></mrow></msub></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>C</mi><mrow><mn>4</mn></mrow></msub></mrow></mstyle></math>.</p>@
qu.8.14.algorithm=@
qu.8.14.uid=4a59bffb-af18-4cd3-a9d8-cdc89f721163@
qu.8.14.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Capacitors in Series and Parallel;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Diagram;
  Features=Algorithmic;
  Features=Algebraic;
@

qu.8.15.question=<p>If the potential difference between the plates of the capacitor in the diagram is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$V</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>V</mi></mrow></mstyle></math>, what is magnitude<br />
of the total charge accumulated on each plate?</p>
<p align="center">&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="500" height="560">
<param name="image" value="__BASE_URI__img/Capacitance/Capacitance-ParallelPlate/Diagram.png" />
<param name="size" value="3" />
<param name="label.1.x" value="280" />
<param name="label.1.y" value="20" />
<param name="label.1.text" value="$d mm" />
<param name="label.2.x" value="130" />
<param name="label.2.y" value="385" />
<param name="label.2.text" value="$x cm" />
<param name="label.3.x" value="305" />
<param name="label.3.y" value="480" />
<param name="label.3.text" value="$y cm" /></applet></p>
<p align="center">&nbsp;</p>
<p align="center">&nbsp;</p>
<p>&nbsp;</p>@
qu.8.15.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.8.15.allow2d=0@
qu.8.15.maple_answer=SigFigs[roundToSigFigs]($ans,3)*C@
qu.8.15.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.8.15.type=maple@
qu.8.15.mode=Maple@
qu.8.15.name=Capacitor Geometry - Parallel Plate - Find Charge ~ PGc@
qu.8.15.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.8.15.editing=useHTML@
qu.8.15.solution=<p>The relationship between charge, capacitance and voltage for a capacitor is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>Q</mi><mrow><mi>V</mi></mrow></mfrac></mrow></mrow></mstyle></math>.&nbsp; We know <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi></mrow></mstyle></math>, so we need to find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;in order to calculate <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Q</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>The capacitance of a parallel-plate capacitor is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>&epsilon;</mi><mrow><mn>0</mn></mrow></msub><mfrac><mi>A</mi><mrow><mi>d</mi></mrow></mfrac></mrow></mstyle></math>.</p>
<p>Therefore:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>&epsilon;</mi><mrow><mn>0</mn></mrow></msub><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>cm</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$y</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>cm</mi></mrow></mfenced></mrow><mrow><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mfrac></mrow></mstyle></math>.</p>
<p>Now, we can calculate <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Q</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>@
qu.8.15.algorithm=$x=rand(10.0,99.9,3);
$y=rand(10.0,99.9,3);
$d=rand(.10,3.0,3);
$V=rand(1,9.99,3);
$C=($x*10^(-2))*($y*10^(-2))*(8.85*10^(-12))/($d*10^(-3));
$ans=$V*$C;@
qu.8.15.uid=fde18af8-8c01-4885-a668-61c0253c4898@
qu.8.15.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Geometry of a Capacitor;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.8.16.question=<p>A cylindrical capacitor has inner and outer radii as labelled in following diagram.</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="400" height="420">
<param name="image" value="__BASE_URI__img/Capacitance/Capacitance-Coaxial/Diagram.png" />
<param name="size" value="4" />
<param name="label.1.x" value="83" />
<param name="label.1.y" value="24" />
<param name="label.1.text" value="$rout mm" />
<param name="label.2.x" value="315" />
<param name="label.2.y" value="42" />
<param name="label.2.text" value="$rin mm" />
<param name="label.3.x" value="85" />
<param name="label.3.y" value="150" />
<param name="label.3.text" value="+D" />
<param name="label.4.x" value="310" />
<param name="label.4.y" value="150" />
<param name="label.4.text" value="-D" /></applet></p>
<p>Assuming that the surface charge densities are&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>D</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>D</mi></mrow></mstyle></math>, calculate<br />
the capacitance per unit length&nbsp;of the capacitor.</p>@
qu.8.16.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.8.16.allow2d=0@
qu.8.16.maple_answer=SigFigs[roundToSigFigs](evalf(2*Pi*8.854187817*10^(-12)/ln($rout/$rin)),3)*(F/m);@
qu.8.16.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.8.16.type=maple@
qu.8.16.mode=Maple@
qu.8.16.name=Capacitance - Coaxial - Find C/L ~ PG@
qu.8.16.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.8.16.editing=useHTML@
qu.8.16.solution=<p>The capacitance per unit length for a cylindrical capacitor is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mfrac><mi>C</mi><mrow><mi>L</mi></mrow></mfrac></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mrow><msub><mi>&epsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow><mrow><mi mathvariant='normal'>ln</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><msub><mi>r</mi><mrow><mi>out</mi></mrow></msub><mrow><msub><mi>r</mi><mrow><mi>inner</mi></mrow></msub></mrow></mfrac></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>.</p>
<p>Therefore:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mfrac><mi>C</mi><mrow><mi>L</mi></mrow></mfrac></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mrow><msub><mi>&epsilon;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow><mrow><mi mathvariant='normal'>ln</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$rout</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow><mrow><mi>$rin</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mfrac></mrow></mfenced></mrow></mfrac></mrow></mstyle></math></p>@
qu.8.16.algorithm=$rin=rand(0.2,15,3);
$rout=rand($rin,25,3);@
qu.8.16.uid=2f2f791c-c9b8-4dea-ba69-ce873ac723b7@
qu.8.16.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Capacitance;
  Difficulty=Medium;
  Features=Algorithmic;
  Features=Diagram;
  Features=Partial Grading;
@

qu.8.17.question=<p>A spherical capacitor has&nbsp;a capacitance of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$C</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$Cex</mi></mrow></msup><mi>F</mi></mrow></mstyle></math>.&nbsp; When&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$U</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>J</mi></mrow></mstyle></math>&nbsp;of energy are stored in the&nbsp;<br />
capacitor, how many&nbsp;electrons are on the&nbsp;$wPlate plate?</p>
<p>&nbsp;</p>
<p>&nbsp;</p>@
qu.8.17.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.8.17.allow2d=0@
qu.8.17.maple_answer=SigFigs[roundToSigFigs]($ans,3)@
qu.8.17.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.8.17.type=maple@
qu.8.17.mode=Maple@
qu.8.17.name=Energy Stored in Capacitor - Find Electrons ~ PGc@
qu.8.17.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.8.17.editing=useHTML@
qu.8.17.solution=<p>The energy stored in a capacitor can be expressed as:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>U</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><msup><mi>Q</mi><mrow><mn>2</mn></mrow></msup><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>C</mi></mrow></mfrac></mrow></mstyle></math>.</p>
<p>We are given <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>U</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$U</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>J</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$C</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$Cex</mi></mrow></msup><mi>F</mi></mrow></mfenced></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>.</p>
<p>The total charge can then be divided by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>e</mi></mrow></mstyle></math>&nbsp;to find the number of electrons.</p>@
qu.8.17.algorithm=$U=rand(0.1,0.999,3);
$C=rand(1.00,9.99,3);
$Cex=range(5,9);
$ans=sqrt(2*($C*10^(-$Cex))*$U)/(1.60*10^(-19));
$wPlate=switch(rint(2),'positive','negative');@
qu.8.17.uid=93fbb0ef-8637-4585-ab54-12f23cba1646@
qu.8.17.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Energy Stored in a Capacitor;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Algorithmic;
@

qu.9.topic=Circuits@

qu.9.1.question=<p><img alt="" align="middle" width="609" height="150" src="__BASE_URI__img/EMF/Battery-Terminal-Voltage-2/Diagram.png" /></p>
<p><span><br />
A battery with&nbsp;internal resistance&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$r</mi></mrow><mrow><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&Omega;</mi></mrow></mstyle></math>&nbsp;and terminal voltage <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$Vterm</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>V</mi></mrow></mstyle></math>&nbsp;has a current&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mstyle></math>&nbsp;passing<br />
&nbsp;through it in the direction indicated.&nbsp; What is the internal voltage of the battery?</span></p>
<p>&nbsp;</p>@
qu.9.1.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.9.1.allow2d=0@
qu.9.1.maple_answer=SigFigs[roundToSigFigs]($V,3)*V@
qu.9.1.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.9.1.type=maple@
qu.9.1.mode=Maple@
qu.9.1.name=Battery - Terminal Voltage 2 - Find EMF ~ PGc@
qu.9.1.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.9.1.editing=useHTML@
qu.9.1.solution=<p>The terminal voltage is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>V</mi><mrow><mi>AB</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>V</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>Ir</mi></mrow></mstyle></math>.</p>
<p>Thus:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>V</mi><mrow><mi>AB</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>Ir</mi></mrow></mstyle></math>.</p>@
qu.9.1.algorithm=$I=rand(0.300,0.999,3);
$r=rand(2,8,3);
$Vterm=rand(5,15,3);
$V=$Vterm-$r*$I;@
qu.9.1.uid=51edea24-abce-4cf5-b25e-051297c85526@
qu.9.1.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Terminal Voltage;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.9.2.question=<p><img alt="" align="middle" width="609" height="150" src="__BASE_URI__img/EMF/Battery-Terminal-Voltage-1/Diagram.png" /></p>
<p><span><br />
A battery with internal voltage <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$V</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>V</mi></mrow></mstyle></math>&nbsp;and internal resistance <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$r</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mi>&Omega;</mi></mrow></mrow></mstyle></math>&nbsp;has a current&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mstyle></math>&nbsp;passing<br />
&nbsp;through it in the direction indicated.&nbsp; What is the terminal voltage <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>V</mi><mrow><mi>AB</mi></mrow></msub></mrow></mstyle></math>&nbsp;of the battery?</span></p>@
qu.9.2.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.9.2.allow2d=0@
qu.9.2.maple_answer=SigFigs[roundToSigFigs]($ans,3)*V@
qu.9.2.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.9.2.type=maple@
qu.9.2.mode=Maple@
qu.9.2.name=Battery - Terminal Voltage 1 - Find V_term ~ PGc@
qu.9.2.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.9.2.editing=useHTML@
qu.9.2.solution=<p>The terminal voltage is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>V</mi><mrow><mi>AB</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>V</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>Ir</mi></mrow></mstyle></math>.</p>@
qu.9.2.algorithm=$r=rand(1.00,9.99,3);
$I=rand(0.300,0.999,3);
$V=rand(10.0,19.9,3);
$ans=$V-$I*$r;@
qu.9.2.uid=969e6cc0-9af1-4094-a3d3-185694b13701@
qu.9.2.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Terminal Voltage;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.9.3.question=<p><img alt="" align="middle" width="609" height="150" src="__BASE_URI__img/EMF/Battery-Terminal-Voltage-2/Diagram.png" /></p>
<p><span><br />
A battery with internal voltage <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$V</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>V</mi></mrow></mstyle></math>&nbsp;and terminal voltage <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$Vterm</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>V</mi></mrow></mstyle></math>&nbsp;has a current&nbsp;passing<br />
&nbsp;through it in the direction indicated.&nbsp; If the internal resistance of the battery is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$r</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mi>&Omega;</mi></mrow></mrow></mstyle></math>, what is the<br />
&nbsp;magnitude of the current?</span></p>@
qu.9.3.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.9.3.allow2d=0@
qu.9.3.maple_answer=SigFigs[roundToSigFigs]($ans,3)*A@
qu.9.3.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.9.3.type=maple@
qu.9.3.mode=Maple@
qu.9.3.name=Battery - Terminal Voltage 2 - Find Current ~ PGc@
qu.9.3.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.9.3.editing=useHTML@
qu.9.3.solution=<p>The terminal voltage is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>V</mi><mrow><mi>AB</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>V</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>Ir</mi></mrow></mstyle></math>.</p>
<p>Thus:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>V</mi><mrow><mi>AB</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>V</mi></mrow><mrow><mi>r</mi></mrow></mfrac></mrow></mstyle></math>.</p>@
qu.9.3.algorithm=$Vterm=rand(10.0,19.9,3);
$V=rand(5,($Vterm-0.5),3);
$r=rand(2.0,9.0,3);
$ans=(-$V+$Vterm)/$r;@
qu.9.3.uid=5712fad8-8384-43d3-8806-0025d6bf7482@
qu.9.3.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Terminal Voltage;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.9.4.mode=Inline@
qu.9.4.name=Kirchhoff's Laws - 3 Loops - Numeric - 1@
qu.9.4.comment=<ol>
    <li>Kirchhoff's current (or node) law states that the sum of all currents entering a node is equal to the sum of all currents leaving the node.</li>
    <li>Kirchhoff's voltage (or loop) law states that the sum of all voltage gains and drops around a closed loop is equal to zero.</li>
    <li>Remember that choosing one direction around a loop is important. It does not matter <em>which</em> direction you choose to follow, but you must be consistent.</li>
    <li>When following a loop: Passing through a battery is a positive if you are going from negative to positive terminal and a negative if going from positive to negative.</li>
    <li>When following a loop: Passing through a resistor is dependant both on the direction you are following the loop<em> and the assumed direction of the current passing through the resistor</em>. If you are following the loop in the same direction as the current through the resistor, the resistor provides a negative term. If you are following the loop against the current, the resistor provides a positive term.</li>
</ol>@
qu.9.4.editing=useHTML@
qu.9.4.hint.1=Kirchhoff's current (or node) law states that the sum of all currents entering a node is equal to the sum of all currents leaving the node.@
qu.9.4.hint.2=Kirchhoff's voltage (or loop) law states that the sum of all voltage gains and drops around a closed loop is equal to zero.@
qu.9.4.hint.3=Remember that choosing one direction around a loop is important.&nbsp; It does not matter <em>which</em> direction you choose to follow, but you must be consistent.@
qu.9.4.hint.4=When following a loop:&nbsp; Passing through a battery is a positive if you are going from negative to positive terminal and a negative if going from positive to negative.@
qu.9.4.hint.5=When following a loop:&nbsp; Passing through a resistor is dependant both on the direction you are following the loop<em> and the assumed direction of the current passing through the resistor</em>.&nbsp; If you are following the loop in the same direction as the current through the resistor, the resistor provides a negative term.&nbsp; If you are following the loop against the current, the resistor provides a positive term.@
qu.9.4.solution=<p>We must choose a direction to traverse each loop and a direction for the currents in each branch.</p>
<p>If we cross a voltage source from the negative to the positive terminal, then the potential difference is positive, otherwise it is negative.</p>
<p>If we cross a resistor in the direction of current flow, then the potential difference is negative, otherwise it is positive.</p>
<p>&nbsp;</p>
<p>At a node, the sum of all currents entering must equal the sum of all currents leaving.</p>@
qu.9.4.algorithm=$V1=rand(1.00,20.0,3);
$V2=rand(1.00,20.0,3);
$V3=rand(1.00,20.0,3);
$V4=rand(1.00,20.0,3);
$R1=rand(10.0,999,3);
$R2=rand(10.0,999,3);
$R3=rand(10.0,999,3);
$R4=rand(10.0,999,3);
$m=maple("
assign(solve({I1+I4-I2=0,I1+I5+I6=0,I3+I4-I5=0,$V1-$V2+I6*$R3-I1*$R4=0,$V2+I4*$R1-$V4-$V3=0,$V3+I5*$R2-I6*$R3=0}
));
I1,I2,I3,I4,I5,I6;
");
$I1=switch(0,$m)*1000;
$I2=switch(1,$m)*1000;
$I3=switch(2,$m)*1000;
$I4=switch(3,$m)*1000;
$I5=switch(4,$m)*1000;
$I6=switch(5,$m)*1000;@
qu.9.4.uid=70d76392-a30f-449b-9b24-af19d0f292dd@
qu.9.4.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Difficulty=Medium;
  Topic=Kirchhoff's Laws;
  Features=Algorithmic;
  Features=Diagram;
@
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qu.9.4.question=<p>Given the following circuit, calculate the currents <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I1</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I2</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I3</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I4</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I5</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I6</mi></mrow></mstyle></math>&nbsp;in <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mA</mi></mrow></mstyle></math>.&nbsp;</p><p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="583" height="600"><param name="image" value="__BASE_URI__img/Circuits/Kirchhoffs3L-1-NoLabel-Dir/Diagram.png" /><param name="size" value="22" /><param name="label.1.x" value="50" /><param name="label.1.y" value="330" /><param name="label.1.text" value="A" /><param name="label.2.x" value="50" /><param name="label.2.y" value="60" /><param name="label.2.text" value="B" /><param name="label.3.x" value="265" /><param name="label.3.y" value="60" /><param name="label.3.text" value="C" /><param name="label.4.x" value="530" /><param name="label.4.y" value="60" /><param name="label.4.text" value="D" /><param name="label.5.x" value="530" /><param name="label.5.y" value="330" /><param name="label.5.text" value="E" /><param name="label.6.x" value="530" /><param name="label.6.y" value="550" /><param name="label.6.text" value="F" /><param name="label.7.x" value="50" /><param name="label.7.y" value="550" /><param name="label.7.text" value="G" /><param name="label.8.x" value="265" /><param name="label.8.y" value="345" /><param name="label.8.text" value="H" /><param name="label.9.x" value="110" /><param name="label.9.y" value="195" /><param name="label.9.text" value="$R1 Ohm" /><param name="label.10.x" value="398" /><param name="label.10.y" value="50" /><param name="label.10.text" value="$R2 Ohm" /><param name="label.11.x" value="398" /><param name="label.11.y" value="300" /><param name="label.11.text" value="$R3 Ohm" /><param name="label.12.x" value="480" /><param name="label.12.y" value="430" /><param name="label.12.text" value="$R4 Ohm" /><param name="label.13.x" value="160" /><param name="label.13.y" value="570" /><param name="label.13.text" value="$V1 V" /><param name="label.14.x" value="160" /><param name="label.14.y" value="290" /><param name="label.14.text" value="$V2 V" /><param name="label.15.x" value="315" /><param name="label.15.y" value="200" /><param name="label.15.text" value="$V3 V" /><param name="label.16.x" value="160" /><param name="label.16.y" value="110" /><param name="label.16.text" value="$V4 V" /><param name="label.17.x" value="225" /><param name="label.17.y" value="510" /><param name="label.17.text" value="I1" /><param name="label.18.x" value="100" /><param name="label.18.y" value="350" /><param name="label.18.text" value="I2" /><param name="label.19.x" value="240" /><param name="label.19.y" value="130" /><param name="label.19.text" value="I3" /><param name="label.20.x" value="220" /><param name="label.20.y" value="90" /><param name="label.20.text" value="I4" /><param name="label.21.x" value="490" /><param name="label.21.y" value="90" /><param name="label.21.text" value="I5" /><param name="label.22.x" value="490" /><param name="label.22.y" value="350" /><param name="label.22.text" value="I6" /></applet></p><p align="left"><strong>(a)</strong>&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I1</mi></mrow></mstyle></math>&nbsp;&nbsp;<span>&nbsp;</span><1><span>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mA</mi></mrow></mstyle></math></span></p><p><span><strong>(b)</strong>&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I2</mi></mrow></mstyle></math>&nbsp;&nbsp;<span>&nbsp;</span><2><span>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mA</mi></mrow></mstyle></math></span></span></p><p><span><span><strong>(c)</strong>&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I3</mi></mrow></mstyle></math>&nbsp;&nbsp;<span>&nbsp;</span><3><span>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mA</mi></mrow></mstyle></math></span></span></span></p><p><span><span><span><strong>(d)</strong>&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I4</mi></mrow></mstyle></math>&nbsp;&nbsp;<span>&nbsp;</span><4><span>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mA</mi></mrow></mstyle></math></span></span></span></span></p><p><span><span><span><span><strong>(e)</strong>&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I5</mi></mrow></mstyle></math>&nbsp;&nbsp;<span>&nbsp;</span><5><span>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mA</mi></mrow></mstyle></math></span></span></span></span></span></p><p><span><span><span><span><span><strong>(f)</strong>&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I6</mi></mrow></mstyle></math>&nbsp;&nbsp;<span>&nbsp;</span><6><span>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mA</mi></mrow></mstyle></math></span></span></span></span></span></span></p>@

qu.9.5.question=<p align="center"><img alt="" align="middle" width="438" height="400" src="__BASE_URI__img/Capacitance/CapacitorsSeriesParallel2/Diagram.png" /></p>
<p>&nbsp;</p>
<p>If <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C1</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$C1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$C1ex</mi></mrow></msup><mi>F</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C2</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$C2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$C2ex</mi></mrow></msup><mi>F</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C3</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$C3</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$C3ex</mi></mrow></msup><mi>F</mi></mrow></mstyle></math><br />
and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C4</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$C4</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$C4ex</mi></mrow></msup><mi>F</mi></mrow></mstyle></math>, what is the total capacitance of the circuit between <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>?</p>@
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qu.9.5.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.9.5.editing=useHTML@
qu.9.5.solution=<p>For the bottom arm, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>C</mi><mrow><mi>eq</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><msub><mi>C</mi><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mn>1</mn><mrow><msub><mi>C</mi><mrow><mn>2</mn></mrow></msub></mrow></mfrac></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup></mrow></mstyle></math>.</p>
<p>The total is then:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>C</mi><mrow><mi>AB</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>C</mi><mrow><mi>eq</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>C</mi><mrow><mn>3</mn></mrow></msub></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>C</mi><mrow><mn>4</mn></mrow></msub></mrow></mstyle></math>.</p>@
qu.9.5.algorithm=$C1=rand(1.00,9.99,3);
$C2=rand(1.00,9.99,3);
$C3=rand(1.00,9.99,3);
$C4=rand(1.00,9.99,3);
$C1ex=range(5,7);
$C2ex=range(5,7);
$C3ex=range(5,7);
$C4ex=range(5,7);@
qu.9.5.uid=3455e9a3-80bc-4d05-b452-f2963576da0e@
qu.9.5.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Capacitors in Series and Parallel;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Diagram;
  Features=Algorithmic;
@

qu.9.6.question=<p><img alt="" align="middle" width="609" height="150" src="__BASE_URI__img/EMF/Battery-Terminal-Voltage-1/Diagram.png" /></p>
<p><span><br />
A battery with internal voltage <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$V</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>V</mi></mrow></mstyle></math>&nbsp;and terminal voltage <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$Vterm</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>V</mi></mrow></mstyle></math>&nbsp;has a current&nbsp;passing<br />
&nbsp;through it in the direction indicated.&nbsp; If the internal resistance of the battery is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$r</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mi>&Omega;</mi></mrow></mrow></mstyle></math>, what is the<br />
&nbsp;magnitude of the current?</span></p>@
qu.9.6.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.9.6.allow2d=0@
qu.9.6.maple_answer=SigFigs[roundToSigFigs]($ans,3)*A@
qu.9.6.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.9.6.type=maple@
qu.9.6.mode=Maple@
qu.9.6.name=Battery - Terminal Voltage 1 - Find Current ~ PGc@
qu.9.6.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.9.6.editing=useHTML@
qu.9.6.solution=<p>The terminal voltage is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>V</mi><mrow><mi>AB</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>V</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>Ir</mi></mrow></mstyle></math>.</p>@
qu.9.6.algorithm=$V=rand(10.0,19.9,3);
$Vterm=rand(5,($V-0.5),3);
$r=rand(2.0,9.0,3);
$ans=($V-$Vterm)/$r;@
qu.9.6.uid=10351c5e-7b23-4218-9203-aa657bd7c8d2@
qu.9.6.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Terminal Voltage;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.9.7.mode=Inline@
qu.9.7.name=Kirchhoff's Laws - 2 Loops - 1@
qu.9.7.comment=<ol>
    <li>Kirchhoff's current (or node) law states that the sum of all currents entering a node is equal to the sum of all currents leaving the node.</li>
    <li>Kirchhoff's voltage (or loop) law states that the sum of all voltage gains and drops around a closed loop is equal to zero.</li>
    <li>Remember that choosing one direction around a loop is important. It does not matter <em>which</em> direction you choose to follow, but you must be consistent.</li>
    <li>When following a loop: Passing through a battery is a positive if you are going from negative to positive terminal and a negative if going from positive to negative.</li>
    <li>When following a loop: Passing through a resistor is dependant both on the direction you are following the loop<em> and the assumed direction of the current passing through the resistor</em>. If you are following the loop in the same direction as the current through the resistor, the resistor provides a negative term. If you are following the loop against the current, the resistor provides a positive term.</li>
</ol>@
qu.9.7.editing=useHTML@
qu.9.7.hint.1=Kirchhoff's current (or node) law states that the sum of all currents entering a node is equal to the sum of all currents leaving the node.@
qu.9.7.hint.2=Kirchhoff's voltage (or loop) law states that the sum of all voltage gains and drops around a closed loop is equal to zero.@
qu.9.7.hint.3=Remember that choosing one direction around a loop is important.&nbsp; It does not matter <em>which</em> direction you choose to follow, but you must be consistent.@
qu.9.7.hint.4=When following a loop:&nbsp; Passing through a battery is a positive if you are going from negative to positive terminal and a negative if going from positive to negative.@
qu.9.7.hint.5=When following a loop:&nbsp; Passing through a resistor is dependant both on the direction you are following the loop<em> and the assumed direction of the current passing through the resistor</em>.&nbsp; If you are following the loop in the same direction as the current through the resistor, the resistor provides a negative term.&nbsp; If you are following the loop against the current, the resistor provides a positive term.@
qu.9.7.solution=<p>We must choose a direction to traverse each loop and a direction for the currents in each branch.</p>
<p>If we cross a voltage source from the negative to the positive terminal, then the potential difference is positive, otherwise it is negative.</p>
<p>If we cross a resistor in the direction of current flow, then the potential difference is&nbsp;negative, otherwise it is positive.</p>
<p>&nbsp;</p>
<p>At a node, the sum of all currents entering must equal the sum of all currents leaving.</p>@
qu.9.7.algorithm=$top=rint(1,6);
$m=maple("randomize();
b:=RandomTools[Generate](choose({1,2,3,4,5}minus {$top})):
c:=RandomTools[Generate](choose({1,2,3,4,5}minus {$top,b})):
d:=RandomTools[Generate](choose({1,2,3,4,5}minus {$top,b,c})):
e:=RandomTools[Generate](choose({1,2,3,4,5}minus {$top,b,c,d})):
b,c,d,e;
");
$bot=switch(0,$m);
$mid=switch(1,$m);
$lef=switch(2,$m);
$rig=switch(3,$m);
$wNode=switch(rint(2),A,B);
$idx=rint(3);
$wLoop=switch($idx,'top','bottom','outer');
$ansNode='I$top+I$mid+I$bot=0';
$ansLoop=switch($idx,'-V$top+I$top*R$top-R$mid*I$mid+V$mid+I$top*R$lef=0','-V$mid+I$mid*R$mid-R$rig*I$bot+V$bot=0','V$bot+I$top*R$lef-V$top+I$top*R$top-I$bot*R$rig=0');@
qu.9.7.uid=aae28ec4-5e44-4720-95ae-f9422f882b44@
qu.9.7.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Difficulty=Medium;
  Topic=Kirchhoff's Laws;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Diagram;
@
qu.9.7.weighting=1,1@
qu.9.7.numbering=alpha@
qu.9.7.part.1.name=sro_id_1@
qu.9.7.part.1.maple_answer=$ansNode@
qu.9.7.part.1.editing=useHTML@
qu.9.7.part.1.question=(Unset)@
qu.9.7.part.1.libname=@
qu.9.7.part.1.mode=Maple@
qu.9.7.part.1.allow2d=0@
qu.9.7.part.1.plot=@
qu.9.7.part.1.maple=is((solve($ANSWER,I1))=(solve($RESPONSE,I1)) );@
qu.9.7.part.1.type=maple@
qu.9.7.part.2.name=sro_id_2@
qu.9.7.part.2.maple_answer=$ansLoop@
qu.9.7.part.2.editing=useHTML@
qu.9.7.part.2.question=(Unset)@
qu.9.7.part.2.libname=@
qu.9.7.part.2.mode=Maple@
qu.9.7.part.2.allow2d=0@
qu.9.7.part.2.plot=@
qu.9.7.part.2.maple=is(solve($ANSWER,V2)-solve($RESPONSE,V2) = 0);@
qu.9.7.part.2.type=maple@
qu.9.7.question=<p>Consider the following circuit where the symbols have their usual meaning.&nbsp;&nbsp;State your answers in terms of the given variables and directions in the diagram.</p><p>&nbsp;</p><p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="384" height="400"><param name="image" value="__BASE_URI__img/Circuits/KirchhoffsLaws-2Loops/Diagram.png" /><param name="size" value="12" /><param name="label.1.x" value="268" /><param name="label.1.y" value="10" /><param name="label.1.text" value="R$top" /><param name="label.2.x" value="100" /><param name="label.2.y" value="10" /><param name="label.2.text" value="V$top" /><param name="label.3.x" value="268" /><param name="label.3.y" value="190" /><param name="label.3.text" value="R$mid" /><param name="label.4.x" value="100" /><param name="label.4.y" value="190" /><param name="label.4.text" value="V$mid" /><param name="label.5.x" value="100" /><param name="label.5.y" value="340" /><param name="label.5.text" value="V$bot" /><param name="label.6.x" value="377" /><param name="label.6.y" value="296" /><param name="label.6.text" value="R$rig" /><param name="label.7.x" value="8" /><param name="label.7.y" value="128" /><param name="label.7.text" value="R$lef" /><param name="label.8.x" value="143" /><param name="label.8.y" value="55" /><param name="label.8.text" value="I$top" /><param name="label.9.x" value="143" /><param name="label.9.y" value="235" /><param name="label.9.text" value="I$mid" /><param name="label.10.x" value="143" /><param name="label.10.y" value="385" /><param name="label.10.text" value="I$bot" /><param name="label.11.x" value="8" /><param name="label.11.y" value="220" /><param name="label.11.text" value="A" /><param name="label.12.x" value="377" /><param name="label.12.y" value="220" /><param name="label.12.text" value="B" /></applet></p><p><strong>(a)</strong>&nbsp; Use Kirchhoff's current rule to write an equation for the algebraic sum of the currents&nbsp;flowing into node $wNode.&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p><span><strong>(b)</strong>&nbsp; Use Kirchhoff's voltage rule to write&nbsp;an equation for the algebraic&nbsp;sum of the potential differences&nbsp;in the $wLoop loop of the circuit.</span></p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p>@

qu.9.8.mode=Inline@
qu.9.8.name=Kirchhoff's Laws - 2 Loops - 2@
qu.9.8.comment=<ol>
    <li>Kirchhoff's current (or node) law states that the sum of all currents entering a node is equal to the sum of all currents leaving the node.</li>
    <li>Kirchhoff's voltage (or loop) law states that the sum of all voltage gains and drops around a closed loop is equal to zero.</li>
    <li>Remember that choosing one direction around a loop is important. It does not matter <em>which</em> direction you choose to follow, but you must be consistent.</li>
    <li>When following a loop: Passing through a battery is a positive if you are going from negative to positive terminal and a negative if going from positive to negative.</li>
    <li>When following a loop: Passing through a resistor is dependant both on the direction you are following the loop<em> and the assumed direction of the current passing through the resistor</em>. If you are following the loop in the same direction as the current through the resistor, the resistor provides a negative term. If you are following the loop against the current, the resistor provides a positive term.</li>
</ol>@
qu.9.8.editing=useHTML@
qu.9.8.hint.1=Kirchhoff's current (or node) law states that the sum of all currents entering a node is equal to the sum of all currents leaving the node.@
qu.9.8.hint.2=Kirchhoff's voltage (or loop) law states that the sum of all voltage gains and drops around a closed loop is equal to zero.@
qu.9.8.hint.3=Remember that choosing one direction around a loop is important.&nbsp; It does not matter <em>which</em> direction you choose to follow, but you must be consistent.@
qu.9.8.hint.4=When following a loop:&nbsp; Passing through a battery is a positive if you are going from negative to positive terminal and a negative if going from positive to negative.@
qu.9.8.hint.5=When following a loop:&nbsp; Passing through a resistor is dependant both on the direction you are following the loop<em> and the assumed direction of the current passing through the resistor</em>.&nbsp; If you are following the loop in the same direction as the current through the resistor, the resistor provides a negative term.&nbsp; If you are following the loop against the current, the resistor provides a positive term.@
qu.9.8.solution=<p>We must choose a direction to traverse each loop and a direction for the currents in each branch.</p>
<p>If we cross a voltage source from the negative to the positive terminal, then the potential difference is positive, otherwise it is negative.</p>
<p>If we cross a resistor in the direction of current flow, then the potential difference is negative, otherwise it is positive.</p>
<p>&nbsp;</p>
<p>At a node, the sum of all currents entering must equal the sum of all currents leaving.</p>@
qu.9.8.algorithm=$r1=rint(1,8);
$v1=rint(1,3);
$i1=rint(1,4);
$m=maple("randomize();
b:=RandomTools[Generate](choose({1,2,3,4,5,6,7}minus {$r1})):
c:=RandomTools[Generate](choose({1,2,3,4,5,6,7}minus {$top,b})):
d:=RandomTools[Generate](choose({1,2,3,4,5,6,7}minus {$top,b,c})):
e:=RandomTools[Generate](choose({1,2,3,4,5,6,7}minus {$top,b,c,d})):
f:=RandomTools[Generate](choose({1,2,3,4,5,6,7}minus {$top,b,c,d,e})):
g:=RandomTools[Generate](choose({1,2,3,4,5,6,7}minus {$top,b,c,d,e,f})):
bb:=RandomTools[Generate](choose({1,2}minus {$v1})):
bbb:=RandomTools[Generate](choose({1,2,3}minus {$i1})):
ccc:=RandomTools[Generate](choose({1,2,3}minus {$i1,bbb})):
b,c,d,e,f,g,bb,bbb,ccc;
");
$r2=switch(0,$m);
$r3=switch(1,$m);
$r4=switch(2,$m);
$r5=switch(3,$m);
$r6=switch(4,$m);
$r7=switch(5,$m);
$v2=switch(6,$m);
$i2=switch(7,$m);
$i3=switch(8,$m);
$idxNode=rint(2);
$wNode=switch($idxNode,A,B);
$ansNode=switch($idxNode,'I$i1=I$i2+I$i3','I$i1=I$i2+I$i3');
$idxLoop=rint(3);
$wLoop=switch($idxLoop,'left','right','outer');
$ansLoop=switch($idxLoop,'V$v1-I$i1*R$r1-I$i1*(1/(1/R$r2+1/R$r3))+V$v2-I$i2*R$r4-I$i1*R$r5=0','V$v2-I$i2*R$r4+I$i3*(1/(1/R$r6+1/R$r7))=0','$v1-I$i1*R$r1-I$i1*(1/(1/R$r2+1/R$r3))-I$i3*(1/(1/R$r6+1/R$r7))=0');@
qu.9.8.uid=c2e8f05c-d51b-4046-9021-11db49381a14@
qu.9.8.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Difficulty=Medium;
  Topic=Kirchhoff's Laws;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Diagram;
@
qu.9.8.weighting=1,1@
qu.9.8.numbering=alpha@
qu.9.8.part.1.name=sro_id_1@
qu.9.8.part.1.maple_answer=$ansNode@
qu.9.8.part.1.editing=useHTML@
qu.9.8.part.1.question=(Unset)@
qu.9.8.part.1.libname=@
qu.9.8.part.1.mode=Maple@
qu.9.8.part.1.allow2d=0@
qu.9.8.part.1.plot=@
qu.9.8.part.1.maple=is((solve($ANSWER,I1))=(solve($RESPONSE,I1)) );@
qu.9.8.part.1.type=maple@
qu.9.8.part.2.name=sro_id_2@
qu.9.8.part.2.maple_answer=$ansLoop@
qu.9.8.part.2.editing=useHTML@
qu.9.8.part.2.question=(Unset)@
qu.9.8.part.2.libname=@
qu.9.8.part.2.mode=Maple@
qu.9.8.part.2.allow2d=0@
qu.9.8.part.2.plot=@
qu.9.8.part.2.maple=is(solve($ANSWER,V2)-solve($RESPONSE,V2) = 0);@
qu.9.8.part.2.type=maple@
qu.9.8.question=<p>Consider the following circuit where the symbols have their usual meaning.&nbsp;&nbsp;State your answers in terms of the given variables and directions in the diagram.</p><p>&nbsp;</p><p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="675" height="506"><param name="image" value="__BASE_URI__img/Circuits/Kirchhoffs2L-2-NoLabel-Dir/Diagram.png" /><param name="size" value="14" /><param name="label.1.x" value="45" /><param name="label.1.y" value="150" /><param name="label.1.text" value="R$r3" /><param name="label.2.x" value="175" /><param name="label.2.y" value="150" /><param name="label.2.text" value="R$r2" /><param name="label.3.x" value="45" /><param name="label.3.y" value="250" /><param name="label.3.text" value="R$r1" /><param name="label.4.x" value="160" /><param name="label.4.y" value="365" /><param name="label.4.text" value="V$v1" /><param name="label.5.x" value="245" /><param name="label.5.y" value="365" /><param name="label.5.text" value="R$r5" /><param name="label.6.x" value="185" /><param name="label.6.y" value="320" /><param name="label.6.text" value="I$i1" /><param name="label.7.x" value="390" /><param name="label.7.y" value="250" /><param name="label.7.text" value="R$r4" /><param name="label.8.x" value="390" /><param name="label.8.y" value="170" /><param name="label.8.text" value="V$v2" /><param name="label.9.x" value="500" /><param name="label.9.y" value="220" /><param name="label.9.text" value="R$r6" /><param name="label.10.x" value="500" /><param name="label.10.y" value="390" /><param name="label.10.text" value="R$r7" /><param name="label.11.x" value="345" /><param name="label.11.y" value="140" /><param name="label.11.text" value="I$i2" /><param name="label.12.x" value="590" /><param name="label.12.y" value="205" /><param name="label.12.text" value="I$i3" /><param name="label.13.x" value="360" /><param name="label.13.y" value="350" /><param name="label.13.text" value="A" /><param name="label.14.x" value="360" /><param name="label.14.y" value="70" /><param name="label.14.text" value="B" /></applet></p><p><strong>(a)</strong>&nbsp; Use Kirchhoff's current rule to write an equation for the algebraic sum of the currents&nbsp;flowing into node $wNode.&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p><span><strong>(b)</strong>&nbsp; Use Kirchhoff's voltage rule to write&nbsp;an equation for the algebraic&nbsp;sum of the potential differences&nbsp;in the $wLoop loop of the circuit.</span></p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p>@

qu.9.9.mode=Inline@
qu.9.9.name=Kirchhoff's Laws - 3 Loops - 1@
qu.9.9.comment=<ol>
    <li>Kirchhoff's current (or node) law states that the sum of all currents entering a node is equal to the sum of all currents leaving the node.</li>
    <li>Kirchhoff's voltage (or loop) law states that the sum of all voltage gains and drops around a closed loop is equal to zero.</li>
    <li>Remember that choosing one direction around a loop is important. It does not matter <em>which</em> direction you choose to follow, but you must be consistent.</li>
    <li>When following a loop: Passing through a battery is a positive if you are going from negative to positive terminal and a negative if going from positive to negative.</li>
    <li>When following a loop: Passing through a resistor is dependant both on the direction you are following the loop<em> and the assumed direction of the current passing through the resistor</em>. If you are following the loop in the same direction as the current through the resistor, the resistor provides a negative term. If you are following the loop against the current, the resistor provides a positive term.</li>
</ol>@
qu.9.9.editing=useHTML@
qu.9.9.hint.1=Kirchhoff's current (or node) law states that the sum of all currents entering a node is equal to the sum of all currents leaving the node.@
qu.9.9.hint.2=Kirchhoff's voltage (or loop) law states that the sum of all voltage gains and drops around a closed loop is equal to zero.@
qu.9.9.hint.3=Remember that choosing one direction around a loop is important.&nbsp; It does not matter <em>which</em> direction you choose to follow, but you must be consistent.@
qu.9.9.hint.4=When following a loop:&nbsp; Passing through a battery is a positive if you are going from negative to positive terminal and a negative if going from positive to negative.@
qu.9.9.hint.5=When following a loop:&nbsp; Passing through a resistor is dependant both on the direction you are following the loop<em> and the assumed direction of the current passing through the resistor</em>.&nbsp; If you are following the loop in the same direction as the current through the resistor, the resistor provides a negative term.&nbsp; If you are following the loop against the current, the resistor provides a positive term.@
qu.9.9.solution=<p>We must choose a direction to traverse each loop and a direction for the currents in each branch.</p>
<p>If we cross a voltage source from the negative to the positive terminal, then the potential difference is positive, otherwise it is negative.</p>
<p>If we cross a resistor in the direction of current flow, then the potential difference is negative, otherwise it is positive.</p>
<p>&nbsp;</p>
<p>At a node, the sum of all currents entering must equal the sum of all currents leaving.</p>@
qu.9.9.algorithm=$r1=range(1,4);
$v1=range(1,4);
$i1=range(1,6);
$m=maple("randomize();
b:=RandomTools[Generate](choose({1,2,3,4}minus {$r1})):
c:=RandomTools[Generate](choose({1,2,3,4}minus {$r1,b})):
d:=RandomTools[Generate](choose({1,2,3,4}minus {$r1,b,c})):
bb:=RandomTools[Generate](choose({1,2,3,4}minus {$v1})):
cc:=RandomTools[Generate](choose({1,2,3,4}minus {$v1,bb})):
dd:=RandomTools[Generate](choose({1,2,3,4}minus {$v1,bb,cc})):
bbb:=RandomTools[Generate](choose({1,2,3,4,5,6}minus {$i1})):
ccc:=RandomTools[Generate](choose({1,2,3,4,5,6}minus {$i1,bbb})):
ddd:=RandomTools[Generate](choose({1,2,3,4,5,6}minus {$i1,bbb,ccc})):
eee:=RandomTools[Generate](choose({1,2,3,4,5,6}minus {$i1,bbb,ccc,ddd})):
fff:=RandomTools[Generate](choose({1,2,3,4,5,6}minus {$i1,bbb,ccc,ddd,eee})):
b,c,d,bb,cc,dd,bbb,ccc,ddd,eee,fff;
");
$r2=switch(0,$m);
$r3=switch(1,$m);
$r4=switch(2,$m);
$v2=switch(3,$m);
$v3=switch(4,$m);
$v4=switch(5,$m);
$i2=switch(6,$m);
$i3=switch(7,$m);
$i4=switch(8,$m);
$i5=switch(9,$m);
$i6=switch(10,$m);
$idxNode=rint(4);
$wNode=switch($idxNode,A,H,E,C);
$ansNode=switch($idxNode,'I$i1+I$i4=I$i2','I$i6+I$i2+I$i3=0','I$i5+I$i6+I$i1=0','I$i5-I$i3-I$i4=0');
$idxLoop=rint(7);
$wLoop=switch($idxLoop,'AHEFGA','ABCHA','CDEHC','ABCDEHA','ABCDEFGA','ABCHEFGA','CDEFGAHC');
$ansLoop=switch($idxLoop,'-V$v2+I$i6*R$r3-I$i1*R$r4+V$v1=0','I$i4*R$r1-V$v4-V$v3+V$v2=0','I$i5*R$r2-I$i6*R$r3+V$v3=0','I$i4*R$r1-V$v4+I$i5*R$r2-I$i6*R$r3+V$v2=0','I$i4*R$r1-V$v4+I$i5*R$r2-I$i1*R$r4+V$v1=0','I$i4*R$r1-V$v4-V$v3+I$i6*R$r3-I$i1*R$r4+V$v1=0','I$i5*R$r2-I$i1*R$r4+V$v1-V$v2+V$v3=0');@
qu.9.9.uid=b05a5595-13e9-4f78-b6fe-702dea6d16c9@
qu.9.9.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Difficulty=Medium;
  Topic=Kirchhoff's Laws;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Diagram;
@
qu.9.9.weighting=1,1@
qu.9.9.numbering=alpha@
qu.9.9.part.1.name=sro_id_1@
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qu.9.9.part.1.libname=@
qu.9.9.part.1.mode=Maple@
qu.9.9.part.1.allow2d=0@
qu.9.9.part.1.plot=@
qu.9.9.part.1.maple=is((solve($ANSWER,I1))=(solve($RESPONSE,I1)) );@
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qu.9.9.part.2.name=sro_id_2@
qu.9.9.part.2.maple_answer=$ansLoop@
qu.9.9.part.2.editing=useHTML@
qu.9.9.part.2.question=(Unset)@
qu.9.9.part.2.libname=@
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qu.9.9.part.2.maple=is(solve($ANSWER,V2)-solve($RESPONSE,V2) = 0);@
qu.9.9.part.2.type=maple@
qu.9.9.question=<p>Consider the following circuit where the symbols have their usual meaning.&nbsp;&nbsp;State your answers in terms of the given variables and directions in the diagram.</p><p>&nbsp;</p><p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="583" height="600"><param name="image" value="__BASE_URI__img/Circuits/Kirchhoffs3L-1-NoLabel-Dir/Diagram.png" /><param name="size" value="22" /><param name="label.1.x" value="50" /><param name="label.1.y" value="330" /><param name="label.1.text" value="A" /><param name="label.2.x" value="50" /><param name="label.2.y" value="60" /><param name="label.2.text" value="B" /><param name="label.3.x" value="265" /><param name="label.3.y" value="60" /><param name="label.3.text" value="C" /><param name="label.4.x" value="530" /><param name="label.4.y" value="60" /><param name="label.4.text" value="D" /><param name="label.5.x" value="530" /><param name="label.5.y" value="330" /><param name="label.5.text" value="E" /><param name="label.6.x" value="530" /><param name="label.6.y" value="550" /><param name="label.6.text" value="F" /><param name="label.7.x" value="50" /><param name="label.7.y" value="550" /><param name="label.7.text" value="G" /><param name="label.8.x" value="265" /><param name="label.8.y" value="345" /><param name="label.8.text" value="H" /><param name="label.9.x" value="40" /><param name="label.9.y" value="195" /><param name="label.9.text" value="R$r1" /><param name="label.10.x" value="398" /><param name="label.10.y" value="50" /><param name="label.10.text" value="R$r2" /><param name="label.11.x" value="398" /><param name="label.11.y" value="300" /><param name="label.11.text" value="R$r3" /><param name="label.12.x" value="555" /><param name="label.12.y" value="430" /><param name="label.12.text" value="R$r4" /><param name="label.13.x" value="160" /><param name="label.13.y" value="570" /><param name="label.13.text" value="V$v1" /><param name="label.14.x" value="160" /><param name="label.14.y" value="290" /><param name="label.14.text" value="V$v2" /><param name="label.15.x" value="310" /><param name="label.15.y" value="200" /><param name="label.15.text" value="V$v3" /><param name="label.16.x" value="160" /><param name="label.16.y" value="40" /><param name="label.16.text" value="V$v4" /><param name="label.17.x" value="225" /><param name="label.17.y" value="510" /><param name="label.17.text" value="I$i1" /><param name="label.18.x" value="100" /><param name="label.18.y" value="350" /><param name="label.18.text" value="I$i2" /><param name="label.19.x" value="240" /><param name="label.19.y" value="130" /><param name="label.19.text" value="I$i3" /><param name="label.20.x" value="220" /><param name="label.20.y" value="90" /><param name="label.20.text" value="I$i4" /><param name="label.21.x" value="490" /><param name="label.21.y" value="90" /><param name="label.21.text" value="I$i5" /><param name="label.22.x" value="490" /><param name="label.22.y" value="350" /><param name="label.22.text" value="I$i6" /></applet></p><p><strong>(a)</strong>&nbsp; Use Kirchhoff's current rule to write an equation for the algebraic sum of the currents&nbsp;flowing into node $wNode.&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p><span><strong>(b)</strong>&nbsp; Use Kirchhoff's voltage rule to write&nbsp;an equation for the algebraic&nbsp;sum of the potential differences&nbsp;in the $wLoop loop of the circuit.</span></p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p>@

qu.9.10.question=<p><img alt="" align="middle" width="438" height="400" src="__BASE_URI__img/Capacitance/CapacitorsSeriesParallel1/Diagram.png" /></p>
<p>&nbsp;</p>
<p>If <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C1</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$C1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$C1ex</mi></mrow></msup><mi>F</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C2</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$C2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$C2ex</mi></mrow></msup><mi>F</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C3</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$C3</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$C3ex</mi></mrow></msup><mi>F</mi></mrow></mstyle></math>,<br />
what is the total capacitance of the circuit between <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>?</p>@
qu.9.10.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN)@
qu.9.10.allow2d=0@
qu.9.10.maple_answer=SigFigs[roundToSigFigs](((1/($C1*10^(-$C1ex))+1/($C2*10^(-$C2ex)))^(-1)+($C3*10^(-$C3ex))),3)*F;@
qu.9.10.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.9.10.type=maple@
qu.9.10.mode=Maple@
qu.9.10.name=Capacitors in Series and Parallel 1 - Numeric ~ PG@
qu.9.10.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.9.10.editing=useHTML@
qu.9.10.solution=<p>For the bottom arm, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>C</mi><mrow><mi>eq</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><msub><mi>C</mi><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mn>1</mn><mrow><msub><mi>C</mi><mrow><mn>2</mn></mrow></msub></mrow></mfrac></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup></mrow></mstyle></math>.</p>
<p>The total is then:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>C</mi><mrow><mi>AB</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>C</mi><mrow><mi>eq</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>C</mi><mrow><mn>3</mn></mrow></msub></mrow></mstyle></math>.</p>@
qu.9.10.algorithm=$C1=rand(1.00,9.99,3);
$C2=rand(1.00,9.99,3);
$C3=rand(1.00,9.99,3);
$C1ex=range(5,7);
$C2ex=range(5,7);
$C3ex=range(5,7);@
qu.9.10.uid=b122aa02-0f9d-44ef-b4bd-e9d7ca81be1c@
qu.9.10.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Capacitors in Series and Parallel;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Diagram;
  Features=Algorithmic;
@

qu.9.11.question=<p><img alt="" align="middle" width="438" height="400" src="__BASE_URI__img/Capacitance/CapacitorsSeriesParallel1/Diagram.png" /></p>
<p>&nbsp;</p>
<p>What is the total capacitance of the circuit between <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>?</p>@
qu.9.11.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=algebraicScalar)@
qu.9.11.allow2d=0@
qu.9.11.maple_answer=(1/C1+1/C2)^(-1)+C3@
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qu.9.11.editing=useHTML@
qu.9.11.solution=<p>For the bottom arm, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>C</mi><mrow><mi>eq</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><msub><mi>C</mi><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mn>1</mn><mrow><msub><mi>C</mi><mrow><mn>2</mn></mrow></msub></mrow></mfrac></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup></mrow></mstyle></math>.</p>
<p>The total is then:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>C</mi><mrow><mi>AB</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>C</mi><mrow><mi>eq</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>C</mi><mrow><mn>3</mn></mrow></msub></mrow></mstyle></math>.</p>@
qu.9.11.algorithm=@
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qu.9.11.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Capacitors in Series and Parallel;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Diagram;
  Features=Algorithmic;
  Features=Algebraic;
@

qu.9.12.question=<p><img alt="" align="middle" width="609" height="150" src="__BASE_URI__img/EMF/Battery-Terminal-Voltage-1/Diagram.png" /></p>
<p><span><br />
A battery with&nbsp;internal resistance&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$r</mi></mrow><mrow><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&Omega;</mi></mrow></mstyle></math>&nbsp;and terminal voltage <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$Vterm</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>V</mi></mrow></mstyle></math>&nbsp;has a current&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mstyle></math>&nbsp;passing<br />
&nbsp;through it in the direction indicated.&nbsp; What is the internal voltage of the battery?</span></p>@
qu.9.12.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.9.12.allow2d=0@
qu.9.12.maple_answer=SigFigs[roundToSigFigs]($V,3)*V@
qu.9.12.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.9.12.type=maple@
qu.9.12.mode=Maple@
qu.9.12.name=Battery - Terminal Voltage 1 - Find EMF ~ PGc@
qu.9.12.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.9.12.editing=useHTML@
qu.9.12.solution=<p>The terminal voltage is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>V</mi><mrow><mi>AB</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>V</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>Ir</mi></mrow></mstyle></math>.</p>
<p>Thus:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>V</mi><mrow><mi>AB</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>Ir</mi></mrow></mstyle></math>.</p>@
qu.9.12.algorithm=$I=rand(0.300,0.999,3);
$r=rand(2,8,3);
$Vterm=rand(5,15,3);
$V=$Vterm+$r*$I;@
qu.9.12.uid=c560d2cb-81c7-4766-9615-5889714734b5@
qu.9.12.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Terminal Voltage;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.9.13.mode=Inline@
qu.9.13.name=Kirchhoff's Laws - 3 Loops - 2@
qu.9.13.comment=<ol>
    <li>Kirchhoff's current (or node) law states that the sum of all currents entering a node is equal to the sum of all currents leaving the node.</li>
    <li>Kirchhoff's voltage (or loop) law states that the sum of all voltage gains and drops around a closed loop is equal to zero.</li>
    <li>Remember that choosing one direction around a loop is important. It does not matter <em>which</em> direction you choose to follow, but you must be consistent.</li>
    <li>When following a loop: Passing through a battery is a positive if you are going from negative to positive terminal and a negative if going from positive to negative.</li>
    <li>When following a loop: Passing through a resistor is dependant both on the direction you are following the loop<em> and the assumed direction of the current passing through the resistor</em>. If you are following the loop in the same direction as the current through the resistor, the resistor provides a negative term. If you are following the loop against the current, the resistor provides a positive term.</li>
</ol>@
qu.9.13.editing=useHTML@
qu.9.13.hint.1=Kirchhoff's current (or node) law states that the sum of all currents entering a node is equal to the sum of all currents leaving the node.@
qu.9.13.hint.2=Kirchhoff's voltage (or loop) law states that the sum of all voltage gains and drops around a closed loop is equal to zero.@
qu.9.13.hint.3=Remember that choosing one direction around a loop is important.&nbsp; It does not matter <em>which</em> direction you choose to follow, but you must be consistent.@
qu.9.13.hint.4=When following a loop:&nbsp; Passing through a battery is a positive if you are going from negative to positive terminal and a negative if going from positive to negative.@
qu.9.13.hint.5=When following a loop:&nbsp; Passing through a resistor is dependant both on the direction you are following the loop<em> and the assumed direction of the current passing through the resistor</em>.&nbsp; If you are following the loop in the same direction as the current through the resistor, the resistor provides a negative term.&nbsp; If you are following the loop against the current, the resistor provides a positive term.@
qu.9.13.solution=<p>We must choose a direction to traverse each loop and a direction for the currents in each branch.</p>
<p>If we cross a voltage source from the negative to the positive terminal, then the potential difference is positive, otherwise it is negative.</p>
<p>If we cross a resistor in the direction of current flow, then the potential difference is negative, otherwise it is positive.</p>
<p>&nbsp;</p>
<p>At a node, the sum of all currents entering must equal the sum of all currents leaving.</p>@
qu.9.13.algorithm=$r1=range(1,5);
$v1=range(1,3);
$i1=range(1,6);
$m=maple("randomize();
b:=RandomTools[Generate](choose({1,2,3,4,5}minus {$r1})):
c:=RandomTools[Generate](choose({1,2,3,4,5}minus {$r1,b})):
d:=RandomTools[Generate](choose({1,2,3,4,5}minus {$r1,b,c})):
e:=RandomTools[Generate](choose({1,2,3,4,5}minus {$r1,b,c,d})):
bb:=RandomTools[Generate](choose({1,2,3}minus {$v1})):
cc:=RandomTools[Generate](choose({1,2,3}minus {$v1,bb})):
bbb:=RandomTools[Generate](choose({1,2,3,4,5,6}minus {$i1})):
ccc:=RandomTools[Generate](choose({1,2,3,4,5,6}minus {$i1,bbb})):
ddd:=RandomTools[Generate](choose({1,2,3,4,5,6}minus {$i1,bbb,ccc})):
eee:=RandomTools[Generate](choose({1,2,3,4,5,6}minus {$i1,bbb,ccc,ddd})):
fff:=RandomTools[Generate](choose({1,2,3,4,5,6}minus {$i1,bbb,ccc,ddd,eee})):
b,c,d,e,bb,cc,bbb,ccc,ddd,eee,fff;
");
$r2=switch(0,$m);
$r3=switch(1,$m);
$r4=switch(2,$m);
$r5=switch(3,$m);
$v2=switch(4,$m);
$v3=switch(5,$m);
$i2=switch(6,$m);
$i3=switch(7,$m);
$i4=switch(8,$m);
$i5=switch(9,$m);
$i6=switch(10,$m);
$idxNode=rint(4);
$wNode=switch($idxNode,A,H,E,C);
$ansNode=switch($idxNode,'I$i1+I$i4+I$i2=0','I$i6=I$i2+I$i3','I$i5+I$i6+I$i1=0','I$i5+I$i3-I$i4=0');
$idxLoop=rint(7);
$wLoop=switch($idxLoop,'AHEFGA','ABCHA','CDEHC','ABCDEHA','ABCDEFGA','ABCHEFGA','CDEFGAHC');
$ansLoop=switch($idxLoop,'I$i2*R$r5+I$i6*R$r3-I$i1*R$r4+V$v1=0','I$i4*R$r1-V$v2+V$v3-I$i2*R$r5=0','I$i5*R$r2-I$i6*R$r3-V$v3=0','I$i4*R$r1-V$v2+I$i5*R$r2-I$i6*R$r3-I$i2*R$r5=0','I$i4*R$r1-V$v2+I$i5*R$r2-I$i1*R$r4+V$v1=0','I$i4*R$r1-V$v2+V$v3+I$i6*R$r3-I$i1*R$r4+V$v1=0','I$i5*R$r2-I$i1*R$r4+V$v1+I$i2*R$r5-V$v3=0');@
qu.9.13.uid=329565ec-6a68-4b4a-8b5b-3a2a89195816@
qu.9.13.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Difficulty=Medium;
  Topic=Kirchhoff's Laws;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Diagram;
@
qu.9.13.weighting=1,1@
qu.9.13.numbering=alpha@
qu.9.13.part.1.name=sro_id_1@
qu.9.13.part.1.maple_answer=$ansNode@
qu.9.13.part.1.editing=useHTML@
qu.9.13.part.1.question=(Unset)@
qu.9.13.part.1.libname=@
qu.9.13.part.1.mode=Maple@
qu.9.13.part.1.allow2d=0@
qu.9.13.part.1.plot=@
qu.9.13.part.1.maple=is((solve($ANSWER,I1))=(solve($RESPONSE,I1)) );@
qu.9.13.part.1.type=maple@
qu.9.13.part.2.name=sro_id_2@
qu.9.13.part.2.maple_answer=$ansLoop@
qu.9.13.part.2.editing=useHTML@
qu.9.13.part.2.question=(Unset)@
qu.9.13.part.2.libname=@
qu.9.13.part.2.mode=Maple@
qu.9.13.part.2.allow2d=0@
qu.9.13.part.2.plot=@
qu.9.13.part.2.maple=is(solve($ANSWER,V2)-solve($RESPONSE,V2) = 0);@
qu.9.13.part.2.type=maple@
qu.9.13.question=<p>Consider the following circuit where the symbols have their usual meaning.&nbsp;&nbsp;State your answers in terms of the given variables and directions in the diagram.</p><p>&nbsp;</p><p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="583" height="600"><param name="image" value="__BASE_URI__img/Circuits/Kirchhoffs3L-2-NoLabel-Dir/Diagram.png" /><param name="size" value="22" /><param name="label.1.x" value="50" /><param name="label.1.y" value="330" /><param name="label.1.text" value="A" /><param name="label.2.x" value="50" /><param name="label.2.y" value="60" /><param name="label.2.text" value="B" /><param name="label.3.x" value="265" /><param name="label.3.y" value="60" /><param name="label.3.text" value="C" /><param name="label.4.x" value="530" /><param name="label.4.y" value="60" /><param name="label.4.text" value="D" /><param name="label.5.x" value="530" /><param name="label.5.y" value="330" /><param name="label.5.text" value="E" /><param name="label.6.x" value="530" /><param name="label.6.y" value="550" /><param name="label.6.text" value="F" /><param name="label.7.x" value="50" /><param name="label.7.y" value="550" /><param name="label.7.text" value="G" /><param name="label.8.x" value="265" /><param name="label.8.y" value="345" /><param name="label.8.text" value="H" /><param name="label.9.x" value="40" /><param name="label.9.y" value="195" /><param name="label.9.text" value="R$r1" /><param name="label.10.x" value="398" /><param name="label.10.y" value="50" /><param name="label.10.text" value="R$r2" /><param name="label.11.x" value="398" /><param name="label.11.y" value="300" /><param name="label.11.text" value="R$r3" /><param name="label.12.x" value="555" /><param name="label.12.y" value="430" /><param name="label.12.text" value="R$r4" /><param name="label.13.x" value="160" /><param name="label.13.y" value="570" /><param name="label.13.text" value="V$v1" /><param name="label.14.x" value="160" /><param name="label.14.y" value="290" /><param name="label.14.text" value="R$r5" /><param name="label.15.x" value="310" /><param name="label.15.y" value="200" /><param name="label.15.text" value="V$v3" /><param name="label.16.x" value="160" /><param name="label.16.y" value="40" /><param name="label.16.text" value="V$v2" /><param name="label.17.x" value="225" /><param name="label.17.y" value="510" /><param name="label.17.text" value="I$i1" /><param name="label.18.x" value="240" /><param name="label.18.y" value="350" /><param name="label.18.text" value="I$i2" /><param name="label.19.x" value="245" /><param name="label.19.y" value="270" /><param name="label.19.text" value="I$i3" /><param name="label.20.x" value="220" /><param name="label.20.y" value="90" /><param name="label.20.text" value="I$i4" /><param name="label.21.x" value="490" /><param name="label.21.y" value="90" /><param name="label.21.text" value="I$i5" /><param name="label.22.x" value="490" /><param name="label.22.y" value="350" /><param name="label.22.text" value="I$i6" /></applet></p><p><strong>(a)</strong>&nbsp; Use Kirchhoff's current rule to write an equation for the algebraic sum of the currents&nbsp;flowing into node $wNode.&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p><span><strong>(b)</strong>&nbsp; Use Kirchhoff's voltage rule to write&nbsp;an equation for the algebraic&nbsp;sum of the potential differences&nbsp;in the $wLoop loop of the circuit.</span></p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p>@

qu.9.14.question=<p><img alt="" align="middle" width="609" height="150" src="__BASE_URI__img/EMF/Battery-Terminal-Voltage-2/Diagram.png" /></p>
<p><span><br />
A battery with internal voltage <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$V</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>V</mi></mrow></mstyle></math>&nbsp;and terminal voltage <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$Vterm</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>V</mi></mrow></mstyle></math>&nbsp;has a current&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mstyle></math>&nbsp;passing<br />
&nbsp;through it in the direction indicated.&nbsp; What is the internal resistance&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$r</mi></mrow></mstyle></math>&nbsp;of the battery?</span></p>
<p><span><span><em>Note:&nbsp; Enter <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ohm</mi></mrow></mstyle></math>&nbsp;for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&Omega;</mi></mrow></mstyle></math>.</em></span></span></p>@
qu.9.14.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.9.14.allow2d=0@
qu.9.14.maple_answer=SigFigs[roundToSigFigs]($ans,3)*ohm@
qu.9.14.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.9.14.type=maple@
qu.9.14.mode=Maple@
qu.9.14.name=Battery - Terminal Voltage 2 - Find r ~ PGc@
qu.9.14.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.9.14.editing=useHTML@
qu.9.14.solution=<p>The terminal voltage is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>V</mi><mrow><mi>AB</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>V</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>Ir</mi></mrow></mstyle></math>.</p>
<p>Thus:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>V</mi><mrow><mi>AB</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>V</mi></mrow><mrow><mi>I</mi></mrow></mfrac></mrow></mstyle></math>.</p>@
qu.9.14.algorithm=$I=rand(0.300,0.999,3);
$Vterm=rand(10.0,19.9,3);
$V=rand(5,($Vterm-0.5),3);
$ans=(-$V+$Vterm)/$I;@
qu.9.14.uid=e01369ad-0d29-4e7a-bdf6-a3f3a238947d@
qu.9.14.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Terminal Voltage;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.9.15.question=<p><img alt="" align="middle" width="609" height="150" src="__BASE_URI__img/EMF/Battery-Terminal-Voltage-2/Diagram.png" /></p>
<p><span><br />
A battery with internal voltage <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$V</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>V</mi></mrow></mstyle></math>&nbsp;and internal resistance <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$r</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mi>&Omega;</mi></mrow></mrow></mstyle></math>&nbsp;has a current&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mstyle></math>&nbsp;passing<br />
&nbsp;through it in the direction indicated.&nbsp; What is the terminal voltage <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>V</mi><mrow><mi>AB</mi></mrow></msub></mrow></mstyle></math>&nbsp;of the battery?</span></p>@
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qu.9.15.maple_answer=SigFigs[roundToSigFigs]($ans,3)*V@
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qu.9.15.mode=Maple@
qu.9.15.name=Battery - Terminal Voltage 2 - Find V_term ~ PGc@
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qu.9.15.editing=useHTML@
qu.9.15.solution=<p>The terminal voltage is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>V</mi><mrow><mi>AB</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>V</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>Ir</mi></mrow></mstyle></math>.</p>@
qu.9.15.algorithm=$r=rand(1.00,9.99,3);
$I=rand(0.300,0.999,3);
$V=rand(10.0,19.9,3);
$ans=$V+$I*$r;@
qu.9.15.uid=90bf862f-76db-4c8b-ada2-82a32d630641@
qu.9.15.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Terminal Voltage;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.9.16.mode=Inline@
qu.9.16.name=Kirchhoff's Laws - 2 Loops - Numeric - 1@
qu.9.16.comment=<ol>
    <li>Kirchhoff's current (or node) law states that the sum of all currents entering a node is equal to the sum of all currents leaving the node.</li>
    <li>Kirchhoff's voltage (or loop) law states that the sum of all voltage gains and drops around a closed loop is equal to zero.</li>
    <li>Remember that choosing one direction around a loop is important. It does not matter <em>which</em> direction you choose to follow, but you must be consistent.</li>
    <li>When following a loop: Passing through a battery is a positive if you are going from negative to positive terminal and a negative if going from positive to negative.</li>
    <li>When following a loop: Passing through a resistor is dependant both on the direction you are following the loop<em> and the assumed direction of the current passing through the resistor</em>. If you are following the loop in the same direction as the current through the resistor, the resistor provides a negative term. If you are following the loop against the current, the resistor provides a positive term.</li>
</ol>@
qu.9.16.editing=useHTML@
qu.9.16.hint.1=Kirchhoff's current (or node) law states that the sum of all currents entering a node is equal to the sum of all currents leaving the node.@
qu.9.16.hint.2=Kirchhoff's voltage (or loop) law states that the sum of all voltage gains and drops around a closed loop is equal to zero.@
qu.9.16.hint.3=Remember that choosing one direction around a loop is important.&nbsp; It does not matter <em>which</em> direction you choose to follow, but you must be consistent.@
qu.9.16.hint.4=When following a loop:&nbsp; Passing through a battery is a positive if you are going from negative to positive terminal and a negative if going from positive to negative.@
qu.9.16.hint.5=When following a loop:&nbsp; Passing through a resistor is dependant both on the direction you are following the loop<em> and the assumed direction of the current passing through the resistor</em>.&nbsp; If you are following the loop in the same direction as the current through the resistor, the resistor provides a negative term.&nbsp; If you are following the loop against the current, the resistor provides a positive term.@
qu.9.16.solution=<p>We must choose a direction to traverse each loop and a direction for the currents in each branch.</p>
<p>If we cross a voltage source from the negative to the positive terminal, then the potential difference is positive, otherwise it is negative.</p>
<p>If we cross a resistor in the direction of current flow, then the potential difference is negative, otherwise it is positive.</p>
<p>&nbsp;</p>
<p>At a node, the sum of all currents entering must equal the sum of all currents leaving.</p>@
qu.9.16.algorithm=$V1=rand(1.00,20.0,3);
$V2=rand(1.00,20.0,3);
$V3=rand(1.00,20.0,3);
$R1=rand(10.0,999,3);
$R2=rand(10.0,999,3);
$R3=rand(10.0,999,3);
$R4=rand(10.0,999,3);
$m=maple("
assign(solve({I1+I2+I3=0,$V1-I1*$R1-$V2+I2*$R3-I1*$R2=0,$V2-$V3+I3*$R4-I2*$R3=0})):
I1,I2,I3;
");
$I1=switch(0,$m)*1000;
$I2=switch(1,$m)*1000;
$I3=switch(2,$m)*1000;@
qu.9.16.uid=4933af50-23d1-4e99-ba99-0c026b91c1c4@
qu.9.16.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Difficulty=Medium;
  Topic=Kirchhoff's Laws;
  Features=Algorithmic;
  Features=Diagram;
@
qu.9.16.weighting=1,1,1@
qu.9.16.numbering=alpha@
qu.9.16.part.1.name=sro_id_1@
qu.9.16.part.1.answer.units=@
qu.9.16.part.1.numStyle= scientific  @
qu.9.16.part.1.editing=useHTML@
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qu.9.16.part.1.question=(Unset)@
qu.9.16.part.1.mode=Numeric@
qu.9.16.part.1.grading=exact_sigd@
qu.9.16.part.1.negStyle=both@
qu.9.16.part.1.digit=3@
qu.9.16.part.1.answer.num=$I1@
qu.9.16.part.2.name=sro_id_2@
qu.9.16.part.2.answer.units=@
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qu.9.16.part.2.editing=useHTML@
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qu.9.16.part.2.digit=3@
qu.9.16.part.2.answer.num=$I2@
qu.9.16.part.3.name=sro_id_3@
qu.9.16.part.3.answer.units=@
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qu.9.16.part.3.answer.num=$I3@
qu.9.16.question=<p>Given the following circuit, calculate the currents <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I1</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I2</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I3</mi></mrow></mstyle></math>, in <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mA</mi></mrow></mstyle></math>.</p><p>&nbsp;</p><p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="384" height="400"><param name="image" value="__BASE_URI__img/Circuits/KirchhoffsLaws-2Loops/Diagram.png" /><param name="size" value="10" /><param name="label.1.x" value="268" /><param name="label.1.y" value="10" /><param name="label.1.text" value="$R2 Ohm" /><param name="label.2.x" value="100" /><param name="label.2.y" value="10" /><param name="label.2.text" value="$V1 V" /><param name="label.3.x" value="268" /><param name="label.3.y" value="190" /><param name="label.3.text" value="$R3 Ohm" /><param name="label.4.x" value="100" /><param name="label.4.y" value="190" /><param name="label.4.text" value="$V2 V" /><param name="label.5.x" value="100" /><param name="label.5.y" value="340" /><param name="label.5.text" value="$V3 V" /><param name="label.6.x" value="310" /><param name="label.6.y" value="296" /><param name="label.6.text" value="$R4 Ohm" /><param name="label.7.x" value="70" /><param name="label.7.y" value="128" /><param name="label.7.text" value="$R1 Ohm" /><param name="label.8.x" value="143" /><param name="label.8.y" value="55" /><param name="label.8.text" value="I1" /><param name="label.9.x" value="143" /><param name="label.9.y" value="235" /><param name="label.9.text" value="I2" /><param name="label.10.x" value="143" /><param name="label.10.y" value="385" /><param name="label.10.text" value="I3" /></applet></p><p><strong>(a)</strong>&nbsp;&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I1</mi></mrow></mstyle></math><span>&nbsp; </span><1><span>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mA</mi></mrow></mstyle></math></span></p><p><span><strong>(b)&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I2</mi></mrow></mstyle></math>&nbsp;&nbsp;</strong><span>&nbsp;</span><2><span>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mA</mi></mrow></mstyle></math></span></span></p><p><span><span><strong>(c)</strong>&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I3</mi></mrow></mstyle></math>&nbsp;&nbsp;<span>&nbsp;</span><3><span>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mA</mi></mrow></mstyle></math></span></span></span></p>@

qu.9.17.question=<p><img alt="" align="middle" width="561" height="500" src="__BASE_URI__img/Resistors/ResistorsInSeriesAndParallel2/Diagram.png" /></p>
<p>&nbsp;</p>
<p>What is the total resistance of the circuit between <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>?</p>@
qu.9.17.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=algebraicScalar)@
qu.9.17.allow2d=0@
qu.9.17.maple_answer=1/((1/(R1+R2))+(1/R3)+(1/R4))@
qu.9.17.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.9.17.type=maple@
qu.9.17.mode=Maple@
qu.9.17.name=Resistors in Series and Parallel 2 ~ PG@
qu.9.17.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=algebraicScalar,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.9.17.editing=useHTML@
qu.9.17.solution=<p>For the bottom branch, the equivalent resistance is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>R</mi><mrow><mi>eq</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>R</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>R</mi><mrow><mn>2</mn></mrow></msub></mrow></mstyle></math>.</p>
<p>Therefore,</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>R</mi><mrow><mi>AB</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><msub><mi>R</mi><mrow><mi>eq</mi></mrow></msub></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mn>1</mn><mrow><msub><mi>R</mi><mrow><mn>3</mn></mrow></msub></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn>1</mn><mrow><msub><mi>R</mi><mrow><mn>4</mn></mrow></msub></mrow></mfrac></mrow></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup></mrow></mstyle></math>.</p>@
qu.9.17.algorithm=@
qu.9.17.uid=9cddf04b-71dd-4ba8-ac2b-057925c536a0@
qu.9.17.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Resistors in Series and Parallel;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Diagram;
  Features=Algebraic;
@

qu.9.18.question=<p><img alt="" align="middle" width="561" height="500" src="__BASE_URI__img/Resistors/ResistorsInSeriesAndParallel1/Diagram.png" /></p>
<p>&nbsp;</p>
<p>What is the total resistance of the circuit between <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>?</p>@
qu.9.18.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=algebraicScalar)@
qu.9.18.allow2d=0@
qu.9.18.maple_answer=1/((1/(R1+R2))+(1/R3))@
qu.9.18.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.9.18.type=maple@
qu.9.18.mode=Maple@
qu.9.18.name=Resistors in Series and Parallel 1 ~ PG@
qu.9.18.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=algebraicScalar,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.9.18.editing=useHTML@
qu.9.18.solution=<p>For the bottom branch, the equivalent resistance is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>R</mi><mrow><mi>eq</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>R</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>R</mi><mrow><mn>2</mn></mrow></msub></mrow></mstyle></math>.</p>
<p>Therefore,</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>R</mi><mrow><mi>AB</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><msub><mi>R</mi><mrow><mi>eq</mi></mrow></msub></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mn>1</mn><mrow><msub><mi>R</mi><mrow><mn>3</mn></mrow></msub></mrow></mfrac></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup></mrow></mstyle></math>.</p>@
qu.9.18.algorithm=@
qu.9.18.uid=87f5900a-fe18-44b8-afcc-f1ba0d519572@
qu.9.18.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Resistors in Series and Parallel;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Diagram;
  Features=Algebraic;
@

qu.9.19.question=<p><img alt="" align="middle" width="609" height="150" src="__BASE_URI__img/EMF/Battery-Terminal-Voltage-1/Diagram.png" /></p>
<p><span><br />
A battery with internal voltage <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$V</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>V</mi></mrow></mstyle></math>&nbsp;and terminal voltage <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$Vterm</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>V</mi></mrow></mstyle></math>&nbsp;has a current&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mstyle></math>&nbsp;passing<br />
&nbsp;through it in the direction indicated.&nbsp; What is the internal resistance&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$r</mi></mrow></mstyle></math>&nbsp;of the battery?</span></p>
<p><span><em>Note:&nbsp; Enter <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ohm</mi></mrow></mstyle></math>&nbsp;for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&Omega;</mi></mrow></mstyle></math>.</em></span></p>@
qu.9.19.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.9.19.allow2d=0@
qu.9.19.maple_answer=SigFigs[roundToSigFigs]($ans,3)*ohm@
qu.9.19.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.9.19.type=maple@
qu.9.19.mode=Maple@
qu.9.19.name=Battery - Terminal Voltage 1 - Find r ~ PGc@
qu.9.19.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.9.19.editing=useHTML@
qu.9.19.solution=<p>The terminal voltage is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>V</mi><mrow><mi>AB</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>V</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>Ir</mi></mrow></mstyle></math>.</p>
<p>Thus:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>V</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mi>V</mi><mrow><mi>AB</mi></mrow></msub></mrow><mrow><mi>I</mi></mrow></mfrac></mrow></mstyle></math></p>@
qu.9.19.algorithm=$I=rand(0.300,0.999,3);
$V=rand(10.0,19.9,3);
$Vterm=rand(5,($V-0.5),3);
$ans=($V-$Vterm)/$I;@
qu.9.19.uid=d3d54c36-a9e7-49ba-9ef1-13653371b74a@
qu.9.19.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Terminal Voltage;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.9.20.mode=Inline@
qu.9.20.name=Kirchhoff's Laws - 3 Loops - Numeric - 2@
qu.9.20.comment=<ol>
    <li>Kirchhoff's current (or node) law states that the sum of all currents entering a node is equal to the sum of all currents leaving the node.</li>
    <li>Kirchhoff's voltage (or loop) law states that the sum of all voltage gains and drops around a closed loop is equal to zero.</li>
    <li>Remember that choosing one direction around a loop is important. It does not matter <em>which</em> direction you choose to follow, but you must be consistent.</li>
    <li>When following a loop: Passing through a battery is a positive if you are going from negative to positive terminal and a negative if going from positive to negative.</li>
    <li>When following a loop: Passing through a resistor is dependant both on the direction you are following the loop<em> and the assumed direction of the current passing through the resistor</em>. If you are following the loop in the same direction as the current through the resistor, the resistor provides a negative term. If you are following the loop against the current, the resistor provides a positive term.</li>
</ol>@
qu.9.20.editing=useHTML@
qu.9.20.hint.1=Kirchhoff's current (or node) law states that the sum of all currents entering a node is equal to the sum of all currents leaving the node.@
qu.9.20.hint.2=Kirchhoff's voltage (or loop) law states that the sum of all voltage gains and drops around a closed loop is equal to zero.@
qu.9.20.hint.3=Remember that choosing one direction around a loop is important.&nbsp; It does not matter <em>which</em> direction you choose to follow, but you must be consistent.@
qu.9.20.hint.4=When following a loop:&nbsp; Passing through a battery is a positive if you are going from negative to positive terminal and a negative if going from positive to negative.@
qu.9.20.hint.5=When following a loop:&nbsp; Passing through a resistor is dependant both on the direction you are following the loop<em> and the assumed direction of the current passing through the resistor</em>.&nbsp; If you are following the loop in the same direction as the current through the resistor, the resistor provides a negative term.&nbsp; If you are following the loop against the current, the resistor provides a positive term.@
qu.9.20.solution=<p>We must choose a direction to traverse each loop and a direction for the currents in each branch.</p>
<p>If we cross a voltage source from the negative to the positive terminal, then the potential difference is positive, otherwise it is negative.</p>
<p>If we cross a resistor in the direction of current flow, then the potential difference is negative, otherwise it is positive.</p>
<p>&nbsp;</p>
<p>At a node, the sum of all currents entering must equal the sum of all currents leaving.</p>@
qu.9.20.algorithm=$V1=rand(1.00,20.0,3);
$V2=rand(1.00,20.0,3);
$V3=rand(1.00,20.0,3);
$V4=rand(1.00,20.0,3);
$R1=rand(10.0,999,3);
$R2=rand(10.0,999,3);
$R3=rand(10.0,999,3);
$R4=rand(10.0,999,3);
$m=maple("
assign(solve({I1+I4+I2=0,I1+I5+I6=0,I4-I3-I5=0,$V1-$V2+I6*$R3-I1*$R4=0,$V2+I4*$R1-$V4-$V3=0,$V3+I5*$R2-I6*$R3=0}
));
I1,I2,I3,I4,I5,I6;
");
$I1=switch(0,$m)*1000;
$I2=switch(1,$m)*1000;
$I3=switch(2,$m)*1000;
$I4=switch(3,$m)*1000;
$I5=switch(4,$m)*1000;
$I6=switch(5,$m)*1000;@
qu.9.20.uid=38772a5f-9874-455c-8c9d-d847cdd58eea@
qu.9.20.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Difficulty=Medium;
  Topic=Kirchhoff's Laws;
  Features=Algorithmic;
  Features=Diagram;
@
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qu.9.20.question=<p>Given the following circuit, calculate the currents <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I1</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I2</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I3</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I4</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I5</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I6</mi></mrow></mstyle></math>&nbsp;in <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mA</mi></mrow></mstyle></math>.&nbsp;</p><p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="583" height="600"><param name="image" value="__BASE_URI__img/Circuits/Kirchhoffs3L-2-NoLabel-Dir/Diagram.png" /><param name="size" value="14" /><param name="label.1.x" value="110" /><param name="label.1.y" value="195" /><param name="label.1.text" value="$R1 Ohm" /><param name="label.2.x" value="398" /><param name="label.2.y" value="50" /><param name="label.2.text" value="$R2 Ohm" /><param name="label.3.x" value="398" /><param name="label.3.y" value="300" /><param name="label.3.text" value="$R3 Ohm" /><param name="label.4.x" value="480" /><param name="label.4.y" value="430" /><param name="label.4.text" value="$R4 Ohm" /><param name="label.5.x" value="160" /><param name="label.5.y" value="570" /><param name="label.5.text" value="$V1 V" /><param name="label.6.x" value="160" /><param name="label.6.y" value="290" /><param name="label.6.text" value="$V2 V" /><param name="label.7.x" value="315" /><param name="label.7.y" value="200" /><param name="label.7.text" value="$V3 V" /><param name="label.8.x" value="160" /><param name="label.8.y" value="110" /><param name="label.8.text" value="$V4 V" /><param name="label.9.x" value="225" /><param name="label.9.y" value="510" /><param name="label.9.text" value="I1" /><param name="label.10.x" value="240" /><param name="label.10.y" value="350" /><param name="label.10.text" value="I2" /><param name="label.11.x" value="245" /><param name="label.11.y" value="275" /><param name="label.11.text" value="I3" /><param name="label.12.x" value="220" /><param name="label.12.y" value="90" /><param name="label.12.text" value="I4" /><param name="label.13.x" value="490" /><param name="label.13.y" value="90" /><param name="label.13.text" value="I5" /><param name="label.14.x" value="490" /><param name="label.14.y" value="350" /><param name="label.14.text" value="I6" /></applet></p><p align="left"><strong>(a)</strong>&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I1</mi></mrow></mstyle></math>&nbsp;&nbsp;<span>&nbsp;</span><1><span>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mA</mi></mrow></mstyle></math></span></p><p><span><strong>(b)</strong>&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I2</mi></mrow></mstyle></math>&nbsp;&nbsp;<span>&nbsp;</span><2><span>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mA</mi></mrow></mstyle></math></span></span></p><p><span><span><strong>(c)</strong>&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I3</mi></mrow></mstyle></math>&nbsp;&nbsp;<span>&nbsp;</span><3><span>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mA</mi></mrow></mstyle></math></span></span></span></p><p><span><span><span><strong>(d)</strong>&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I4</mi></mrow></mstyle></math>&nbsp;&nbsp;<span>&nbsp;</span><4><span>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mA</mi></mrow></mstyle></math></span></span></span></span></p><p><span><span><span><span><strong>(e)</strong>&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I5</mi></mrow></mstyle></math>&nbsp;&nbsp;<span>&nbsp;</span><5><span>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mA</mi></mrow></mstyle></math></span></span></span></span></span></p><p><span><span><span><span><span><strong>(f)</strong>&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I6</mi></mrow></mstyle></math>&nbsp;&nbsp;<span>&nbsp;</span><6><span>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mA</mi></mrow></mstyle></math></span></span></span></span></span></span></p>@

qu.9.21.mode=Inline@
qu.9.21.name=Kirchhoff's Laws - 2 Loops - Numeric - 2@
qu.9.21.comment=<ol>
    <li>Kirchhoff's current (or node) law states that the sum of all currents entering a node is equal to the sum of all currents leaving the node.</li>
    <li>Kirchhoff's voltage (or loop) law states that the sum of all voltage gains and drops around a closed loop is equal to zero.</li>
    <li>Remember that choosing one direction around a loop is important. It does not matter <em>which</em> direction you choose to follow, but you must be consistent.</li>
    <li>When following a loop: Passing through a battery is a positive if you are going from negative to positive terminal and a negative if going from positive to negative.</li>
    <li>When following a loop: Passing through a resistor is dependant both on the direction you are following the loop<em> and the assumed direction of the current passing through the resistor</em>. If you are following the loop in the same direction as the current through the resistor, the resistor provides a negative term. If you are following the loop against the current, the resistor provides a positive term.</li>
</ol>@
qu.9.21.editing=useHTML@
qu.9.21.hint.1=Kirchhoff's current (or node) law states that the sum of all currents entering a node is equal to the sum of all currents leaving the node.@
qu.9.21.hint.2=Kirchhoff's voltage (or loop) law states that the sum of all voltage gains and drops around a closed loop is equal to zero.@
qu.9.21.hint.3=Remember that choosing one direction around a loop is important.&nbsp; It does not matter <em>which</em> direction you choose to follow, but you must be consistent.@
qu.9.21.hint.4=When following a loop:&nbsp; Passing through a battery is a positive if you are going from negative to positive terminal and a negative if going from positive to negative.@
qu.9.21.hint.5=When following a loop:&nbsp; Passing through a resistor is dependant both on the direction you are following the loop<em> and the assumed direction of the current passing through the resistor</em>.&nbsp; If you are following the loop in the same direction as the current through the resistor, the resistor provides a negative term.&nbsp; If you are following the loop against the current, the resistor provides a positive term.@
qu.9.21.solution=<p>We must choose a direction to traverse each loop and a direction for the currents in each branch.</p>
<p>If we cross a voltage source from the negative to the positive terminal, then the potential difference is positive, otherwise it is negative.</p>
<p>If we cross a resistor in the direction of current flow, then the potential difference is negative, otherwise it is positive.</p>
<p>&nbsp;</p>
<p>At a node, the sum of all currents entering must equal the sum of all currents leaving.</p>@
qu.9.21.algorithm=$i1=1;
$i2=2;
$i3=3;
$V1=rand(1.00,20.0,3);
$V2=rand(1.00,20.0,3);
$V3=rand(1.00,20.0,3);
$R1=rand(10.0,999,3);
$R2=rand(10.0,999,3);
$R3=rand(10.0,999,3);
$R4=rand(10.0,999,3);
$R5=rand(10.0,999,3);
$R6=rand(10.0,999,3);
$R7=rand(10.0,999,3);
$m=maple("
assign(solve({I1=I2+I3,$V1-$R1*I1-I1*(1/(1/$R2+1/$R3))+$V2-I2*$R4-I1*$R5=0,$V2-I2*$R4+I3*(1/(1/$R6+1/$R7))=0}));
I1,I2,I3;
");
$I1=switch(0,$m)*1000;
$I2=switch(1,$m)*1000;
$I3=switch(2,$m)*1000;@
qu.9.21.uid=6ed3d363-171a-412a-b90e-42763b83ffc3@
qu.9.21.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Difficulty=Medium;
  Topic=Kirchhoff's Laws;
  Features=Algorithmic;
  Features=Diagram;
@
qu.9.21.weighting=1,1,1@
qu.9.21.numbering=alpha@
qu.9.21.part.1.name=sro_id_1@
qu.9.21.part.1.answer.units=@
qu.9.21.part.1.numStyle= scientific  @
qu.9.21.part.1.editing=useHTML@
qu.9.21.part.1.showUnits=false@
qu.9.21.part.1.question=(Unset)@
qu.9.21.part.1.mode=Numeric@
qu.9.21.part.1.grading=exact_sigd@
qu.9.21.part.1.negStyle=both@
qu.9.21.part.1.digit=3@
qu.9.21.part.1.answer.num=$I1@
qu.9.21.part.2.name=sro_id_2@
qu.9.21.part.2.answer.units=@
qu.9.21.part.2.numStyle= scientific  @
qu.9.21.part.2.editing=useHTML@
qu.9.21.part.2.showUnits=false@
qu.9.21.part.2.question=(Unset)@
qu.9.21.part.2.mode=Numeric@
qu.9.21.part.2.grading=exact_sigd@
qu.9.21.part.2.negStyle=both@
qu.9.21.part.2.digit=3@
qu.9.21.part.2.answer.num=$I2@
qu.9.21.part.3.name=sro_id_3@
qu.9.21.part.3.answer.units=@
qu.9.21.part.3.numStyle= scientific  @
qu.9.21.part.3.editing=useHTML@
qu.9.21.part.3.showUnits=false@
qu.9.21.part.3.question=(Unset)@
qu.9.21.part.3.mode=Numeric@
qu.9.21.part.3.grading=exact_sigd@
qu.9.21.part.3.negStyle=both@
qu.9.21.part.3.digit=3@
qu.9.21.part.3.answer.num=$I3@
qu.9.21.question=<p>Given the following circuit, calculate the currents <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I1</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I2</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I3</mi></mrow></mstyle></math>&nbsp;in <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mA</mi></mrow></mstyle></math>.&nbsp;</p><p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="675" height="506"><param name="image" value="__BASE_URI__img/Circuits/Kirchhoffs2L-2-NoLabel-Dir/Diagram.png" /><param name="size" value="12" /><param name="label.1.x" value="30" /><param name="label.1.y" value="150" /><param name="label.1.text" value="$R3 Ohm" /><param name="label.2.x" value="185" /><param name="label.2.y" value="150" /><param name="label.2.text" value="$R2 Ohm" /><param name="label.3.x" value="30" /><param name="label.3.y" value="250" /><param name="label.3.text" value="$R1 Ohm" /><param name="label.4.x" value="160" /><param name="label.4.y" value="365" /><param name="label.4.text" value="$V1 V" /><param name="label.5.x" value="245" /><param name="label.5.y" value="365" /><param name="label.5.text" value="$R5 Ohm" /><param name="label.6.x" value="185" /><param name="label.6.y" value="320" /><param name="label.6.text" value="I1" /><param name="label.7.x" value="285" /><param name="label.7.y" value="250" /><param name="label.7.text" value="$R4 Ohm" /><param name="label.8.x" value="425" /><param name="label.8.y" value="170" /><param name="label.8.text" value="$V2 V" /><param name="label.9.x" value="500" /><param name="label.9.y" value="220" /><param name="label.9.text" value="$R6 Ohm" /><param name="label.10.x" value="500" /><param name="label.10.y" value="390" /><param name="label.10.text" value="$R7 Ohm" /><param name="label.11.x" value="345" /><param name="label.11.y" value="140" /><param name="label.11.text" value="I2" /><param name="label.12.x" value="590" /><param name="label.12.y" value="205" /><param name="label.12.text" value="I3" /></applet></p><p align="left"><strong>(a)</strong>&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I1</mi></mrow></mstyle></math>&nbsp;&nbsp;<span>&nbsp;</span><1><span>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mA</mi></mrow></mstyle></math></span></p><p><span><strong>(b)</strong>&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I2</mi></mrow></mstyle></math>&nbsp;&nbsp;<span>&nbsp;</span><2><span>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mA</mi></mrow></mstyle></math></span></span></p><p><span><span><strong>(c)</strong>&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I3</mi></mrow></mstyle></math>&nbsp;&nbsp;<span>&nbsp;</span><3><span>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mA</mi></mrow></mstyle></math></span></span></span></p>@

qu.9.22.question=<p><img alt="" align="middle" width="561" height="500" src="__BASE_URI__img/Resistors/ResistorsInSeriesAndParallel2/Diagram.png" /></p>
<p>&nbsp;</p>
<p>If <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>R1</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$R1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mi>&Omega;</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>R2</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$R2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mi>&Omega;</mi></mrow></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>R3</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$R3</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&Omega;</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>R4</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$R4</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mi>&Omega;</mi></mrow></mrow></mstyle></math>, what is the total<br />
resistance of the circuit between <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>?</p>@
qu.9.22.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN)@
qu.9.22.allow2d=0@
qu.9.22.maple_answer=SigFigs[roundToSigFigs](1/((1/($R1+$R2))+(1/$R3)+(1/$R4)),3)*ohms;@
qu.9.22.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.9.22.type=maple@
qu.9.22.mode=Maple@
qu.9.22.name=Resistors in Series and Parallel 2 - Numeric ~ PG@
qu.9.22.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.9.22.editing=useHTML@
qu.9.22.solution=<p>For the bottom branch, the equivalent resistance is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>R</mi><mrow><mi>eq</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>R</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>R</mi><mrow><mn>2</mn></mrow></msub></mrow></mstyle></math>.</p>
<p>Therefore,</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>R</mi><mrow><mi>AB</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><msub><mi>R</mi><mrow><mi>eq</mi></mrow></msub></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mn>1</mn><mrow><msub><mi>R</mi><mrow><mn>3</mn></mrow></msub></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn>1</mn><mrow><msub><mi>R</mi><mrow><mn>4</mn></mrow></msub></mrow></mfrac></mrow></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>.</p>@
qu.9.22.algorithm=$R1=rand(1.00,100,3);
$R2=rand(1.00,100,3);
$R3=rand(1.00,100,3);
$R4=rand(1.00,100,3);@
qu.9.22.uid=6d8596b0-e517-4afb-b5aa-ffdade6b8cac@
qu.9.22.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Resistors in Series and Parallel;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Diagram;
@

qu.9.23.question=<p><img alt="" align="middle" width="561" height="500" src="__BASE_URI__img/Resistors/ResistorsInSeriesAndParallel1/Diagram.png" /></p>
<p>&nbsp;</p>
<p>If <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>R1</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$R1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mi>&Omega;</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>R2</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$R2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mi>&Omega;</mi></mrow></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>R3</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$R3</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&Omega;</mi></mrow></mstyle></math>, what is the total resistance of the circuit between <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>?</p>@
qu.9.23.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN)@
qu.9.23.allow2d=0@
qu.9.23.maple_answer=SigFigs[roundToSigFigs](1/((1/($R1+$R2))+(1/$R3)),3)*ohm;@
qu.9.23.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.9.23.type=maple@
qu.9.23.mode=Maple@
qu.9.23.name=Resistors in Series and Parallel 1 - Numeric ~ PG@
qu.9.23.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.9.23.editing=useHTML@
qu.9.23.solution=<p>For the bottom branch, the equivalent resistance is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>R</mi><mrow><mi>eq</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>R</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>R</mi><mrow><mn>2</mn></mrow></msub></mrow></mstyle></math>.</p>
<p>Therefore,</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>R</mi><mrow><mi>AB</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><msub><mi>R</mi><mrow><mi>eq</mi></mrow></msub></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mn>1</mn><mrow><msub><mi>R</mi><mrow><mn>3</mn></mrow></msub></mrow></mfrac></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup></mrow></mstyle></math>.</p>@
qu.9.23.algorithm=$R1=rand(1.00,100,3);
$R2=rand(1.00,100,3);
$R3=rand(1.00,100,3);@
qu.9.23.uid=307e76f6-b994-4689-9fe1-2232fcc78b5a@
qu.9.23.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Resistors in Series and Parallel;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Diagram;
@

qu.9.24.question=<p><img alt="" align="middle" width="438" height="400" src="__BASE_URI__img/Capacitance/CapacitorsSeriesParallel2/Diagram.png" /></p>
<p>&nbsp;</p>
<p>What is the total capacitance of the circuit between <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>?</p>@
qu.9.24.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=algebraicScalar)@
qu.9.24.allow2d=0@
qu.9.24.maple_answer=(1/C1+1/C2)^(-1)+C3+C4@
qu.9.24.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.9.24.type=maple@
qu.9.24.mode=Maple@
qu.9.24.name=Capacitors in Series and Parallel 2 ~ PG@
qu.9.24.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,subPackage=algebraicScalar,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.9.24.editing=useHTML@
qu.9.24.solution=<p>For the bottom arm, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>C</mi><mrow><mi>eq</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><msub><mi>C</mi><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mn>1</mn><mrow><msub><mi>C</mi><mrow><mn>2</mn></mrow></msub></mrow></mfrac></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup></mrow></mstyle></math>.</p>
<p>The total is then:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>C</mi><mrow><mi>AB</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>C</mi><mrow><mi>eq</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>C</mi><mrow><mn>3</mn></mrow></msub></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>C</mi><mrow><mn>4</mn></mrow></msub></mrow></mstyle></math>.</p>@
qu.9.24.algorithm=@
qu.9.24.uid=4a59bffb-af18-4cd3-a9d8-cdc89f721163@
qu.9.24.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Capacitors in Series and Parallel;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Diagram;
  Features=Algorithmic;
  Features=Algebraic;
@

qu.10.topic=Wires@

qu.10.1.question=<p>The wires <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>&nbsp;are carrying a current <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;is carrying current <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$cur2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>, each in the&nbsp;<br />
directions indicated in the diagram.&nbsp; In&nbsp;terms of the given parameters, provide a vector&nbsp;expression for&nbsp;<br />
the&nbsp;force per unit length exerted by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>&nbsp;on <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>, for&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;placed at&nbsp;any point on the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></mstyle></math>axis.</p>
<p>&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="500" height="505">
<param name="image" value="__BASE_URI__img/MagneticFields/CurrentCarryingWiresVertical/Diagram3Wires2-2.png" />
<param name="size" value="3" />
<param name="label.1.x" value="232" />
<param name="label.1.y" value="215" />
<param name="label.1.text" value="$mag$var" />
<param name="label.2.x" value="232" />
<param name="label.2.y" value="315" />
<param name="label.2.text" value="$mag$var" />
<param name="label.3.x" value="315" />
<param name="label.3.y" value="257" />
<param name="label.3.text" value="x" /></applet></p>
<p>&nbsp;</p>
<p><em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>xhat</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>yhat</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>z</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>zhat</mi></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
Enter your response in terms of the given parameters and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>, entering&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi></mrow></mstyle></math> instead of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>.</p>@
qu.10.1.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents);@
qu.10.1.allow2d=0@
qu.10.1.maple_answer=(-xhat)*($cur2*I)*2*(u*$cur*I/(2*Pi*sqrt(x^2+($mag*$var)^2)))*((x)/sqrt(x^2+($mag*$var)^2));@
qu.10.1.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.10.1.type=maple@
qu.10.1.mode=Maple@
qu.10.1.name=Current Carrying Wires - Vertical - 3 Wires - 2-2@
qu.10.1.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.10.1.editing=useHTML@
qu.10.1.solution=<p>We will calculate the magnetic field due to wires <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>, and then the force due to this magnetic field on <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>.</p>
<p>The magnitude of the magnetic field due to an infinite current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>&pi;r</mi></mrow></mfrac></mrow></mstyle></math>.&nbsp; The direction of the magnetic field is found using the right-hand rule, where the thumb points in the direction of current flow and the curl of the fingers represents the direction of the magnetic field.&nbsp; If we draw a line from the wire to the point in question, then the magnetic field will always be perpendicular to that line.</p>
<p>&nbsp;</p>
<p>At point&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math> the magnitude of the magnetic field due to each of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math> and&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math> will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow></mfrac></mrow></mstyle></math>.</p>
<p>Based on the configuration, the horizontal components will cancel.&nbsp; Thus, we need only calculate the&nbsp;vertical components, which are:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>B</mi><mrow><mi>y</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>, and therefore the total magnetic field at&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mn>2</mn><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac><mover><mrow><mi>y</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>Finally, the force due to a magnetic field on a current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>I</mi><mrow><mover><mrow><mi>l</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math>.</p>
<p>The&nbsp;current&nbsp;flows perpendicular to the magnetic field, so:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>F</mi><mrow><mi>l</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>IB</mi></mrow></mstyle></math>.&nbsp; Using the right-hand rule, we get the direction of the force, thus:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mrow><mi>l</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><mn>2</mn><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac><mover><mrow><mi>x</mi></mrow><mi>&#x005e;</mi></mover></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>.</p>@
qu.10.1.algorithm=$var=switch($rint(5),'a','b','c','d');
$mag=range(1,9);
$cur=range(1,9);
$cur2=range(1,9);@
qu.10.1.uid=5f648921-ea06-4857-b7c8-13aaeb622922@
qu.10.1.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Force Between Current Carrying Wires;
  Difficulty=Hard;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Partial Grading;
@

qu.10.2.question=<p>The wires <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>&nbsp;are carrying a current <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;is carrying current <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$cur2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>, each in the&nbsp;<br />
directions indicated in the diagram.&nbsp; In&nbsp;terms of the given parameters, provide a vector&nbsp;expression for&nbsp;<br />
the&nbsp;force per unit length exerted by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>&nbsp;on <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>, for&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;placed at&nbsp;any point on the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></mstyle></math>axis.</p>
<p>&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="500" height="505">
<param name="image" value="__BASE_URI__img/MagneticFields/CurrentCarryingWiresVertical/Diagram3Wires4-2.png" />
<param name="size" value="3" />
<param name="label.1.x" value="232" />
<param name="label.1.y" value="215" />
<param name="label.1.text" value="$mag$var" />
<param name="label.2.x" value="232" />
<param name="label.2.y" value="315" />
<param name="label.2.text" value="$mag$var" />
<param name="label.3.x" value="315" />
<param name="label.3.y" value="257" />
<param name="label.3.text" value="x" /></applet></p>
<p>&nbsp;</p>
<p><em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>xhat</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>yhat</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>z</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>zhat</mi></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
Enter your response in terms of the given parameters and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>, entering&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi></mrow></mstyle></math> instead of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>.</p>@
qu.10.2.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents);@
qu.10.2.allow2d=0@
qu.10.2.maple_answer=+xhat*($cur2*I)*2*(u*$cur*I/(2*Pi*sqrt(x^2+($mag*$var)^2)))*((x)/sqrt(x^2+($mag*$var)^2));@
qu.10.2.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.10.2.type=maple@
qu.10.2.mode=Maple@
qu.10.2.name=Current Carrying Wires - Vertical - 3 Wires - 4-2@
qu.10.2.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.10.2.editing=useHTML@
qu.10.2.solution=<p>We will calculate the magnetic field due to wires <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>, and then the force due to this magnetic field on <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>.</p>
<p>The magnitude of the magnetic field due to an infinite current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>&pi;r</mi></mrow></mfrac></mrow></mstyle></math>.&nbsp; The direction of the magnetic field is found using the right-hand rule, where the thumb points in the direction of current flow and the curl of the fingers represents the direction of the magnetic field.&nbsp; If we draw a line from the wire to the point in question, then the magnetic field will always be perpendicular to that line.</p>
<p>&nbsp;</p>
<p>At point&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math> the magnitude of the magnetic field due to each of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math> and&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math> will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow></mfrac></mrow></mstyle></math>.</p>
<p>Based on the configuration, the horizontal components will cancel.&nbsp; Thus, we need only calculate the&nbsp;vertical components, which are:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>B</mi><mrow><mi>y</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>-&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>, and therefore the total magnetic field at&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><mn>2</mn><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac><mover><mrow><mi>y</mi></mrow><mi>&#x005e;</mi></mover></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>Finally, the force due to a magnetic field on a current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>I</mi><mrow><mover><mrow><mi>l</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math>.</p>
<p>The&nbsp;current&nbsp;flows perpendicular to the magnetic field, so:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>F</mi><mrow><mi>l</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>IB</mi></mrow></mstyle></math>.&nbsp; Using the right-hand rule, we get the direction of the force, thus:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mrow><mi>l</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mn>2</mn><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac><mover><mrow><mi>x</mi></mrow><mi>&#x005e;</mi></mover></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>.</p>@
qu.10.2.algorithm=$var=switch($rint(5),'a','b','c','d');
$mag=range(1,9);
$cur=range(1,9);
$cur2=range(1,9);@
qu.10.2.uid=329abc1f-36bd-4417-b75d-a77ca2dbbd3e@
qu.10.2.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Force Between Current Carrying Wires;
  Difficulty=Hard;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Partial Grading;
@

qu.10.3.question=<p>The wires <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>&nbsp;are both carrying a current <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>&nbsp;in the directions indicated in the diagram.&nbsp; In<br />
&nbsp;terms of the given parameters, provide an equation for the magnetic field at any point on the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></mstyle></math>axis.</p>
<p>&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="500" height="505">
<param name="image" value="__BASE_URI__img/MagneticFields/CurrentCarryingWiresVertical/Diagram2Wires3.png" />
<param name="size" value="3" />
<param name="label.1.x" value="232" />
<param name="label.1.y" value="215" />
<param name="label.1.text" value="$mag$var" />
<param name="label.2.x" value="232" />
<param name="label.2.y" value="315" />
<param name="label.2.text" value="$mag$var" />
<param name="label.3.x" value="315" />
<param name="label.3.y" value="257" />
<param name="label.3.text" value="x" /></applet></p>
<p>&nbsp;</p>
<p><em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>xhat</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>yhat</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>z</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>zhat</mi></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
Enter your response in terms of the given parameters and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>, entering&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi></mrow></mstyle></math> instead of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>.</p>@
qu.10.3.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents);@
qu.10.3.allow2d=0@
qu.10.3.maple_answer=xhat*2*(u*$cur*I/(2*Pi*sqrt(x^2+($mag*$var)^2)))*(($mag*$var)/sqrt(x^2+($mag*$var)^2));@
qu.10.3.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.10.3.type=maple@
qu.10.3.mode=Maple@
qu.10.3.name=Current Carrying Wires - Vertical - 2 Wires - 3@
qu.10.3.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.10.3.editing=useHTML@
qu.10.3.solution=<p>The magnitude of the magnetic field due to an infinite current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>&pi;r</mi></mrow></mfrac></mrow></mstyle></math>.&nbsp; The direction of the magnetic field is found using the right-hand rule, where the thumb points in the direction of current flow and the curl of the fingers represents the direction of the magnetic field.&nbsp; If we draw a line from the wire to the point in question, then the magnetic field will always be perpendicular to that line.</p>
<p>&nbsp;</p>
<p>At point&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math> the magnitude of the magnetic field due to each of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math> and&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math> will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow></mfrac></mrow></mstyle></math>.</p>
<p>Based on the configuration, the vertical components will cancel.&nbsp; Thus, we need only calculate the horizontal components, which are:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>B</mi><mrow><mi>x</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>, and therefore the total magnetic field at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math>&nbsp;will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mn>2</mn><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac><mover><mrow><mi>x</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math>.</p>@
qu.10.3.algorithm=$var=switch($rint(5),'a','b','c','d');
$mag=range(1,9);
$cur=range(1,9);@
qu.10.3.uid=d91792a8-0393-4ec9-9d16-6464fcf9b785@
qu.10.3.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Magnetic Field Due to Current Carrying Wires;
  Difficulty=Hard;
  Features=Algebraic;
  Features=Algorithmic;
  Features=Diagram;
@

qu.10.4.question=<p>The wires <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>&nbsp;are carrying a current <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;is carrying current <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$cur2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>, each in the&nbsp;<br />
directions indicated in the diagram.&nbsp; In&nbsp;terms of the given parameters, provide a vector&nbsp;expression for&nbsp;<br />
the&nbsp;force per unit length exerted by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>&nbsp;on <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>, for&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;placed at&nbsp;any point on the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></mstyle></math>axis.</p>
<p>&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="500" height="505">
<param name="image" value="__BASE_URI__img/MagneticFields/CurrentCarryingWiresVertical/Diagram3Wires4-1.png" />
<param name="size" value="3" />
<param name="label.1.x" value="232" />
<param name="label.1.y" value="215" />
<param name="label.1.text" value="$mag$var" />
<param name="label.2.x" value="232" />
<param name="label.2.y" value="315" />
<param name="label.2.text" value="$mag$var" />
<param name="label.3.x" value="315" />
<param name="label.3.y" value="257" />
<param name="label.3.text" value="x" /></applet></p>
<p>&nbsp;</p>
<p><em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>xhat</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>yhat</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>z</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>zhat</mi></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
Enter your response in terms of the given parameters and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>, entering&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi></mrow></mstyle></math> instead of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>.</p>@
qu.10.4.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents);@
qu.10.4.allow2d=0@
qu.10.4.maple_answer=(-xhat)*($cur2*I)*2*(u*$cur*I/(2*Pi*sqrt(x^2+($mag*$var)^2)))*((x)/sqrt(x^2+($mag*$var)^2));@
qu.10.4.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.10.4.type=maple@
qu.10.4.mode=Maple@
qu.10.4.name=Current Carrying Wires - Vertical - 3 Wires - 4-1@
qu.10.4.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.10.4.editing=useHTML@
qu.10.4.solution=<p>We will calculate the magnetic field due to wires <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>, and then the force due to this magnetic field on <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>.</p>
<p>The magnitude of the magnetic field due to an infinite current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>&pi;r</mi></mrow></mfrac></mrow></mstyle></math>.&nbsp; The direction of the magnetic field is found using the right-hand rule, where the thumb points in the direction of current flow and the curl of the fingers represents the direction of the magnetic field.&nbsp; If we draw a line from the wire to the point in question, then the magnetic field will always be perpendicular to that line.</p>
<p>&nbsp;</p>
<p>At point&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math> the magnitude of the magnetic field due to each of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math> and&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math> will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow></mfrac></mrow></mstyle></math>.</p>
<p>Based on the configuration, the horizontal components will cancel.&nbsp; Thus, we need only calculate the&nbsp;vertical components, which are:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>B</mi><mrow><mi>y</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>, and therefore the total magnetic field at&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mn>2</mn><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac><mover><mrow><mi>y</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>Finally, the force due to a magnetic field on a current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>I</mi><mrow><mover><mrow><mi>l</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math>.</p>
<p>The&nbsp;current&nbsp;flows perpendicular to the magnetic field, so:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>F</mi><mrow><mi>l</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>IB</mi></mrow></mstyle></math>.&nbsp; Using the right-hand rule, we get the direction of the force, thus:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mrow><mi>l</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mn>2</mn><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac><mover><mrow><mi>x</mi></mrow><mi>&#x005e;</mi></mover></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>.</p>@
qu.10.4.algorithm=$var=switch($rint(5),'a','b','c','d');
$mag=range(1,9);
$cur=range(1,9);
$cur2=range(1,9);@
qu.10.4.uid=b8b849a0-c781-4d00-9fa1-bd740af4272a@
qu.10.4.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Force Between Current Carrying Wires;
  Difficulty=Hard;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Partial Grading;
@

qu.10.5.question=<p>The wires <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>&nbsp;are both carrying a current <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>&nbsp;in the directions indicated in the diagram.&nbsp; In<br />
&nbsp;terms of the given parameters, provide an equation for the magnetic field at any point on the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></mstyle></math>axis.</p>
<p>&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="500" height="505">
<param name="image" value="__BASE_URI__img/MagneticFields/CurrentCarryingWiresVertical/Diagram2Wires2.png" />
<param name="size" value="3" />
<param name="label.1.x" value="232" />
<param name="label.1.y" value="215" />
<param name="label.1.text" value="$mag$var" />
<param name="label.2.x" value="232" />
<param name="label.2.y" value="315" />
<param name="label.2.text" value="$mag$var" />
<param name="label.3.x" value="315" />
<param name="label.3.y" value="257" />
<param name="label.3.text" value="x" /></applet></p>
<p>&nbsp;</p>
<p><em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>xhat</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>yhat</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>z</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>zhat</mi></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
Enter your response in terms of the given parameters and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>, entering&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi></mrow></mstyle></math> instead of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>.</p>@
qu.10.5.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents);@
qu.10.5.allow2d=0@
qu.10.5.maple_answer=yhat*2*(u*$cur*I/(2*Pi*sqrt(x^2+($mag*$var)^2)))*(x/sqrt(x^2+($mag*$var)^2));@
qu.10.5.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.10.5.type=maple@
qu.10.5.mode=Maple@
qu.10.5.name=Current Carrying Wires - Vertical - 2 Wires - 2@
qu.10.5.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.10.5.editing=useHTML@
qu.10.5.solution=<p>The magnitude of the magnetic field due to an infinite current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>&pi;r</mi></mrow></mfrac></mrow></mstyle></math>.&nbsp; The direction of the magnetic field is found using the right-hand rule, where the thumb points in the direction of current flow and the curl of the fingers represents the direction of the magnetic field.&nbsp; If we draw a line from the wire to the point in question, then the magnetic field will always be perpendicular to that line.</p>
<p>&nbsp;</p>
<p>At point&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math> the magnitude of the magnetic field due to each of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math> and&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math> will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow></mfrac></mrow></mstyle></math>.</p>
<p>Based on the configuration, the horizontal components will cancel.&nbsp; Thus, we need only calculate the vertical components, which are:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>B</mi><mrow><mi>y</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>, and therefore the total magnetic field at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math>&nbsp;will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mn>2</mn><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac></mrow><mrow><mover><mrow><mi>y</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math>.</p>@
qu.10.5.algorithm=$var=switch($rint(5),'a','b','c','d');
$mag=range(1,9);
$cur=range(1,9);@
qu.10.5.uid=e6033809-8d58-4ca0-b705-a030c12883ec@
qu.10.5.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Magnetic Field Due to Current Carrying Wires;
  Difficulty=Hard;
  Features=Algebraic;
  Features=Algorithmic;
  Features=Diagram;
@

qu.10.6.question=<p>Four current carrying wires are arranged as seen in the diagram, where the arrows indicate the direction of current flow.<br />
The wires are each separated by a distance <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$w</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mstyle></math>. Calculate the magnetic field vector at the center point, given the following information:<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>B</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$B</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>C</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$C</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi mathvariant='normal'>D</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$D</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mstyle></math>.</p>
<p><img alt="" align="middle" width="753" height="500" src="__BASE_URI__img/MagneticFields/CurrentCarryingWires-1/Diagram2.png" /></p>
<p>&nbsp;<em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>xhat</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>yhat</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>z</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>zhat</mi></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.</p>@
qu.10.6.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uCT,mCT=0.5,uVS,mVS=0.5,uSM,mSM=0.8,subPackage=vectorNumericComponents);@
qu.10.6.allow2d=0@
qu.10.6.maple_answer=((-$A-$B-$C+$D)*(4*Pi*10^(-7))*zhat*T)/(Pi*($w/1000))@
qu.10.6.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.10.6.type=maple@
qu.10.6.mode=Maple@
qu.10.6.name=Magnetic Fields Due to Current Carrying Wires - 2 - Numeric@
qu.10.6.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT,mCT=0.5,uVS,mVS=0.5,uSM,mSM=0.8,subPackage=vectorNumericComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.10.6.editing=useHTML@
qu.10.6.solution=<p>The magnitude of the magnetic field due to an infinite current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>&pi;r</mi></mrow></mfrac></mrow></mstyle></math>.&nbsp; The direction of the magnetic field is found using the right-hand rule, where the thumb points in the direction of current flow and the curl of the fingers represents the direction of the magnetic field.&nbsp; When we consider the&nbsp;wires from the side, at any point on the screen, the magnetic field will either be directly into or out of the screen.&nbsp;</p>
<p>At the center point, for each wire, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mi>$w</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mstyle></math>.&nbsp; We are given the current in each wire and its direction.</p>
<p>For each wire, we need then to identify whether their resultant magnetic field is into or out of the screen, then add each together.</p>@
qu.10.6.algorithm=$A=rand(1.00,9.99,3);
$B=rand(1.00,9.99,3);
$C=rand(1.00,9.99,3);
$D=rand(1.00,9.99,3);
$w=rand(1.00,9.99,3);@
qu.10.6.uid=5b974ca0-8030-402a-85f6-1d6fce91bf08@
qu.10.6.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Magnetic Fields Due to Current Carrying Wires;
  Difficulty=Medium;
  Features=Algorithmic;
  Features=Diagram;
  Features=Partial Grading;
@

qu.10.7.question=<p>Four current carrying wires are arranged as seen in the diagram, where the arrows indicate the direction of current flow.<br />
The wires are each separated by a distance <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>w</mi></mrow></mstyle></math>. Calculate the magnetic field vector at the center point, given the following information:<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>B</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$B</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>C</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$C</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi mathvariant='normal'>D</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$D</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>.</p>
<p><img alt="" align="middle" width="753" height="500" src="__BASE_URI__img/MagneticFields/CurrentCarryingWires-1/Diagram4.png" /></p>
<p>&nbsp;<em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>xhat</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>yhat</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>z</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>zhat</mi></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
Enter your response in terms of the given parameters and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>, entering <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi></mrow></mstyle></math>&nbsp;instead of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>.</p>@
qu.10.7.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT,mCT=0.5,uVS,mVS=0.5,uSM,mSM=0.8,subPackage=vectorAlgebraicComponents);@
qu.10.7.allow2d=0@
qu.10.7.maple_answer=(($A+$B-$C-$D)*I*u*zhat)/(Pi*w)@
qu.10.7.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.10.7.type=maple@
qu.10.7.mode=Maple@
qu.10.7.name=Magnetic Fields Due to Current Carrying Wires - 4@
qu.10.7.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT,mCT=0.5,uVS,mVS=0.5,uSM,mSM=0.8,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.10.7.editing=useHTML@
qu.10.7.solution=<p>The magnitude of the magnetic field due to an infinite current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>&pi;r</mi></mrow></mfrac></mrow></mstyle></math>.&nbsp; The direction of the magnetic field is found using the right-hand rule, where the thumb points in the direction of current flow and the curl of the fingers represents the direction of the magnetic field.&nbsp; When we consider the&nbsp;wires from the side, at any point on the screen, the magnetic field will either be directly into or out of the screen.&nbsp;</p>
<p>At the center point, for each wire, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>w</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mstyle></math>.&nbsp; We are given the current in each wire and its direction.</p>
<p>For each wire, we need then to identify whether their resultant magnetic field is into or out of the screen, then add each together.</p>@
qu.10.7.algorithm=$A=range(1,9);
$B=range(1,9);
$C=range(1,9);
$D=range(1,9);@
qu.10.7.uid=c7c96511-3def-4698-a392-90cbf3358dad@
qu.10.7.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Magnetic Fields Due to Current Carrying Wires;
  Difficulty=Medium;
  Features=Algorithmic;
  Features=Algebraic;
@

qu.10.8.question=<p>The wires <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>&nbsp;are both carrying a current <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>&nbsp;in the directions indicated in the diagram.&nbsp; In<br />
&nbsp;terms of the given parameters, provide an equation for the magnetic field at any point on the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></mstyle></math>axis.</p>
<p>&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="500" height="505">
<param name="image" value="__BASE_URI__img/MagneticFields/CurrentCarryingWiresVertical/Diagram2Wires1.png" />
<param name="size" value="3" />
<param name="label.1.x" value="232" />
<param name="label.1.y" value="215" />
<param name="label.1.text" value="$mag$var" />
<param name="label.2.x" value="232" />
<param name="label.2.y" value="315" />
<param name="label.2.text" value="$mag$var" />
<param name="label.3.x" value="315" />
<param name="label.3.y" value="257" />
<param name="label.3.text" value="x" /></applet></p>
<p>&nbsp;</p>
<p><em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>xhat</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>yhat</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>z</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>zhat</mi></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
Enter your response in terms of the given parameters and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>, entering&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi></mrow></mstyle></math> instead of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>.</p>@
qu.10.8.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents)@
qu.10.8.allow2d=0@
qu.10.8.maple_answer=(-xhat)*2*(u*$cur*I/(2*Pi*sqrt(x^2+($mag*$var)^2)))*(($mag*$var)/sqrt(x^2+($mag*$var)^2))@
qu.10.8.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.10.8.type=maple@
qu.10.8.mode=Maple@
qu.10.8.name=Current Carrying Wires - Vertical - 2 Wires - 1@
qu.10.8.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.10.8.editing=useHTML@
qu.10.8.solution=<p>The magnitude of the magnetic field due to an infinite current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>&pi;r</mi></mrow></mfrac></mrow></mstyle></math>.&nbsp; The direction of the magnetic field is found using the right-hand rule, where the thumb points in the direction of current flow and the curl of the fingers represents the direction of the magnetic field.&nbsp; If we draw a line from the wire to the point in question, then the magnetic field will always be perpendicular to that line.</p>
<p>&nbsp;</p>
<p>At point&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math> the magnitude of the magnetic field due to each of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math> and&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math> will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow></mfrac></mrow></mstyle></math>.</p>
<p>Based on the configuration, the vertical components will cancel.&nbsp; Thus, we need only calculate the horizontal components, which are:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>B</mi><mrow><mi>x</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>, and therefore the total magnetic field at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math>&nbsp;will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mn>2</mn><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac><mrow><mover><mrow><mi>x</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.</p>@
qu.10.8.algorithm=$var=switch($rint(5),'a','b','c','d');
$mag=range(1,9);
$cur=range(1,9);@
qu.10.8.uid=f3e52422-abc7-440b-b9a7-72cb76cb39ac@
qu.10.8.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Magnetic Field Due to Current Carrying Wires;
  Difficulty=Hard;
  Features=Algebraic;
  Features=Algorithmic;
  Features=Diagram;
@

qu.10.9.question=<p>The wires <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>&nbsp;are carrying a current <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;is carrying current <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$cur2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>, each in the&nbsp;<br />
directions indicated in the diagram.&nbsp; In&nbsp;terms of the given parameters, provide a vector&nbsp;expression for&nbsp;<br />
the&nbsp;force per unit length exerted by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>&nbsp;on <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>, for&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;placed at&nbsp;any point on the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></mstyle></math>axis.</p>
<p>&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="500" height="505">
<param name="image" value="__BASE_URI__img/MagneticFields/CurrentCarryingWiresVertical/Diagram3Wires2-1.png" />
<param name="size" value="3" />
<param name="label.1.x" value="232" />
<param name="label.1.y" value="215" />
<param name="label.1.text" value="$mag$var" />
<param name="label.2.x" value="232" />
<param name="label.2.y" value="315" />
<param name="label.2.text" value="$mag$var" />
<param name="label.3.x" value="315" />
<param name="label.3.y" value="257" />
<param name="label.3.text" value="x" /></applet></p>
<p>&nbsp;</p>
<p><em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>xhat</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>yhat</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>z</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>zhat</mi></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
Enter your response in terms of the given parameters and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>, entering&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi></mrow></mstyle></math> instead of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>.</p>@
qu.10.9.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents);@
qu.10.9.allow2d=0@
qu.10.9.maple_answer=+xhat*($cur2*I)*2*(u*$cur*I/(2*Pi*sqrt(x^2+($mag*$var)^2)))*((x)/sqrt(x^2+($mag*$var)^2));@
qu.10.9.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.10.9.type=maple@
qu.10.9.mode=Maple@
qu.10.9.name=Current Carrying Wires - Vertical - 3 Wires - 2-1@
qu.10.9.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.10.9.editing=useHTML@
qu.10.9.solution=<p>We will calculate the magnetic field due to wires <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>, and then the force due to this magnetic field on <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>.</p>
<p>The magnitude of the magnetic field due to an infinite current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>&pi;r</mi></mrow></mfrac></mrow></mstyle></math>.&nbsp; The direction of the magnetic field is found using the right-hand rule, where the thumb points in the direction of current flow and the curl of the fingers represents the direction of the magnetic field.&nbsp; If we draw a line from the wire to the point in question, then the magnetic field will always be perpendicular to that line.</p>
<p>&nbsp;</p>
<p>At point&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math> the magnitude of the magnetic field due to each of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math> and&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math> will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow></mfrac></mrow></mstyle></math>.</p>
<p>Based on the configuration, the horizontal components will cancel.&nbsp; Thus, we need only calculate the&nbsp;vertical components, which are:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>B</mi><mrow><mi>y</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>, and therefore the total magnetic field at&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mn>2</mn><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac><mover><mrow><mi>y</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>Finally, the force due to a magnetic field on a current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>I</mi><mrow><mover><mrow><mi>l</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math>.</p>
<p>The&nbsp;current&nbsp;flows perpendicular to the magnetic field, so:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>F</mi><mrow><mi>l</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>IB</mi></mrow></mstyle></math>.&nbsp; Using the right-hand rule, we get the direction of the force, thus:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mrow><mi>l</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mn>2</mn><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac><mover><mrow><mi>x</mi></mrow><mi>&#x005e;</mi></mover></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>.</p>@
qu.10.9.algorithm=$var=switch($rint(5),'a','b','c','d');
$mag=range(1,9);
$cur=range(1,9);
$cur2=range(1,9);@
qu.10.9.uid=051f6e21-6d12-41ff-a4c4-c8c295e0eb60@
qu.10.9.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Force Between Current Carrying Wires;
  Difficulty=Hard;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Partial Grading;
@

qu.10.10.question=<p>A region of space contains a magnetic field of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$B1</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$B2</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow><mrow><mi></mi></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow><mrow><mi>T</mi></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>.&nbsp; A current of&nbsp;<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mA</mi></mrow></mstyle></math>is travelling&nbsp;through a&nbsp;wire of displacement<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>l</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$l1</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$l2</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow><mrow><mi></mi></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow></mrow><mrow></mrow><mrow></mrow><mrow><mi>cm</mi></mrow></mstyle></math>&nbsp;in this region.&nbsp;<br />
What is the&nbsp;force on the wire due to the magnetic field?&nbsp;</p>
<p>&nbsp;</p>
<p><em>Entry Notes</em>:&nbsp;<br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ihat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>jhat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>&nbsp; and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>khat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp;<br />
Ensure that you explicitly enter the multiplication symbol * between terms.</p>@
qu.10.10.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uVS=false,uUN);@
qu.10.10.allow2d=0@
qu.10.10.maple_answer=with(Physics[Vectors]);
l:=(($l1)*_i+($l2)*_j)*10^(-2);
B:=(($B1)*_i+($B2)*_j);
temp:=($I*10^(-3))*(l &x B);
SigFigs[roundToSigFigs](Component(temp,1),3)*N*ihat+SigFigs[roundToSigFigs](Component(temp,2),3)*N*jhat+SigFigs[roundToSigFigs](Component(temp,3),3)*N*khat;@
qu.10.10.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.10.10.type=maple@
qu.10.10.mode=Maple@
qu.10.10.name=Magnetic Force On A Current Carrying Wire - 2D@
qu.10.10.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uVS=false,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.10.10.editing=useHTML@
qu.10.10.hint.1=Set up a determinant.&nbsp; The first row should be <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp; The second row should be the components of the first vector and the third row should be the components of the second vector.&nbsp; Evaluate the determinant.@
qu.10.10.solution=<p>The force due to a magnetic field on a current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>I</mi><mrow><mover><mrow><mi>l</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math>.</p>
<p align="left">We are given <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>l</mi></mrow><mi>&rarr;</mi></mover></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow></mstyle></math>.</p>@
qu.10.10.algorithm=$B1=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$B2=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$l1=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$l2=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$I=rand(1,9.99,3);@
qu.10.10.uid=76cdc495-8281-4b6f-ae5d-e914afcc5760@
qu.10.10.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Magnetic Force Due to Current Carrying Wire;
  Difficulty=Hard;
  Features=Algorithmic;
  Features=Partial Grading;
@

qu.10.11.question=<p>Four current carrying wires are arranged as seen in the diagram, where the arrows indicate the direction of current flow.<br />
The wires are each separated by a distance <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$w</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mstyle></math>. Calculate the magnetic field vector at the center point, given the following information:<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>B</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$B</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>C</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$C</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi mathvariant='normal'>D</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$D</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mstyle></math>.</p>
<p><img alt="" align="middle" width="753" height="500" src="__BASE_URI__img/MagneticFields/CurrentCarryingWires-1/Diagram4.png" /></p>
<p>&nbsp;<em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>xhat</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>yhat</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>z</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>zhat</mi></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.</p>@
qu.10.11.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uCT,mCT=0.5,uVS,mVS=0.5,uSM,mSM=0.8,subPackage=vectorNumericComponents);@
qu.10.11.allow2d=0@
qu.10.11.maple_answer=(($A+$B-$C-$D)*(4*Pi*10^(-7))*zhat*T)/(Pi*($w/1000))@
qu.10.11.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.10.11.type=maple@
qu.10.11.mode=Maple@
qu.10.11.name=Magnetic Fields Due to Current Carrying Wires - 4 - Numeric@
qu.10.11.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT,mCT=0.5,uVS,mVS=0.5,uSM,mSM=0.8,subPackage=vectorNumericComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.10.11.editing=useHTML@
qu.10.11.solution=<p>The magnitude of the magnetic field due to an infinite current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>&pi;r</mi></mrow></mfrac></mrow></mstyle></math>.&nbsp; The direction of the magnetic field is found using the right-hand rule, where the thumb points in the direction of current flow and the curl of the fingers represents the direction of the magnetic field.&nbsp; When we consider the&nbsp;wires from the side, at any point on the screen, the magnetic field will either be directly into or out of the screen.&nbsp;</p>
<p>At the center point, for each wire, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mi>$w</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mstyle></math>.&nbsp; We are given the current in each wire and its direction.</p>
<p>For each wire, we need then to identify whether their resultant magnetic field is into or out of the screen, then add each together.</p>@
qu.10.11.algorithm=$A=rand(1.00,9.99,3);
$B=rand(1.00,9.99,3);
$C=rand(1.00,9.99,3);
$D=rand(1.00,9.99,3);
$w=rand(1.00,9.99,3);@
qu.10.11.uid=c07c5ae1-1140-4025-9bbd-0acb4ea78c31@
qu.10.11.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Magnetic Fields Due to Current Carrying Wires;
  Difficulty=Medium;
  Features=Algorithmic;
  Features=Diagram;
  Features=Partial Grading;
@

qu.10.12.question=<p>The wires <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>&nbsp;are carrying a current <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;is carrying current <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$cur2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>, each in the&nbsp;<br />
directions indicated in the diagram.&nbsp; In&nbsp;terms of the given parameters, provide a vector&nbsp;expression for&nbsp;<br />
the&nbsp;force per unit length exerted by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>&nbsp;on <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>, for&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;placed at&nbsp;any point on the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></mstyle></math>axis.</p>
<p>&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="500" height="505">
<param name="image" value="__BASE_URI__img/MagneticFields/CurrentCarryingWiresVertical/Diagram3Wires3-2.png" />
<param name="size" value="3" />
<param name="label.1.x" value="232" />
<param name="label.1.y" value="215" />
<param name="label.1.text" value="$mag$var" />
<param name="label.2.x" value="232" />
<param name="label.2.y" value="315" />
<param name="label.2.text" value="$mag$var" />
<param name="label.3.x" value="315" />
<param name="label.3.y" value="257" />
<param name="label.3.text" value="x" /></applet></p>
<p>&nbsp;</p>
<p><em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>xhat</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>yhat</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>z</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>zhat</mi></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
Enter your response in terms of the given parameters and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>, entering&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi></mrow></mstyle></math> instead of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>.</p>@
qu.10.12.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents);@
qu.10.12.allow2d=0@
qu.10.12.maple_answer=+yhat*($cur2*I)*2*(u*$cur*I/(2*Pi*sqrt(x^2+($mag*$var)^2)))*(($mag*$var)/sqrt(x^2+($mag*$var)^2));@
qu.10.12.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.10.12.type=maple@
qu.10.12.mode=Maple@
qu.10.12.name=Current Carrying Wires - Vertical - 3 Wires - 3-2@
qu.10.12.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.10.12.editing=useHTML@
qu.10.12.solution=<p>We will calculate the magnetic field due to wires <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>, and then the force due to this magnetic field on <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>.</p>
<p>The magnitude of the magnetic field due to an infinite current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>&pi;r</mi></mrow></mfrac></mrow></mstyle></math>.&nbsp; The direction of the magnetic field is found using the right-hand rule, where the thumb points in the direction of current flow and the curl of the fingers represents the direction of the magnetic field.&nbsp; If we draw a line from the wire to the point in question, then the magnetic field will always be perpendicular to that line.</p>
<p>&nbsp;</p>
<p>At point&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math> the magnitude of the magnetic field due to each of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math> and&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math> will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow></mfrac></mrow></mstyle></math>.</p>
<p>Based on the configuration, the vertical components will cancel.&nbsp; Thus, we need only calculate the horizontal components, which are:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>B</mi><mrow><mi>x</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>, and therefore the total magnetic field at&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mn>2</mn><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac><mover><mrow><mi>x</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>Finally, the force due to a magnetic field on a current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>I</mi><mrow><mover><mrow><mi>l</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math>.</p>
<p>The&nbsp;current&nbsp;flows perpendicular to the magnetic field, so:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>F</mi><mrow><mi>l</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>IB</mi></mrow></mstyle></math>.&nbsp; Using the right-hand rule, we get the direction of the force, thus:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mrow><mi>l</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mn>2</mn><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac><mover><mrow><mi>y</mi></mrow><mi>&#x005e;</mi></mover></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>.</p>@
qu.10.12.algorithm=$var=switch($rint(5),'a','b','c','d');
$mag=range(1,9);
$cur=range(1,9);
$cur2=range(1,9);@
qu.10.12.uid=77d01e8d-0b3f-4e6c-b0a6-4a25ded4ab98@
qu.10.12.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Force Between Current Carrying Wires;
  Difficulty=Hard;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Partial Grading;
@

qu.10.13.question=<p>The wires <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>&nbsp;are both carrying a current <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>&nbsp;in the directions indicated in the diagram.&nbsp; In<br />
&nbsp;terms of the given parameters, provide an equation for the magnetic field at any point on the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></mstyle></math>axis.</p>
<p>&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="500" height="505">
<param name="image" value="__BASE_URI__img/MagneticFields/CurrentCarryingWiresVertical/Diagram2Wires4.png" />
<param name="size" value="3" />
<param name="label.1.x" value="232" />
<param name="label.1.y" value="215" />
<param name="label.1.text" value="$mag$var" />
<param name="label.2.x" value="232" />
<param name="label.2.y" value="315" />
<param name="label.2.text" value="$mag$var" />
<param name="label.3.x" value="315" />
<param name="label.3.y" value="257" />
<param name="label.3.text" value="x" /></applet></p>
<p>&nbsp;</p>
<p><em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>xhat</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>yhat</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>z</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>zhat</mi></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
Enter your response in terms of the given parameters and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>, entering&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi></mrow></mstyle></math> instead of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>.</p>@
qu.10.13.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents);@
qu.10.13.allow2d=0@
qu.10.13.maple_answer=(-yhat)*2*(u*$cur*I/(2*Pi*sqrt(x^2+($mag*$var)^2)))*(x/sqrt(x^2+($mag*$var)^2));@
qu.10.13.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.10.13.type=maple@
qu.10.13.mode=Maple@
qu.10.13.name=Current Carrying Wires - Vertical - 2 Wires - 4@
qu.10.13.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.10.13.editing=useHTML@
qu.10.13.solution=<p>The magnitude of the magnetic field due to an infinite current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>&pi;r</mi></mrow></mfrac></mrow></mstyle></math>.&nbsp; The direction of the magnetic field is found using the right-hand rule, where the thumb points in the direction of current flow and the curl of the fingers represents the direction of the magnetic field.&nbsp; If we draw a line from the wire to the point in question, then the magnetic field will always be perpendicular to that line.</p>
<p>&nbsp;</p>
<p>At point&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math> the magnitude of the magnetic field due to each of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math> and&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math> will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow></mfrac></mrow></mstyle></math>.</p>
<p>Based on the configuration, the horizontal components will cancel.&nbsp; Thus, we need only calculate the&nbsp;vertical&nbsp;components, which are:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>B</mi><mrow><mi>y</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>, and therefore the total magnetic field at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math>&nbsp;will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mn>2</mn><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac><mrow><mover><mrow><mi>y</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.</p>@
qu.10.13.algorithm=$var=switch($rint(5),'a','b','c','d');
$mag=range(1,9);
$cur=range(1,9);@
qu.10.13.uid=2b879887-2cb8-4bdd-adb1-0afa6ababeb8@
qu.10.13.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Magnetic Field Due to Current Carrying Wires;
  Difficulty=Hard;
  Features=Algebraic;
  Features=Algorithmic;
  Features=Diagram;
@

qu.10.14.question=<p>The wires <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>&nbsp;are carrying a current <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;is carrying current <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$cur2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>, each in the&nbsp;<br />
directions indicated in the diagram.&nbsp; In&nbsp;terms of the given parameters, provide a vector&nbsp;expression for&nbsp;<br />
the&nbsp;force per unit length exerted by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>&nbsp;on <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>, for&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;placed at&nbsp;any point on the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></mstyle></math>axis.</p>
<p>&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="500" height="505">
<param name="image" value="__BASE_URI__img/MagneticFields/CurrentCarryingWiresVertical/Diagram3Wires1-1.png" />
<param name="size" value="3" />
<param name="label.1.x" value="232" />
<param name="label.1.y" value="215" />
<param name="label.1.text" value="$mag$var" />
<param name="label.2.x" value="232" />
<param name="label.2.y" value="315" />
<param name="label.2.text" value="$mag$var" />
<param name="label.3.x" value="315" />
<param name="label.3.y" value="257" />
<param name="label.3.text" value="x" /></applet></p>
<p>&nbsp;</p>
<p><em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>xhat</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>yhat</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>z</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>zhat</mi></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
Enter your response in terms of the given parameters and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>, entering&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi></mrow></mstyle></math> instead of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>.</p>@
qu.10.14.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents);@
qu.10.14.allow2d=0@
qu.10.14.maple_answer=+yhat*($cur2*I)*2*(u*$cur*I/(2*Pi*sqrt(x^2+($mag*$var)^2)))*(($mag*$var)/sqrt(x^2+($mag*$var)^2));@
qu.10.14.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.10.14.type=maple@
qu.10.14.mode=Maple@
qu.10.14.name=Current Carrying Wires - Vertical - 3 Wires - 1-1@
qu.10.14.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.10.14.editing=useHTML@
qu.10.14.solution=<p>We will calculate the magnetic field due to wires <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>, and then the force due to this magnetic field on <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>.</p>
<p>The magnitude of the magnetic field due to an infinite current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>&pi;r</mi></mrow></mfrac></mrow></mstyle></math>.&nbsp; The direction of the magnetic field is found using the right-hand rule, where the thumb points in the direction of current flow and the curl of the fingers represents the direction of the magnetic field.&nbsp; If we draw a line from the wire to the point in question, then the magnetic field will always be perpendicular to that line.</p>
<p>&nbsp;</p>
<p>At point&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math> the magnitude of the magnetic field due to each of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math> and&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math> will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow></mfrac></mrow></mstyle></math>.</p>
<p>Based on the configuration, the vertical components will cancel.&nbsp; Thus, we need only calculate the horizontal components, which are:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>B</mi><mrow><mi>x</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>, and therefore the total magnetic field at&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mn>2</mn><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac><mrow><mover><mrow><mi>x</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>Finally, the force due to a magnetic field on a current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>I</mi><mrow><mover><mrow><mi>l</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math>.</p>
<p>The&nbsp;current&nbsp;flows perpendicular to the magnetic field, so:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>F</mi><mrow><mi>l</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>IB</mi></mrow></mstyle></math>.&nbsp; Using the right-hand rule, we get the direction of the force, thus:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mrow><mi>l</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mn>2</mn><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac><mover><mrow><mi>y</mi></mrow><mi>&#x005e;</mi></mover></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>.</p>@
qu.10.14.algorithm=$var=switch($rint(5),'a','b','c','d');
$mag=range(1,9);
$cur=range(1,9);
$cur2=range(1,9);@
qu.10.14.uid=89bb2d5a-b698-47d1-ab83-9e52f288dacd@
qu.10.14.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Force Between Current Carrying Wires;
  Difficulty=Hard;
  Features=Algebraic;
  Features=Algorithmic;
  Features=Diagram;
@

qu.10.15.question=<p>Four current carrying wires are arranged as seen in the diagram, where the arrows indicate the direction of current flow.<br />
The wires are each separated by a distance <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$w</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mstyle></math>. Calculate the magnetic field vector at the center point, given the following information:<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>B</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$B</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>C</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$C</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi mathvariant='normal'>D</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$D</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mstyle></math>.</p>
<p><img alt="" align="middle" width="753" height="500" src="__BASE_URI__img/MagneticFields/CurrentCarryingWires-1/Diagram5.png" /></p>
<p>&nbsp;<em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>xhat</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>yhat</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>z</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>zhat</mi></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.</p>@
qu.10.15.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uCT,mCT=0.5,uVS,mVS=0.5,uSM,mSM=0.8,subPackage=vectorNumericComponents);@
qu.10.15.allow2d=0@
qu.10.15.maple_answer=(($A+$B+$C+$D)*(4*Pi*10^(-7))*zhat*T)/(Pi*($w/1000))@
qu.10.15.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.10.15.type=maple@
qu.10.15.mode=Maple@
qu.10.15.name=Magnetic Fields Due to Current Carrying Wires - 5 - Numeric@
qu.10.15.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT,mCT=0.5,uVS,mVS=0.5,uSM,mSM=0.8,subPackage=vectorNumericComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.10.15.editing=useHTML@
qu.10.15.solution=<p>The magnitude of the magnetic field due to an infinite current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>&pi;r</mi></mrow></mfrac></mrow></mstyle></math>.&nbsp; The direction of the magnetic field is found using the right-hand rule, where the thumb points in the direction of current flow and the curl of the fingers represents the direction of the magnetic field.&nbsp; When we consider the&nbsp;wires from the side, at any point on the screen, the magnetic field will either be directly into or out of the screen.&nbsp;</p>
<p>At the center point, for each wire, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mi>$w</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mstyle></math>.&nbsp; We are given the current in each wire and its direction.</p>
<p>For each wire, we need then to identify whether their resultant magnetic field is into or out of the screen, then add each together.</p>@
qu.10.15.algorithm=$A=rand(1.00,9.99,3);
$B=rand(1.00,9.99,3);
$C=rand(1.00,9.99,3);
$D=rand(1.00,9.99,3);
$w=rand(1.00,9.99,3);@
qu.10.15.uid=b1ec392c-417c-4d7a-bb26-defbdeb6f9ad@
qu.10.15.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Magnetic Fields Due to Current Carrying Wires;
  Difficulty=Medium;
  Features=Algorithmic;
  Features=Diagram;
  Features=Partial Grading;
@

qu.10.16.question=<p>Four current carrying wires are arranged as seen in the diagram, where the arrows indicate the direction of current flow.<br />
The wires are each separated by a distance <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>w</mi></mrow></mstyle></math>. Calculate the magnetic field vector at the center point, given the following information:<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>B</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$B</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>C</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$C</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi mathvariant='normal'>D</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$D</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>.</p>
<p><img alt="" align="middle" width="753" height="500" src="__BASE_URI__img/MagneticFields/CurrentCarryingWires-1/Diagram2.png" /></p>
<p>&nbsp;<em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>xhat</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>yhat</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>z</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>zhat</mi></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
Enter your response in terms of the given parameters and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>, entering <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi></mrow></mstyle></math>&nbsp;instead of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>.</p>@
qu.10.16.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT,mCT=0.5,uVS,mVS=0.5,uSM,mSM=0.8,subPackage=vectorAlgebraicComponents);@
qu.10.16.allow2d=0@
qu.10.16.maple_answer=((-$A-$B-$C+$D)*I*u*zhat)/(Pi*w)@
qu.10.16.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.10.16.type=maple@
qu.10.16.mode=Maple@
qu.10.16.name=Magnetic Fields Due to Current Carrying Wires - 2@
qu.10.16.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT,mCT=0.5,uVS,mVS=0.5,uSM,mSM=0.8,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.10.16.editing=useHTML@
qu.10.16.solution=<p>The magnitude of the magnetic field due to an infinite current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>&pi;r</mi></mrow></mfrac></mrow></mstyle></math>.&nbsp; The direction of the magnetic field is found using the right-hand rule, where the thumb points in the direction of current flow and the curl of the fingers represents the direction of the magnetic field.&nbsp; When we consider the&nbsp;wires from the side, at any point on the screen, the magnetic field will either be directly into or out of the screen.&nbsp;</p>
<p>At the center point, for each wire, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>w</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mstyle></math>.&nbsp; We are given the current in each wire and its direction.</p>
<p>For each wire, we need then to identify whether their resultant magnetic field is into or out of the screen, then add each together.</p>@
qu.10.16.algorithm=$A=range(1,9);
$B=range(1,9);
$C=range(1,9);
$D=range(1,9);@
qu.10.16.uid=8d1ef18a-0f89-476d-8396-9a20f1fbe897@
qu.10.16.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Magnetic Fields Due to Current Carrying Wires;
  Difficulty=Medium;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Diagram;
  Features=Partial Grading;
@

qu.10.17.question=<p>Four current carrying wires are arranged as seen in the diagram, where the arrows indicate the direction of current flow.<br />
The wires are each separated by a distance <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>w</mi></mrow></mstyle></math>. Calculate the magnetic field vector at the center point, given the following information:<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>B</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$B</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>C</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$C</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi mathvariant='normal'>D</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$D</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>.</p>
<p><img alt="" align="middle" width="753" height="500" src="__BASE_URI__img/MagneticFields/CurrentCarryingWires-1/Diagram1.png" /></p>
<p>&nbsp;<em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>xhat</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>yhat</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>z</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>zhat</mi></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
Enter your response in terms of the given parameters and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>, entering <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi></mrow></mstyle></math>&nbsp;instead of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>.</p>@
qu.10.17.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT,mCT=0.5,uVS,mVS=0.5,uSM,mSM=0.8,subPackage=vectorAlgebraicComponents);@
qu.10.17.allow2d=0@
qu.10.17.maple_answer=((-$A+$B-$C+$D)*I*u*zhat)/(Pi*w)@
qu.10.17.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.10.17.type=maple@
qu.10.17.mode=Maple@
qu.10.17.name=Magnetic Fields Due to Current Carrying Wires - 1@
qu.10.17.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT,mCT=0.5,uVS,mVS=0.5,uSM,mSM=0.8,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.10.17.editing=useHTML@
qu.10.17.solution=<p>The magnitude of the magnetic field due to an infinite current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>&pi;r</mi></mrow></mfrac></mrow></mstyle></math>.&nbsp; The direction of the magnetic field is found using the right-hand rule, where the thumb points in the direction of current flow and the curl of the fingers represents the direction of the magnetic field.&nbsp; When we consider the&nbsp;wires from the side, at any point on the screen, the magnetic field will either be directly into or out of the screen.&nbsp;</p>
<p>At the center point, for each wire, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>w</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mstyle></math>.&nbsp; We are given the current in each wire and its direction.</p>
<p>For each wire, we need then to identify whether their resultant magnetic field is into or out of the screen, then add each together.</p>@
qu.10.17.algorithm=$A=range(1,9);
$B=range(1,9);
$C=range(1,9);
$D=range(1,9);@
qu.10.17.uid=d8c0fcd7-1a5d-4ce1-a058-5c50c2d3d1f8@
qu.10.17.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Magnetic Fields Due to Current Carrying Wires;
  Difficulty=Medium;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Diagram;
  Features=Partial Grading;
@

qu.10.18.question=<p>Four current carrying wires are arranged as seen in the diagram, where the arrows indicate the direction of current flow.<br />
The wires are each separated by a distance <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>w</mi></mrow></mstyle></math>. Calculate the magnetic field vector at the center point, given the following information:<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>B</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$B</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>C</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$C</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi mathvariant='normal'>D</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$D</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>.</p>
<p><img alt="" align="middle" width="753" height="500" src="__BASE_URI__img/MagneticFields/CurrentCarryingWires-1/Diagram5.png" /></p>
<p>&nbsp;<em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>xhat</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>yhat</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>z</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>zhat</mi></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
Enter your response in terms of the given parameters and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>, entering <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi></mrow></mstyle></math>&nbsp;instead of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>.</p>@
qu.10.18.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT,mCT=0.5,uVS,mVS=0.5,uSM,mSM=0.8,subPackage=vectorAlgebraicComponents);@
qu.10.18.allow2d=0@
qu.10.18.maple_answer=((+$A+$B+$C+$D)*I*u*zhat)/(Pi*w)@
qu.10.18.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.10.18.type=maple@
qu.10.18.mode=Maple@
qu.10.18.name=Magnetic Fields Due to Current Carrying Wires - 5@
qu.10.18.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT,mCT=0.5,uVS,mVS=0.5,uSM,mSM=0.8,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.10.18.editing=useHTML@
qu.10.18.solution=<p>The magnitude of the magnetic field due to an infinite current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>&pi;r</mi></mrow></mfrac></mrow></mstyle></math>.&nbsp; The direction of the magnetic field is found using the right-hand rule, where the thumb points in the direction of current flow and the curl of the fingers represents the direction of the magnetic field.&nbsp; When we consider the&nbsp;wires from the side, at any point on the screen, the magnetic field will either be directly into or out of the screen.&nbsp;</p>
<p>At the center point, for each wire, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>w</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mstyle></math>.&nbsp; We are given the current in each wire and its direction.</p>
<p>For each wire, we need then to identify whether their resultant magnetic field is into or out of the screen, then add each together.</p>@
qu.10.18.algorithm=$A=range(1,9);
$B=range(1,9);
$C=range(1,9);
$D=range(1,9);@
qu.10.18.uid=1f636b43-0f2d-42cb-b49e-50699ab1d54c@
qu.10.18.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Magnetic Fields Due to Current Carrying Wires;
  Difficulty=Medium;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Diagram;
  Features=Partial Grading;
@

qu.10.19.question=<p>A current of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mA</mi></mrow></mstyle></math>&nbsp;is travelling with through a wire of displacement <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mover><mi>l</mi><mi>&rarr;</mi></mover></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$l1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>cm</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mi>$ldir</mi><mi>&#x005e;</mi></mover></mrow></mstyle></math>.&nbsp; If the region&nbsp;<br />
has a uniform magnetic field of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>B</mi><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$B1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>T</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mover><mrow><mi>$Bdir</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>, what is the force on the wires?&nbsp;</p>
<p>&nbsp;</p>
<p><em>Entry Notes</em>:&nbsp;<br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ihat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>jhat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>&nbsp; and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>khat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp;<br />
Ensure that you explicitly enter the multiplication symbol * between terms.</p>@
qu.10.19.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uVS=false,uUN);@
qu.10.19.allow2d=0@
qu.10.19.maple_answer=with(Physics[Vectors]);
B:=($B1)*subs({i=_i,j=_j,k=_k},$Bdir);
l:=($l1*10^(-2))*subs({i=_i,j=_j,k=_k},$ldir);
temp:=($I*10^(-3))*(l &x B);
SigFigs[roundToSigFigs](Component(temp,1),3)*N*ihat+SigFigs[roundToSigFigs](Component(temp,2),3)*N*jhat+SigFigs[roundToSigFigs](Component(temp,3),3)*N*khat;@
qu.10.19.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.10.19.type=maple@
qu.10.19.mode=Maple@
qu.10.19.name=Magnetic Force On A Current Carrying Wire - 1D@
qu.10.19.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uVS=false,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.10.19.editing=useHTML@
qu.10.19.hint.1=Set up a determinant.&nbsp; The first row should be <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp; The second row should be the components of the first vector and the third row should be the components of the second vector.&nbsp; Evaluate the determinant.@
qu.10.19.solution=<p>The force due to a magnetic field on a current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>I</mi><mrow><mover><mrow><mi>l</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math>.</p>
<p align="left">We are given <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>l</mi></mrow><mi>&rarr;</mi></mover></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow></mstyle></math>.</p>@
qu.10.19.algorithm=$B1=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$l1=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$I=rand(1.00,9.99,3);
$Bdir=switch(rint(0,3),'i','j','k');
$ldir=switch(rint(0,3),'i','j','k');
$BDisp='


<math><mrow><mn>$B1</mn><mover><mi>$Bdir</mi><mo>&Hat;</mo></mover></mrow></math>';
$lDisp='


<math><mrow><mn>$l1</mn><mover><mi>$ldir</mi><mo>&Hat;</mo></mover></mrow></math>';@
qu.10.19.uid=9c279d8e-3388-4e13-bdf5-5b01f37d4358@
qu.10.19.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Magnetic Force On Current Carrying Wire;
  Difficulty=Hard;
  Features=Algorithmic;
  Features=Partial Grading;
@

qu.10.20.question=<p>Four current carrying wires are arranged as seen in the diagram, where the arrows indicate the direction of current flow.<br />
The wires are each separated by a distance <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>w</mi></mrow></mstyle></math>. Calculate the magnetic field vector at the center point, given the following information:<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>B</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$B</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>C</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$C</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi mathvariant='normal'>D</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$D</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>.</p>
<p><img alt="" align="middle" width="753" height="500" src="__BASE_URI__img/MagneticFields/CurrentCarryingWires-1/Diagram3.png" /></p>
<p>&nbsp;<em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>xhat</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>yhat</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>z</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>zhat</mi></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
Enter your response in terms of the given parameters and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>, entering <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi></mrow></mstyle></math>&nbsp;instead of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>.</p>@
qu.10.20.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT,mCT=0.5,uVS,mVS=0.5,uSM,mSM=0.8,subPackage=vectorAlgebraicComponents);@
qu.10.20.allow2d=0@
qu.10.20.maple_answer=(($A-$B-$C+$D)*I*u*zhat)/(Pi*w)@
qu.10.20.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.10.20.type=maple@
qu.10.20.mode=Maple@
qu.10.20.name=Magnetic Fields Due to Current Carrying Wires - 3@
qu.10.20.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT,mCT=0.5,uVS,mVS=0.5,uSM,mSM=0.8,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.10.20.editing=useHTML@
qu.10.20.solution=<p>The magnitude of the magnetic field due to an infinite current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>&pi;r</mi></mrow></mfrac></mrow></mstyle></math>.&nbsp; The direction of the magnetic field is found using the right-hand rule, where the thumb points in the direction of current flow and the curl of the fingers represents the direction of the magnetic field.&nbsp; When we consider the&nbsp;wires from the side, at any point on the screen, the magnetic field will either be directly into or out of the screen.&nbsp;</p>
<p>At the center point, for each wire, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>w</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mstyle></math>.&nbsp; We are given the current in each wire and its direction.</p>
<p>For each wire, we need then to identify whether their resultant magnetic field is into or out of the screen, then add each together.</p>@
qu.10.20.algorithm=$A=range(1,9);
$B=range(1,9);
$C=range(1,9);
$D=range(1,9);@
qu.10.20.uid=cad43adf-31b4-4c84-a9eb-c67d140817d5@
qu.10.20.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Magnetic Fields Due to Current Carrying Wires;
  Difficulty=Medium;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Diagram;
  Features=Partial Grading;
@

qu.10.21.question=<p>The wires <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>&nbsp;are carrying a current <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;is carrying current <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$cur2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>, each in the&nbsp;<br />
directions indicated in the diagram.&nbsp; In&nbsp;terms of the given parameters, provide a vector&nbsp;expression for&nbsp;<br />
the&nbsp;force per unit length exerted by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>&nbsp;on <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>, for&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;placed at&nbsp;any point on the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></mstyle></math>axis.</p>
<p>&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="500" height="505">
<param name="image" value="__BASE_URI__img/MagneticFields/CurrentCarryingWiresVertical/Diagram3Wires1-2.png" />
<param name="size" value="3" />
<param name="label.1.x" value="232" />
<param name="label.1.y" value="215" />
<param name="label.1.text" value="$mag$var" />
<param name="label.2.x" value="232" />
<param name="label.2.y" value="315" />
<param name="label.2.text" value="$mag$var" />
<param name="label.3.x" value="315" />
<param name="label.3.y" value="257" />
<param name="label.3.text" value="x" /></applet></p>
<p>&nbsp;</p>
<p><em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>xhat</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>yhat</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>z</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>zhat</mi></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
Enter your response in terms of the given parameters and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>, entering&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi></mrow></mstyle></math> instead of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>.</p>@
qu.10.21.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents);@
qu.10.21.allow2d=0@
qu.10.21.maple_answer=(-yhat)*($cur2*I)*2*(u*$cur*I/(2*Pi*sqrt(x^2+($mag*$var)^2)))*(($mag*$var)/sqrt(x^2+($mag*$var)^2));@
qu.10.21.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.10.21.type=maple@
qu.10.21.mode=Maple@
qu.10.21.name=Current Carrying Wires - Vertical - 3 Wires - 1-2@
qu.10.21.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.10.21.editing=useHTML@
qu.10.21.solution=<p>We will calculate the magnetic field due to wires <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>, and then the force due to this magnetic field on <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>.</p>
<p>The magnitude of the magnetic field due to an infinite current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>&pi;r</mi></mrow></mfrac></mrow></mstyle></math>.&nbsp; The direction of the magnetic field is found using the right-hand rule, where the thumb points in the direction of current flow and the curl of the fingers represents the direction of the magnetic field.&nbsp; If we draw a line from the wire to the point in question, then the magnetic field will always be perpendicular to that line.</p>
<p>&nbsp;</p>
<p>At point&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math> the magnitude of the magnetic field due to each of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math> and&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math> will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow></mfrac></mrow></mstyle></math>.</p>
<p>Based on the configuration, the vertical components will cancel.&nbsp; Thus, we need only calculate the horizontal components, which are:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>B</mi><mrow><mi>x</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>, and therefore the total magnetic field at&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mn>2</mn><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac><mrow><mover><mrow><mi>x</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>Finally, the force due to a magnetic field on a current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>I</mi><mrow><mover><mrow><mi>l</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math>.</p>
<p>The&nbsp;current&nbsp;flows perpendicular to the magnetic field, so:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>F</mi><mrow><mi>l</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>IB</mi></mrow></mstyle></math>.&nbsp; Using the right-hand rule, we get the direction of the force, thus:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mrow><mi>l</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mn>2</mn><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac><mover><mrow><mi>y</mi></mrow><mi>&#x005e;</mi></mover></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>.</p>@
qu.10.21.algorithm=$var=switch($rint(5),'a','b','c','d');
$mag=range(1,9);
$cur=range(1,9);
$cur2=range(1,9);@
qu.10.21.uid=1ca78ee2-6622-4092-b527-4a6bce3d82d1@
qu.10.21.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Force Between Current Carrying Wires;
  Difficulty=Hard;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Partial Grading;
@

qu.10.22.question=<p>Four current carrying wires are arranged as seen in the diagram, where the arrows indicate the direction of current flow.<br />
The wires are each separated by a distance <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$w</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mstyle></math>. Calculate the magnetic field vector at the center point, given the following information:<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>B</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$B</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>C</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$C</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi mathvariant='normal'>D</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$D</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mstyle></math>.</p>
<p><img alt="" align="middle" width="753" height="500" src="__BASE_URI__img/MagneticFields/CurrentCarryingWires-1/Diagram1.png" /></p>
<p>&nbsp;<em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>xhat</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>yhat</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>z</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>zhat</mi></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.</p>@
qu.10.22.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uCT,mCT=0.5,uVS,mVS=0.5,uSM,mSM=0.8,subPackage=vectorNumericComponents);@
qu.10.22.allow2d=0@
qu.10.22.maple_answer=((-$A+$B-$C+$D)*(4*Pi*10^(-7))*zhat*T)/(Pi*($w/1000))@
qu.10.22.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.10.22.type=maple@
qu.10.22.mode=Maple@
qu.10.22.name=Magnetic Fields Due to Current Carrying Wires - 1 - Numeric@
qu.10.22.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT,mCT=0.5,uVS,mVS=0.5,uSM,mSM=0.8,subPackage=vectorNumericComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.10.22.editing=useHTML@
qu.10.22.solution=<p>The magnitude of the magnetic field due to an infinite current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>&pi;r</mi></mrow></mfrac></mrow></mstyle></math>.&nbsp; The direction of the magnetic field is found using the right-hand rule, where the thumb points in the direction of current flow and the curl of the fingers represents the direction of the magnetic field.&nbsp; When we consider the&nbsp;wires from the side, at any point on the screen, the magnetic field will either be directly into or out of the screen.&nbsp;</p>
<p>At the center point, for each wire, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mi>$w</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mstyle></math>.&nbsp; We are given the current in each wire and its direction.</p>
<p>For each wire, we need then to identify whether their resultant magnetic field is into or out of the screen, then add each together.</p>@
qu.10.22.algorithm=$A=rand(1.00,9.99,3);
$B=rand(1.00,9.99,3);
$C=rand(1.00,9.99,3);
$D=rand(1.00,9.99,3);
$w=rand(1.00,9.99,3);@
qu.10.22.uid=a1619aa3-e351-4b10-813d-f33b76373887@
qu.10.22.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Magnetic Fields Due to Current Carrying Wires;
  Difficulty=Medium;
  Features=Algorithmic;
  Features=Diagram;
  Features=Partial Grading;
@

qu.10.23.question=<p>The wires <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>&nbsp;are carrying a current <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;is carrying current <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$cur2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>, each in the&nbsp;<br />
directions indicated in the diagram.&nbsp; In&nbsp;terms of the given parameters, provide a vector&nbsp;expression for&nbsp;<br />
the&nbsp;force per unit length exerted by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>&nbsp;on <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>, for&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;placed at&nbsp;any point on the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></mstyle></math>axis.</p>
<p>&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="500" height="505">
<param name="image" value="__BASE_URI__img/MagneticFields/CurrentCarryingWiresVertical/Diagram3Wires3-1.png" />
<param name="size" value="3" />
<param name="label.1.x" value="232" />
<param name="label.1.y" value="215" />
<param name="label.1.text" value="$mag$var" />
<param name="label.2.x" value="232" />
<param name="label.2.y" value="315" />
<param name="label.2.text" value="$mag$var" />
<param name="label.3.x" value="315" />
<param name="label.3.y" value="257" />
<param name="label.3.text" value="x" /></applet></p>
<p>&nbsp;</p>
<p><em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>xhat</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>yhat</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>z</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>zhat</mi></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
Enter your response in terms of the given parameters and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>, entering&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi></mrow></mstyle></math> instead of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>.</p>@
qu.10.23.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents);@
qu.10.23.allow2d=0@
qu.10.23.maple_answer=(-yhat)*($cur2*I)*2*(u*$cur*I/(2*Pi*sqrt(x^2+($mag*$var)^2)))*(($mag*$var)/sqrt(x^2+($mag*$var)^2));@
qu.10.23.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.10.23.type=maple@
qu.10.23.mode=Maple@
qu.10.23.name=Current Carrying Wires - Vertical - 3 Wires - 3-1@
qu.10.23.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.10.23.editing=useHTML@
qu.10.23.solution=<p>We will calculate the magnetic field due to wires <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>, and then the force due to this magnetic field on <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>.</p>
<p>The magnitude of the magnetic field due to an infinite current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>&pi;r</mi></mrow></mfrac></mrow></mstyle></math>.&nbsp; The direction of the magnetic field is found using the right-hand rule, where the thumb points in the direction of current flow and the curl of the fingers represents the direction of the magnetic field.&nbsp; If we draw a line from the wire to the point in question, then the magnetic field will always be perpendicular to that line.</p>
<p>&nbsp;</p>
<p>At point&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math> the magnitude of the magnetic field due to each of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math> and&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math> will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow></mfrac></mrow></mstyle></math>.</p>
<p>Based on the configuration, the vertical components will cancel.&nbsp; Thus, we need only calculate the horizontal components, which are:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>B</mi><mrow><mi>x</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>, and therefore the total magnetic field at&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mn>2</mn><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac><mover><mrow><mi>x</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>Finally, the force due to a magnetic field on a current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>I</mi><mrow><mover><mrow><mi>l</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math>.</p>
<p>The&nbsp;current&nbsp;flows perpendicular to the magnetic field, so:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>F</mi><mrow><mi>l</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>IB</mi></mrow></mstyle></math>.&nbsp; Using the right-hand rule, we get the direction of the force, thus:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mrow><mi>l</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><mn>2</mn><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac><mover><mrow><mi>y</mi></mrow><mi>&#x005e;</mi></mover></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>.</p>@
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qu.10.23.uid=188d8388-a987-4ebe-b067-02769fbdc869@
qu.10.23.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Force Between Current Carrying Wires;
  Difficulty=Hard;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Partial Grading;
@

qu.10.24.question=<p>What is the maximum force that a <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$B</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>T</mi></mrow></mstyle></math>&nbsp;magnetic field can cause on a wire of length <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$l</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>cm</mi></mrow></mstyle></math>&nbsp;that&nbsp;is&nbsp;carrying<br />
a current of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mstyle></math>?</p>@
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qu.10.24.name=Force on Current Carrying Wire - Max Force ~ PGc@
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qu.10.24.editing=useHTML@
qu.10.24.solution=<p>The force due to a magnetic field on a current carrying wire is given by:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mover><mi>F</mi><mi>&rarr;</mi></mover></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>I</mi><mrow><mover><mrow><mi>l</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math>.</p>
<p>This force is at a maximum when the direction of the current flow is perpendicular to the magnetic field, so that:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>IlB</mi></mrow></mstyle></math>.</p>
<p>We are given the current, magnetic field and the length of the wire.</p>@
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qu.10.24.uid=cbf24828-79c3-4f32-acba-250e2e4b3ad9@
qu.10.24.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Force on Current Carrying Wire in Magnetic Field;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Algorithmic;
@

qu.10.25.question=<p>Four current carrying wires are arranged as seen in the diagram, where the arrows indicate the direction of current flow.<br />
The wires are each separated by a distance <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$w</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mstyle></math>. Calculate the magnetic field vector at the center point, given the following information:<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>B</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$B</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>C</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$C</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi mathvariant='normal'>D</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$D</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mstyle></math>.</p>
<p><img alt="" align="middle" width="753" height="500" src="__BASE_URI__img/MagneticFields/CurrentCarryingWires-1/Diagram3.png" /></p>
<p>&nbsp;<em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>xhat</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>yhat</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>z</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>zhat</mi></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.</p>@
qu.10.25.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uCT,mCT=0.5,uVS,mVS=0.5,uSM,mSM=0.8,subPackage=vectorNumericComponents);@
qu.10.25.allow2d=0@
qu.10.25.maple_answer=(($A-$B-$C+$D)*(4*Pi*10^(-7))*zhat*T)/(Pi*($w/1000))@
qu.10.25.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.10.25.type=maple@
qu.10.25.mode=Maple@
qu.10.25.name=Magnetic Fields Due to Current Carrying Wires - 3 - Numeric@
qu.10.25.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT,mCT=0.5,uVS,mVS=0.5,uSM,mSM=0.8,subPackage=vectorNumericComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.10.25.editing=useHTML@
qu.10.25.solution=<p>The magnitude of the magnetic field due to an infinite current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>&pi;r</mi></mrow></mfrac></mrow></mstyle></math>.&nbsp; The direction of the magnetic field is found using the right-hand rule, where the thumb points in the direction of current flow and the curl of the fingers represents the direction of the magnetic field.&nbsp; When we consider the&nbsp;wires from the side, at any point on the screen, the magnetic field will either be directly into or out of the screen.&nbsp;</p>
<p>At the center point, for each wire, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mi>$w</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mstyle></math>.&nbsp; We are given the current in each wire and its direction.</p>
<p>For each wire, we need then to identify whether their resultant magnetic field is into or out of the screen, then add each together.</p>@
qu.10.25.algorithm=$A=rand(1.00,9.99,3);
$B=rand(1.00,9.99,3);
$C=rand(1.00,9.99,3);
$D=rand(1.00,9.99,3);
$w=rand(1.00,9.99,3);@
qu.10.25.uid=a77d3544-9fa6-44fa-8e21-1bcedef8ee8a@
qu.10.25.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Magnetic Fields Due to Current Carrying Wires;
  Difficulty=Medium;
  Features=Algorithmic;
  Features=Diagram;
  Features=Partial Grading;
@

qu.10.26.question=<p>A region of space contains a magnetic field of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$B1</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$B2</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$B3</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo></mrow><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow><mrow><mi>T</mi></mrow></mstyle></math>.&nbsp; A current of&nbsp;<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mA</mi></mrow></mstyle></math>is travelling&nbsp;through a&nbsp;wire of displacement<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>l</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$l1</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$l2</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$l3</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo></mrow><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mi mathvariant='normal'></mi></mrow><mrow><mi mathvariant='normal'></mi></mrow><mrow><mi mathvariant='normal'></mi></mrow><mrow><mi>cm</mi></mrow></mstyle></math>&nbsp;in this region.&nbsp;<br />
What is the&nbsp;force on the wire due to the magnetic field?&nbsp;</p>
<p>&nbsp;</p>
<p><em>Entry Notes</em>:&nbsp;<br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ihat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>jhat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>&nbsp; and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>khat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp;<br />
Ensure that you explicitly enter the multiplication symbol * between terms.</p>@
qu.10.26.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uVS=false,uUN);@
qu.10.26.allow2d=0@
qu.10.26.maple_answer=with(Physics[Vectors]);
l:=(($l1)*_i+($l2)*_j+($l3)*_k)*10^(-2);
B:=(($B1)*_i+($B2)*_j+($B3)*_k);
temp:=($I*10^(-3))*(l &x B);
SigFigs[roundToSigFigs](Component(temp,1),3)*N*ihat+SigFigs[roundToSigFigs](Component(temp,2),3)*N*jhat+SigFigs[roundToSigFigs](Component(temp,3),3)*N*khat;@
qu.10.26.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.10.26.type=maple@
qu.10.26.mode=Maple@
qu.10.26.name=Magnetic Force On A Current Carrying Wire - 3D@
qu.10.26.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uVS=false,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.10.26.editing=useHTML@
qu.10.26.hint.1=Set up a determinant.&nbsp; The first row should be <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp; The second row should be the components of the first vector and the third row should be the components of the second vector.&nbsp; Evaluate the determinant.@
qu.10.26.solution=<p>The force due to a magnetic field on a current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>I</mi><mrow><mover><mrow><mi>l</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math>.</p>
<p align="left">We are given <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>l</mi></mrow><mi>&rarr;</mi></mover></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow></mstyle></math>.</p>@
qu.10.26.algorithm=$B1=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$B2=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$B3=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$l1=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$l2=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$l3=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$I=rand(1,9.99,3);@
qu.10.26.uid=f4e9837c-810f-4a0c-93ed-87f117ef53e0@
qu.10.26.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Magnetic Force Due to Current Carrying Wire;
  Difficulty=Hard;
  Features=Algorithmic;
  Features=Partial Grading;
@

qu.11.topic=Charge in B-Field@

qu.11.1.question=<p>A velocity selector is set up as in the diagram with uniform electric and magnetic fields. If the electric <br />
field is of magnitude&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$E</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$Eex</mi></mrow></msup><mfrac><mi>V</mi><mrow><mi>m</mi></mrow></mfrac></mrow></mstyle></math> and&nbsp;a charged particle of speed <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$v</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$vex</mi></mrow></msup><mfrac><mrow><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mstyle></math>&nbsp;will pass<br />
through undeflected, what is the magnitude and direction of the magnetic field?</p>
<p><img alt="" align="middle" width="600" height="171" src="__BASE_URI__img/MagneticFields/VelocitySelector/DiagramNoB1.png" /></p>
<p>&nbsp;</p>
<p>&nbsp;</p>@
qu.11.1.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.11.1.allow2d=0@
qu.11.1.maple_answer=SigFigs[roundToSigFigs]($ans,3)*T*zhat@
qu.11.1.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.11.1.type=maple@
qu.11.1.mode=Maple@
qu.11.1.name=Velocity Selector - Find B - 1 ~ PGc@
qu.11.1.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.11.1.editing=useHTML@
qu.11.1.solution=<p>In a velocity selector, the forces due to a magnetic field and an electric field cancel eachother, for one velocity of charged particle:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>qE</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>qvB</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mi>v</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>E</mi><mrow><mi>B</mi></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>&nbsp;We are given the electric field and the velocity, so we can find the magnitude of the magnetic field.</p>
<p>Assume that the charge is positive and determine which direction the electric force will be - the magnetic force must be in the opposite direction.&nbsp; Still assuming the charge is positive, use <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>q</mi><mrow><mover><mrow><mi>v</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math>&nbsp;to determine the direction of the magnetic field.</p>
<p>&nbsp;</p>@
qu.11.1.algorithm=$E=rand(1.00,9.99,3);
$Eex=range(5,13);
$v=rand(1.00,9.99,3);
$vex=range(2,7);
$ans=($E*10^($Eex))/($v*10^($vex));@
qu.11.1.uid=8cdeaeb0-9e72-4b28-a95e-fcecdc21df06@
qu.11.1.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Motion of Charged Particles in Electromagnetic Fields;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.11.2.question=<p>A charged particle, moving at a speed&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$v</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$vex</mi></mrow></msup></mrow><mrow><mfrac><mi>m</mi><mrow><mi>s</mi></mrow></mfrac></mrow></mstyle></math>,&nbsp;is experiencing circular motion in a magnetic&nbsp;<br />
field of strength <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$B</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>T</mi></mrow></mstyle></math>.&nbsp; If the particle has a charge of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$q</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$qex</mi></mrow></msup><mi>C</mi></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>&nbsp;and mass <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$m</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$mex</mi></mrow></msup><mi>kg</mi></mrow></mstyle></math>,<br />
what is radius of its motion?</p>@
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qu.11.2.allow2d=0@
qu.11.2.maple_answer=SigFigs[roundToSigFigs]($R,3)*cm@
qu.11.2.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.11.2.type=maple@
qu.11.2.mode=Maple@
qu.11.2.name=Charged Particle in Magnetic Field - Find Radius  ~ PGc@
qu.11.2.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.11.2.editing=useHTML@
qu.11.2.solution=<p>If a charge is experiencing circular motion due solely to a magnetic field, then the magnitude of the magnetic field must equal the magnitude of the centripetal force:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='|' close='|' separators=','><mrow><mi>q</mi></mrow></mfenced><mi>vB</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><msup><mi>mv</mi><mrow><mn>2</mn></mrow></msup><mrow><mi>R</mi></mrow></mfrac></mrow></mstyle></math>.</p>
<p>We are given <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>q</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>v</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>m</mi></mrow></mstyle></math>.</p>@
qu.11.2.algorithm=$q=rand(1.00,9.99,3);
$qex=range(10,15);
$B=rand(0.1,0.999,3);
$v=rand(1.00,9.99,3);
$vex=range(3,5);
$m=rand(1.00,9.99,3);
$mex=range(10,15);
$R=(100*$m*10^(-$mex))*($v*10^(-$vex))/(($q*10^(-$qex))*$B);@
qu.11.2.uid=86a9ce7f-ba93-4c72-b59c-51d3ef75b1c6@
qu.11.2.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Charged Particle in a Magnetic Field;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Algorithmic;
@

qu.11.3.mode=Inline@
qu.11.3.name=Velocity Selector - Find Direction of E - 2@
qu.11.3.comment=@
qu.11.3.editing=useHTML@
qu.11.3.solution=<p>In a velocity selector, the forces due to a magnetic field and an electric field cancel eachother, for one velocity of charged particle:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>qE</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>qvB</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mi>v</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>E</mi><mrow><mi>B</mi></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>Assuming the charge is positive, use <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>q</mi><mrow><mover><mrow><mi>v</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math>&nbsp;to determine the direction of the magnetic force - the electric <em>force</em> must be in the opposite direction.&nbsp; Again, assume that the charge is positive and determine the direction of the electric <em>field.</em></p>
<p>&nbsp;</p>@
qu.11.3.algorithm=@
qu.11.3.uid=f67f93f7-ec2e-479b-9a66-dac7329e4675@
qu.11.3.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Motion of Charged Particles in Electromagnetic Fields;
  Difficulty=Easy;
  Features=Diagram;
@
qu.11.3.weighting=1@
qu.11.3.numbering=alpha@
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qu.11.3.part.1.answer.4=- y@
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qu.11.3.part.1.answer.2=- x@
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qu.11.3.part.1.question=(Unset)@
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qu.11.3.question=<p>A velocity selector is set up as in the diagram with uniform electric and magnetic fields.&nbsp; If the magnetic&nbsp;<br />field is directed as shown, in what direction must the electric field point so that a charged particle of a&nbsp;<br />particular velocity can pass through undeflected?</p><p><img alt="" align="middle" width="600" height="178" src="__BASE_URI__img/MagneticFields/VelocitySelector/DiagramNoE2.png" /></p><p>&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.11.4.question=<p>A velocity selector is set up as in the diagram with uniform electric and magnetic fields. If the electric <br />
field is of magnitude&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$E</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$Eex</mi></mrow></msup><mfrac><mi>V</mi><mrow><mi>m</mi></mrow></mfrac></mrow></mstyle></math> and&nbsp;a charged particle of speed <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$v</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$vex</mi></mrow></msup><mfrac><mrow><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mstyle></math>&nbsp;will pass<br />
through undeflected, what is the magnitude and direction of the magnetic field?</p>
<p><img alt="" align="middle" width="600" height="171" src="__BASE_URI__img/MagneticFields/VelocitySelector/DiagramNoB2.png" /></p>
<p>&nbsp;</p>@
qu.11.4.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.11.4.allow2d=0@
qu.11.4.maple_answer=SigFigs[roundToSigFigs]($ans,3)*T*(-zhat)@
qu.11.4.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.11.4.type=maple@
qu.11.4.mode=Maple@
qu.11.4.name=Velocity Selector - Find B - 2 ~ PGc@
qu.11.4.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.11.4.editing=useHTML@
qu.11.4.solution=<p>In a velocity selector, the forces due to a magnetic field and an electric field cancel eachother, for one velocity of charged particle:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>qE</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>qvB</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mi>v</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>E</mi><mrow><mi>B</mi></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>&nbsp;We are given the electric field and the velocity, so we can find the magnitude of the magnetic field.</p>
<p>Assume that the charge is positive and determine which direction the electric force will be - the magnetic force must be in the opposite direction.&nbsp; Still assuming the charge is positive, use <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>q</mi><mrow><mover><mrow><mi>v</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math>&nbsp;to determine the direction of the magnetic field.</p>
<p>&nbsp;</p>@
qu.11.4.algorithm=$E=rand(1.00,9.99,3);
$Eex=range(5,13);
$v=rand(1.00,9.99,3);
$vex=range(2,7);
$ans=($E*10^($Eex))/($v*10^($vex));@
qu.11.4.uid=1f45e28a-1bbc-47ad-a1f3-e5dfb9933a08@
qu.11.4.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Motion of Charged Particles in Electromagnetic Fields;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.11.5.question=<p>A charge of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$q</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>C</mi></mrow></mstyle></math> is moving with velocity&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>v</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$v1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$v2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>j</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$v3</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mfenced></mrow></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>m</mi></mrow><mrow><mo lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&sol;</mo><mi>s</mi></mrow><mrow><mi></mi></mrow></mstyle></math>&nbsp;through a magnetic&nbsp;<br />
field&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>B</mi><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$b1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mi>i</mi><mi>&#x005e;</mi></mover></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$b2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$b3</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mstyle></math>&nbsp;and an electric field <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$e1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mi>i</mi><mi>&#x005e;</mi></mover></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$e2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$e3</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mi>V</mi><mrow><mi>m</mi></mrow></mfrac></mrow></mfenced></mrow></mstyle></math>.&nbsp;<br />
What is the force acting on the charge due to the magnetic field?</p>
<p>&nbsp;</p>
<p><em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ihat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mi>i</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>jhat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>khat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.</p>@
qu.11.5.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT=false,uVS=false,uUN);@
qu.11.5.allow2d=0@
qu.11.5.maple_answer=with(Physics[Vectors]);
with(SigFigs);
v:=($v1)*_i+($v2)*_j+($v3)*_k;
B:=($b1)*_j+($b2)*_j+($b3)*_k;
E:=($e1)*_i+($e2)*_j+($e3)*_k;
temp:=(($q)*((v &x B)+E));
roundToSigFigs(Component(temp,1),3)*N*ihat+roundToSigFigs(Component(temp,2),3)*N*jhat+roundToSigFigs(Component(temp,3),3)*N*khat@
qu.11.5.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.11.5.type=maple@
qu.11.5.mode=Maple@
qu.11.5.name=Force on Charges in Electro-Magnetic Field - 3D@
qu.11.5.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT=false,uVS=false,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.11.5.editing=useHTML@
qu.11.5.hint.1=Use the right-hand rule to find the direction, then multiply the coefficients.@
qu.11.5.hint.2=Set up a determinant.&nbsp; The first row should be <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp; The second row should be the components of the first vector and the third row should be the components of the second vector.&nbsp; Evaluate the determinant.@
qu.11.5.solution=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>F</mi><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>q</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mover><mi>v</mi><mi>&rarr;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><mover><mi>B</mi><mi>&rarr;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$q</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>C</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo></mrow><mrow><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mfenced open='[' close=']' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mover><mi>i</mi><mi>&#x005e;</mi></mover></mrow></mtd><mtd><mrow><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow></mtd><mtd><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mi>$v1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mtd><mtd><mrow><mfrac><mrow><mi>$v2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mtd><mtd><mrow><mfrac><mrow><mi>$v3</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mtd></mtr><mtr><mtd><mrow><mi>$b1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mtd><mtd><mrow><mi>$b2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mtd><mtd><mrow><mi>$b3</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mtd></mtr></mtable></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mrow></mfenced></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$q</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>C</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$e1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mi>i</mi><mi>&#x005e;</mi></mover></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$e2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$e3</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mi>V</mi><mrow><mi>m</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$q</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>C</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$v2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$b3</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$v3</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$b2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$v1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$b3</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$v3</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$b1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$v1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$b2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$v2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$b1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mfenced></mrow></mfenced></mrow></mfenced></mrow><mrow><mi mathvariant='normal'></mi></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$q</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>C</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$e1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mi>i</mi><mi>&#x005e;</mi></mover></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$e2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$e3</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mi>V</mi><mrow><mi>m</mi></mrow></mfrac></mrow></mfenced></mrow></mstyle></math></p>@
qu.11.5.algorithm=$v1=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$v2=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$v3=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$b1=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$b2=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$b3=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$e1=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$e2=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$e3=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$q=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));@
qu.11.5.uid=30d65bf1-2b7d-4402-8723-63b2ff2c71b6@
qu.11.5.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Force on Moving Charge in Electro-Magnetic Field;
  Difficulty=Hard;
  Features=Algorithmic;
  Features=Partial Grading;
@

qu.11.6.question=<p>A charge of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$q</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>C</mi></mrow></mstyle></math> is moving with velocity&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>v</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$v1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$v2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow></mfenced></mrow></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>m</mi></mrow><mrow><mo lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&sol;</mo><mi>s</mi></mrow></mstyle></math>&nbsp;through a magnetic field&nbsp;<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>B</mi><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$b1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>T</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math>.&nbsp; What is the force acting on the charge due to the magnetic field?</p>
<p>&nbsp;</p>
<p><em>Entry Notes</em>:&nbsp;<br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ihat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>jhat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>&nbsp; and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>khat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp;<br />
Ensure that you explicitly enter the multiplication symbol * between terms.</p>@
qu.11.6.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT=false,uVS=false,uUN);@
qu.11.6.allow2d=0@
qu.11.6.maple_answer=with(Physics[Vectors]);
with(SigFigs);
v:=($v1)*_i+($v2)*_j;
B:=($b1)*_k;
temp:=(($q)*(v &x B));
roundToSigFigs(Component(temp,1),3)*N*ihat+roundToSigFigs(Component(temp,2),3)*N*jhat@
qu.11.6.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.11.6.type=maple@
qu.11.6.mode=Maple@
qu.11.6.name=Force on Charges in Magnetic Field - 2D@
qu.11.6.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT=false,uVS=false,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.11.6.editing=useHTML@
qu.11.6.hint.1=Use the right-hand rule to find the direction, then multiply the coefficients.@
qu.11.6.hint.2=Set up a determinant.&nbsp; The first row should be <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp; The second row should be the components of the first vector and the third row should be the components of the second vector.&nbsp; Evaluate the determinant.@
qu.11.6.solution=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>F</mi><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>q</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mrow><mover><mi>v</mi><mi>&rarr;</mi></mover></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><mrow><mover><mi>B</mi><mi>&rarr;</mi></mover></mrow></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$q</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>C</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo></mrow><mrow><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mfenced open='[' close=']' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mover><mi>i</mi><mi>&#x005e;</mi></mover></mrow></mtd><mtd><mrow><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow></mtd><mtd><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mi>$v1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mtd><mtd><mrow><mfrac><mrow><mi>$v2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mtd><mtd><mrow><mn>0</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mi>$b1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mtd></mtr></mtable></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mrow></mfenced></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$q</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>C</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mfenced open='(' close=')' separators=','><mrow><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$v2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$b1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>0</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mn>0</mn></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$v1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$b1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>0</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mn>0</mn></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$v1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mn>0</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><mi>$v2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mn>0</mn></mrow></mfenced></mrow></mfenced></mrow></mrow></mstyle></math></p>@
qu.11.6.algorithm=$v1=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$v2=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$b1=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$b2=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$q=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));@
qu.11.6.uid=8908f121-1365-449c-bfea-f1a383730a62@
qu.11.6.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Force on Moving Charge in Magnetic Field;
  Difficulty=Medium;
  Features=Algorithmic;
  Features=Partial Grading;
@

qu.11.7.question=<p>A velocity selector is set up as in the diagram with uniform electric and magnetic fields. If the electric <br />
field is of magnitude&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$E</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$Eex</mi></mrow></msup><mfrac><mi>V</mi><mrow><mi>m</mi></mrow></mfrac></mrow></mstyle></math> and the magnetic field is of magnitude <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$B</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$Bex</mi></mrow></msup><mi>T</mi></mrow></mstyle></math>, what <br />
speed of charged particle will pass through undeflected?</p>
<p><img alt="" align="middle" width="600" height="171" src="__BASE_URI__img/MagneticFields/VelocitySelector/Diagram2.png" /></p>
<p>&nbsp;</p>@
qu.11.7.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.11.7.allow2d=0@
qu.11.7.maple_answer=SigFigs[roundToSigFigs]($ans,3)*m/s@
qu.11.7.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.11.7.type=maple@
qu.11.7.mode=Maple@
qu.11.7.name=Velocity Selector - Find v - 2 ~ PGc@
qu.11.7.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.11.7.editing=useHTML@
qu.11.7.solution=<p>In a velocity selector, the forces due to a magnetic field and an electric field cancel eachother, for one velocity of charged particle:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>qE</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>qvB</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mi>v</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>E</mi><mrow><mi>B</mi></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>We are&nbsp;given the electric and magnetic fields.&nbsp;</p>@
qu.11.7.algorithm=$E=rand(1.00,9.99,3);
$Eex=range(5,13);
$v=range(10000,299790000);
$B=rand(1.00,9.99,3);
$Bex=int(log($v))-1;
$ans=($E*10^($Eex))/($B*10^($Bex));@
qu.11.7.uid=1d87bdc8-4a24-4df1-908d-977e61e451f4@
qu.11.7.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Motion of Charged Particles in Electromagnetic Fields;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.11.8.mode=Inline@
qu.11.8.name=Velocity Selector - Find Direction of E - 1@
qu.11.8.comment=@
qu.11.8.editing=useHTML@
qu.11.8.solution=<p>In a velocity selector, the forces due to a magnetic field and an electric field cancel eachother, for one velocity of charged particle:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>qE</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>qvB</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mi>v</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>E</mi><mrow><mi>B</mi></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>Assuming the charge is positive, use <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>q</mi><mrow><mover><mrow><mi>v</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math>&nbsp;to determine the direction of the magnetic force - the electric <em>force</em> must be in the opposite direction.&nbsp; Again, assume that the charge is positive and determine the direction of the electric <em>field.</em></p>
<p>&nbsp;</p>@
qu.11.8.algorithm=@
qu.11.8.uid=c3d4799e-c316-4722-a242-6d5970621ac3@
qu.11.8.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Motion of Charged Particles in Electromagnetic Fields;
  Difficulty=Easy;
  Features=Diagram;
@
qu.11.8.weighting=1@
qu.11.8.numbering=alpha@
qu.11.8.part.1.answer.6=- z@
qu.11.8.part.1.answer.5=+ z@
qu.11.8.part.1.answer.4=- y@
qu.11.8.part.1.editing=useHTML@
qu.11.8.part.1.answer.3=+ y@
qu.11.8.part.1.answer.2=- x@
qu.11.8.part.1.answer.1=+ x@
qu.11.8.part.1.credit.6=0.0@
qu.11.8.part.1.credit.5=0.0@
qu.11.8.part.1.question=(Unset)@
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qu.11.8.part.1.credit.3=0.0@
qu.11.8.part.1.mode=List@
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qu.11.8.part.1.grader=exact@
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qu.11.8.part.1.name=sro_id_1@
qu.11.8.part.1.display.permute=false@
qu.11.8.question=<p>A velocity selector is set up as in the diagram with uniform electric and magnetic fields.&nbsp; If the magnetic&nbsp;<br />field is directed as shown, in what direction must the electric field point so that a charged particle of a&nbsp;<br />particular velocity can pass through undeflected?</p><p><img alt="" align="middle" width="600" height="178" src="__BASE_URI__img/MagneticFields/VelocitySelector/DiagramNoE1.png" /></p><p>&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.11.9.question=<p>What is the maximum force that a <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$B</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>T</mi></mrow></mstyle></math>&nbsp;magnetic field can cause on a wire of length <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$l</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>cm</mi></mrow></mstyle></math>&nbsp;that&nbsp;is&nbsp;carrying<br />
a current of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mstyle></math>?</p>@
qu.11.9.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.11.9.allow2d=0@
qu.11.9.maple_answer=SigFigs[roundToSigFigs]($ans,3)*N@
qu.11.9.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.11.9.type=maple@
qu.11.9.mode=Maple@
qu.11.9.name=Force on Current Carrying Wire - Max Force ~ PGc@
qu.11.9.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.11.9.editing=useHTML@
qu.11.9.solution=<p>The force due to a magnetic field on a current carrying wire is given by:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mover><mi>F</mi><mi>&rarr;</mi></mover></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>I</mi><mrow><mover><mrow><mi>l</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math>.</p>
<p>This force is at a maximum when the direction of the current flow is perpendicular to the magnetic field, so that:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>IlB</mi></mrow></mstyle></math>.</p>
<p>We are given the current, magnetic field and the length of the wire.</p>@
qu.11.9.algorithm=$I=rand(1.00,9.99,3);
$B=rand(0.1,0.999,3);
$l=rand(2,7,3);
$ans=($l*10^(-2))*$B*$I;@
qu.11.9.uid=cbf24828-79c3-4f32-acba-250e2e4b3ad9@
qu.11.9.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Force on Current Carrying Wire in Magnetic Field;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Algorithmic;
@

qu.11.10.question=<p>A charge of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$q</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>C</mi></mrow></mstyle></math> is moving with velocity&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>v</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$v1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$v2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow></mfenced></mrow></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>m</mi></mrow><mrow><mo lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&sol;</mo><mi>s</mi></mrow></mstyle></math>&nbsp;through a magnetic field<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>B</mi><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$b1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>T</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math>&nbsp;and an electric field <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$e</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>V</mi></mrow><mrow><mi>m</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp; What is the force acting on the charge due to&nbsp;<br />
the magnetic field?</p>
<p>&nbsp;</p>
<p><em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ihat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>jhat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>khat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.</p>@
qu.11.10.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT=false,uVS=false,uUN);@
qu.11.10.allow2d=0@
qu.11.10.maple_answer=with(Physics[Vectors]);
with(SigFigs);
v:=($v1)*_i+($v2)*_j;
B:=($b1)*_k;
E:=($e)*_k;
temp:=(($q)*((v &x B)+E));
roundToSigFigs(Component(temp,1),3)*N*ihat+roundToSigFigs(Component(temp,2),3)*N*jhat+roundToSigFigs(Component(temp,3),3)*N*khat@
qu.11.10.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.11.10.type=maple@
qu.11.10.mode=Maple@
qu.11.10.name=Force on Charges in Electro-Magnetic Field - Helix@
qu.11.10.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT=false,uVS=false,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.11.10.editing=useHTML@
qu.11.10.hint.1=Use the right-hand rule to find the direction, then multiply the coefficients.@
qu.11.10.hint.2=Set up a determinant.&nbsp; The first row should be <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp; The second row should be the components of the first vector and the third row should be the components of the second vector.&nbsp; Evaluate the determinant.@
qu.11.10.solution=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>F</mi><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>q</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mover><mi>v</mi><mi>&rarr;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><mover><mi>B</mi><mi>&rarr;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mover><mi>E</mi><mi>&rarr;</mi></mover></mrow></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$q</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>C</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo></mrow><mrow><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mfenced open='[' close=']' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mover><mi>i</mi><mi>&#x005e;</mi></mover></mrow></mtd><mtd><mrow><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow></mtd><mtd><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mi>$v1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mtd><mtd><mrow><mfrac><mrow><mi>$v2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mtd><mtd><mrow><mn>0</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mi>$b1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mtd></mtr></mtable></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mrow></mfenced></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$q</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>C</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$e</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>V</mi></mrow><mrow><mi>m</mi></mrow></mfrac><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$q</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>C</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$v2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$b1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>0</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mn>0</mn></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$v1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$b1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>0</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mn>0</mn></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$v1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mn>0</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><mi>$v2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mn>0</mn></mrow></mfenced></mrow></mfenced></mrow><mrow></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$q</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>C</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$e</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>V</mi></mrow><mrow><mi>m</mi></mrow></mfrac><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'></mo></mrow></mstyle></math></p>@
qu.11.10.algorithm=$v1=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$v2=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$b1=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$b2=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$e=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$q=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));@
qu.11.10.uid=34044ffb-e074-4b22-b407-0e5a7c1075dc@
qu.11.10.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Force on Moving Charge in Electro-Magnetic Field;
  Difficulty=Medium;
  Features=Algorithmic;
  Features=Partial Grading;
@

qu.11.11.question=<p>A charged particle, moving at a speed&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$v</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$vex</mi></mrow></msup></mrow><mrow><mfrac><mi>m</mi><mrow><mi>s</mi></mrow></mfrac></mrow></mstyle></math>,&nbsp;is experiencing circular motion, with radius&nbsp;<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$R</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>cm</mi></mrow></mstyle></math>,&nbsp;in a magnetic&nbsp;field.&nbsp; If the particle has a charge of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$q</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$qex</mi></mrow></msup><mi>C</mi></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>&nbsp;and mass&nbsp;<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$m</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$mex</mi></mrow></msup><mi>kg</mi></mrow></mstyle></math>, what is magnitude of the magnetic field?</p>@
qu.11.11.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.11.11.allow2d=0@
qu.11.11.maple_answer=SigFigs[roundToSigFigs]($B,3)*T@
qu.11.11.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.11.11.type=maple@
qu.11.11.mode=Maple@
qu.11.11.name=Charged Particle in Magnetic Field - Find Field ~ PGc@
qu.11.11.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.11.11.editing=useHTML@
qu.11.11.solution=<p>If a charge is experiencing circular motion due solely to a magnetic field, then the magnitude of the magnetic field must equal the magnitude of the centripetal force:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='|' close='|' separators=','><mrow><mi>q</mi></mrow></mfenced><mi>vB</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><msup><mi>mv</mi><mrow><mn>2</mn></mrow></msup><mrow><mi>R</mi></mrow></mfrac></mrow></mstyle></math>.</p>
<p>We are given <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>q</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>v</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>R</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>m</mi></mrow></mstyle></math>.</p>@
qu.11.11.algorithm=$q=rand(1.00,9.99,3);
$qex=range(10,15);
$R=rand(0.1,0.999,3);
$v=rand(1.00,9.99,3);
$vex=range(3,5);
$m=rand(1.00,9.99,3);
$mex=range(10,15);
$B=($m*10^(-$mex))*($v*10^(-$vex))/(($q*10^(-$qex))*($R/100));@
qu.11.11.uid=4216f68b-8a0f-40a4-8361-a00371041509@
qu.11.11.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Charged Particle in a Magnetic Field;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Algorithmic;
@

qu.11.12.question=<p>A charged particle, moving at a speed&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$v</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$vex</mi></mrow></msup></mrow><mrow><mfrac><mi>m</mi><mrow><mi>s</mi></mrow></mfrac></mrow></mstyle></math>,&nbsp;is experiencing circular motion in a magnetic&nbsp;<br />
field of strength <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$B</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>T</mi></mrow></mstyle></math>.&nbsp; If the particle has a charge of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$q</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$qex</mi></mrow></msup><mi>C</mi></mrow></mstyle></math>&nbsp;and the radius of its motion is&nbsp;<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$R</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>cm</mi></mrow></mstyle></math>, what is the mass of the particle?</p>@
qu.11.12.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.11.12.allow2d=0@
qu.11.12.maple_answer=SigFigs[roundToSigFigs]($ans,3)*kg@
qu.11.12.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.11.12.type=maple@
qu.11.12.mode=Maple@
qu.11.12.name=Charged Particle in Magnetic Field - Find Mass ~ PGc@
qu.11.12.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.11.12.editing=useHTML@
qu.11.12.solution=<p>If a charge is experiencing circular motion due solely to a magnetic field, then the magnitude of the magnetic field must equal the magnitude of the centripetal force:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='|' close='|' separators=','><mrow><mi>q</mi></mrow></mfenced><mi>vB</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><msup><mi>mv</mi><mrow><mn>2</mn></mrow></msup><mrow><mi>R</mi></mrow></mfrac></mrow></mstyle></math>.</p>
<p>We are given <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>q</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>v</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>R</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>.</p>@
qu.11.12.algorithm=$R=rand(10.0,19.9,3);
$q=rand(1.00,9.99,3);
$qex=range(10,15);
$B=rand(0.1,0.999,3);
$v=rand(1.00,9.99,3);
$vex=range(3,5);
$ans=($R*10^(-2))*($q*10^(-$qex))*$B/($v*10^($vex));@
qu.11.12.uid=137c669f-1236-48bf-b178-0c3bc2b6efab@
qu.11.12.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Charged Particle in a Magnetic Field;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Algorithmic;
@

qu.11.13.question=<p>A velocity selector is set up as in the diagram with uniform electric and magnetic fields. If the magnetic<br />
field is of magnitude&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$B</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$Bex</mi></mrow></msup><mi>T</mi></mrow></mstyle></math> and&nbsp;a charged particle of speed <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$v</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$vex</mi></mrow></msup><mfrac><mrow><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mstyle></math>&nbsp;will pass<br />
through undeflected, what is the magnitude and direction of the magnetic field?</p>
<p><img alt="" align="middle" width="600" height="171" src="__BASE_URI__img/MagneticFields/VelocitySelector/DiagramNoE2.png" /></p>
<p>&nbsp;</p>@
qu.11.13.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.11.13.allow2d=0@
qu.11.13.maple_answer=SigFigs[roundToSigFigs]($ans,3)*(V/m)*(yhat)@
qu.11.13.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.11.13.type=maple@
qu.11.13.mode=Maple@
qu.11.13.name=Velocity Selector - Find E - 2 ~ PGc@
qu.11.13.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.11.13.editing=useHTML@
qu.11.13.solution=<p>In a velocity selector, the forces due to a magnetic field and an electric field cancel eachother, for one velocity of charged particle:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>qE</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>qvB</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mi>v</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>E</mi><mrow><mi>B</mi></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>We are given the velocity and the magnetic field, which we can use to calculate the magnitude of the electric field.</p>
<p>Assuming the charge is positive, use <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>q</mi><mrow><mover><mrow><mi>v</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math>&nbsp;to determine the direction of the magnetic force - the electric <em>force</em> must be in the opposite direction.&nbsp; Again, assume that the charge is positive and determine the direction of the electric <em>field.</em></p>
<p>&nbsp;</p>@
qu.11.13.algorithm=$B=rand(1.00,9.99,3);
$Bex=range(2,10);
$v=rand(1.00,9.99,3);
$vex=range(2,7);
$ans=($B*10^($Bex))*($v*10^($vex));@
qu.11.13.uid=bfdb0318-c090-45db-9615-338dfa4f12dc@
qu.11.13.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Motion of Charged Particles in Electromagnetic Fields;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.11.14.question=<p>A velocity selector is set up as in the diagram with uniform electric and magnetic fields. If the magnetic<br />
field is of magnitude&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$B</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$Bex</mi></mrow></msup><mi>T</mi></mrow></mstyle></math> and&nbsp;a charged particle of speed <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$v</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$vex</mi></mrow></msup><mfrac><mrow><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mstyle></math>&nbsp;will pass<br />
through undeflected, what is the magnitude and direction of the magnetic field?</p>
<p><img alt="" align="middle" width="600" height="171" src="__BASE_URI__img/MagneticFields/VelocitySelector/DiagramNoE1.png" /></p>
<p>&nbsp;</p>@
qu.11.14.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.11.14.allow2d=0@
qu.11.14.maple_answer=SigFigs[roundToSigFigs]($ans,3)*(V/m)*(-yhat)@
qu.11.14.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.11.14.type=maple@
qu.11.14.mode=Maple@
qu.11.14.name=Velocity Selector - Find E - 1 ~ PGc@
qu.11.14.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.11.14.editing=useHTML@
qu.11.14.solution=<p>In a velocity selector, the forces due to a magnetic field and an electric field cancel eachother, for one velocity of charged particle:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>qE</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>qvB</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mi>v</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>E</mi><mrow><mi>B</mi></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>We are given the magnetic field and the velocity to calculate the magnitude of the electric field.</p>
<p>Assuming the charge is positive, use <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>q</mi><mrow><mover><mrow><mi>v</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math>&nbsp;to determine the direction of the magnetic force - the electric <em>force</em> must be in the opposite direction.&nbsp; Again, assume that the charge is positive and determine the direction of the electric <em>field.</em></p>
<p>&nbsp;</p>@
qu.11.14.algorithm=$B=rand(1.00,9.99,3);
$Bex=range(2,10);
$v=rand(1.00,9.99,3);
$vex=range(2,7);
$ans=($B*10^($Bex))*($v*10^($vex));@
qu.11.14.uid=a1360cb3-9d93-40c4-92a1-66f4fbf2e764@
qu.11.14.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Motion of Charged Particles in Electromagnetic Fields;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.11.15.question=<p>A charge&nbsp;of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$q</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>C</mi></mrow></mstyle></math> is moving with velocity&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>v</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$v1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$v2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$v3</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mfenced></mrow></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>m</mi></mrow><mrow><mo lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&sol;</mo><mi>s</mi></mrow><mrow><mi></mi></mrow></mstyle></math>&nbsp;through<br />
a magnetic field&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>B</mi><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$b1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$b2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$b3</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mstyle></math>.<br />
&nbsp;<br />
<br />
What&nbsp;is the force acting on the charge due to the magnetic field?</p>
<p>&nbsp;</p>@
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qu.11.15.allow2d=0@
qu.11.15.maple_answer=with(Physics[Vectors]);
with(SigFigs);
v:=($v1)*_i+($v2)*_j+($v3)*_k;
B:=($b1)*_i+($b2)*_j+($b3)*_k;
temp:=(($q)*(v &x B));
roundToSigFigs(Component(temp,1),3)*N*ihat+roundToSigFigs(Component(temp,2),3)*N*jhat+roundToSigFigs(Component(temp,3),3)*N*khat@
qu.11.15.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.11.15.type=maple@
qu.11.15.mode=Maple@
qu.11.15.name=Force on Charges in Magnetic Field - 3D@
qu.11.15.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT=false,uVS=false,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.11.15.editing=useHTML@
qu.11.15.hint.1=Set up a determinant.&nbsp; The first row should be <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp; The second row should be the components of the first vector and the third row should be the components of the second vector.&nbsp; Evaluate the determinant.@
qu.11.15.solution=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mover><mi>F</mi><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>q</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mover><mi>v</mi><mi>&rarr;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><mrow><mover><mi>B</mi><mi>&rarr;</mi></mover></mrow></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$q</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>C</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo></mrow><mi></mi></mover></mrow><mrow><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mfenced open='[' close=']' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mover><mi>i</mi><mi>&#x005e;</mi></mover></mrow></mtd><mtd><mrow><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow></mtd><mtd><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mi>$v1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mtd><mtd><mrow><mfrac><mrow><mi>$v2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mtd><mtd><mrow><mfrac><mrow><mi>$v3</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mtd></mtr><mtr><mtd><mrow><mi>$b1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mtd><mtd><mrow><mi>$b2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mtd><mtd><mrow><mi>$b3</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mtd></mtr></mtable></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mrow></mfenced></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$q</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>C</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mfenced open='(' close=')' separators=','><mrow><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$v2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$b3</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$v3</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$b2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$v1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$b3</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$v3</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$b1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$v1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$b2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$v2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$b1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>T</mi></mrow></mfenced></mrow></mfenced></mrow></mfenced></mrow></mrow><mrow></mrow></mstyle></math></p>@
qu.11.15.algorithm=$v1=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$v2=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$v3=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$b1=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$b2=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$b3=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$q=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));@
qu.11.15.uid=452bd2b4-3866-40e9-835a-d74fd6aada4d@
qu.11.15.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Force on Moving Charge in Magnetic Field;
  Difficulty=Hard;
  Features=Algorithmic;
  Features=Partial Grading;
@

qu.11.16.question=<p>A velocity selector is set up as in the diagram with uniform electric and magnetic fields. If the electric <br />
field is of magnitude&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$E</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$Eex</mi></mrow></msup><mfrac><mi>V</mi><mrow><mi>m</mi></mrow></mfrac></mrow></mstyle></math> and the magnetic field is of magnitude <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$B</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$Bex</mi></mrow></msup><mi>T</mi></mrow></mstyle></math>, what <br />
speed of charged particle will pass through undeflected?</p>
<p><img alt="" align="middle" width="600" height="171" src="__BASE_URI__img/MagneticFields/VelocitySelector/Diagram1.png" /></p>
<p>&nbsp;</p>
<p><span>&nbsp;</span></p>@
qu.11.16.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.11.16.allow2d=0@
qu.11.16.maple_answer=SigFigs[roundToSigFigs]($ans,3)*m/s@
qu.11.16.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.11.16.type=maple@
qu.11.16.mode=Maple@
qu.11.16.name=Velocity Selector - Find v - 1 ~ PGc@
qu.11.16.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.11.16.editing=useHTML@
qu.11.16.solution=<p>In a velocity selector, the forces due to a magnetic field and an electric field cancel eachother, for one velocity of charged particle:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>qE</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>qvB</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mi>v</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>E</mi><mrow><mi>B</mi></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>We are given the electric and magnetic fields.</p>@
qu.11.16.algorithm=$E=rand(1.00,9.99,3);
$Eex=range(5,13);
$v=range(10000,299790000);
$B=rand(1.00,9.99,3);
$Bex=int(log($v))-1;
$ans=($E*10^($Eex))/($B*10^($Bex));@
qu.11.16.uid=b8b96b46-23ce-41d7-a62a-d0745fd50c56@
qu.11.16.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Motion of Charged Particles in Electromagnetic Fields;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.11.17.mode=Inline@
qu.11.17.name=Velocity Selector - Find Direction of B - 1@
qu.11.17.comment=@
qu.11.17.editing=useHTML@
qu.11.17.solution=<p>In a velocity selector, the forces due to a magnetic field and an electric field cancel eachother, for one velocity of charged particle:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>qE</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>qvB</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mi>v</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>E</mi><mrow><mi>B</mi></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>Assume that the charge is positive and determine which direction the electric force will be - the magnetic force must be in the opposite direction.&nbsp; Still assuming the charge is positive, use <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>q</mi><mrow><mover><mrow><mi>v</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math>&nbsp;to determine the direction of the magnetic field.</p>
<p>&nbsp;</p>@
qu.11.17.algorithm=@
qu.11.17.uid=21f1dbb0-5e96-4ca7-89a2-69e870d6a207@
qu.11.17.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Motion of Charged Particles in Electromagnetic Fields;
  Difficulty=Easy;
@
qu.11.17.weighting=1@
qu.11.17.numbering=alpha@
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qu.11.17.part.1.answer.5=+ z@
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qu.11.17.part.1.editing=useHTML@
qu.11.17.part.1.answer.3=+ y@
qu.11.17.part.1.answer.2=- x@
qu.11.17.part.1.answer.1=+ x@
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qu.11.17.part.1.question=(Unset)@
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qu.11.17.part.1.mode=List@
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qu.11.17.part.1.grader=exact@
qu.11.17.part.1.display=menu@
qu.11.17.part.1.name=sro_id_1@
qu.11.17.part.1.display.permute=false@
qu.11.17.question=<p>A velocity selector is set up as in the diagram with uniform electric and magnetic fields.&nbsp; If the magnetic&nbsp;<br />field is directed as shown, in what direction must the electric field point so that a charged particle of a&nbsp;<br />particular velocity can pass through undeflected?</p><p><img alt="" align="middle" width="600" height="178" src="__BASE_URI__img/MagneticFields/VelocitySelector/DiagramNoB1.png" /></p><p>&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.11.18.mode=Inline@
qu.11.18.name=Velocity Selector - Find Direction of B - 2@
qu.11.18.comment=@
qu.11.18.editing=useHTML@
qu.11.18.solution=<p>In a velocity selector, the forces due to a magnetic field and an electric field cancel eachother, for one velocity of charged particle:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>qE</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>qvB</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mi>v</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>E</mi><mrow><mi>B</mi></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>Assume that the charge is positive and determine which direction the electric force will be - the magnetic force must be in the opposite direction.&nbsp; Still assuming the charge is positive, use <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>q</mi><mrow><mover><mrow><mi>v</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math>&nbsp;to determine the direction of the magnetic field.</p>
<p>&nbsp;</p>@
qu.11.18.algorithm=@
qu.11.18.uid=702b1a2b-8b58-452b-97fd-062f7ead199a@
qu.11.18.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Motion of Charged Particles in Electromagnetic Fields;
  Difficulty=Easy;
  Features=Diagram;
@
qu.11.18.weighting=1@
qu.11.18.numbering=alpha@
qu.11.18.part.1.answer.6=- z@
qu.11.18.part.1.answer.5=+ z@
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qu.11.18.part.1.editing=useHTML@
qu.11.18.part.1.answer.3=+ y@
qu.11.18.part.1.answer.2=- x@
qu.11.18.part.1.answer.1=+ x@
qu.11.18.part.1.credit.6=1.0@
qu.11.18.part.1.credit.5=0.0@
qu.11.18.part.1.question=(Unset)@
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qu.11.18.part.1.mode=List@
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qu.11.18.part.1.grader=exact@
qu.11.18.part.1.display=menu@
qu.11.18.part.1.name=sro_id_1@
qu.11.18.part.1.display.permute=false@
qu.11.18.question=<p>A velocity selector is set up as in the diagram with uniform electric and magnetic fields.&nbsp; If the magnetic&nbsp;<br />field is directed as shown, in what direction must the electric field point so that a charged particle of a&nbsp;<br />particular velocity can pass through undeflected?</p><p><img alt="" align="middle" width="600" height="178" src="__BASE_URI__img/MagneticFields/VelocitySelector/DiagramNoB2.png" /></p><p>&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.12.topic=Resistance@

qu.12.1.question=<p><img alt="" align="middle" width="561" height="500" src="__BASE_URI__img/Resistors/ResistorsInSeriesAndParallel2/Diagram.png" /></p>
<p>&nbsp;</p>
<p>If <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>R1</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$R1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mi>&Omega;</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>R2</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$R2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mi>&Omega;</mi></mrow></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>R3</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$R3</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&Omega;</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>R4</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$R4</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mi>&Omega;</mi></mrow></mrow></mstyle></math>, what is the total<br />
resistance of the circuit between <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>?</p>@
qu.12.1.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN)@
qu.12.1.allow2d=0@
qu.12.1.maple_answer=SigFigs[roundToSigFigs](1/((1/($R1+$R2))+(1/$R3)+(1/$R4)),3)*ohms;@
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qu.12.1.mode=Maple@
qu.12.1.name=Resistors in Series and Parallel 2 - Numeric ~ PG@
qu.12.1.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.12.1.editing=useHTML@
qu.12.1.solution=<p>For the bottom branch, the equivalent resistance is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>R</mi><mrow><mi>eq</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>R</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>R</mi><mrow><mn>2</mn></mrow></msub></mrow></mstyle></math>.</p>
<p>Therefore,</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>R</mi><mrow><mi>AB</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><msub><mi>R</mi><mrow><mi>eq</mi></mrow></msub></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mn>1</mn><mrow><msub><mi>R</mi><mrow><mn>3</mn></mrow></msub></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn>1</mn><mrow><msub><mi>R</mi><mrow><mn>4</mn></mrow></msub></mrow></mfrac></mrow></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>.</p>@
qu.12.1.algorithm=$R1=rand(1.00,100,3);
$R2=rand(1.00,100,3);
$R3=rand(1.00,100,3);
$R4=rand(1.00,100,3);@
qu.12.1.uid=6d8596b0-e517-4afb-b5aa-ffdade6b8cac@
qu.12.1.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Resistors in Series and Parallel;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Diagram;
@

qu.12.2.question=<p><img alt="" align="middle" width="561" height="500" src="__BASE_URI__img/Resistors/ResistorsInSeriesAndParallel1/Diagram.png" /></p>
<p>&nbsp;</p>
<p>If <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>R1</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$R1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mi>&Omega;</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>R2</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$R2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mi>&Omega;</mi></mrow></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>R3</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$R3</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&Omega;</mi></mrow></mstyle></math>, what is the total resistance of the circuit between <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>?</p>@
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qu.12.2.allow2d=0@
qu.12.2.maple_answer=SigFigs[roundToSigFigs](1/((1/($R1+$R2))+(1/$R3)),3)*ohm;@
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qu.12.2.type=maple@
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qu.12.2.name=Resistors in Series and Parallel 1 - Numeric ~ PG@
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qu.12.2.editing=useHTML@
qu.12.2.solution=<p>For the bottom branch, the equivalent resistance is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>R</mi><mrow><mi>eq</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>R</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>R</mi><mrow><mn>2</mn></mrow></msub></mrow></mstyle></math>.</p>
<p>Therefore,</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>R</mi><mrow><mi>AB</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><msub><mi>R</mi><mrow><mi>eq</mi></mrow></msub></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mn>1</mn><mrow><msub><mi>R</mi><mrow><mn>3</mn></mrow></msub></mrow></mfrac></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup></mrow></mstyle></math>.</p>@
qu.12.2.algorithm=$R1=rand(1.00,100,3);
$R2=rand(1.00,100,3);
$R3=rand(1.00,100,3);@
qu.12.2.uid=307e76f6-b994-4689-9fe1-2232fcc78b5a@
qu.12.2.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Resistors in Series and Parallel;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Diagram;
@

qu.12.3.question=<p align="left">A current of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mstyle></math>&nbsp;is passed through a&nbsp;gold wire&nbsp;of length <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$L</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mstyle></math>, resulting in a potential difference of<br />
&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$V</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mV</mi></mrow></mstyle></math>&nbsp;between the two ends of the wire.&nbsp; If these measurements are made at room temperature,&nbsp;<br />
what is the cross-sectional area of the wire?<br />
<br />
(The resistivity of gold at room temperature is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&rho;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn mathvariant='italic'>2.44</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>8</mn></mrow></msup><mi>&Omega;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow></mstyle></math>.)</p>@
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qu.12.3.maple_answer=SigFigs[roundToSigFigs]($ans,3)*(m^2)@
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qu.12.3.type=maple@
qu.12.3.mode=Maple@
qu.12.3.name=Resistivity - Find Area ~ PGc@
qu.12.3.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.12.3.editing=useHTML@
qu.12.3.solution=<p>Resistance is related to resistivity by:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>R</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>&rho;L</mi><mrow><mi>A</mi></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>We are given the current and the potential difference, thus we can find the resistance using Ohm's Law.</p>
<p>So,</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>&rho;LI</mi><mrow><mi>V</mi></mrow></mfrac></mrow></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$rho</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&Omega;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$L</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mfenced></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$V</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mV</mi></mrow></mfenced></mrow></mfrac></mrow></mstyle></math></p>@
qu.12.3.algorithm=$V=rand(1.00,9.99,3);
$L=rand(1.00,9.99,3);
$rho=2.44*10^(-8);
$I=rand(1.00,9.99,3);
$ans=$rho*($L*10^(-3))*$I/($V*10^(-3));@
qu.12.3.uid=755a60b1-bd03-4ebb-aa9f-91888ef27637@
qu.12.3.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Resistivity;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Algorithmic;
@

qu.12.4.question=<p align="left">A particular wire&nbsp;of length <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$L</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mstyle></math>&nbsp;and cross-sectional area <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi>mm</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>&nbsp;has resistivity <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$rho</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$rhoex</mi></mrow></msup><mi>&Omega;</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow></mstyle></math>.<br />
What potential&nbsp;difference must there be between the ends of the&nbsp;length of wire to cause<br />
a&nbsp;current&nbsp;of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mstyle></math>&nbsp;to pass through it?</p>@
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qu.12.4.maple_answer=SigFigs[roundToSigFigs]($ans,3)*V@
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qu.12.4.type=maple@
qu.12.4.mode=Maple@
qu.12.4.name=Resistivity - Find Voltage ~ PGc@
qu.12.4.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.12.4.editing=useHTML@
qu.12.4.solution=<p>Resistance is related to resistivity by:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>R</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>&rho;L</mi><mrow><mi>A</mi></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>We are given the&nbsp;current, thus we can&nbsp;replace the resistance using Ohm's Law to introduce potential difference.</p>
<p>So,</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>I</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo></mrow><mrow><mfrac><mrow><mi>&rho;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>L</mi></mrow><mrow><mi>A</mi></mrow></mfrac></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$rho</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$rhoex</mi></mrow></msup><mi>&Omega;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$L</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mfenced></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi>mm</mi><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>.</p>@
qu.12.4.algorithm=$L=rand(1.00,9.99,3);
$rho=rand(1.00,9.99,3);
$rhoex=range(6,9);
$I=rand(1.00,9.99,3);
$A=rand(0.100,0.999,3);
$ans=($rho*10^(-$rhoex))*($L*10^(-3))*$I/($A*10^(-6));@
qu.12.4.uid=79132c75-f72c-4019-8e97-ff633730a694@
qu.12.4.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Resistivity;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Algorithmic;
@

qu.12.5.question=<p align="left">A particular Ohmic wire&nbsp;of length <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$L</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mstyle></math>&nbsp;and cross-sectional area <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi>mm</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>&nbsp;has unknown resistivity.<br />
If a&nbsp;potential&nbsp;difference&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$V</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$Vex</mi></mrow></msup><mi>V</mi></mrow></mstyle></math>&nbsp;produces a current of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mstyle></math>, what is the resistivity of the wire?<br />
<br />
<em>Note:&nbsp; Enter <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ohm</mi></mrow></mstyle></math>&nbsp;for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&Omega;</mi></mrow></mstyle></math>.</em></p>@
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qu.12.5.editing=useHTML@
qu.12.5.solution=<p>Resistance is related to resistivity by:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>R</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>&rho;L</mi><mrow><mi>A</mi></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>We are given the&nbsp;potential difference and current thus we can&nbsp;replace the resistance using&nbsp;Ohm's Law</p>
<p>So,</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&rho;</mi></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>V</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>A</mi></mrow><mrow><mi>I</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>L</mi></mrow></mfrac></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mi>&rho;</mi></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$V</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$Vex</mi></mrow></msup><mi>V</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi>mm</mi><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$L</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>.</p>@
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$A=rand(0.100,0.999,3);
$ans=($A*10^(-6))*($V*10^(-$Vex))/(($L*10^(-3))*$I);@
qu.12.5.uid=5a725d92-e8d0-4330-bb2e-65352e1685ba@
qu.12.5.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Resistivity;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Algorithmic;
@

qu.12.6.question=<p><img alt="" align="middle" width="561" height="500" src="__BASE_URI__img/Resistors/ResistorsInSeriesAndParallel2/Diagram.png" /></p>
<p>&nbsp;</p>
<p>What is the total resistance of the circuit between <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>?</p>@
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qu.12.6.allow2d=0@
qu.12.6.maple_answer=1/((1/(R1+R2))+(1/R3)+(1/R4))@
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qu.12.6.editing=useHTML@
qu.12.6.solution=<p>For the bottom branch, the equivalent resistance is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>R</mi><mrow><mi>eq</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>R</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>R</mi><mrow><mn>2</mn></mrow></msub></mrow></mstyle></math>.</p>
<p>Therefore,</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>R</mi><mrow><mi>AB</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><msub><mi>R</mi><mrow><mi>eq</mi></mrow></msub></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mn>1</mn><mrow><msub><mi>R</mi><mrow><mn>3</mn></mrow></msub></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn>1</mn><mrow><msub><mi>R</mi><mrow><mn>4</mn></mrow></msub></mrow></mfrac></mrow></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup></mrow></mstyle></math>.</p>@
qu.12.6.algorithm=@
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qu.12.6.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Resistors in Series and Parallel;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Diagram;
  Features=Algebraic;
@

qu.12.7.question=<p><img alt="" align="middle" width="561" height="500" src="__BASE_URI__img/Resistors/ResistorsInSeriesAndParallel1/Diagram.png" /></p>
<p>&nbsp;</p>
<p>What is the total resistance of the circuit between <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>?</p>@
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qu.12.7.editing=useHTML@
qu.12.7.solution=<p>For the bottom branch, the equivalent resistance is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>R</mi><mrow><mi>eq</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>R</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>R</mi><mrow><mn>2</mn></mrow></msub></mrow></mstyle></math>.</p>
<p>Therefore,</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>R</mi><mrow><mi>AB</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><msub><mi>R</mi><mrow><mi>eq</mi></mrow></msub></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mn>1</mn><mrow><msub><mi>R</mi><mrow><mn>3</mn></mrow></msub></mrow></mfrac></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup></mrow></mstyle></math>.</p>@
qu.12.7.algorithm=@
qu.12.7.uid=87f5900a-fe18-44b8-afcc-f1ba0d519572@
qu.12.7.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Resistors in Series and Parallel;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Diagram;
  Features=Algebraic;
@

qu.12.8.question=<p align="left">A particular wire&nbsp;of length <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$L</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mstyle></math>&nbsp;and cross-sectional area <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi>mm</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>&nbsp;has resistivity <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$rho</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$rhoex</mi></mrow></msup><mi>&Omega;</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow></mstyle></math>.<br />
If a&nbsp;potential&nbsp;difference&nbsp;of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$V</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$Vex</mi></mrow></msup><mi>V</mi></mrow></mstyle></math>&nbsp;is applied across the wire, what is the resulting current?</p>@
qu.12.8.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.12.8.allow2d=0@
qu.12.8.maple_answer=SigFigs[roundToSigFigs]($ans,3)*A@
qu.12.8.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.12.8.type=maple@
qu.12.8.mode=Maple@
qu.12.8.name=Resistivity - Find Current ~ PGc@
qu.12.8.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.12.8.editing=useHTML@
qu.12.8.solution=<p>Resistance is related to resistivity by:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>R</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>&rho;L</mi><mrow><mi>A</mi></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>We are given the&nbsp;potential difference, thus we can&nbsp;replace the resistance using Ohm's Law to introduce current.</p>
<p>So,</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>V</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>A</mi></mrow><mrow><mi>&rho;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>L</mi></mrow></mfrac></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$V</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$Vex</mi></mrow></msup><mi>V</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi>mm</mi><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$rho</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$rhoex</mi></mrow></msup><mi>&Omega;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$L</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>.</p>@
qu.12.8.algorithm=$L=rand(1.00,9.99,3);
$rho=rand(1.00,9.99,3);
$rhoex=range(6,9);
$A=rand(0.100,0.999,3);
$V=rand(1.00,9.99,3);
$Vex=range(1,5);
$ans=($V*10^(-$Vex))*($A*10^(-6))/($rho*10^(-$rhoex))*($L*10^(-3));@
qu.12.8.uid=34c86567-d183-463e-972c-d4cb55cc6158@
qu.12.8.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Resistivity;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Algorithmic;
@

qu.13.topic=Potential Energy@

qu.13.1.question=<p>A charged particle of mass&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$m1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$m1ex</mi></mrow></msup><mi>kg</mi></mrow></mstyle></math> is travelling at&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$v</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$vex</mi></mrow></msup><mfrac><mi>m</mi><mrow><mi>s</mi></mrow></mfrac></mrow></mstyle></math> towards a second<br />
charged particle that is fixed in space. The fixed particle has a charge of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$q2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$q2ex</mi></mrow></msup><mi>C</mi></mrow></mstyle></math>.&nbsp; What&nbsp;<br />
charge must the travelling particle have if it comes to a stop <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$r</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mstyle></math> before it collides with the fixed particle?</p>
<p>&nbsp;</p>@
qu.13.1.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.13.1.allow2d=0@
qu.13.1.maple_answer=SigFigs[roundToSigFigs]($ans,3)*C@
qu.13.1.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.13.1.type=maple@
qu.13.1.mode=Maple@
qu.13.1.name=Electrical Potential Energy - Conservation - Find Charge ~ PGc@
qu.13.1.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.13.1.editing=useHTML@
qu.13.1.solution=<p>Since the Coulomb force changes based on separation, it is easiest to use conservation of energy instead of forces to solve this problem.</p>
<p>&nbsp;Assuming that the moving particle starts at infinity, it will have no electric potential energy between it and the stationary particle.&nbsp; Thus, initially we only have the kinetic energy of the moving particle.&nbsp; At the point of closest approach, the initially moving particle will temporarily have zero kinetic energy - there will only be electric potential energy.&nbsp; As a result, the conservation equation is:</p>
<p>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac><msub><mi>m</mi><mrow><mn>1</mn></mrow></msub><msubsup><mi>v</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>kq</mi><mrow><mn>1</mn></mrow></msub><msub><mi>q</mi><mrow><mn>2</mn></mrow></msub></mrow><mrow><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mstyle></math>.</p>
<p>Solving for the charge of the moving particle, we have:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>q</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac><mfenced open='(' close=')' separators=','><mrow><mi>$m1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$m1ex</mi></mrow></msup><mi>kg</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$v</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$vex</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$r</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><mi>k</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$q2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$q2ex</mi></mrow></msup><mi>C</mi></mrow></mfenced></mrow></mfrac></mrow></mfenced></mrow></mstyle></math>.</p>
<p>&nbsp;</p>@
qu.13.1.algorithm=$m1=rand(1.0,9.99,3);
$m1ex=range(19,25);
$v=rand(1.0,9.99,3);
$vex=range(4,7);
$idx=range(0,1);
$q2=switch($idx,rand(1.0,9.99,3),-rand(1.0,9.99,3));
$q2ex=range(10,15);
$r=rand(1.0,9.99,3);
$ans=($r*10^(-3))*($m1*10^(-$m1ex))*(($v*10^($vex))^2)/(2*($q2*10^(-$q2ex))*(8.98755*10^(9)));@
qu.13.1.uid=7e4e9ac3-7b6f-4b15-bb05-ae9f05f7b5dc@
qu.13.1.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Electrical Potential Energy;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
@

qu.13.2.question=<p>A charged particle of mass&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$m1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$m1ex</mi></mrow></msup><mi>kg</mi></mrow></mstyle></math> is travelling from infinity&nbsp;towards a second&nbsp;<br />
charged particle that is fixed in space. The moving particle has a charge of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$q2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$q2ex</mi></mrow></msup><mi>C</mi></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow></mrow></mstyle></math>, while&nbsp;<br />
the fixed particle has&nbsp;a charge of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$q1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$q1ex</mi></mrow></msup><mi>C</mi></mrow></mstyle></math>.&nbsp;&nbsp;If the particles come within&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$r</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mstyle></math> of&nbsp;<br />
eachother, what was the initial speed of the travelling particle?</p>
<p>&nbsp;</p>@
qu.13.2.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.13.2.allow2d=0@
qu.13.2.maple_answer=SigFigs[roundToSigFigs]($ans,3)*(m/s)@
qu.13.2.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.13.2.type=maple@
qu.13.2.mode=Maple@
qu.13.2.name=Electrical Potential Energy - Conservation - Find Speed ~ PGc@
qu.13.2.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.13.2.editing=useHTML@
qu.13.2.solution=<p>Since the Coulomb force changes based on separation, it is easiest to use conservation of energy instead of forces to solve this problem.</p>
<p>&nbsp;Assuming that the moving particle starts at infinity, it will have no electric potential energy between it and the stationary particle.&nbsp; Thus, initially we only have the kinetic energy of the moving particle.&nbsp; At the point of closest approach, the initially moving particle will temporarily have zero kinetic energy - there will only be electric potential energy.&nbsp; As a result, the conservation equation is:</p>
<p>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac><msub><mi>m</mi><mrow><mn>1</mn></mrow></msub><msubsup><mi>v</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>kq</mi><mrow><mn>1</mn></mrow></msub><msub><mi>q</mi><mrow><mn>2</mn></mrow></msub></mrow><mrow><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mstyle></math>.</p>
<p>Solving for the initial speed of the particle, we have:<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>v</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><msqrt><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>k</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$q1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$q1ex</mi></mrow></msup><mi>C</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$q2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$q2ex</mi></mrow></msup><mi>C</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>$r</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$m1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$m1ex</mi></mrow></msup><mi>kg</mi></mrow></mfenced></mrow></mfrac></mrow></mfenced></mrow></msqrt></mrow></mstyle></math>.</p>
<p>&nbsp;</p>@
qu.13.2.algorithm=$m1=rand(1.0,9.99,3);
$m1ex=range(19,25);
$idx=range(0,1);
$q1=switch($idx,rand(1.0,9.99,3),-rand(1.0,9.99,3));
$q1ex=range(10,15);
$q2=switch($idx,rand(1.0,9.99,3),-rand(1.0,9.99,3));
$q2ex=range(10,15);
$r=rand(1.0,9.99,3);
$ans=sqrt((2*($q2*10^(-$q2ex))*(8.98755*10^(9))*($q1*10^(-$q1ex)))/(($r*10^(-3))*($m1*10^(-$m1ex))));@
qu.13.2.uid=c40e2473-19a4-4af5-b7c5-d80d6237a02d@
qu.13.2.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Electrical Potential Energy;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
@

qu.13.3.question=<p>A charged particle of mass&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$m1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$m1ex</mi></mrow></msup><mi>kg</mi></mrow></mstyle></math> is travelling at&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$v</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$vex</mi></mrow></msup><mfrac><mi>m</mi><mrow><mi>s</mi></mrow></mfrac></mrow></mstyle></math> towards a second&nbsp;<br />
charged particle that is fixed in space. The moving particle has a charge of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$q2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$q2ex</mi></mrow></msup><mi>C</mi></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow></mrow></mstyle></math>.&nbsp;<br />
&nbsp;What charge must the fixed particle have if the particles come within&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$r</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mstyle></math> of eachother?</p>
<p>&nbsp;</p>@
qu.13.3.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.13.3.allow2d=0@
qu.13.3.maple_answer=SigFigs[roundToSigFigs]($ans,3)*C@
qu.13.3.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.13.3.type=maple@
qu.13.3.mode=Maple@
qu.13.3.name=Electrical Potential Energy - Conservation - Find Charge 2 ~ PGc@
qu.13.3.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.13.3.editing=useHTML@
qu.13.3.solution=<p>Since the Coulomb force changes based on separation, it is easiest to use conservation of energy instead of forces to solve this problem.</p>
<p>&nbsp;Assuming that the moving particle starts at infinity, it will have no electric potential energy between it and the stationary particle.&nbsp; Thus, initially we only have the kinetic energy of the moving particle.&nbsp; At the point of closest approach, the initially moving particle will temporarily have zero kinetic energy - there will only be electric potential energy.&nbsp; As a result, the conservation equation is:</p>
<p>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac><msub><mi>m</mi><mrow><mn>1</mn></mrow></msub><msubsup><mi>v</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>kq</mi><mrow><mn>1</mn></mrow></msub><msub><mi>q</mi><mrow><mn>2</mn></mrow></msub></mrow><mrow><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mstyle></math>.</p>
<p>Solving for the charge of the stationary particle, we have:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>q</mi><mrow><mn>2</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac><mfenced open='(' close=')' separators=','><mrow><mi>$m1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$m1ex</mi></mrow></msup><mi>kg</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$v</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$vex</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mi>m</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$r</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><mi>k</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$q2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$q2ex</mi></mrow></msup><mi>C</mi></mrow></mfenced></mrow></mfrac></mrow></mfenced></mrow></mstyle></math>.</p>
<p>&nbsp;</p>@
qu.13.3.algorithm=$m1=rand(1.0,9.99,3);
$m1ex=range(19,25);
$v=rand(1.0,9.99,3);
$vex=range(4,7);
$idx=range(0,1);
$q2=switch($idx,rand(1.0,9.99,3),-rand(1.0,9.99,3));
$q2ex=range(10,15);
$r=rand(1.0,9.99,3);
$ans=($r*10^(-3))*($m1*10^(-$m1ex))*(($v*10^($vex))^2)/(2*($q2*10^(-$q2ex))*(8.98755*10^(9)));@
qu.13.3.uid=9ac53f20-327e-4d9e-917e-3527a4bb5412@
qu.13.3.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Electrical Potential Energy;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
@

qu.14.topic=EM Waves@

qu.14.1.mode=Inline@
qu.14.1.name=Electromagnetic Wave Direction Identification - E@
qu.14.1.comment=@
qu.14.1.editing=useHTML@
qu.14.1.solution=<p>The axis of propagation is the direction variable inside the trigonometric function.&nbsp; If the signs of the direction and time terms are the same, then propagation is in the negative direction, otherwise it is positive.</p>
<p>Substitute the time and position values to find the direction of the first field.</p>
<p>Then, find the direction of the remaining field by:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>E</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&times;</mo><mover><mrow><mi>B</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math>&nbsp;is the propagation direction.</p>@
qu.14.1.algorithm=$idx=rint(2);
$idx2=rint(2);
$ESign=switch($idx,'+','-');
$EOpSign=switch($idx,'-','+');
$BSign=switch($idx2,'+','-');
$BOpSign=switch($idx2,'-','+');
$m=maple("with(Physics[Vectors]);
randomize();
EDir:=RandomTools[Generate](choose({'i','j','k'})):
BDir:=RandomTools[Generate](choose({'i','j','k'} minus {EDir})):
kDir:=RandomTools[Generate](choose({'i','j','k'} minus {EDir,BDir})):
EDirC:=(-1)^($idx)*cat('_',EDir):
BDirC:=(-1)^($idx2)*cat('_',BDir):
kDirC:=EDirC&x BDirC:
kidx:=sign(kDirC)+1:
subs({i=x,j=y,k=z},EDir),subs({i=x,j=y,k=z},BDir),subs({i=x,j=y,k=z},kDir),kidx;
");
$EDir=switch(0,$m);
$BDir=switch(1,$m);
$kDir=switch(2,$m);
$kidx=switch(3,$m);
$kSign=switch($kidx,'-','','+');
$kOpSign=switch($kidx,'+','','-');
$E=rand(1.00,9.99,3);
$Eex=range(3,7);
$w=rand(1.00,9.99,3);
$k=$w/2.9979;
$kex=range(3,5);
$wex=$kex+8;@
qu.14.1.uid=3812cf39-5ca8-401b-b812-181f860c3fa5@
qu.14.1.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Directions of Electromagnetic Wave Field Components;
  Difficulty=Medium;
  Features=Algorithmic;
@
qu.14.1.weighting=1,1,1@
qu.14.1.numbering=alpha@
qu.14.1.part.1.answer.6=-$BDir@
qu.14.1.part.1.answer.5=+$BDir@
qu.14.1.part.1.answer.4=-$EDir@
qu.14.1.part.1.editing=useHTML@
qu.14.1.part.1.answer.3=+$EDir@
qu.14.1.part.1.answer.2=$kOpSign$kDir@
qu.14.1.part.1.answer.1=$kSign$kDir@
qu.14.1.part.1.credit.6=0.0@
qu.14.1.part.1.credit.5=0.0@
qu.14.1.part.1.question=(Unset)@
qu.14.1.part.1.credit.4=0.0@
qu.14.1.part.1.credit.3=0.0@
qu.14.1.part.1.mode=List@
qu.14.1.part.1.credit.2=0.0@
qu.14.1.part.1.credit.1=1.0@
qu.14.1.part.1.grader=exact@
qu.14.1.part.1.display=menu@
qu.14.1.part.1.name=sro_id_1@
qu.14.1.part.1.display.permute=true@
qu.14.1.part.2.answer.6=-$kDir@
qu.14.1.part.2.answer.5=+$kDir@
qu.14.1.part.2.answer.4=-$EDir@
qu.14.1.part.2.editing=useHTML@
qu.14.1.part.2.answer.3=+$EDir@
qu.14.1.part.2.answer.2=$BOpSign$BDir@
qu.14.1.part.2.answer.1=$BSign$BDir@
qu.14.1.part.2.credit.6=0.0@
qu.14.1.part.2.credit.5=0.0@
qu.14.1.part.2.question=(Unset)@
qu.14.1.part.2.credit.4=0.0@
qu.14.1.part.2.credit.3=0.0@
qu.14.1.part.2.mode=List@
qu.14.1.part.2.credit.2=0.0@
qu.14.1.part.2.credit.1=1.0@
qu.14.1.part.2.grader=exact@
qu.14.1.part.2.display=menu@
qu.14.1.part.2.name=sro_id_2@
qu.14.1.part.2.display.permute=true@
qu.14.1.part.3.answer.6=-$kDir@
qu.14.1.part.3.answer.5=+$kDir@
qu.14.1.part.3.answer.4=-$BDir@
qu.14.1.part.3.editing=useHTML@
qu.14.1.part.3.answer.3=+$BDir@
qu.14.1.part.3.answer.2=$EOpSign$EDir@
qu.14.1.part.3.answer.1=$ESign$EDir@
qu.14.1.part.3.credit.6=0.0@
qu.14.1.part.3.credit.5=0.0@
qu.14.1.part.3.question=(Unset)@
qu.14.1.part.3.credit.4=0.0@
qu.14.1.part.3.credit.3=0.0@
qu.14.1.part.3.mode=List@
qu.14.1.part.3.credit.2=0.0@
qu.14.1.part.3.credit.1=1.0@
qu.14.1.part.3.grader=exact@
qu.14.1.part.3.display=menu@
qu.14.1.part.3.name=sro_id_3@
qu.14.1.part.3.display.permute=true@
qu.14.1.question=<p>Consider the following electromagnetic wave:<br /><br /><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>E</mi></mrow><mi>&rarr;</mi></mover><mfenced open='(' close=')' separators=','><mrow><mi>$kDir</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>t</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$ESign</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$E</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$Eex</mi></mrow></msup><mfrac><mi>V</mi><mrow><mi>m</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$k</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$kex</mi></mrow></msup><mfrac><mrow><mi>rad</mi></mrow><mrow><mi>m</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$kDir</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$kOpSign</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$w</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$wex</mi></mrow></msup><mfrac><mrow><mi>rad</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>t</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mrow><mi>$EDir</mi></mrow><mi>&#x005e;</mi></mover></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math></p><p>&nbsp;Identify the following:</p><p>&nbsp;</p><p><span><span><table border="0" cellspacing="1" cellpadding="1" width="90%">    <tbody>        <tr>            <td>Direction of propagation</td>            <td>&nbsp;<1><span> </span></td>        </tr>        <tr>            <td>Magnetic field direction at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$kDir</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mstyle></math></td>            <td><span>&nbsp;</span><2><span>&nbsp;</span></td>        </tr>        <tr>            <td>Electric field direction at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$kDir</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mstyle></math></td>            <td><span>&nbsp;</span><3><span>&nbsp;</span></td>        </tr>    </tbody></table></span></span></p>@

qu.14.2.question=<p>A particular electromagnetic wave has a maximum electric field amplitude of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$E</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$Eex</mi></mrow></msup><mfrac><mrow><mi>N</mi></mrow><mrow><mi>C</mi></mrow></mfrac></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>, what is&nbsp;<br />
the maximum magnetic field amplitude?</p>@
qu.14.2.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.14.2.allow2d=0@
qu.14.2.maple_answer=SigFigs[roundToSigFigs]($ans,3)*T@
qu.14.2.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.14.2.type=maple@
qu.14.2.mode=Maple@
qu.14.2.name=Electromagnetic Wave Amplitude - E to B ~ PGc@
qu.14.2.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.14.2.editing=useHTML@
qu.14.2.solution=<p>For electromagnetic radiation, the magnitudes of the amplitudes of the fields are related by:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>E</mi><mrow><mi>B</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>c</mi></mrow></mstyle></math>.</p>@
qu.14.2.algorithm=$E=rand(1.00,9.99,3);
$Eex=range(12,16);
$ans=($E*10^($Eex))/(3.00*10^(8));@
qu.14.2.uid=32416879-4efb-44d7-a070-e00f0881e084@
qu.14.2.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Electromagnetic Wave Magnitude;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Algorithmic;
@

qu.14.3.mode=Inline@
qu.14.3.name=Electromagnetic Wave Direction Identification - B - Dimensionless@
qu.14.3.comment=@
qu.14.3.editing=useHTML@
qu.14.3.solution=<p>The axis of propagation is the direction variable inside the trigonometric function.&nbsp; If the signs of the direction and time terms are the same, then propagation is in the negative direction, otherwise it is positive.</p>
<p>Substitute the time and position values to find the direction of the first field.</p>
<p>Then, find the direction of the remaining field by:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>E</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&times;</mo><mover><mrow><mi>B</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math>&nbsp;is the propagation direction.</p>@
qu.14.3.algorithm=$w=range(1,9);
$k=range(1,9);
$t=range(1,9);
$x=range(1,9);
$idx=rint(2);
$idx2=rint(2);
$EDispSign=switch($idx,'+','-');
$BSign=switch($idx2,'+','-');
$BOpSign=switch($idx2,'-','+');
$m=maple("with(Physics[Vectors]);
randomize();
EDir:=RandomTools[Generate](choose({'i','j','k'})):
BDir:=RandomTools[Generate](choose({'i','j','k'} minus {EDir})):
kDir:=RandomTools[Generate](choose({'i','j','k'} minus {EDir,BDir})):
EDirC:=(-1)^($idx)*cat('_',EDir):
BDirC:=(-1)^($idx2)*cat('_',BDir):
kDirC:=EDirC&x BDirC:
kidx:=sign(kDirC)+1:
n:=cos(($k*$x+(kidx)*$w*$t)*Pi);
subs({i=x,j=y,k=z},EDir),subs({i=x,j=y,k=z},BDir),subs({i=x,j=y,k=z},kDir),kidx,n;
");
$EDir=switch(0,$m);
$BDir=switch(1,$m);
$kDir=switch(2,$m);
$kidx=switch(3,$m);
$n=switch(4,$m);
$kSign=switch($kidx,'-','','+');
$kOpSign=switch($kidx,'+','','-');
$E=range(1,9);
$idx3=($n*switch($idx,+1,-1)+1);
$ESign=switch($idx3,'-',0,'+');
$EOpSign=switch($idx3,'+',0,'-');@
qu.14.3.uid=0a977197-9e60-4d44-bec2-7aa5a57125bf@
qu.14.3.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Directions of Electromagnetic Wave Field Components;
  Difficulty=Medium;
  Features=Algorithmic;
@
qu.14.3.weighting=1,1,1@
qu.14.3.numbering=alpha@
qu.14.3.part.1.answer.6=-$BDir@
qu.14.3.part.1.answer.5=+$BDir@
qu.14.3.part.1.answer.4=-$EDir@
qu.14.3.part.1.editing=useHTML@
qu.14.3.part.1.answer.3=+$EDir@
qu.14.3.part.1.answer.2=$kOpSign$kDir@
qu.14.3.part.1.answer.1=$kSign$kDir@
qu.14.3.part.1.credit.6=0.0@
qu.14.3.part.1.credit.5=0.0@
qu.14.3.part.1.question=(Unset)@
qu.14.3.part.1.credit.4=0.0@
qu.14.3.part.1.credit.3=0.0@
qu.14.3.part.1.mode=List@
qu.14.3.part.1.credit.2=0.0@
qu.14.3.part.1.credit.1=1.0@
qu.14.3.part.1.grader=exact@
qu.14.3.part.1.display=menu@
qu.14.3.part.1.name=sro_id_1@
qu.14.3.part.1.display.permute=true@
qu.14.3.part.2.answer.6=-$kDir@
qu.14.3.part.2.answer.5=+$kDir@
qu.14.3.part.2.answer.4=-$EDir@
qu.14.3.part.2.editing=useHTML@
qu.14.3.part.2.answer.3=+$EDir@
qu.14.3.part.2.answer.2=$BOpSign$BDir@
qu.14.3.part.2.answer.1=$BSign$BDir@
qu.14.3.part.2.credit.6=0.0@
qu.14.3.part.2.credit.5=0.0@
qu.14.3.part.2.question=(Unset)@
qu.14.3.part.2.credit.4=0.0@
qu.14.3.part.2.credit.3=0.0@
qu.14.3.part.2.mode=List@
qu.14.3.part.2.credit.2=0.0@
qu.14.3.part.2.credit.1=1.0@
qu.14.3.part.2.grader=exact@
qu.14.3.part.2.display=menu@
qu.14.3.part.2.name=sro_id_2@
qu.14.3.part.2.display.permute=true@
qu.14.3.part.3.answer.6=-$kDir@
qu.14.3.part.3.answer.5=+$kDir@
qu.14.3.part.3.answer.4=-$BDir@
qu.14.3.part.3.editing=useHTML@
qu.14.3.part.3.answer.3=+$BDir@
qu.14.3.part.3.answer.2=$EOpSign$EDir@
qu.14.3.part.3.answer.1=$ESign$EDir@
qu.14.3.part.3.credit.6=0.0@
qu.14.3.part.3.credit.5=0.0@
qu.14.3.part.3.question=(Unset)@
qu.14.3.part.3.credit.4=0.0@
qu.14.3.part.3.credit.3=0.0@
qu.14.3.part.3.mode=List@
qu.14.3.part.3.credit.2=0.0@
qu.14.3.part.3.credit.1=1.0@
qu.14.3.part.3.grader=exact@
qu.14.3.part.3.display=menu@
qu.14.3.part.3.name=sro_id_3@
qu.14.3.part.3.display.permute=true@
qu.14.3.question=<p align="left">Consider the following electromagnetic wave, in dimensionless units:</p><p align="center"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover><mfenced open='(' close=')' separators=','><mrow><mi>$kDir</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>t</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$EDispSign</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$E</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>$k</mi><mrow><mi>&pi;</mi></mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$kDir</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$kOpSign</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$w</mi><mrow><mi>&pi;</mi></mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>t</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mrow><mi>$BDir</mi></mrow><mi>&#x005e;</mi></mover></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow></mrow></mstyle></math></p><p>&nbsp;Identify the following:</p><p>&nbsp;</p><p><span><span><table border="0" cellspacing="1" cellpadding="1" width="100%">    <tbody>        <tr>            <td>Direction of propagation</td>            <td>&nbsp;<1><span> </span></td>        </tr>        <tr>            <td>Magnetic field direction at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$t</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$kDir</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$x</mi></mrow></mstyle></math></td>            <td><span>&nbsp;</span><2><span>&nbsp;</span></td>        </tr>        <tr>            <td>Electric field direction at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$t</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$kDir</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$x</mi></mrow></mstyle></math></td>            <td><span>&nbsp;</span><3><span>&nbsp;</span></td>        </tr>    </tbody></table></span></span></p>@

qu.14.4.mode=Inline@
qu.14.4.name=Electromagnetic Wave Direction Identification - B@
qu.14.4.comment=@
qu.14.4.editing=useHTML@
qu.14.4.solution=<p>The axis of propagation is the direction variable inside the trigonometric function.&nbsp; If the signs of the direction and time terms are the same, then propagation is in the negative direction, otherwise it is positive.</p>
<p>Substitute the time and position values to find the direction of the first field.</p>
<p>Then, find the direction of the remaining field by:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>E</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&times;</mo><mover><mrow><mi>B</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math>&nbsp;is the propagation direction.</p>@
qu.14.4.algorithm=$idx=rint(2);
$idx2=rint(2);
$ESign=switch($idx,'+','-');
$EOpSign=switch($idx,'-','+');
$BSign=switch($idx2,'+','-');
$BOpSign=switch($idx2,'-','+');
$m=maple("with(Physics[Vectors]);
randomize();
EDir:=RandomTools[Generate](choose({'i','j','k'})):
BDir:=RandomTools[Generate](choose({'i','j','k'} minus {EDir})):
kDir:=RandomTools[Generate](choose({'i','j','k'} minus {EDir,BDir})):
EDirC:=(-1)^($idx)*cat('_',EDir):
BDirC:=(-1)^($idx2)*cat('_',BDir):
kDirC:=EDirC&x BDirC:
kidx:=sign(kDirC)+1:
subs({i=x,j=y,k=z},EDir),subs({i=x,j=y,k=z},BDir),subs({i=x,j=y,k=z},kDir),kidx;
");
$EDir=switch(0,$m);
$BDir=switch(1,$m);
$kDir=switch(2,$m);
$kidx=switch(3,$m);
$kSign=switch($kidx,'-','','+');
$kOpSign=switch($kidx,'+','','-');
$E=rand(1.00,9.99,3);
$Eex=range(3,7);
$w=rand(1.00,9.99,3);
$k=$w/2.9979;
$kex=range(3,5);
$wex=$kex+8;@
qu.14.4.uid=f0f11c64-70ca-4357-9ca2-75bea378ec64@
qu.14.4.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Directions of Electromagnetic Wave Field Components;
  Difficulty=Medium;
  Features=Algorithmic;
@
qu.14.4.weighting=1,1,1@
qu.14.4.numbering=alpha@
qu.14.4.part.1.answer.6=-$BDir@
qu.14.4.part.1.answer.5=+$BDir@
qu.14.4.part.1.answer.4=-$EDir@
qu.14.4.part.1.editing=useHTML@
qu.14.4.part.1.answer.3=+$EDir@
qu.14.4.part.1.answer.2=$kOpSign$kDir@
qu.14.4.part.1.answer.1=$kSign$kDir@
qu.14.4.part.1.credit.6=0.0@
qu.14.4.part.1.credit.5=0.0@
qu.14.4.part.1.question=(Unset)@
qu.14.4.part.1.credit.4=0.0@
qu.14.4.part.1.credit.3=0.0@
qu.14.4.part.1.mode=List@
qu.14.4.part.1.credit.2=0.0@
qu.14.4.part.1.credit.1=1.0@
qu.14.4.part.1.grader=exact@
qu.14.4.part.1.display=menu@
qu.14.4.part.1.name=sro_id_1@
qu.14.4.part.1.display.permute=true@
qu.14.4.part.2.answer.6=-$kDir@
qu.14.4.part.2.answer.5=+$kDir@
qu.14.4.part.2.answer.4=-$EDir@
qu.14.4.part.2.editing=useHTML@
qu.14.4.part.2.answer.3=+$EDir@
qu.14.4.part.2.answer.2=$BOpSign$BDir@
qu.14.4.part.2.answer.1=$BSign$BDir@
qu.14.4.part.2.credit.6=0.0@
qu.14.4.part.2.credit.5=0.0@
qu.14.4.part.2.question=(Unset)@
qu.14.4.part.2.credit.4=0.0@
qu.14.4.part.2.credit.3=0.0@
qu.14.4.part.2.mode=List@
qu.14.4.part.2.credit.2=0.0@
qu.14.4.part.2.credit.1=1.0@
qu.14.4.part.2.grader=exact@
qu.14.4.part.2.display=menu@
qu.14.4.part.2.name=sro_id_2@
qu.14.4.part.2.display.permute=true@
qu.14.4.part.3.answer.6=-$kDir@
qu.14.4.part.3.answer.5=+$kDir@
qu.14.4.part.3.answer.4=-$BDir@
qu.14.4.part.3.editing=useHTML@
qu.14.4.part.3.answer.3=+$BDir@
qu.14.4.part.3.answer.2=$EOpSign$EDir@
qu.14.4.part.3.answer.1=$ESign$EDir@
qu.14.4.part.3.credit.6=0.0@
qu.14.4.part.3.credit.5=0.0@
qu.14.4.part.3.question=(Unset)@
qu.14.4.part.3.credit.4=0.0@
qu.14.4.part.3.credit.3=0.0@
qu.14.4.part.3.mode=List@
qu.14.4.part.3.credit.2=0.0@
qu.14.4.part.3.credit.1=1.0@
qu.14.4.part.3.grader=exact@
qu.14.4.part.3.display=menu@
qu.14.4.part.3.name=sro_id_3@
qu.14.4.part.3.display.permute=true@
qu.14.4.question=<p>Consider the following electromagnetic wave:<br /><br /><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover><mfenced open='(' close=')' separators=','><mrow><mi>$kDir</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>t</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$BSign</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$E</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$Eex</mi></mrow></msup><mi>T</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$k</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$kex</mi></mrow></msup><mfrac><mrow><mi>rad</mi></mrow><mrow><mi>m</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$kDir</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$kOpSign</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$w</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$wex</mi></mrow></msup><mfrac><mrow><mi>rad</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>t</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mrow><mi>$BDir</mi></mrow><mi>&#x005e;</mi></mover></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math></p><p>&nbsp;Identify the following:</p><p>&nbsp;</p><p><span><span><table border="0" cellspacing="1" cellpadding="1" width="90%">    <tbody>        <tr>            <td>Direction of propagation</td>            <td>&nbsp;<1><span> </span></td>        </tr>        <tr>            <td>Magnetic field direction at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$kDir</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mstyle></math></td>            <td><span>&nbsp;</span><2><span>&nbsp;</span></td>        </tr>        <tr>            <td>Electric field direction at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$kDir</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mstyle></math></td>            <td><span>&nbsp;</span><3><span>&nbsp;</span></td>        </tr>    </tbody></table></span></span></p>@

qu.14.5.mode=Inline@
qu.14.5.name=Electromagnetic Wave Direction Identification - E - Dimensionless@
qu.14.5.comment=@
qu.14.5.editing=useHTML@
qu.14.5.solution=<p>The axis of propagation is the direction variable inside the trigonometric function.&nbsp; If the signs of the direction and time terms are the same, then propagation is in the negative direction, otherwise it is positive.</p>
<p>Substitute the time and position values to find the direction of the first field.</p>
<p>Then, find the direction of the remaining field by:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>E</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&times;</mo><mover><mrow><mi>B</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math>&nbsp;is the propagation direction.</p>@
qu.14.5.algorithm=$w=range(1,9);
$k=range(1,9);
$t=range(1,9);
$x=range(1,9);
$idx=rint(2);
$idx2=rint(2);
$EDispSign=switch($idx,'+','-');
$BSign=switch($idx2,'+','-');
$BOpSign=switch($idx2,'-','+');
$m=maple("with(Physics[Vectors]);
randomize();
EDir:=RandomTools[Generate](choose({'i','j','k'})):
BDir:=RandomTools[Generate](choose({'i','j','k'} minus {EDir})):
kDir:=RandomTools[Generate](choose({'i','j','k'} minus {EDir,BDir})):
EDirC:=(-1)^($idx)*cat('_',EDir):
BDirC:=(-1)^($idx2)*cat('_',BDir):
kDirC:=EDirC&x BDirC:
kidx:=sign(kDirC)+1:
n:=cos(($k*$x+(kidx)*$w*$t)*Pi);
subs({i=x,j=y,k=z},EDir),subs({i=x,j=y,k=z},BDir),subs({i=x,j=y,k=z},kDir),kidx,n;
");
$EDir=switch(0,$m);
$BDir=switch(1,$m);
$kDir=switch(2,$m);
$kidx=switch(3,$m);
$n=switch(4,$m);
$kSign=switch($kidx,'-','','+');
$kOpSign=switch($kidx,'+','','-');
$E=range(1,9);
$idx3=($n*switch($idx,+1,-1)+1);
$ESign=switch($idx3,'-',0,'+');
$EOpSign=switch($idx3,'+',0,'-');@
qu.14.5.uid=7661997e-ca31-477c-9543-810fe56fcbc5@
qu.14.5.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Directions of Electromagnetic Wave Field Components;
  Difficulty=Medium;
  Features=Algorithmic;
@
qu.14.5.weighting=1,1,1@
qu.14.5.numbering=alpha@
qu.14.5.part.1.answer.6=-$BDir@
qu.14.5.part.1.answer.5=+$BDir@
qu.14.5.part.1.answer.4=-$EDir@
qu.14.5.part.1.editing=useHTML@
qu.14.5.part.1.answer.3=+$EDir@
qu.14.5.part.1.answer.2=$kOpSign$kDir@
qu.14.5.part.1.answer.1=$kSign$kDir@
qu.14.5.part.1.credit.6=0.0@
qu.14.5.part.1.credit.5=0.0@
qu.14.5.part.1.question=(Unset)@
qu.14.5.part.1.credit.4=0.0@
qu.14.5.part.1.credit.3=0.0@
qu.14.5.part.1.mode=List@
qu.14.5.part.1.credit.2=0.0@
qu.14.5.part.1.credit.1=1.0@
qu.14.5.part.1.grader=exact@
qu.14.5.part.1.display=menu@
qu.14.5.part.1.name=sro_id_1@
qu.14.5.part.1.display.permute=true@
qu.14.5.part.2.answer.6=-$kDir@
qu.14.5.part.2.answer.5=+$kDir@
qu.14.5.part.2.answer.4=-$EDir@
qu.14.5.part.2.editing=useHTML@
qu.14.5.part.2.answer.3=+$EDir@
qu.14.5.part.2.answer.2=$BOpSign$BDir@
qu.14.5.part.2.answer.1=$BSign$BDir@
qu.14.5.part.2.credit.6=0.0@
qu.14.5.part.2.credit.5=0.0@
qu.14.5.part.2.question=(Unset)@
qu.14.5.part.2.credit.4=0.0@
qu.14.5.part.2.credit.3=0.0@
qu.14.5.part.2.mode=List@
qu.14.5.part.2.credit.2=0.0@
qu.14.5.part.2.credit.1=1.0@
qu.14.5.part.2.grader=exact@
qu.14.5.part.2.display=menu@
qu.14.5.part.2.name=sro_id_2@
qu.14.5.part.2.display.permute=true@
qu.14.5.part.3.answer.6=-$kDir@
qu.14.5.part.3.answer.5=+$kDir@
qu.14.5.part.3.answer.4=-$BDir@
qu.14.5.part.3.editing=useHTML@
qu.14.5.part.3.answer.3=+$BDir@
qu.14.5.part.3.answer.2=$EOpSign$EDir@
qu.14.5.part.3.answer.1=$ESign$EDir@
qu.14.5.part.3.credit.6=0.0@
qu.14.5.part.3.credit.5=0.0@
qu.14.5.part.3.question=(Unset)@
qu.14.5.part.3.credit.4=0.0@
qu.14.5.part.3.credit.3=0.0@
qu.14.5.part.3.mode=List@
qu.14.5.part.3.credit.2=0.0@
qu.14.5.part.3.credit.1=1.0@
qu.14.5.part.3.grader=exact@
qu.14.5.part.3.display=menu@
qu.14.5.part.3.name=sro_id_3@
qu.14.5.part.3.display.permute=true@
qu.14.5.question=<p align="left">Consider the following electromagnetic wave, in dimensionless units:</p><p align="center"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>E</mi></mrow><mi>&rarr;</mi></mover><mfenced open='(' close=')' separators=','><mrow><mi>$kDir</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>t</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$EDispSign</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$E</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>$k</mi><mrow><mi>&pi;</mi></mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$kDir</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$kOpSign</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$w</mi><mrow><mi>&pi;</mi></mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>t</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mrow><mi>$EDir</mi></mrow><mi>&#x005e;</mi></mover></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math></p><p>&nbsp;Identify the following:</p><p>&nbsp;</p><p><span><span><table border="0" cellspacing="1" cellpadding="1" width="100%">    <tbody>        <tr>            <td>Direction of propagation</td>            <td>&nbsp;<1><span> </span></td>        </tr>        <tr>            <td>Magnetic field direction at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$t</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$kDir</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$x</mi></mrow></mstyle></math></td>            <td><span>&nbsp;</span><2><span>&nbsp;</span></td>        </tr>        <tr>            <td>Electric field direction at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$t</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$kDir</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$x</mi></mrow></mstyle></math></td>            <td><span>&nbsp;</span><3><span>&nbsp;</span></td>        </tr>    </tbody></table></span></span></p>@

qu.14.6.question=<p>A particular electromagnetic wave has a maximum magnetic field amplitude of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$B</mi></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$Bex</mi></mrow></msup><mi>T</mi></mrow></mstyle></math>, what is&nbsp;<br />
the maximum electric field amplitude?</p>@
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qu.14.6.editing=useHTML@
qu.14.6.solution=<p>For electromagnetic radiation, the magnitudes of the amplitudes of the fields are related by:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>E</mi><mrow><mi>B</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>c</mi></mrow></mstyle></math>.</p>@
qu.14.6.algorithm=$B=rand(1.00,9.99,3);
$Bex=range(5,10);
$ans=(3.00*10^(8))*$B*10^($Bex);@
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qu.14.6.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Electromagnetic Wave Magnitude;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Algorithmic;
@

qu.15.topic=Power@

qu.15.1.question=<p>If&nbsp;completely charging a particular battery&nbsp;from empty by applying a voltage of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$V</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>V</mi></mrow></mstyle></math>&nbsp;for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$t</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>min</mi></mrow></mstyle></math>&nbsp;drew an&nbsp;<br />
average current of<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mA</mi></mrow></mstyle></math>, what is the total capacity of the battery?</p>@
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qu.15.1.editing=useHTML@
qu.15.1.solution=<p>The average power during charging would be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>P</mi><mrow><mi>av</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>I</mi><mrow><mi>av</mi></mrow></msub><mi>V</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>The total energy used is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>E</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>P</mi><mrow><mi>av</mi></mrow></msub><mi>&Delta;t</mi></mrow></mstyle></math></p>
<p>Thus,</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>E</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mA</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$V</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>V</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$t</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>min</mi></mrow></mfenced></mrow></mstyle></math>.</p>@
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$V=rand(4.60,5.20,3);
$I=rand(400,510,3);
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  Topic=Power Consumption;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
@

qu.15.2.question=<p>Charging an iPhone over a USB cable draws roughly <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$P</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>W</mi></mrow></mstyle></math>.&nbsp; The <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>5</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>V</mi></mrow></mstyle></math>&nbsp;line in the USB cable that&nbsp;<br />
provides the current, is measured in a particular case to be at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$V</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>V</mi></mrow></mstyle></math>.&nbsp; What current is being drawn by&nbsp;<br />
the phone?</p>@
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qu.15.2.editing=useHTML@
qu.15.2.solution=<p>The instantaneous power during charging would be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>P</mi><mrow><mi></mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>I</mi><mrow><mi></mi></mrow></msub><mi>V</mi></mrow></mstyle></math>.</p>
<p>&nbsp;Thus,</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mi>$P</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>W</mi></mrow></mfenced><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$V</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>V</mi></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>.</p>@
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  Difficulty=Easy;
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  Features=Algorithmic;
@

qu.15.3.question=<p>The <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>5</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>V</mi></mrow></mstyle></math>&nbsp;line&nbsp;in a&nbsp;USB cable does not provide exactly <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>5</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>V</mi></mrow></mstyle></math>, but fluctuates around this value.&nbsp; If , while&nbsp;<br />
charging an iPhone over a USB, one measures the power&nbsp;used to be <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$P</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>W</mi></mrow></mstyle></math>, and the current to be<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mA</mi></mrow></mstyle></math>,&nbsp;what is the actual voltage provided by the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>5</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>V</mi></mrow></mstyle></math>&nbsp;line?</p>@
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qu.15.3.editing=useHTML@
qu.15.3.solution=<p>The instantaneous power during charging would be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>P</mi><mrow><mi></mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>I</mi><mrow><mi></mi></mrow></msub><mi>V</mi></mrow></mstyle></math>.</p>
<p>&nbsp;Thus,</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mi>$P</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>W</mi></mrow></mfenced><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mA</mi></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>.</p>@
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qu.15.3.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Power Usage;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Algorithmic;
@

qu.15.4.question=<p>A particular phone battery has a capacity of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$cap</mi></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mn>4</mn></mrow></msup><mi>J</mi></mrow></mstyle></math>.&nbsp; If&nbsp;completely charging it from empty takes&nbsp;<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$t</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>min</mi></mrow></mstyle></math>&nbsp;when a potential difference of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$V</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>V</mi></mrow></mstyle></math>&nbsp;is applied,&nbsp;what is the average current passing through&nbsp;<br />
the&nbsp;battery during this time?</p>@
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qu.15.4.editing=useHTML@
qu.15.4.solution=<p>The average power during charging would be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>P</mi><mrow><mi>av</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>I</mi><mrow><mi>av</mi></mrow></msub><mi>V</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>The total energy used is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>E</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>P</mi><mrow><mi>av</mi></mrow></msub><mi>&Delta;t</mi></mrow></mstyle></math></p>
<p>Thus,</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>av</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mi>$cap</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mn>4</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>J</mi></mrow></mfenced><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$V</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>V</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$t</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>min</mi></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>.</p>@
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qu.15.4.info=  Course=Introductory Electricity and Magnetism;
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qu.15.5.question=<p>The <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>5</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>V</mi></mrow></mstyle></math>&nbsp;line&nbsp;in a&nbsp;USB cable does not provide exactly <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>5</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>V</mi></mrow></mstyle></math>, but fluctuates around this value.&nbsp; If , while&nbsp;<br />
charging an iPhone over a USB cable, one measures the actual voltage to be <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$V</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>V</mi></mrow></mstyle></math> and the current to&nbsp;<br />
be<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mA</mi></mrow></mstyle></math>,&nbsp;what is the power drawn?</p>@
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qu.15.5.editing=useHTML@
qu.15.5.solution=<p>The instantaneous power during charging would be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>P</mi><mrow><mi></mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>I</mi><mrow><mi></mi></mrow></msub><mi>V</mi></mrow></mstyle></math>.</p>
<p>&nbsp;Thus,</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mA</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$V</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>V</mi></mrow></mfenced></mrow></mstyle></math>.</p>@
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qu.15.6.question=<p>A particular phone battery has a capacity of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$cap</mi></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mn>4</mn></mrow></msup><mi>J</mi></mrow></mstyle></math>.&nbsp; If you were to charge it from empty through&nbsp;<br />
a&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$V</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>V</mi></mrow></mstyle></math>&nbsp;USB port and measured the average current to be <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mA</mi></mrow></mstyle></math>,&nbsp;how many hours would it take to&nbsp;<br />
fully&nbsp;charge?</p>@
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qu.15.6.editing=useHTML@
qu.15.6.solution=<p>The average power during charging would be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>P</mi><mrow><mi>av</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>I</mi><mrow><mi>av</mi></mrow></msub><mi>V</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>The total energy used is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>E</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>P</mi><mrow><mi>av</mi></mrow></msub><mi>&Delta;t</mi></mrow></mstyle></math></p>
<p>Thus,</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&Delta;t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mi>$cap</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mn>4</mn></mrow></msup><mi>J</mi></mrow></mfenced><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$V</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>V</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mA</mi></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>.</p>@
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qu.15.7.question=<p>A particular phone battery has a capacity of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$cap</mi></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mn>4</mn></mrow></msup><mi>J</mi></mrow></mstyle></math>.&nbsp; If&nbsp;completely charging it from empty takes&nbsp;<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$t</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>min</mi></mrow></mstyle></math>&nbsp;and draws an average&nbsp;current&nbsp;of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mA</mi></mrow></mstyle></math>,&nbsp;what is the voltage applied across the terminals of&nbsp;<br />
the&nbsp;battery?</p>@
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qu.15.7.editing=useHTML@
qu.15.7.solution=<p>The average power during charging would be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>P</mi><mrow><mi>av</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>I</mi><mrow><mi>av</mi></mrow></msub><mi>V</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>The total energy used is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>E</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>P</mi><mrow><mi>av</mi></mrow></msub><mi>&Delta;t</mi></mrow></mstyle></math></p>
<p>Thus,</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mi>$cap</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mn>4</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>J</mi></mrow></mfenced><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mA</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$t</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>min</mi></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>.</p>@
qu.15.7.algorithm=$cap=rand(1.00,3.00,3);
$I=rand(400,510,3);
$t=rand(30.0,90.0,3);
$ans=($cap*10^(4))/(($t*60)*$I/1000);@
qu.15.7.uid=9f50112c-ce11-4b1b-ab79-7b76590e6eea@
qu.15.7.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Power Consumption;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
@

qu.16.topic=Kirchhoff's Laws@

qu.16.1.mode=Inline@
qu.16.1.name=Kirchhoff's Laws - 3 Loops - 2@
qu.16.1.comment=<ol>
    <li>Kirchhoff's current (or node) law states that the sum of all currents entering a node is equal to the sum of all currents leaving the node.</li>
    <li>Kirchhoff's voltage (or loop) law states that the sum of all voltage gains and drops around a closed loop is equal to zero.</li>
    <li>Remember that choosing one direction around a loop is important. It does not matter <em>which</em> direction you choose to follow, but you must be consistent.</li>
    <li>When following a loop: Passing through a battery is a positive if you are going from negative to positive terminal and a negative if going from positive to negative.</li>
    <li>When following a loop: Passing through a resistor is dependant both on the direction you are following the loop<em> and the assumed direction of the current passing through the resistor</em>. If you are following the loop in the same direction as the current through the resistor, the resistor provides a negative term. If you are following the loop against the current, the resistor provides a positive term.</li>
</ol>@
qu.16.1.editing=useHTML@
qu.16.1.hint.1=Kirchhoff's current (or node) law states that the sum of all currents entering a node is equal to the sum of all currents leaving the node.@
qu.16.1.hint.2=Kirchhoff's voltage (or loop) law states that the sum of all voltage gains and drops around a closed loop is equal to zero.@
qu.16.1.hint.3=Remember that choosing one direction around a loop is important.&nbsp; It does not matter <em>which</em> direction you choose to follow, but you must be consistent.@
qu.16.1.hint.4=When following a loop:&nbsp; Passing through a battery is a positive if you are going from negative to positive terminal and a negative if going from positive to negative.@
qu.16.1.hint.5=When following a loop:&nbsp; Passing through a resistor is dependant both on the direction you are following the loop<em> and the assumed direction of the current passing through the resistor</em>.&nbsp; If you are following the loop in the same direction as the current through the resistor, the resistor provides a negative term.&nbsp; If you are following the loop against the current, the resistor provides a positive term.@
qu.16.1.solution=<p>We must choose a direction to traverse each loop and a direction for the currents in each branch.</p>
<p>If we cross a voltage source from the negative to the positive terminal, then the potential difference is positive, otherwise it is negative.</p>
<p>If we cross a resistor in the direction of current flow, then the potential difference is negative, otherwise it is positive.</p>
<p>&nbsp;</p>
<p>At a node, the sum of all currents entering must equal the sum of all currents leaving.</p>@
qu.16.1.algorithm=$r1=range(1,5);
$v1=range(1,3);
$i1=range(1,6);
$m=maple("randomize();
b:=RandomTools[Generate](choose({1,2,3,4,5}minus {$r1})):
c:=RandomTools[Generate](choose({1,2,3,4,5}minus {$r1,b})):
d:=RandomTools[Generate](choose({1,2,3,4,5}minus {$r1,b,c})):
e:=RandomTools[Generate](choose({1,2,3,4,5}minus {$r1,b,c,d})):
bb:=RandomTools[Generate](choose({1,2,3}minus {$v1})):
cc:=RandomTools[Generate](choose({1,2,3}minus {$v1,bb})):
bbb:=RandomTools[Generate](choose({1,2,3,4,5,6}minus {$i1})):
ccc:=RandomTools[Generate](choose({1,2,3,4,5,6}minus {$i1,bbb})):
ddd:=RandomTools[Generate](choose({1,2,3,4,5,6}minus {$i1,bbb,ccc})):
eee:=RandomTools[Generate](choose({1,2,3,4,5,6}minus {$i1,bbb,ccc,ddd})):
fff:=RandomTools[Generate](choose({1,2,3,4,5,6}minus {$i1,bbb,ccc,ddd,eee})):
b,c,d,e,bb,cc,bbb,ccc,ddd,eee,fff;
");
$r2=switch(0,$m);
$r3=switch(1,$m);
$r4=switch(2,$m);
$r5=switch(3,$m);
$v2=switch(4,$m);
$v3=switch(5,$m);
$i2=switch(6,$m);
$i3=switch(7,$m);
$i4=switch(8,$m);
$i5=switch(9,$m);
$i6=switch(10,$m);
$idxNode=rint(4);
$wNode=switch($idxNode,A,H,E,C);
$ansNode=switch($idxNode,'I$i1+I$i4+I$i2=0','I$i6=I$i2+I$i3','I$i5+I$i6+I$i1=0','I$i5+I$i3-I$i4=0');
$idxLoop=rint(7);
$wLoop=switch($idxLoop,'AHEFGA','ABCHA','CDEHC','ABCDEHA','ABCDEFGA','ABCHEFGA','CDEFGAHC');
$ansLoop=switch($idxLoop,'I$i2*R$r5+I$i6*R$r3-I$i1*R$r4+V$v1=0','I$i4*R$r1-V$v2+V$v3-I$i2*R$r5=0','I$i5*R$r2-I$i6*R$r3-V$v3=0','I$i4*R$r1-V$v2+I$i5*R$r2-I$i6*R$r3-I$i2*R$r5=0','I$i4*R$r1-V$v2+I$i5*R$r2-I$i1*R$r4+V$v1=0','I$i4*R$r1-V$v2+V$v3+I$i6*R$r3-I$i1*R$r4+V$v1=0','I$i5*R$r2-I$i1*R$r4+V$v1+I$i2*R$r5-V$v3=0');@
qu.16.1.uid=329565ec-6a68-4b4a-8b5b-3a2a89195816@
qu.16.1.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Difficulty=Medium;
  Topic=Kirchhoff's Laws;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Diagram;
@
qu.16.1.weighting=1,1@
qu.16.1.numbering=alpha@
qu.16.1.part.1.name=sro_id_1@
qu.16.1.part.1.maple_answer=$ansNode@
qu.16.1.part.1.editing=useHTML@
qu.16.1.part.1.question=(Unset)@
qu.16.1.part.1.libname=@
qu.16.1.part.1.mode=Maple@
qu.16.1.part.1.allow2d=0@
qu.16.1.part.1.plot=@
qu.16.1.part.1.maple=is((solve($ANSWER,I1))=(solve($RESPONSE,I1)) );@
qu.16.1.part.1.type=maple@
qu.16.1.part.2.name=sro_id_2@
qu.16.1.part.2.maple_answer=$ansLoop@
qu.16.1.part.2.editing=useHTML@
qu.16.1.part.2.question=(Unset)@
qu.16.1.part.2.libname=@
qu.16.1.part.2.mode=Maple@
qu.16.1.part.2.allow2d=0@
qu.16.1.part.2.plot=@
qu.16.1.part.2.maple=is(solve($ANSWER,V2)-solve($RESPONSE,V2) = 0);@
qu.16.1.part.2.type=maple@
qu.16.1.question=<p>Consider the following circuit where the symbols have their usual meaning.&nbsp;&nbsp;State your answers in terms of the given variables and directions in the diagram.</p><p>&nbsp;</p><p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="583" height="600"><param name="image" value="__BASE_URI__img/Circuits/Kirchhoffs3L-2-NoLabel-Dir/Diagram.png" /><param name="size" value="22" /><param name="label.1.x" value="50" /><param name="label.1.y" value="330" /><param name="label.1.text" value="A" /><param name="label.2.x" value="50" /><param name="label.2.y" value="60" /><param name="label.2.text" value="B" /><param name="label.3.x" value="265" /><param name="label.3.y" value="60" /><param name="label.3.text" value="C" /><param name="label.4.x" value="530" /><param name="label.4.y" value="60" /><param name="label.4.text" value="D" /><param name="label.5.x" value="530" /><param name="label.5.y" value="330" /><param name="label.5.text" value="E" /><param name="label.6.x" value="530" /><param name="label.6.y" value="550" /><param name="label.6.text" value="F" /><param name="label.7.x" value="50" /><param name="label.7.y" value="550" /><param name="label.7.text" value="G" /><param name="label.8.x" value="265" /><param name="label.8.y" value="345" /><param name="label.8.text" value="H" /><param name="label.9.x" value="40" /><param name="label.9.y" value="195" /><param name="label.9.text" value="R$r1" /><param name="label.10.x" value="398" /><param name="label.10.y" value="50" /><param name="label.10.text" value="R$r2" /><param name="label.11.x" value="398" /><param name="label.11.y" value="300" /><param name="label.11.text" value="R$r3" /><param name="label.12.x" value="555" /><param name="label.12.y" value="430" /><param name="label.12.text" value="R$r4" /><param name="label.13.x" value="160" /><param name="label.13.y" value="570" /><param name="label.13.text" value="V$v1" /><param name="label.14.x" value="160" /><param name="label.14.y" value="290" /><param name="label.14.text" value="R$r5" /><param name="label.15.x" value="310" /><param name="label.15.y" value="200" /><param name="label.15.text" value="V$v3" /><param name="label.16.x" value="160" /><param name="label.16.y" value="40" /><param name="label.16.text" value="V$v2" /><param name="label.17.x" value="225" /><param name="label.17.y" value="510" /><param name="label.17.text" value="I$i1" /><param name="label.18.x" value="240" /><param name="label.18.y" value="350" /><param name="label.18.text" value="I$i2" /><param name="label.19.x" value="245" /><param name="label.19.y" value="270" /><param name="label.19.text" value="I$i3" /><param name="label.20.x" value="220" /><param name="label.20.y" value="90" /><param name="label.20.text" value="I$i4" /><param name="label.21.x" value="490" /><param name="label.21.y" value="90" /><param name="label.21.text" value="I$i5" /><param name="label.22.x" value="490" /><param name="label.22.y" value="350" /><param name="label.22.text" value="I$i6" /></applet></p><p><strong>(a)</strong>&nbsp; Use Kirchhoff's current rule to write an equation for the algebraic sum of the currents&nbsp;flowing into node $wNode.&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p><span><strong>(b)</strong>&nbsp; Use Kirchhoff's voltage rule to write&nbsp;an equation for the algebraic&nbsp;sum of the potential differences&nbsp;in the $wLoop loop of the circuit.</span></p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p>@

qu.16.2.mode=Inline@
qu.16.2.name=Kirchhoff's Laws - 3 Loops - Numeric - 1@
qu.16.2.comment=<ol>
    <li>Kirchhoff's current (or node) law states that the sum of all currents entering a node is equal to the sum of all currents leaving the node.</li>
    <li>Kirchhoff's voltage (or loop) law states that the sum of all voltage gains and drops around a closed loop is equal to zero.</li>
    <li>Remember that choosing one direction around a loop is important. It does not matter <em>which</em> direction you choose to follow, but you must be consistent.</li>
    <li>When following a loop: Passing through a battery is a positive if you are going from negative to positive terminal and a negative if going from positive to negative.</li>
    <li>When following a loop: Passing through a resistor is dependant both on the direction you are following the loop<em> and the assumed direction of the current passing through the resistor</em>. If you are following the loop in the same direction as the current through the resistor, the resistor provides a negative term. If you are following the loop against the current, the resistor provides a positive term.</li>
</ol>@
qu.16.2.editing=useHTML@
qu.16.2.hint.1=Kirchhoff's current (or node) law states that the sum of all currents entering a node is equal to the sum of all currents leaving the node.@
qu.16.2.hint.2=Kirchhoff's voltage (or loop) law states that the sum of all voltage gains and drops around a closed loop is equal to zero.@
qu.16.2.hint.3=Remember that choosing one direction around a loop is important.&nbsp; It does not matter <em>which</em> direction you choose to follow, but you must be consistent.@
qu.16.2.hint.4=When following a loop:&nbsp; Passing through a battery is a positive if you are going from negative to positive terminal and a negative if going from positive to negative.@
qu.16.2.hint.5=When following a loop:&nbsp; Passing through a resistor is dependant both on the direction you are following the loop<em> and the assumed direction of the current passing through the resistor</em>.&nbsp; If you are following the loop in the same direction as the current through the resistor, the resistor provides a negative term.&nbsp; If you are following the loop against the current, the resistor provides a positive term.@
qu.16.2.solution=<p>We must choose a direction to traverse each loop and a direction for the currents in each branch.</p>
<p>If we cross a voltage source from the negative to the positive terminal, then the potential difference is positive, otherwise it is negative.</p>
<p>If we cross a resistor in the direction of current flow, then the potential difference is negative, otherwise it is positive.</p>
<p>&nbsp;</p>
<p>At a node, the sum of all currents entering must equal the sum of all currents leaving.</p>@
qu.16.2.algorithm=$V1=rand(1.00,20.0,3);
$V2=rand(1.00,20.0,3);
$V3=rand(1.00,20.0,3);
$V4=rand(1.00,20.0,3);
$R1=rand(10.0,999,3);
$R2=rand(10.0,999,3);
$R3=rand(10.0,999,3);
$R4=rand(10.0,999,3);
$m=maple("
assign(solve({I1+I4-I2=0,I1+I5+I6=0,I3+I4-I5=0,$V1-$V2+I6*$R3-I1*$R4=0,$V2+I4*$R1-$V4-$V3=0,$V3+I5*$R2-I6*$R3=0}
));
I1,I2,I3,I4,I5,I6;
");
$I1=switch(0,$m)*1000;
$I2=switch(1,$m)*1000;
$I3=switch(2,$m)*1000;
$I4=switch(3,$m)*1000;
$I5=switch(4,$m)*1000;
$I6=switch(5,$m)*1000;@
qu.16.2.uid=70d76392-a30f-449b-9b24-af19d0f292dd@
qu.16.2.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Difficulty=Medium;
  Topic=Kirchhoff's Laws;
  Features=Algorithmic;
  Features=Diagram;
@
qu.16.2.weighting=1,1,1,1,1,1@
qu.16.2.numbering=alpha@
qu.16.2.part.1.name=sro_id_1@
qu.16.2.part.1.answer.units=@
qu.16.2.part.1.numStyle= scientific  @
qu.16.2.part.1.editing=useHTML@
qu.16.2.part.1.showUnits=false@
qu.16.2.part.1.question=(Unset)@
qu.16.2.part.1.mode=Numeric@
qu.16.2.part.1.grading=exact_sigd@
qu.16.2.part.1.negStyle=both@
qu.16.2.part.1.digit=3@
qu.16.2.part.1.answer.num=$I1@
qu.16.2.part.2.name=sro_id_2@
qu.16.2.part.2.answer.units=@
qu.16.2.part.2.numStyle= scientific  @
qu.16.2.part.2.editing=useHTML@
qu.16.2.part.2.showUnits=false@
qu.16.2.part.2.question=(Unset)@
qu.16.2.part.2.mode=Numeric@
qu.16.2.part.2.grading=exact_sigd@
qu.16.2.part.2.negStyle=both@
qu.16.2.part.2.digit=3@
qu.16.2.part.2.answer.num=$I2@
qu.16.2.part.3.name=sro_id_3@
qu.16.2.part.3.answer.units=@
qu.16.2.part.3.numStyle= scientific  @
qu.16.2.part.3.editing=useHTML@
qu.16.2.part.3.showUnits=false@
qu.16.2.part.3.question=(Unset)@
qu.16.2.part.3.mode=Numeric@
qu.16.2.part.3.grading=exact_sigd@
qu.16.2.part.3.negStyle=both@
qu.16.2.part.3.digit=3@
qu.16.2.part.3.answer.num=$I3@
qu.16.2.part.4.name=sro_id_4@
qu.16.2.part.4.answer.units=@
qu.16.2.part.4.numStyle= scientific  @
qu.16.2.part.4.editing=useHTML@
qu.16.2.part.4.showUnits=false@
qu.16.2.part.4.question=(Unset)@
qu.16.2.part.4.mode=Numeric@
qu.16.2.part.4.grading=exact_sigd@
qu.16.2.part.4.negStyle=both@
qu.16.2.part.4.digit=3@
qu.16.2.part.4.answer.num=$I4@
qu.16.2.part.5.name=sro_id_5@
qu.16.2.part.5.answer.units=@
qu.16.2.part.5.numStyle= scientific  @
qu.16.2.part.5.editing=useHTML@
qu.16.2.part.5.showUnits=false@
qu.16.2.part.5.question=(Unset)@
qu.16.2.part.5.mode=Numeric@
qu.16.2.part.5.grading=exact_sigd@
qu.16.2.part.5.negStyle=both@
qu.16.2.part.5.digit=3@
qu.16.2.part.5.answer.num=$I5@
qu.16.2.part.6.name=sro_id_6@
qu.16.2.part.6.answer.units=@
qu.16.2.part.6.numStyle= scientific  @
qu.16.2.part.6.editing=useHTML@
qu.16.2.part.6.showUnits=false@
qu.16.2.part.6.question=(Unset)@
qu.16.2.part.6.mode=Numeric@
qu.16.2.part.6.grading=exact_sigd@
qu.16.2.part.6.negStyle=both@
qu.16.2.part.6.digit=3@
qu.16.2.part.6.answer.num=$I6@
qu.16.2.question=<p>Given the following circuit, calculate the currents <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I1</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I2</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I3</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I4</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I5</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I6</mi></mrow></mstyle></math>&nbsp;in <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mA</mi></mrow></mstyle></math>.&nbsp;</p><p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="583" height="600"><param name="image" value="__BASE_URI__img/Circuits/Kirchhoffs3L-1-NoLabel-Dir/Diagram.png" /><param name="size" value="22" /><param name="label.1.x" value="50" /><param name="label.1.y" value="330" /><param name="label.1.text" value="A" /><param name="label.2.x" value="50" /><param name="label.2.y" value="60" /><param name="label.2.text" value="B" /><param name="label.3.x" value="265" /><param name="label.3.y" value="60" /><param name="label.3.text" value="C" /><param name="label.4.x" value="530" /><param name="label.4.y" value="60" /><param name="label.4.text" value="D" /><param name="label.5.x" value="530" /><param name="label.5.y" value="330" /><param name="label.5.text" value="E" /><param name="label.6.x" value="530" /><param name="label.6.y" value="550" /><param name="label.6.text" value="F" /><param name="label.7.x" value="50" /><param name="label.7.y" value="550" /><param name="label.7.text" value="G" /><param name="label.8.x" value="265" /><param name="label.8.y" value="345" /><param name="label.8.text" value="H" /><param name="label.9.x" value="110" /><param name="label.9.y" value="195" /><param name="label.9.text" value="$R1 Ohm" /><param name="label.10.x" value="398" /><param name="label.10.y" value="50" /><param name="label.10.text" value="$R2 Ohm" /><param name="label.11.x" value="398" /><param name="label.11.y" value="300" /><param name="label.11.text" value="$R3 Ohm" /><param name="label.12.x" value="480" /><param name="label.12.y" value="430" /><param name="label.12.text" value="$R4 Ohm" /><param name="label.13.x" value="160" /><param name="label.13.y" value="570" /><param name="label.13.text" value="$V1 V" /><param name="label.14.x" value="160" /><param name="label.14.y" value="290" /><param name="label.14.text" value="$V2 V" /><param name="label.15.x" value="315" /><param name="label.15.y" value="200" /><param name="label.15.text" value="$V3 V" /><param name="label.16.x" value="160" /><param name="label.16.y" value="110" /><param name="label.16.text" value="$V4 V" /><param name="label.17.x" value="225" /><param name="label.17.y" value="510" /><param name="label.17.text" value="I1" /><param name="label.18.x" value="100" /><param name="label.18.y" value="350" /><param name="label.18.text" value="I2" /><param name="label.19.x" value="240" /><param name="label.19.y" value="130" /><param name="label.19.text" value="I3" /><param name="label.20.x" value="220" /><param name="label.20.y" value="90" /><param name="label.20.text" value="I4" /><param name="label.21.x" value="490" /><param name="label.21.y" value="90" /><param name="label.21.text" value="I5" /><param name="label.22.x" value="490" /><param name="label.22.y" value="350" /><param name="label.22.text" value="I6" /></applet></p><p align="left"><strong>(a)</strong>&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I1</mi></mrow></mstyle></math>&nbsp;&nbsp;<span>&nbsp;</span><1><span>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mA</mi></mrow></mstyle></math></span></p><p><span><strong>(b)</strong>&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I2</mi></mrow></mstyle></math>&nbsp;&nbsp;<span>&nbsp;</span><2><span>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mA</mi></mrow></mstyle></math></span></span></p><p><span><span><strong>(c)</strong>&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I3</mi></mrow></mstyle></math>&nbsp;&nbsp;<span>&nbsp;</span><3><span>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mA</mi></mrow></mstyle></math></span></span></span></p><p><span><span><span><strong>(d)</strong>&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I4</mi></mrow></mstyle></math>&nbsp;&nbsp;<span>&nbsp;</span><4><span>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mA</mi></mrow></mstyle></math></span></span></span></span></p><p><span><span><span><span><strong>(e)</strong>&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I5</mi></mrow></mstyle></math>&nbsp;&nbsp;<span>&nbsp;</span><5><span>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mA</mi></mrow></mstyle></math></span></span></span></span></span></p><p><span><span><span><span><span><strong>(f)</strong>&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I6</mi></mrow></mstyle></math>&nbsp;&nbsp;<span>&nbsp;</span><6><span>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mA</mi></mrow></mstyle></math></span></span></span></span></span></span></p>@

qu.16.3.mode=Inline@
qu.16.3.name=Kirchhoff's Laws - 2 Loops - 2@
qu.16.3.comment=<ol>
    <li>Kirchhoff's current (or node) law states that the sum of all currents entering a node is equal to the sum of all currents leaving the node.</li>
    <li>Kirchhoff's voltage (or loop) law states that the sum of all voltage gains and drops around a closed loop is equal to zero.</li>
    <li>Remember that choosing one direction around a loop is important. It does not matter <em>which</em> direction you choose to follow, but you must be consistent.</li>
    <li>When following a loop: Passing through a battery is a positive if you are going from negative to positive terminal and a negative if going from positive to negative.</li>
    <li>When following a loop: Passing through a resistor is dependant both on the direction you are following the loop<em> and the assumed direction of the current passing through the resistor</em>. If you are following the loop in the same direction as the current through the resistor, the resistor provides a negative term. If you are following the loop against the current, the resistor provides a positive term.</li>
</ol>@
qu.16.3.editing=useHTML@
qu.16.3.hint.1=Kirchhoff's current (or node) law states that the sum of all currents entering a node is equal to the sum of all currents leaving the node.@
qu.16.3.hint.2=Kirchhoff's voltage (or loop) law states that the sum of all voltage gains and drops around a closed loop is equal to zero.@
qu.16.3.hint.3=Remember that choosing one direction around a loop is important.&nbsp; It does not matter <em>which</em> direction you choose to follow, but you must be consistent.@
qu.16.3.hint.4=When following a loop:&nbsp; Passing through a battery is a positive if you are going from negative to positive terminal and a negative if going from positive to negative.@
qu.16.3.hint.5=When following a loop:&nbsp; Passing through a resistor is dependant both on the direction you are following the loop<em> and the assumed direction of the current passing through the resistor</em>.&nbsp; If you are following the loop in the same direction as the current through the resistor, the resistor provides a negative term.&nbsp; If you are following the loop against the current, the resistor provides a positive term.@
qu.16.3.solution=<p>We must choose a direction to traverse each loop and a direction for the currents in each branch.</p>
<p>If we cross a voltage source from the negative to the positive terminal, then the potential difference is positive, otherwise it is negative.</p>
<p>If we cross a resistor in the direction of current flow, then the potential difference is negative, otherwise it is positive.</p>
<p>&nbsp;</p>
<p>At a node, the sum of all currents entering must equal the sum of all currents leaving.</p>@
qu.16.3.algorithm=$r1=rint(1,8);
$v1=rint(1,3);
$i1=rint(1,4);
$m=maple("randomize();
b:=RandomTools[Generate](choose({1,2,3,4,5,6,7}minus {$r1})):
c:=RandomTools[Generate](choose({1,2,3,4,5,6,7}minus {$top,b})):
d:=RandomTools[Generate](choose({1,2,3,4,5,6,7}minus {$top,b,c})):
e:=RandomTools[Generate](choose({1,2,3,4,5,6,7}minus {$top,b,c,d})):
f:=RandomTools[Generate](choose({1,2,3,4,5,6,7}minus {$top,b,c,d,e})):
g:=RandomTools[Generate](choose({1,2,3,4,5,6,7}minus {$top,b,c,d,e,f})):
bb:=RandomTools[Generate](choose({1,2}minus {$v1})):
bbb:=RandomTools[Generate](choose({1,2,3}minus {$i1})):
ccc:=RandomTools[Generate](choose({1,2,3}minus {$i1,bbb})):
b,c,d,e,f,g,bb,bbb,ccc;
");
$r2=switch(0,$m);
$r3=switch(1,$m);
$r4=switch(2,$m);
$r5=switch(3,$m);
$r6=switch(4,$m);
$r7=switch(5,$m);
$v2=switch(6,$m);
$i2=switch(7,$m);
$i3=switch(8,$m);
$idxNode=rint(2);
$wNode=switch($idxNode,A,B);
$ansNode=switch($idxNode,'I$i1=I$i2+I$i3','I$i1=I$i2+I$i3');
$idxLoop=rint(3);
$wLoop=switch($idxLoop,'left','right','outer');
$ansLoop=switch($idxLoop,'V$v1-I$i1*R$r1-I$i1*(1/(1/R$r2+1/R$r3))+V$v2-I$i2*R$r4-I$i1*R$r5=0','V$v2-I$i2*R$r4+I$i3*(1/(1/R$r6+1/R$r7))=0','$v1-I$i1*R$r1-I$i1*(1/(1/R$r2+1/R$r3))-I$i3*(1/(1/R$r6+1/R$r7))=0');@
qu.16.3.uid=c2e8f05c-d51b-4046-9021-11db49381a14@
qu.16.3.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Difficulty=Medium;
  Topic=Kirchhoff's Laws;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Diagram;
@
qu.16.3.weighting=1,1@
qu.16.3.numbering=alpha@
qu.16.3.part.1.name=sro_id_1@
qu.16.3.part.1.maple_answer=$ansNode@
qu.16.3.part.1.editing=useHTML@
qu.16.3.part.1.question=(Unset)@
qu.16.3.part.1.libname=@
qu.16.3.part.1.mode=Maple@
qu.16.3.part.1.allow2d=0@
qu.16.3.part.1.plot=@
qu.16.3.part.1.maple=is((solve($ANSWER,I1))=(solve($RESPONSE,I1)) );@
qu.16.3.part.1.type=maple@
qu.16.3.part.2.name=sro_id_2@
qu.16.3.part.2.maple_answer=$ansLoop@
qu.16.3.part.2.editing=useHTML@
qu.16.3.part.2.question=(Unset)@
qu.16.3.part.2.libname=@
qu.16.3.part.2.mode=Maple@
qu.16.3.part.2.allow2d=0@
qu.16.3.part.2.plot=@
qu.16.3.part.2.maple=is(solve($ANSWER,V2)-solve($RESPONSE,V2) = 0);@
qu.16.3.part.2.type=maple@
qu.16.3.question=<p>Consider the following circuit where the symbols have their usual meaning.&nbsp;&nbsp;State your answers in terms of the given variables and directions in the diagram.</p><p>&nbsp;</p><p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="675" height="506"><param name="image" value="__BASE_URI__img/Circuits/Kirchhoffs2L-2-NoLabel-Dir/Diagram.png" /><param name="size" value="14" /><param name="label.1.x" value="45" /><param name="label.1.y" value="150" /><param name="label.1.text" value="R$r3" /><param name="label.2.x" value="175" /><param name="label.2.y" value="150" /><param name="label.2.text" value="R$r2" /><param name="label.3.x" value="45" /><param name="label.3.y" value="250" /><param name="label.3.text" value="R$r1" /><param name="label.4.x" value="160" /><param name="label.4.y" value="365" /><param name="label.4.text" value="V$v1" /><param name="label.5.x" value="245" /><param name="label.5.y" value="365" /><param name="label.5.text" value="R$r5" /><param name="label.6.x" value="185" /><param name="label.6.y" value="320" /><param name="label.6.text" value="I$i1" /><param name="label.7.x" value="390" /><param name="label.7.y" value="250" /><param name="label.7.text" value="R$r4" /><param name="label.8.x" value="390" /><param name="label.8.y" value="170" /><param name="label.8.text" value="V$v2" /><param name="label.9.x" value="500" /><param name="label.9.y" value="220" /><param name="label.9.text" value="R$r6" /><param name="label.10.x" value="500" /><param name="label.10.y" value="390" /><param name="label.10.text" value="R$r7" /><param name="label.11.x" value="345" /><param name="label.11.y" value="140" /><param name="label.11.text" value="I$i2" /><param name="label.12.x" value="590" /><param name="label.12.y" value="205" /><param name="label.12.text" value="I$i3" /><param name="label.13.x" value="360" /><param name="label.13.y" value="350" /><param name="label.13.text" value="A" /><param name="label.14.x" value="360" /><param name="label.14.y" value="70" /><param name="label.14.text" value="B" /></applet></p><p><strong>(a)</strong>&nbsp; Use Kirchhoff's current rule to write an equation for the algebraic sum of the currents&nbsp;flowing into node $wNode.&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p><span><strong>(b)</strong>&nbsp; Use Kirchhoff's voltage rule to write&nbsp;an equation for the algebraic&nbsp;sum of the potential differences&nbsp;in the $wLoop loop of the circuit.</span></p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p>@

qu.16.4.mode=Inline@
qu.16.4.name=Kirchhoff's Laws - 3 Loops - 1@
qu.16.4.comment=<ol>
    <li>Kirchhoff's current (or node) law states that the sum of all currents entering a node is equal to the sum of all currents leaving the node.</li>
    <li>Kirchhoff's voltage (or loop) law states that the sum of all voltage gains and drops around a closed loop is equal to zero.</li>
    <li>Remember that choosing one direction around a loop is important. It does not matter <em>which</em> direction you choose to follow, but you must be consistent.</li>
    <li>When following a loop: Passing through a battery is a positive if you are going from negative to positive terminal and a negative if going from positive to negative.</li>
    <li>When following a loop: Passing through a resistor is dependant both on the direction you are following the loop<em> and the assumed direction of the current passing through the resistor</em>. If you are following the loop in the same direction as the current through the resistor, the resistor provides a negative term. If you are following the loop against the current, the resistor provides a positive term.</li>
</ol>@
qu.16.4.editing=useHTML@
qu.16.4.hint.1=Kirchhoff's current (or node) law states that the sum of all currents entering a node is equal to the sum of all currents leaving the node.@
qu.16.4.hint.2=Kirchhoff's voltage (or loop) law states that the sum of all voltage gains and drops around a closed loop is equal to zero.@
qu.16.4.hint.3=Remember that choosing one direction around a loop is important.&nbsp; It does not matter <em>which</em> direction you choose to follow, but you must be consistent.@
qu.16.4.hint.4=When following a loop:&nbsp; Passing through a battery is a positive if you are going from negative to positive terminal and a negative if going from positive to negative.@
qu.16.4.hint.5=When following a loop:&nbsp; Passing through a resistor is dependant both on the direction you are following the loop<em> and the assumed direction of the current passing through the resistor</em>.&nbsp; If you are following the loop in the same direction as the current through the resistor, the resistor provides a negative term.&nbsp; If you are following the loop against the current, the resistor provides a positive term.@
qu.16.4.solution=<p>We must choose a direction to traverse each loop and a direction for the currents in each branch.</p>
<p>If we cross a voltage source from the negative to the positive terminal, then the potential difference is positive, otherwise it is negative.</p>
<p>If we cross a resistor in the direction of current flow, then the potential difference is negative, otherwise it is positive.</p>
<p>&nbsp;</p>
<p>At a node, the sum of all currents entering must equal the sum of all currents leaving.</p>@
qu.16.4.algorithm=$r1=range(1,4);
$v1=range(1,4);
$i1=range(1,6);
$m=maple("randomize();
b:=RandomTools[Generate](choose({1,2,3,4}minus {$r1})):
c:=RandomTools[Generate](choose({1,2,3,4}minus {$r1,b})):
d:=RandomTools[Generate](choose({1,2,3,4}minus {$r1,b,c})):
bb:=RandomTools[Generate](choose({1,2,3,4}minus {$v1})):
cc:=RandomTools[Generate](choose({1,2,3,4}minus {$v1,bb})):
dd:=RandomTools[Generate](choose({1,2,3,4}minus {$v1,bb,cc})):
bbb:=RandomTools[Generate](choose({1,2,3,4,5,6}minus {$i1})):
ccc:=RandomTools[Generate](choose({1,2,3,4,5,6}minus {$i1,bbb})):
ddd:=RandomTools[Generate](choose({1,2,3,4,5,6}minus {$i1,bbb,ccc})):
eee:=RandomTools[Generate](choose({1,2,3,4,5,6}minus {$i1,bbb,ccc,ddd})):
fff:=RandomTools[Generate](choose({1,2,3,4,5,6}minus {$i1,bbb,ccc,ddd,eee})):
b,c,d,bb,cc,dd,bbb,ccc,ddd,eee,fff;
");
$r2=switch(0,$m);
$r3=switch(1,$m);
$r4=switch(2,$m);
$v2=switch(3,$m);
$v3=switch(4,$m);
$v4=switch(5,$m);
$i2=switch(6,$m);
$i3=switch(7,$m);
$i4=switch(8,$m);
$i5=switch(9,$m);
$i6=switch(10,$m);
$idxNode=rint(4);
$wNode=switch($idxNode,A,H,E,C);
$ansNode=switch($idxNode,'I$i1+I$i4=I$i2','I$i6+I$i2+I$i3=0','I$i5+I$i6+I$i1=0','I$i5-I$i3-I$i4=0');
$idxLoop=rint(7);
$wLoop=switch($idxLoop,'AHEFGA','ABCHA','CDEHC','ABCDEHA','ABCDEFGA','ABCHEFGA','CDEFGAHC');
$ansLoop=switch($idxLoop,'-V$v2+I$i6*R$r3-I$i1*R$r4+V$v1=0','I$i4*R$r1-V$v4-V$v3+V$v2=0','I$i5*R$r2-I$i6*R$r3+V$v3=0','I$i4*R$r1-V$v4+I$i5*R$r2-I$i6*R$r3+V$v2=0','I$i4*R$r1-V$v4+I$i5*R$r2-I$i1*R$r4+V$v1=0','I$i4*R$r1-V$v4-V$v3+I$i6*R$r3-I$i1*R$r4+V$v1=0','I$i5*R$r2-I$i1*R$r4+V$v1-V$v2+V$v3=0');@
qu.16.4.uid=b05a5595-13e9-4f78-b6fe-702dea6d16c9@
qu.16.4.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Difficulty=Medium;
  Topic=Kirchhoff's Laws;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Diagram;
@
qu.16.4.weighting=1,1@
qu.16.4.numbering=alpha@
qu.16.4.part.1.name=sro_id_1@
qu.16.4.part.1.maple_answer=$ansNode@
qu.16.4.part.1.editing=useHTML@
qu.16.4.part.1.question=(Unset)@
qu.16.4.part.1.libname=@
qu.16.4.part.1.mode=Maple@
qu.16.4.part.1.allow2d=0@
qu.16.4.part.1.plot=@
qu.16.4.part.1.maple=is((solve($ANSWER,I1))=(solve($RESPONSE,I1)) );@
qu.16.4.part.1.type=maple@
qu.16.4.part.2.name=sro_id_2@
qu.16.4.part.2.maple_answer=$ansLoop@
qu.16.4.part.2.editing=useHTML@
qu.16.4.part.2.question=(Unset)@
qu.16.4.part.2.libname=@
qu.16.4.part.2.mode=Maple@
qu.16.4.part.2.allow2d=0@
qu.16.4.part.2.plot=@
qu.16.4.part.2.maple=is(solve($ANSWER,V2)-solve($RESPONSE,V2) = 0);@
qu.16.4.part.2.type=maple@
qu.16.4.question=<p>Consider the following circuit where the symbols have their usual meaning.&nbsp;&nbsp;State your answers in terms of the given variables and directions in the diagram.</p><p>&nbsp;</p><p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="583" height="600"><param name="image" value="__BASE_URI__img/Circuits/Kirchhoffs3L-1-NoLabel-Dir/Diagram.png" /><param name="size" value="22" /><param name="label.1.x" value="50" /><param name="label.1.y" value="330" /><param name="label.1.text" value="A" /><param name="label.2.x" value="50" /><param name="label.2.y" value="60" /><param name="label.2.text" value="B" /><param name="label.3.x" value="265" /><param name="label.3.y" value="60" /><param name="label.3.text" value="C" /><param name="label.4.x" value="530" /><param name="label.4.y" value="60" /><param name="label.4.text" value="D" /><param name="label.5.x" value="530" /><param name="label.5.y" value="330" /><param name="label.5.text" value="E" /><param name="label.6.x" value="530" /><param name="label.6.y" value="550" /><param name="label.6.text" value="F" /><param name="label.7.x" value="50" /><param name="label.7.y" value="550" /><param name="label.7.text" value="G" /><param name="label.8.x" value="265" /><param name="label.8.y" value="345" /><param name="label.8.text" value="H" /><param name="label.9.x" value="40" /><param name="label.9.y" value="195" /><param name="label.9.text" value="R$r1" /><param name="label.10.x" value="398" /><param name="label.10.y" value="50" /><param name="label.10.text" value="R$r2" /><param name="label.11.x" value="398" /><param name="label.11.y" value="300" /><param name="label.11.text" value="R$r3" /><param name="label.12.x" value="555" /><param name="label.12.y" value="430" /><param name="label.12.text" value="R$r4" /><param name="label.13.x" value="160" /><param name="label.13.y" value="570" /><param name="label.13.text" value="V$v1" /><param name="label.14.x" value="160" /><param name="label.14.y" value="290" /><param name="label.14.text" value="V$v2" /><param name="label.15.x" value="310" /><param name="label.15.y" value="200" /><param name="label.15.text" value="V$v3" /><param name="label.16.x" value="160" /><param name="label.16.y" value="40" /><param name="label.16.text" value="V$v4" /><param name="label.17.x" value="225" /><param name="label.17.y" value="510" /><param name="label.17.text" value="I$i1" /><param name="label.18.x" value="100" /><param name="label.18.y" value="350" /><param name="label.18.text" value="I$i2" /><param name="label.19.x" value="240" /><param name="label.19.y" value="130" /><param name="label.19.text" value="I$i3" /><param name="label.20.x" value="220" /><param name="label.20.y" value="90" /><param name="label.20.text" value="I$i4" /><param name="label.21.x" value="490" /><param name="label.21.y" value="90" /><param name="label.21.text" value="I$i5" /><param name="label.22.x" value="490" /><param name="label.22.y" value="350" /><param name="label.22.text" value="I$i6" /></applet></p><p><strong>(a)</strong>&nbsp; Use Kirchhoff's current rule to write an equation for the algebraic sum of the currents&nbsp;flowing into node $wNode.&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p><span><strong>(b)</strong>&nbsp; Use Kirchhoff's voltage rule to write&nbsp;an equation for the algebraic&nbsp;sum of the potential differences&nbsp;in the $wLoop loop of the circuit.</span></p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p>@

qu.16.5.mode=Inline@
qu.16.5.name=Kirchhoff's Laws - 2 Loops - 1@
qu.16.5.comment=<ol>
    <li>Kirchhoff's current (or node) law states that the sum of all currents entering a node is equal to the sum of all currents leaving the node.</li>
    <li>Kirchhoff's voltage (or loop) law states that the sum of all voltage gains and drops around a closed loop is equal to zero.</li>
    <li>Remember that choosing one direction around a loop is important. It does not matter <em>which</em> direction you choose to follow, but you must be consistent.</li>
    <li>When following a loop: Passing through a battery is a positive if you are going from negative to positive terminal and a negative if going from positive to negative.</li>
    <li>When following a loop: Passing through a resistor is dependant both on the direction you are following the loop<em> and the assumed direction of the current passing through the resistor</em>. If you are following the loop in the same direction as the current through the resistor, the resistor provides a negative term. If you are following the loop against the current, the resistor provides a positive term.</li>
</ol>@
qu.16.5.editing=useHTML@
qu.16.5.hint.1=Kirchhoff's current (or node) law states that the sum of all currents entering a node is equal to the sum of all currents leaving the node.@
qu.16.5.hint.2=Kirchhoff's voltage (or loop) law states that the sum of all voltage gains and drops around a closed loop is equal to zero.@
qu.16.5.hint.3=Remember that choosing one direction around a loop is important.&nbsp; It does not matter <em>which</em> direction you choose to follow, but you must be consistent.@
qu.16.5.hint.4=When following a loop:&nbsp; Passing through a battery is a positive if you are going from negative to positive terminal and a negative if going from positive to negative.@
qu.16.5.hint.5=When following a loop:&nbsp; Passing through a resistor is dependant both on the direction you are following the loop<em> and the assumed direction of the current passing through the resistor</em>.&nbsp; If you are following the loop in the same direction as the current through the resistor, the resistor provides a negative term.&nbsp; If you are following the loop against the current, the resistor provides a positive term.@
qu.16.5.solution=<p>We must choose a direction to traverse each loop and a direction for the currents in each branch.</p>
<p>If we cross a voltage source from the negative to the positive terminal, then the potential difference is positive, otherwise it is negative.</p>
<p>If we cross a resistor in the direction of current flow, then the potential difference is&nbsp;negative, otherwise it is positive.</p>
<p>&nbsp;</p>
<p>At a node, the sum of all currents entering must equal the sum of all currents leaving.</p>@
qu.16.5.algorithm=$top=rint(1,6);
$m=maple("randomize();
b:=RandomTools[Generate](choose({1,2,3,4,5}minus {$top})):
c:=RandomTools[Generate](choose({1,2,3,4,5}minus {$top,b})):
d:=RandomTools[Generate](choose({1,2,3,4,5}minus {$top,b,c})):
e:=RandomTools[Generate](choose({1,2,3,4,5}minus {$top,b,c,d})):
b,c,d,e;
");
$bot=switch(0,$m);
$mid=switch(1,$m);
$lef=switch(2,$m);
$rig=switch(3,$m);
$wNode=switch(rint(2),A,B);
$idx=rint(3);
$wLoop=switch($idx,'top','bottom','outer');
$ansNode='I$top+I$mid+I$bot=0';
$ansLoop=switch($idx,'-V$top+I$top*R$top-R$mid*I$mid+V$mid+I$top*R$lef=0','-V$mid+I$mid*R$mid-R$rig*I$bot+V$bot=0','V$bot+I$top*R$lef-V$top+I$top*R$top-I$bot*R$rig=0');@
qu.16.5.uid=aae28ec4-5e44-4720-95ae-f9422f882b44@
qu.16.5.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Difficulty=Medium;
  Topic=Kirchhoff's Laws;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Diagram;
@
qu.16.5.weighting=1,1@
qu.16.5.numbering=alpha@
qu.16.5.part.1.name=sro_id_1@
qu.16.5.part.1.maple_answer=$ansNode@
qu.16.5.part.1.editing=useHTML@
qu.16.5.part.1.question=(Unset)@
qu.16.5.part.1.libname=@
qu.16.5.part.1.mode=Maple@
qu.16.5.part.1.allow2d=0@
qu.16.5.part.1.plot=@
qu.16.5.part.1.maple=is((solve($ANSWER,I1))=(solve($RESPONSE,I1)) );@
qu.16.5.part.1.type=maple@
qu.16.5.part.2.name=sro_id_2@
qu.16.5.part.2.maple_answer=$ansLoop@
qu.16.5.part.2.editing=useHTML@
qu.16.5.part.2.question=(Unset)@
qu.16.5.part.2.libname=@
qu.16.5.part.2.mode=Maple@
qu.16.5.part.2.allow2d=0@
qu.16.5.part.2.plot=@
qu.16.5.part.2.maple=is(solve($ANSWER,V2)-solve($RESPONSE,V2) = 0);@
qu.16.5.part.2.type=maple@
qu.16.5.question=<p>Consider the following circuit where the symbols have their usual meaning.&nbsp;&nbsp;State your answers in terms of the given variables and directions in the diagram.</p><p>&nbsp;</p><p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="384" height="400"><param name="image" value="__BASE_URI__img/Circuits/KirchhoffsLaws-2Loops/Diagram.png" /><param name="size" value="12" /><param name="label.1.x" value="268" /><param name="label.1.y" value="10" /><param name="label.1.text" value="R$top" /><param name="label.2.x" value="100" /><param name="label.2.y" value="10" /><param name="label.2.text" value="V$top" /><param name="label.3.x" value="268" /><param name="label.3.y" value="190" /><param name="label.3.text" value="R$mid" /><param name="label.4.x" value="100" /><param name="label.4.y" value="190" /><param name="label.4.text" value="V$mid" /><param name="label.5.x" value="100" /><param name="label.5.y" value="340" /><param name="label.5.text" value="V$bot" /><param name="label.6.x" value="377" /><param name="label.6.y" value="296" /><param name="label.6.text" value="R$rig" /><param name="label.7.x" value="8" /><param name="label.7.y" value="128" /><param name="label.7.text" value="R$lef" /><param name="label.8.x" value="143" /><param name="label.8.y" value="55" /><param name="label.8.text" value="I$top" /><param name="label.9.x" value="143" /><param name="label.9.y" value="235" /><param name="label.9.text" value="I$mid" /><param name="label.10.x" value="143" /><param name="label.10.y" value="385" /><param name="label.10.text" value="I$bot" /><param name="label.11.x" value="8" /><param name="label.11.y" value="220" /><param name="label.11.text" value="A" /><param name="label.12.x" value="377" /><param name="label.12.y" value="220" /><param name="label.12.text" value="B" /></applet></p><p><strong>(a)</strong>&nbsp; Use Kirchhoff's current rule to write an equation for the algebraic sum of the currents&nbsp;flowing into node $wNode.&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p><span><strong>(b)</strong>&nbsp; Use Kirchhoff's voltage rule to write&nbsp;an equation for the algebraic&nbsp;sum of the potential differences&nbsp;in the $wLoop loop of the circuit.</span></p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p>@

qu.16.6.mode=Inline@
qu.16.6.name=Kirchhoff's Laws - 2 Loops - Numeric - 1@
qu.16.6.comment=<ol>
    <li>Kirchhoff's current (or node) law states that the sum of all currents entering a node is equal to the sum of all currents leaving the node.</li>
    <li>Kirchhoff's voltage (or loop) law states that the sum of all voltage gains and drops around a closed loop is equal to zero.</li>
    <li>Remember that choosing one direction around a loop is important. It does not matter <em>which</em> direction you choose to follow, but you must be consistent.</li>
    <li>When following a loop: Passing through a battery is a positive if you are going from negative to positive terminal and a negative if going from positive to negative.</li>
    <li>When following a loop: Passing through a resistor is dependant both on the direction you are following the loop<em> and the assumed direction of the current passing through the resistor</em>. If you are following the loop in the same direction as the current through the resistor, the resistor provides a negative term. If you are following the loop against the current, the resistor provides a positive term.</li>
</ol>@
qu.16.6.editing=useHTML@
qu.16.6.hint.1=Kirchhoff's current (or node) law states that the sum of all currents entering a node is equal to the sum of all currents leaving the node.@
qu.16.6.hint.2=Kirchhoff's voltage (or loop) law states that the sum of all voltage gains and drops around a closed loop is equal to zero.@
qu.16.6.hint.3=Remember that choosing one direction around a loop is important.&nbsp; It does not matter <em>which</em> direction you choose to follow, but you must be consistent.@
qu.16.6.hint.4=When following a loop:&nbsp; Passing through a battery is a positive if you are going from negative to positive terminal and a negative if going from positive to negative.@
qu.16.6.hint.5=When following a loop:&nbsp; Passing through a resistor is dependant both on the direction you are following the loop<em> and the assumed direction of the current passing through the resistor</em>.&nbsp; If you are following the loop in the same direction as the current through the resistor, the resistor provides a negative term.&nbsp; If you are following the loop against the current, the resistor provides a positive term.@
qu.16.6.solution=<p>We must choose a direction to traverse each loop and a direction for the currents in each branch.</p>
<p>If we cross a voltage source from the negative to the positive terminal, then the potential difference is positive, otherwise it is negative.</p>
<p>If we cross a resistor in the direction of current flow, then the potential difference is negative, otherwise it is positive.</p>
<p>&nbsp;</p>
<p>At a node, the sum of all currents entering must equal the sum of all currents leaving.</p>@
qu.16.6.algorithm=$V1=rand(1.00,20.0,3);
$V2=rand(1.00,20.0,3);
$V3=rand(1.00,20.0,3);
$R1=rand(10.0,999,3);
$R2=rand(10.0,999,3);
$R3=rand(10.0,999,3);
$R4=rand(10.0,999,3);
$m=maple("
assign(solve({I1+I2+I3=0,$V1-I1*$R1-$V2+I2*$R3-I1*$R2=0,$V2-$V3+I3*$R4-I2*$R3=0})):
I1,I2,I3;
");
$I1=switch(0,$m)*1000;
$I2=switch(1,$m)*1000;
$I3=switch(2,$m)*1000;@
qu.16.6.uid=4933af50-23d1-4e99-ba99-0c026b91c1c4@
qu.16.6.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Difficulty=Medium;
  Topic=Kirchhoff's Laws;
  Features=Algorithmic;
  Features=Diagram;
@
qu.16.6.weighting=1,1,1@
qu.16.6.numbering=alpha@
qu.16.6.part.1.name=sro_id_1@
qu.16.6.part.1.answer.units=@
qu.16.6.part.1.numStyle= scientific  @
qu.16.6.part.1.editing=useHTML@
qu.16.6.part.1.showUnits=false@
qu.16.6.part.1.question=(Unset)@
qu.16.6.part.1.mode=Numeric@
qu.16.6.part.1.grading=exact_sigd@
qu.16.6.part.1.negStyle=both@
qu.16.6.part.1.digit=3@
qu.16.6.part.1.answer.num=$I1@
qu.16.6.part.2.name=sro_id_2@
qu.16.6.part.2.answer.units=@
qu.16.6.part.2.numStyle= scientific  @
qu.16.6.part.2.editing=useHTML@
qu.16.6.part.2.showUnits=false@
qu.16.6.part.2.question=(Unset)@
qu.16.6.part.2.mode=Numeric@
qu.16.6.part.2.grading=exact_sigd@
qu.16.6.part.2.negStyle=both@
qu.16.6.part.2.digit=3@
qu.16.6.part.2.answer.num=$I2@
qu.16.6.part.3.name=sro_id_3@
qu.16.6.part.3.answer.units=@
qu.16.6.part.3.numStyle= scientific  @
qu.16.6.part.3.editing=useHTML@
qu.16.6.part.3.showUnits=false@
qu.16.6.part.3.question=(Unset)@
qu.16.6.part.3.mode=Numeric@
qu.16.6.part.3.grading=exact_sigd@
qu.16.6.part.3.negStyle=both@
qu.16.6.part.3.digit=3@
qu.16.6.part.3.answer.num=$I3@
qu.16.6.question=<p>Given the following circuit, calculate the currents <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I1</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I2</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I3</mi></mrow></mstyle></math>, in <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mA</mi></mrow></mstyle></math>.</p><p>&nbsp;</p><p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="384" height="400"><param name="image" value="__BASE_URI__img/Circuits/KirchhoffsLaws-2Loops/Diagram.png" /><param name="size" value="10" /><param name="label.1.x" value="268" /><param name="label.1.y" value="10" /><param name="label.1.text" value="$R2 Ohm" /><param name="label.2.x" value="100" /><param name="label.2.y" value="10" /><param name="label.2.text" value="$V1 V" /><param name="label.3.x" value="268" /><param name="label.3.y" value="190" /><param name="label.3.text" value="$R3 Ohm" /><param name="label.4.x" value="100" /><param name="label.4.y" value="190" /><param name="label.4.text" value="$V2 V" /><param name="label.5.x" value="100" /><param name="label.5.y" value="340" /><param name="label.5.text" value="$V3 V" /><param name="label.6.x" value="310" /><param name="label.6.y" value="296" /><param name="label.6.text" value="$R4 Ohm" /><param name="label.7.x" value="70" /><param name="label.7.y" value="128" /><param name="label.7.text" value="$R1 Ohm" /><param name="label.8.x" value="143" /><param name="label.8.y" value="55" /><param name="label.8.text" value="I1" /><param name="label.9.x" value="143" /><param name="label.9.y" value="235" /><param name="label.9.text" value="I2" /><param name="label.10.x" value="143" /><param name="label.10.y" value="385" /><param name="label.10.text" value="I3" /></applet></p><p><strong>(a)</strong>&nbsp;&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I1</mi></mrow></mstyle></math><span>&nbsp; </span><1><span>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mA</mi></mrow></mstyle></math></span></p><p><span><strong>(b)&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I2</mi></mrow></mstyle></math>&nbsp;&nbsp;</strong><span>&nbsp;</span><2><span>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mA</mi></mrow></mstyle></math></span></span></p><p><span><span><strong>(c)</strong>&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I3</mi></mrow></mstyle></math>&nbsp;&nbsp;<span>&nbsp;</span><3><span>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mA</mi></mrow></mstyle></math></span></span></span></p>@

qu.16.7.mode=Inline@
qu.16.7.name=Kirchhoff's Laws - 3 Loops - Numeric - 2@
qu.16.7.comment=<ol>
    <li>Kirchhoff's current (or node) law states that the sum of all currents entering a node is equal to the sum of all currents leaving the node.</li>
    <li>Kirchhoff's voltage (or loop) law states that the sum of all voltage gains and drops around a closed loop is equal to zero.</li>
    <li>Remember that choosing one direction around a loop is important. It does not matter <em>which</em> direction you choose to follow, but you must be consistent.</li>
    <li>When following a loop: Passing through a battery is a positive if you are going from negative to positive terminal and a negative if going from positive to negative.</li>
    <li>When following a loop: Passing through a resistor is dependant both on the direction you are following the loop<em> and the assumed direction of the current passing through the resistor</em>. If you are following the loop in the same direction as the current through the resistor, the resistor provides a negative term. If you are following the loop against the current, the resistor provides a positive term.</li>
</ol>@
qu.16.7.editing=useHTML@
qu.16.7.hint.1=Kirchhoff's current (or node) law states that the sum of all currents entering a node is equal to the sum of all currents leaving the node.@
qu.16.7.hint.2=Kirchhoff's voltage (or loop) law states that the sum of all voltage gains and drops around a closed loop is equal to zero.@
qu.16.7.hint.3=Remember that choosing one direction around a loop is important.&nbsp; It does not matter <em>which</em> direction you choose to follow, but you must be consistent.@
qu.16.7.hint.4=When following a loop:&nbsp; Passing through a battery is a positive if you are going from negative to positive terminal and a negative if going from positive to negative.@
qu.16.7.hint.5=When following a loop:&nbsp; Passing through a resistor is dependant both on the direction you are following the loop<em> and the assumed direction of the current passing through the resistor</em>.&nbsp; If you are following the loop in the same direction as the current through the resistor, the resistor provides a negative term.&nbsp; If you are following the loop against the current, the resistor provides a positive term.@
qu.16.7.solution=<p>We must choose a direction to traverse each loop and a direction for the currents in each branch.</p>
<p>If we cross a voltage source from the negative to the positive terminal, then the potential difference is positive, otherwise it is negative.</p>
<p>If we cross a resistor in the direction of current flow, then the potential difference is negative, otherwise it is positive.</p>
<p>&nbsp;</p>
<p>At a node, the sum of all currents entering must equal the sum of all currents leaving.</p>@
qu.16.7.algorithm=$V1=rand(1.00,20.0,3);
$V2=rand(1.00,20.0,3);
$V3=rand(1.00,20.0,3);
$V4=rand(1.00,20.0,3);
$R1=rand(10.0,999,3);
$R2=rand(10.0,999,3);
$R3=rand(10.0,999,3);
$R4=rand(10.0,999,3);
$m=maple("
assign(solve({I1+I4+I2=0,I1+I5+I6=0,I4-I3-I5=0,$V1-$V2+I6*$R3-I1*$R4=0,$V2+I4*$R1-$V4-$V3=0,$V3+I5*$R2-I6*$R3=0}
));
I1,I2,I3,I4,I5,I6;
");
$I1=switch(0,$m)*1000;
$I2=switch(1,$m)*1000;
$I3=switch(2,$m)*1000;
$I4=switch(3,$m)*1000;
$I5=switch(4,$m)*1000;
$I6=switch(5,$m)*1000;@
qu.16.7.uid=38772a5f-9874-455c-8c9d-d847cdd58eea@
qu.16.7.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Difficulty=Medium;
  Topic=Kirchhoff's Laws;
  Features=Algorithmic;
  Features=Diagram;
@
qu.16.7.weighting=1,1,1,1,1,1@
qu.16.7.numbering=alpha@
qu.16.7.part.1.name=sro_id_1@
qu.16.7.part.1.answer.units=@
qu.16.7.part.1.numStyle= scientific  @
qu.16.7.part.1.editing=useHTML@
qu.16.7.part.1.showUnits=false@
qu.16.7.part.1.question=(Unset)@
qu.16.7.part.1.mode=Numeric@
qu.16.7.part.1.grading=exact_sigd@
qu.16.7.part.1.negStyle=both@
qu.16.7.part.1.digit=3@
qu.16.7.part.1.answer.num=$I1@
qu.16.7.part.2.name=sro_id_2@
qu.16.7.part.2.answer.units=@
qu.16.7.part.2.numStyle= scientific  @
qu.16.7.part.2.editing=useHTML@
qu.16.7.part.2.showUnits=false@
qu.16.7.part.2.question=(Unset)@
qu.16.7.part.2.mode=Numeric@
qu.16.7.part.2.grading=exact_sigd@
qu.16.7.part.2.negStyle=both@
qu.16.7.part.2.digit=3@
qu.16.7.part.2.answer.num=$I2@
qu.16.7.part.3.name=sro_id_3@
qu.16.7.part.3.answer.units=@
qu.16.7.part.3.numStyle= scientific  @
qu.16.7.part.3.editing=useHTML@
qu.16.7.part.3.showUnits=false@
qu.16.7.part.3.question=(Unset)@
qu.16.7.part.3.mode=Numeric@
qu.16.7.part.3.grading=exact_sigd@
qu.16.7.part.3.negStyle=both@
qu.16.7.part.3.digit=3@
qu.16.7.part.3.answer.num=$I3@
qu.16.7.part.4.name=sro_id_4@
qu.16.7.part.4.answer.units=@
qu.16.7.part.4.numStyle= scientific  @
qu.16.7.part.4.editing=useHTML@
qu.16.7.part.4.showUnits=false@
qu.16.7.part.4.question=(Unset)@
qu.16.7.part.4.mode=Numeric@
qu.16.7.part.4.grading=exact_sigd@
qu.16.7.part.4.negStyle=both@
qu.16.7.part.4.digit=3@
qu.16.7.part.4.answer.num=$I4@
qu.16.7.part.5.name=sro_id_5@
qu.16.7.part.5.answer.units=@
qu.16.7.part.5.numStyle= scientific  @
qu.16.7.part.5.editing=useHTML@
qu.16.7.part.5.showUnits=false@
qu.16.7.part.5.question=(Unset)@
qu.16.7.part.5.mode=Numeric@
qu.16.7.part.5.grading=exact_sigd@
qu.16.7.part.5.negStyle=both@
qu.16.7.part.5.digit=3@
qu.16.7.part.5.answer.num=$I5@
qu.16.7.part.6.name=sro_id_6@
qu.16.7.part.6.answer.units=@
qu.16.7.part.6.numStyle= scientific  @
qu.16.7.part.6.editing=useHTML@
qu.16.7.part.6.showUnits=false@
qu.16.7.part.6.question=(Unset)@
qu.16.7.part.6.mode=Numeric@
qu.16.7.part.6.grading=exact_sigd@
qu.16.7.part.6.negStyle=both@
qu.16.7.part.6.digit=3@
qu.16.7.part.6.answer.num=$I6@
qu.16.7.question=<p>Given the following circuit, calculate the currents <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I1</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I2</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I3</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I4</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I5</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I6</mi></mrow></mstyle></math>&nbsp;in <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mA</mi></mrow></mstyle></math>.&nbsp;</p><p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="583" height="600"><param name="image" value="__BASE_URI__img/Circuits/Kirchhoffs3L-2-NoLabel-Dir/Diagram.png" /><param name="size" value="14" /><param name="label.1.x" value="110" /><param name="label.1.y" value="195" /><param name="label.1.text" value="$R1 Ohm" /><param name="label.2.x" value="398" /><param name="label.2.y" value="50" /><param name="label.2.text" value="$R2 Ohm" /><param name="label.3.x" value="398" /><param name="label.3.y" value="300" /><param name="label.3.text" value="$R3 Ohm" /><param name="label.4.x" value="480" /><param name="label.4.y" value="430" /><param name="label.4.text" value="$R4 Ohm" /><param name="label.5.x" value="160" /><param name="label.5.y" value="570" /><param name="label.5.text" value="$V1 V" /><param name="label.6.x" value="160" /><param name="label.6.y" value="290" /><param name="label.6.text" value="$V2 V" /><param name="label.7.x" value="315" /><param name="label.7.y" value="200" /><param name="label.7.text" value="$V3 V" /><param name="label.8.x" value="160" /><param name="label.8.y" value="110" /><param name="label.8.text" value="$V4 V" /><param name="label.9.x" value="225" /><param name="label.9.y" value="510" /><param name="label.9.text" value="I1" /><param name="label.10.x" value="240" /><param name="label.10.y" value="350" /><param name="label.10.text" value="I2" /><param name="label.11.x" value="245" /><param name="label.11.y" value="275" /><param name="label.11.text" value="I3" /><param name="label.12.x" value="220" /><param name="label.12.y" value="90" /><param name="label.12.text" value="I4" /><param name="label.13.x" value="490" /><param name="label.13.y" value="90" /><param name="label.13.text" value="I5" /><param name="label.14.x" value="490" /><param name="label.14.y" value="350" /><param name="label.14.text" value="I6" /></applet></p><p align="left"><strong>(a)</strong>&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I1</mi></mrow></mstyle></math>&nbsp;&nbsp;<span>&nbsp;</span><1><span>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mA</mi></mrow></mstyle></math></span></p><p><span><strong>(b)</strong>&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I2</mi></mrow></mstyle></math>&nbsp;&nbsp;<span>&nbsp;</span><2><span>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mA</mi></mrow></mstyle></math></span></span></p><p><span><span><strong>(c)</strong>&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I3</mi></mrow></mstyle></math>&nbsp;&nbsp;<span>&nbsp;</span><3><span>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mA</mi></mrow></mstyle></math></span></span></span></p><p><span><span><span><strong>(d)</strong>&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I4</mi></mrow></mstyle></math>&nbsp;&nbsp;<span>&nbsp;</span><4><span>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mA</mi></mrow></mstyle></math></span></span></span></span></p><p><span><span><span><span><strong>(e)</strong>&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I5</mi></mrow></mstyle></math>&nbsp;&nbsp;<span>&nbsp;</span><5><span>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mA</mi></mrow></mstyle></math></span></span></span></span></span></p><p><span><span><span><span><span><strong>(f)</strong>&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I6</mi></mrow></mstyle></math>&nbsp;&nbsp;<span>&nbsp;</span><6><span>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mA</mi></mrow></mstyle></math></span></span></span></span></span></span></p>@

qu.16.8.mode=Inline@
qu.16.8.name=Kirchhoff's Laws - 2 Loops - Numeric - 2@
qu.16.8.comment=<ol>
    <li>Kirchhoff's current (or node) law states that the sum of all currents entering a node is equal to the sum of all currents leaving the node.</li>
    <li>Kirchhoff's voltage (or loop) law states that the sum of all voltage gains and drops around a closed loop is equal to zero.</li>
    <li>Remember that choosing one direction around a loop is important. It does not matter <em>which</em> direction you choose to follow, but you must be consistent.</li>
    <li>When following a loop: Passing through a battery is a positive if you are going from negative to positive terminal and a negative if going from positive to negative.</li>
    <li>When following a loop: Passing through a resistor is dependant both on the direction you are following the loop<em> and the assumed direction of the current passing through the resistor</em>. If you are following the loop in the same direction as the current through the resistor, the resistor provides a negative term. If you are following the loop against the current, the resistor provides a positive term.</li>
</ol>@
qu.16.8.editing=useHTML@
qu.16.8.hint.1=Kirchhoff's current (or node) law states that the sum of all currents entering a node is equal to the sum of all currents leaving the node.@
qu.16.8.hint.2=Kirchhoff's voltage (or loop) law states that the sum of all voltage gains and drops around a closed loop is equal to zero.@
qu.16.8.hint.3=Remember that choosing one direction around a loop is important.&nbsp; It does not matter <em>which</em> direction you choose to follow, but you must be consistent.@
qu.16.8.hint.4=When following a loop:&nbsp; Passing through a battery is a positive if you are going from negative to positive terminal and a negative if going from positive to negative.@
qu.16.8.hint.5=When following a loop:&nbsp; Passing through a resistor is dependant both on the direction you are following the loop<em> and the assumed direction of the current passing through the resistor</em>.&nbsp; If you are following the loop in the same direction as the current through the resistor, the resistor provides a negative term.&nbsp; If you are following the loop against the current, the resistor provides a positive term.@
qu.16.8.solution=<p>We must choose a direction to traverse each loop and a direction for the currents in each branch.</p>
<p>If we cross a voltage source from the negative to the positive terminal, then the potential difference is positive, otherwise it is negative.</p>
<p>If we cross a resistor in the direction of current flow, then the potential difference is negative, otherwise it is positive.</p>
<p>&nbsp;</p>
<p>At a node, the sum of all currents entering must equal the sum of all currents leaving.</p>@
qu.16.8.algorithm=$i1=1;
$i2=2;
$i3=3;
$V1=rand(1.00,20.0,3);
$V2=rand(1.00,20.0,3);
$V3=rand(1.00,20.0,3);
$R1=rand(10.0,999,3);
$R2=rand(10.0,999,3);
$R3=rand(10.0,999,3);
$R4=rand(10.0,999,3);
$R5=rand(10.0,999,3);
$R6=rand(10.0,999,3);
$R7=rand(10.0,999,3);
$m=maple("
assign(solve({I1=I2+I3,$V1-$R1*I1-I1*(1/(1/$R2+1/$R3))+$V2-I2*$R4-I1*$R5=0,$V2-I2*$R4+I3*(1/(1/$R6+1/$R7))=0}));
I1,I2,I3;
");
$I1=switch(0,$m)*1000;
$I2=switch(1,$m)*1000;
$I3=switch(2,$m)*1000;@
qu.16.8.uid=6ed3d363-171a-412a-b90e-42763b83ffc3@
qu.16.8.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Difficulty=Medium;
  Topic=Kirchhoff's Laws;
  Features=Algorithmic;
  Features=Diagram;
@
qu.16.8.weighting=1,1,1@
qu.16.8.numbering=alpha@
qu.16.8.part.1.name=sro_id_1@
qu.16.8.part.1.answer.units=@
qu.16.8.part.1.numStyle= scientific  @
qu.16.8.part.1.editing=useHTML@
qu.16.8.part.1.showUnits=false@
qu.16.8.part.1.question=(Unset)@
qu.16.8.part.1.mode=Numeric@
qu.16.8.part.1.grading=exact_sigd@
qu.16.8.part.1.negStyle=both@
qu.16.8.part.1.digit=3@
qu.16.8.part.1.answer.num=$I1@
qu.16.8.part.2.name=sro_id_2@
qu.16.8.part.2.answer.units=@
qu.16.8.part.2.numStyle= scientific  @
qu.16.8.part.2.editing=useHTML@
qu.16.8.part.2.showUnits=false@
qu.16.8.part.2.question=(Unset)@
qu.16.8.part.2.mode=Numeric@
qu.16.8.part.2.grading=exact_sigd@
qu.16.8.part.2.negStyle=both@
qu.16.8.part.2.digit=3@
qu.16.8.part.2.answer.num=$I2@
qu.16.8.part.3.name=sro_id_3@
qu.16.8.part.3.answer.units=@
qu.16.8.part.3.numStyle= scientific  @
qu.16.8.part.3.editing=useHTML@
qu.16.8.part.3.showUnits=false@
qu.16.8.part.3.question=(Unset)@
qu.16.8.part.3.mode=Numeric@
qu.16.8.part.3.grading=exact_sigd@
qu.16.8.part.3.negStyle=both@
qu.16.8.part.3.digit=3@
qu.16.8.part.3.answer.num=$I3@
qu.16.8.question=<p>Given the following circuit, calculate the currents <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I1</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I2</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I3</mi></mrow></mstyle></math>&nbsp;in <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mA</mi></mrow></mstyle></math>.&nbsp;</p><p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="675" height="506"><param name="image" value="__BASE_URI__img/Circuits/Kirchhoffs2L-2-NoLabel-Dir/Diagram.png" /><param name="size" value="12" /><param name="label.1.x" value="30" /><param name="label.1.y" value="150" /><param name="label.1.text" value="$R3 Ohm" /><param name="label.2.x" value="185" /><param name="label.2.y" value="150" /><param name="label.2.text" value="$R2 Ohm" /><param name="label.3.x" value="30" /><param name="label.3.y" value="250" /><param name="label.3.text" value="$R1 Ohm" /><param name="label.4.x" value="160" /><param name="label.4.y" value="365" /><param name="label.4.text" value="$V1 V" /><param name="label.5.x" value="245" /><param name="label.5.y" value="365" /><param name="label.5.text" value="$R5 Ohm" /><param name="label.6.x" value="185" /><param name="label.6.y" value="320" /><param name="label.6.text" value="I1" /><param name="label.7.x" value="285" /><param name="label.7.y" value="250" /><param name="label.7.text" value="$R4 Ohm" /><param name="label.8.x" value="425" /><param name="label.8.y" value="170" /><param name="label.8.text" value="$V2 V" /><param name="label.9.x" value="500" /><param name="label.9.y" value="220" /><param name="label.9.text" value="$R6 Ohm" /><param name="label.10.x" value="500" /><param name="label.10.y" value="390" /><param name="label.10.text" value="$R7 Ohm" /><param name="label.11.x" value="345" /><param name="label.11.y" value="140" /><param name="label.11.text" value="I2" /><param name="label.12.x" value="590" /><param name="label.12.y" value="205" /><param name="label.12.text" value="I3" /></applet></p><p align="left"><strong>(a)</strong>&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I1</mi></mrow></mstyle></math>&nbsp;&nbsp;<span>&nbsp;</span><1><span>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mA</mi></mrow></mstyle></math></span></p><p><span><strong>(b)</strong>&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I2</mi></mrow></mstyle></math>&nbsp;&nbsp;<span>&nbsp;</span><2><span>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mA</mi></mrow></mstyle></math></span></span></p><p><span><span><strong>(c)</strong>&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I3</mi></mrow></mstyle></math>&nbsp;&nbsp;<span>&nbsp;</span><3><span>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>mA</mi></mrow></mstyle></math></span></span></span></p>@

qu.17.topic=EMF@

qu.17.1.question=<p><img alt="" align="middle" width="609" height="150" src="__BASE_URI__img/EMF/Battery-Terminal-Voltage-2/Diagram.png" /></p>
<p><span><br />
A battery with&nbsp;internal resistance&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$r</mi></mrow><mrow><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&Omega;</mi></mrow></mstyle></math>&nbsp;and terminal voltage <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$Vterm</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>V</mi></mrow></mstyle></math>&nbsp;has a current&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mstyle></math>&nbsp;passing<br />
&nbsp;through it in the direction indicated.&nbsp; What is the internal voltage of the battery?</span></p>
<p>&nbsp;</p>@
qu.17.1.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.17.1.allow2d=0@
qu.17.1.maple_answer=SigFigs[roundToSigFigs]($V,3)*V@
qu.17.1.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.17.1.type=maple@
qu.17.1.mode=Maple@
qu.17.1.name=Battery - Terminal Voltage 2 - Find EMF ~ PGc@
qu.17.1.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.17.1.editing=useHTML@
qu.17.1.solution=<p>The terminal voltage is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>V</mi><mrow><mi>AB</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>V</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>Ir</mi></mrow></mstyle></math>.</p>
<p>Thus:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>V</mi><mrow><mi>AB</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>Ir</mi></mrow></mstyle></math>.</p>@
qu.17.1.algorithm=$I=rand(0.300,0.999,3);
$r=rand(2,8,3);
$Vterm=rand(5,15,3);
$V=$Vterm-$r*$I;@
qu.17.1.uid=51edea24-abce-4cf5-b25e-051297c85526@
qu.17.1.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Terminal Voltage;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.17.2.question=<p><img alt="" align="middle" width="609" height="150" src="__BASE_URI__img/EMF/Battery-Terminal-Voltage-1/Diagram.png" /></p>
<p><span><br />
A battery with internal voltage <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$V</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>V</mi></mrow></mstyle></math>&nbsp;and internal resistance <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$r</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mi>&Omega;</mi></mrow></mrow></mstyle></math>&nbsp;has a current&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mstyle></math>&nbsp;passing<br />
&nbsp;through it in the direction indicated.&nbsp; What is the terminal voltage <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>V</mi><mrow><mi>AB</mi></mrow></msub></mrow></mstyle></math>&nbsp;of the battery?</span></p>@
qu.17.2.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.17.2.allow2d=0@
qu.17.2.maple_answer=SigFigs[roundToSigFigs]($ans,3)*V@
qu.17.2.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.17.2.type=maple@
qu.17.2.mode=Maple@
qu.17.2.name=Battery - Terminal Voltage 1 - Find V_term ~ PGc@
qu.17.2.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.17.2.editing=useHTML@
qu.17.2.solution=<p>The terminal voltage is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>V</mi><mrow><mi>AB</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>V</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>Ir</mi></mrow></mstyle></math>.</p>@
qu.17.2.algorithm=$r=rand(1.00,9.99,3);
$I=rand(0.300,0.999,3);
$V=rand(10.0,19.9,3);
$ans=$V-$I*$r;@
qu.17.2.uid=969e6cc0-9af1-4094-a3d3-185694b13701@
qu.17.2.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Terminal Voltage;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.17.3.question=<p><img alt="" align="middle" width="609" height="150" src="__BASE_URI__img/EMF/Battery-Terminal-Voltage-1/Diagram.png" /></p>
<p><span><br />
A battery with&nbsp;internal resistance&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$r</mi></mrow><mrow><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&Omega;</mi></mrow></mstyle></math>&nbsp;and terminal voltage <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$Vterm</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>V</mi></mrow></mstyle></math>&nbsp;has a current&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mstyle></math>&nbsp;passing<br />
&nbsp;through it in the direction indicated.&nbsp; What is the internal voltage of the battery?</span></p>@
qu.17.3.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.17.3.allow2d=0@
qu.17.3.maple_answer=SigFigs[roundToSigFigs]($V,3)*V@
qu.17.3.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.17.3.type=maple@
qu.17.3.mode=Maple@
qu.17.3.name=Battery - Terminal Voltage 1 - Find EMF ~ PGc@
qu.17.3.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.17.3.editing=useHTML@
qu.17.3.solution=<p>The terminal voltage is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>V</mi><mrow><mi>AB</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>V</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>Ir</mi></mrow></mstyle></math>.</p>
<p>Thus:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>V</mi><mrow><mi>AB</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>Ir</mi></mrow></mstyle></math>.</p>@
qu.17.3.algorithm=$I=rand(0.300,0.999,3);
$r=rand(2,8,3);
$Vterm=rand(5,15,3);
$V=$Vterm+$r*$I;@
qu.17.3.uid=c560d2cb-81c7-4766-9615-5889714734b5@
qu.17.3.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Terminal Voltage;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.17.4.question=<p><img alt="" align="middle" width="609" height="150" src="__BASE_URI__img/EMF/Battery-Terminal-Voltage-2/Diagram.png" /></p>
<p><span><br />
A battery with internal voltage <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$V</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>V</mi></mrow></mstyle></math>&nbsp;and terminal voltage <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$Vterm</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>V</mi></mrow></mstyle></math>&nbsp;has a current&nbsp;passing<br />
&nbsp;through it in the direction indicated.&nbsp; If the internal resistance of the battery is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$r</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mi>&Omega;</mi></mrow></mrow></mstyle></math>, what is the<br />
&nbsp;magnitude of the current?</span></p>@
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qu.17.4.allow2d=0@
qu.17.4.maple_answer=SigFigs[roundToSigFigs]($ans,3)*A@
qu.17.4.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.17.4.type=maple@
qu.17.4.mode=Maple@
qu.17.4.name=Battery - Terminal Voltage 2 - Find Current ~ PGc@
qu.17.4.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.17.4.editing=useHTML@
qu.17.4.solution=<p>The terminal voltage is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>V</mi><mrow><mi>AB</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>V</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>Ir</mi></mrow></mstyle></math>.</p>
<p>Thus:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>V</mi><mrow><mi>AB</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>V</mi></mrow><mrow><mi>r</mi></mrow></mfrac></mrow></mstyle></math>.</p>@
qu.17.4.algorithm=$Vterm=rand(10.0,19.9,3);
$V=rand(5,($Vterm-0.5),3);
$r=rand(2.0,9.0,3);
$ans=(-$V+$Vterm)/$r;@
qu.17.4.uid=5712fad8-8384-43d3-8806-0025d6bf7482@
qu.17.4.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Terminal Voltage;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.17.5.question=<p><img alt="" align="middle" width="609" height="150" src="__BASE_URI__img/EMF/Battery-Terminal-Voltage-2/Diagram.png" /></p>
<p><span><br />
A battery with internal voltage <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$V</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>V</mi></mrow></mstyle></math>&nbsp;and terminal voltage <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$Vterm</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>V</mi></mrow></mstyle></math>&nbsp;has a current&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mstyle></math>&nbsp;passing<br />
&nbsp;through it in the direction indicated.&nbsp; What is the internal resistance&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$r</mi></mrow></mstyle></math>&nbsp;of the battery?</span></p>
<p><span><span><em>Note:&nbsp; Enter <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ohm</mi></mrow></mstyle></math>&nbsp;for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&Omega;</mi></mrow></mstyle></math>.</em></span></span></p>@
qu.17.5.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.17.5.allow2d=0@
qu.17.5.maple_answer=SigFigs[roundToSigFigs]($ans,3)*ohm@
qu.17.5.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.17.5.type=maple@
qu.17.5.mode=Maple@
qu.17.5.name=Battery - Terminal Voltage 2 - Find r ~ PGc@
qu.17.5.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.17.5.editing=useHTML@
qu.17.5.solution=<p>The terminal voltage is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>V</mi><mrow><mi>AB</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>V</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>Ir</mi></mrow></mstyle></math>.</p>
<p>Thus:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>V</mi><mrow><mi>AB</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>V</mi></mrow><mrow><mi>I</mi></mrow></mfrac></mrow></mstyle></math>.</p>@
qu.17.5.algorithm=$I=rand(0.300,0.999,3);
$Vterm=rand(10.0,19.9,3);
$V=rand(5,($Vterm-0.5),3);
$ans=(-$V+$Vterm)/$I;@
qu.17.5.uid=e01369ad-0d29-4e7a-bdf6-a3f3a238947d@
qu.17.5.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Terminal Voltage;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.17.6.question=<p><img alt="" align="middle" width="609" height="150" src="__BASE_URI__img/EMF/Battery-Terminal-Voltage-2/Diagram.png" /></p>
<p><span><br />
A battery with internal voltage <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$V</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>V</mi></mrow></mstyle></math>&nbsp;and internal resistance <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$r</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mi>&Omega;</mi></mrow></mrow></mstyle></math>&nbsp;has a current&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mstyle></math>&nbsp;passing<br />
&nbsp;through it in the direction indicated.&nbsp; What is the terminal voltage <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>V</mi><mrow><mi>AB</mi></mrow></msub></mrow></mstyle></math>&nbsp;of the battery?</span></p>@
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qu.17.6.allow2d=0@
qu.17.6.maple_answer=SigFigs[roundToSigFigs]($ans,3)*V@
qu.17.6.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.17.6.type=maple@
qu.17.6.mode=Maple@
qu.17.6.name=Battery - Terminal Voltage 2 - Find V_term ~ PGc@
qu.17.6.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.17.6.editing=useHTML@
qu.17.6.solution=<p>The terminal voltage is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>V</mi><mrow><mi>AB</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>V</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>Ir</mi></mrow></mstyle></math>.</p>@
qu.17.6.algorithm=$r=rand(1.00,9.99,3);
$I=rand(0.300,0.999,3);
$V=rand(10.0,19.9,3);
$ans=$V+$I*$r;@
qu.17.6.uid=90bf862f-76db-4c8b-ada2-82a32d630641@
qu.17.6.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Terminal Voltage;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.17.7.question=<p><img alt="" align="middle" width="609" height="150" src="__BASE_URI__img/EMF/Battery-Terminal-Voltage-1/Diagram.png" /></p>
<p><span><br />
A battery with internal voltage <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$V</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>V</mi></mrow></mstyle></math>&nbsp;and terminal voltage <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$Vterm</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>V</mi></mrow></mstyle></math>&nbsp;has a current&nbsp;passing<br />
&nbsp;through it in the direction indicated.&nbsp; If the internal resistance of the battery is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$r</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mi>&Omega;</mi></mrow></mrow></mstyle></math>, what is the<br />
&nbsp;magnitude of the current?</span></p>@
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qu.17.7.name=Battery - Terminal Voltage 1 - Find Current ~ PGc@
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qu.17.7.editing=useHTML@
qu.17.7.solution=<p>The terminal voltage is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>V</mi><mrow><mi>AB</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>V</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>Ir</mi></mrow></mstyle></math>.</p>@
qu.17.7.algorithm=$V=rand(10.0,19.9,3);
$Vterm=rand(5,($V-0.5),3);
$r=rand(2.0,9.0,3);
$ans=($V-$Vterm)/$r;@
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qu.17.7.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Terminal Voltage;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.17.8.question=<p><img alt="" align="middle" width="609" height="150" src="__BASE_URI__img/EMF/Battery-Terminal-Voltage-1/Diagram.png" /></p>
<p><span><br />
A battery with internal voltage <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$V</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>V</mi></mrow></mstyle></math>&nbsp;and terminal voltage <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$Vterm</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>V</mi></mrow></mstyle></math>&nbsp;has a current&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mstyle></math>&nbsp;passing<br />
&nbsp;through it in the direction indicated.&nbsp; What is the internal resistance&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$r</mi></mrow></mstyle></math>&nbsp;of the battery?</span></p>
<p><span><em>Note:&nbsp; Enter <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ohm</mi></mrow></mstyle></math>&nbsp;for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&Omega;</mi></mrow></mstyle></math>.</em></span></p>@
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qu.17.8.allow2d=0@
qu.17.8.maple_answer=SigFigs[roundToSigFigs]($ans,3)*ohm@
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qu.17.8.type=maple@
qu.17.8.mode=Maple@
qu.17.8.name=Battery - Terminal Voltage 1 - Find r ~ PGc@
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qu.17.8.editing=useHTML@
qu.17.8.solution=<p>The terminal voltage is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>V</mi><mrow><mi>AB</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>V</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>Ir</mi></mrow></mstyle></math>.</p>
<p>Thus:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>V</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mi>V</mi><mrow><mi>AB</mi></mrow></msub></mrow><mrow><mi>I</mi></mrow></mfrac></mrow></mstyle></math></p>@
qu.17.8.algorithm=$I=rand(0.300,0.999,3);
$V=rand(10.0,19.9,3);
$Vterm=rand(5,($V-0.5),3);
$ans=($V-$Vterm)/$I;@
qu.17.8.uid=d3d54c36-a9e7-49ba-9ef1-13653371b74a@
qu.17.8.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Terminal Voltage;
  Difficulty=Easy;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.18.topic=Magnetic Fields@

qu.18.1.question=<p>A charge&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$q</mi></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$qex</mi></mrow></msup><mi>C</mi></mrow></mstyle></math>&nbsp;is travelling with velocity <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>v</mi><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$v1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$vex</mi></mrow></msup><mfrac><mi>m</mi><mrow><mi>s</mi></mrow></mfrac></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mover><mi>$vdir</mi><mi>&#x005e;</mi></mover></mrow></mstyle></math>.&nbsp;<br />
What is the magnetic field at a point&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>r</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$r1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>cm</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mover><mi>$rdir</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>&nbsp;from the&nbsp;location of the charge?&nbsp;</p>
<p>&nbsp;</p>
<p>&nbsp;</p>
<p><em>Entry Notes</em>:&nbsp;<br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ihat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>jhat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>&nbsp; and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>khat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp;<br />
Ensure that you explicitly enter the multiplication symbol * between terms.</p>@
qu.18.1.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uVS=false,uUN);@
qu.18.1.allow2d=0@
qu.18.1.maple_answer=with(Physics[Vectors]);
v:=($v1*10^($vex))*subs({i=_i,j=_j,k=_k},$vdir);
rr:=($r1*10^(-2))*subs({i=_i,j=_j,k=_k},$rdir);
temp:=(10^(-7))*($q*10^($qex))*(v &x rr)/(Norm(rr)^3);
SigFigs[roundToSigFigs](Component(temp,1),3)*N*ihat+SigFigs[roundToSigFigs](Component(temp,2),3)*N*jhat+SigFigs[roundToSigFigs](Component(temp,3),3)*N*khat;@
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qu.18.1.type=maple@
qu.18.1.mode=Maple@
qu.18.1.name=Magnetic Field Due to Moving Charge - 1D@
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qu.18.1.editing=useHTML@
qu.18.1.hint.1=Set up a determinant.&nbsp; The first row should be <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp; The second row should be the components of the first vector and the third row should be the components of the second vector.&nbsp; Evaluate the determinant.@
qu.18.1.solution=<p>The magnetic field due to a moving charged particle is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mrow><mn>4</mn><mrow><mi>&pi;</mi></mrow></mrow></mfrac><mfrac><mrow><mi>q</mi><mrow><mover><mrow><mi>v</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow><mrow><mover><mrow><mi></mi></mrow><mi></mi></mover></mrow><mrow><mover><mrow><mi>r</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow><mrow><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mstyle></math>.</p>
<p>We are given <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>q</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>v</mi></mrow><mi>&rarr;</mi></mover></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>r</mi></mrow><mi>&rarr;</mi></mover></mrow></mstyle></math>.&nbsp; Use the fact that <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>r</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mover><mrow><mi>r</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mfenced open='|' close='|' separators=','><mrow><mover><mrow><mi>r</mi></mrow><mi>&rarr;</mi></mover></mrow></mfenced></mrow></mfrac></mrow></mrow></mstyle></math>, and that <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='|' close='|' separators=','><mrow><mover><mrow><mi>r</mi></mrow><mi>&rarr;</mi></mover></mrow></mfenced></mrow></mstyle></math>.</p>@
qu.18.1.algorithm=$v1=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$r1=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$vex=range(2,7);
$q=switch(rint(2),rand(1.00,9.99,3),-rand(1.00,9.99,3));
$qex=range(-19,-5);
$vdir=switch(rint(0,3),'i','j','k');
$rdir=switch(rint(0,3),'i','j','k');
$vDisp='

<math><mrow><mn>$v1</mn><mover><mi>$vdir</mi><mo>&Hat;</mo></mover></mrow></math>';
$rDisp='

<math><mrow><mn>$r1</mn><mover><mi>$rdir</mi><mo>&Hat;</mo></mover></mrow></math>';@
qu.18.1.uid=03a0c1d3-5f5d-475a-a36e-e0239193b92d@
qu.18.1.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Magnetic Field Due to Moving Charges;
  Difficulty=Hard;
  Features=Algorithmic;
  Features=Partial Grading;
@

qu.18.2.question=<p>The wires <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>&nbsp;are carrying a current <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;is carrying current <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$cur2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>, each in the&nbsp;<br />
directions indicated in the diagram.&nbsp; In&nbsp;terms of the given parameters, provide a vector&nbsp;expression for&nbsp;<br />
the&nbsp;force per unit length exerted by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>&nbsp;on <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>, for&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;placed at&nbsp;any point on the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></mstyle></math>axis.</p>
<p>&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="500" height="505">
<param name="image" value="__BASE_URI__img/MagneticFields/CurrentCarryingWiresVertical/Diagram3Wires2-2.png" />
<param name="size" value="3" />
<param name="label.1.x" value="232" />
<param name="label.1.y" value="215" />
<param name="label.1.text" value="$mag$var" />
<param name="label.2.x" value="232" />
<param name="label.2.y" value="315" />
<param name="label.2.text" value="$mag$var" />
<param name="label.3.x" value="315" />
<param name="label.3.y" value="257" />
<param name="label.3.text" value="x" /></applet></p>
<p>&nbsp;</p>
<p><em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>xhat</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>yhat</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>z</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>zhat</mi></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
Enter your response in terms of the given parameters and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>, entering&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi></mrow></mstyle></math> instead of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>.</p>@
qu.18.2.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents);@
qu.18.2.allow2d=0@
qu.18.2.maple_answer=(-xhat)*($cur2*I)*2*(u*$cur*I/(2*Pi*sqrt(x^2+($mag*$var)^2)))*((x)/sqrt(x^2+($mag*$var)^2));@
qu.18.2.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.18.2.type=maple@
qu.18.2.mode=Maple@
qu.18.2.name=Current Carrying Wires - Vertical - 3 Wires - 2-2@
qu.18.2.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.18.2.editing=useHTML@
qu.18.2.solution=<p>We will calculate the magnetic field due to wires <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>, and then the force due to this magnetic field on <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>.</p>
<p>The magnitude of the magnetic field due to an infinite current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>&pi;r</mi></mrow></mfrac></mrow></mstyle></math>.&nbsp; The direction of the magnetic field is found using the right-hand rule, where the thumb points in the direction of current flow and the curl of the fingers represents the direction of the magnetic field.&nbsp; If we draw a line from the wire to the point in question, then the magnetic field will always be perpendicular to that line.</p>
<p>&nbsp;</p>
<p>At point&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math> the magnitude of the magnetic field due to each of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math> and&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math> will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow></mfrac></mrow></mstyle></math>.</p>
<p>Based on the configuration, the horizontal components will cancel.&nbsp; Thus, we need only calculate the&nbsp;vertical components, which are:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>B</mi><mrow><mi>y</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>, and therefore the total magnetic field at&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mn>2</mn><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac><mover><mrow><mi>y</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>Finally, the force due to a magnetic field on a current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>I</mi><mrow><mover><mrow><mi>l</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math>.</p>
<p>The&nbsp;current&nbsp;flows perpendicular to the magnetic field, so:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>F</mi><mrow><mi>l</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>IB</mi></mrow></mstyle></math>.&nbsp; Using the right-hand rule, we get the direction of the force, thus:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mrow><mi>l</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><mn>2</mn><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac><mover><mrow><mi>x</mi></mrow><mi>&#x005e;</mi></mover></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>.</p>@
qu.18.2.algorithm=$var=switch($rint(5),'a','b','c','d');
$mag=range(1,9);
$cur=range(1,9);
$cur2=range(1,9);@
qu.18.2.uid=5f648921-ea06-4857-b7c8-13aaeb622922@
qu.18.2.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Force Between Current Carrying Wires;
  Difficulty=Hard;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Partial Grading;
@

qu.18.3.question=<p>The wires <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>&nbsp;are carrying a current <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;is carrying current <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$cur2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>, each in the&nbsp;<br />
directions indicated in the diagram.&nbsp; In&nbsp;terms of the given parameters, provide a vector&nbsp;expression for&nbsp;<br />
the&nbsp;force per unit length exerted by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>&nbsp;on <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>, for&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;placed at&nbsp;any point on the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></mstyle></math>axis.</p>
<p>&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="500" height="505">
<param name="image" value="__BASE_URI__img/MagneticFields/CurrentCarryingWiresVertical/Diagram3Wires4-2.png" />
<param name="size" value="3" />
<param name="label.1.x" value="232" />
<param name="label.1.y" value="215" />
<param name="label.1.text" value="$mag$var" />
<param name="label.2.x" value="232" />
<param name="label.2.y" value="315" />
<param name="label.2.text" value="$mag$var" />
<param name="label.3.x" value="315" />
<param name="label.3.y" value="257" />
<param name="label.3.text" value="x" /></applet></p>
<p>&nbsp;</p>
<p><em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>xhat</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>yhat</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>z</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>zhat</mi></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
Enter your response in terms of the given parameters and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>, entering&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi></mrow></mstyle></math> instead of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>.</p>@
qu.18.3.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents);@
qu.18.3.allow2d=0@
qu.18.3.maple_answer=+xhat*($cur2*I)*2*(u*$cur*I/(2*Pi*sqrt(x^2+($mag*$var)^2)))*((x)/sqrt(x^2+($mag*$var)^2));@
qu.18.3.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.18.3.type=maple@
qu.18.3.mode=Maple@
qu.18.3.name=Current Carrying Wires - Vertical - 3 Wires - 4-2@
qu.18.3.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.18.3.editing=useHTML@
qu.18.3.solution=<p>We will calculate the magnetic field due to wires <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>, and then the force due to this magnetic field on <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>.</p>
<p>The magnitude of the magnetic field due to an infinite current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>&pi;r</mi></mrow></mfrac></mrow></mstyle></math>.&nbsp; The direction of the magnetic field is found using the right-hand rule, where the thumb points in the direction of current flow and the curl of the fingers represents the direction of the magnetic field.&nbsp; If we draw a line from the wire to the point in question, then the magnetic field will always be perpendicular to that line.</p>
<p>&nbsp;</p>
<p>At point&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math> the magnitude of the magnetic field due to each of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math> and&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math> will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow></mfrac></mrow></mstyle></math>.</p>
<p>Based on the configuration, the horizontal components will cancel.&nbsp; Thus, we need only calculate the&nbsp;vertical components, which are:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>B</mi><mrow><mi>y</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>-&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>, and therefore the total magnetic field at&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><mn>2</mn><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac><mover><mrow><mi>y</mi></mrow><mi>&#x005e;</mi></mover></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>Finally, the force due to a magnetic field on a current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>I</mi><mrow><mover><mrow><mi>l</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math>.</p>
<p>The&nbsp;current&nbsp;flows perpendicular to the magnetic field, so:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>F</mi><mrow><mi>l</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>IB</mi></mrow></mstyle></math>.&nbsp; Using the right-hand rule, we get the direction of the force, thus:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mrow><mi>l</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mn>2</mn><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac><mover><mrow><mi>x</mi></mrow><mi>&#x005e;</mi></mover></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>.</p>@
qu.18.3.algorithm=$var=switch($rint(5),'a','b','c','d');
$mag=range(1,9);
$cur=range(1,9);
$cur2=range(1,9);@
qu.18.3.uid=329abc1f-36bd-4417-b75d-a77ca2dbbd3e@
qu.18.3.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Force Between Current Carrying Wires;
  Difficulty=Hard;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Partial Grading;
@

qu.18.4.question=<p>The wires <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>&nbsp;are both carrying a current <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>&nbsp;in the directions indicated in the diagram.&nbsp; In<br />
&nbsp;terms of the given parameters, provide an equation for the magnetic field at any point on the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></mstyle></math>axis.</p>
<p>&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="500" height="505">
<param name="image" value="__BASE_URI__img/MagneticFields/CurrentCarryingWiresVertical/Diagram2Wires3.png" />
<param name="size" value="3" />
<param name="label.1.x" value="232" />
<param name="label.1.y" value="215" />
<param name="label.1.text" value="$mag$var" />
<param name="label.2.x" value="232" />
<param name="label.2.y" value="315" />
<param name="label.2.text" value="$mag$var" />
<param name="label.3.x" value="315" />
<param name="label.3.y" value="257" />
<param name="label.3.text" value="x" /></applet></p>
<p>&nbsp;</p>
<p><em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>xhat</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>yhat</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>z</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>zhat</mi></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
Enter your response in terms of the given parameters and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>, entering&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi></mrow></mstyle></math> instead of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>.</p>@
qu.18.4.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents);@
qu.18.4.allow2d=0@
qu.18.4.maple_answer=xhat*2*(u*$cur*I/(2*Pi*sqrt(x^2+($mag*$var)^2)))*(($mag*$var)/sqrt(x^2+($mag*$var)^2));@
qu.18.4.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.18.4.type=maple@
qu.18.4.mode=Maple@
qu.18.4.name=Current Carrying Wires - Vertical - 2 Wires - 3@
qu.18.4.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.18.4.editing=useHTML@
qu.18.4.solution=<p>The magnitude of the magnetic field due to an infinite current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>&pi;r</mi></mrow></mfrac></mrow></mstyle></math>.&nbsp; The direction of the magnetic field is found using the right-hand rule, where the thumb points in the direction of current flow and the curl of the fingers represents the direction of the magnetic field.&nbsp; If we draw a line from the wire to the point in question, then the magnetic field will always be perpendicular to that line.</p>
<p>&nbsp;</p>
<p>At point&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math> the magnitude of the magnetic field due to each of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math> and&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math> will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow></mfrac></mrow></mstyle></math>.</p>
<p>Based on the configuration, the vertical components will cancel.&nbsp; Thus, we need only calculate the horizontal components, which are:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>B</mi><mrow><mi>x</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>, and therefore the total magnetic field at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math>&nbsp;will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mn>2</mn><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac><mover><mrow><mi>x</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math>.</p>@
qu.18.4.algorithm=$var=switch($rint(5),'a','b','c','d');
$mag=range(1,9);
$cur=range(1,9);@
qu.18.4.uid=d91792a8-0393-4ec9-9d16-6464fcf9b785@
qu.18.4.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Magnetic Field Due to Current Carrying Wires;
  Difficulty=Hard;
  Features=Algebraic;
  Features=Algorithmic;
  Features=Diagram;
@

qu.18.5.question=<p>The wires <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>&nbsp;are carrying a current <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;is carrying current <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$cur2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>, each in the&nbsp;<br />
directions indicated in the diagram.&nbsp; In&nbsp;terms of the given parameters, provide a vector&nbsp;expression for&nbsp;<br />
the&nbsp;force per unit length exerted by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>&nbsp;on <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>, for&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;placed at&nbsp;any point on the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></mstyle></math>axis.</p>
<p>&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="500" height="505">
<param name="image" value="__BASE_URI__img/MagneticFields/CurrentCarryingWiresVertical/Diagram3Wires4-1.png" />
<param name="size" value="3" />
<param name="label.1.x" value="232" />
<param name="label.1.y" value="215" />
<param name="label.1.text" value="$mag$var" />
<param name="label.2.x" value="232" />
<param name="label.2.y" value="315" />
<param name="label.2.text" value="$mag$var" />
<param name="label.3.x" value="315" />
<param name="label.3.y" value="257" />
<param name="label.3.text" value="x" /></applet></p>
<p>&nbsp;</p>
<p><em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>xhat</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>yhat</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>z</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>zhat</mi></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
Enter your response in terms of the given parameters and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>, entering&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi></mrow></mstyle></math> instead of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>.</p>@
qu.18.5.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents);@
qu.18.5.allow2d=0@
qu.18.5.maple_answer=(-xhat)*($cur2*I)*2*(u*$cur*I/(2*Pi*sqrt(x^2+($mag*$var)^2)))*((x)/sqrt(x^2+($mag*$var)^2));@
qu.18.5.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.18.5.type=maple@
qu.18.5.mode=Maple@
qu.18.5.name=Current Carrying Wires - Vertical - 3 Wires - 4-1@
qu.18.5.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.18.5.editing=useHTML@
qu.18.5.solution=<p>We will calculate the magnetic field due to wires <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>, and then the force due to this magnetic field on <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>.</p>
<p>The magnitude of the magnetic field due to an infinite current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>&pi;r</mi></mrow></mfrac></mrow></mstyle></math>.&nbsp; The direction of the magnetic field is found using the right-hand rule, where the thumb points in the direction of current flow and the curl of the fingers represents the direction of the magnetic field.&nbsp; If we draw a line from the wire to the point in question, then the magnetic field will always be perpendicular to that line.</p>
<p>&nbsp;</p>
<p>At point&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math> the magnitude of the magnetic field due to each of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math> and&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math> will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow></mfrac></mrow></mstyle></math>.</p>
<p>Based on the configuration, the horizontal components will cancel.&nbsp; Thus, we need only calculate the&nbsp;vertical components, which are:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>B</mi><mrow><mi>y</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>, and therefore the total magnetic field at&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mn>2</mn><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac><mover><mrow><mi>y</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>Finally, the force due to a magnetic field on a current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>I</mi><mrow><mover><mrow><mi>l</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math>.</p>
<p>The&nbsp;current&nbsp;flows perpendicular to the magnetic field, so:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>F</mi><mrow><mi>l</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>IB</mi></mrow></mstyle></math>.&nbsp; Using the right-hand rule, we get the direction of the force, thus:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mrow><mi>l</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mn>2</mn><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac><mover><mrow><mi>x</mi></mrow><mi>&#x005e;</mi></mover></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>.</p>@
qu.18.5.algorithm=$var=switch($rint(5),'a','b','c','d');
$mag=range(1,9);
$cur=range(1,9);
$cur2=range(1,9);@
qu.18.5.uid=b8b849a0-c781-4d00-9fa1-bd740af4272a@
qu.18.5.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Force Between Current Carrying Wires;
  Difficulty=Hard;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Partial Grading;
@

qu.18.6.question=<p>The wires <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>&nbsp;are both carrying a current <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>&nbsp;in the directions indicated in the diagram.&nbsp; In<br />
&nbsp;terms of the given parameters, provide an equation for the magnetic field at any point on the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></mstyle></math>axis.</p>
<p>&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="500" height="505">
<param name="image" value="__BASE_URI__img/MagneticFields/CurrentCarryingWiresVertical/Diagram2Wires2.png" />
<param name="size" value="3" />
<param name="label.1.x" value="232" />
<param name="label.1.y" value="215" />
<param name="label.1.text" value="$mag$var" />
<param name="label.2.x" value="232" />
<param name="label.2.y" value="315" />
<param name="label.2.text" value="$mag$var" />
<param name="label.3.x" value="315" />
<param name="label.3.y" value="257" />
<param name="label.3.text" value="x" /></applet></p>
<p>&nbsp;</p>
<p><em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>xhat</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>yhat</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>z</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>zhat</mi></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
Enter your response in terms of the given parameters and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>, entering&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi></mrow></mstyle></math> instead of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>.</p>@
qu.18.6.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents);@
qu.18.6.allow2d=0@
qu.18.6.maple_answer=yhat*2*(u*$cur*I/(2*Pi*sqrt(x^2+($mag*$var)^2)))*(x/sqrt(x^2+($mag*$var)^2));@
qu.18.6.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.18.6.type=maple@
qu.18.6.mode=Maple@
qu.18.6.name=Current Carrying Wires - Vertical - 2 Wires - 2@
qu.18.6.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.18.6.editing=useHTML@
qu.18.6.solution=<p>The magnitude of the magnetic field due to an infinite current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>&pi;r</mi></mrow></mfrac></mrow></mstyle></math>.&nbsp; The direction of the magnetic field is found using the right-hand rule, where the thumb points in the direction of current flow and the curl of the fingers represents the direction of the magnetic field.&nbsp; If we draw a line from the wire to the point in question, then the magnetic field will always be perpendicular to that line.</p>
<p>&nbsp;</p>
<p>At point&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math> the magnitude of the magnetic field due to each of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math> and&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math> will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow></mfrac></mrow></mstyle></math>.</p>
<p>Based on the configuration, the horizontal components will cancel.&nbsp; Thus, we need only calculate the vertical components, which are:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>B</mi><mrow><mi>y</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>, and therefore the total magnetic field at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math>&nbsp;will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mn>2</mn><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac></mrow><mrow><mover><mrow><mi>y</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math>.</p>@
qu.18.6.algorithm=$var=switch($rint(5),'a','b','c','d');
$mag=range(1,9);
$cur=range(1,9);@
qu.18.6.uid=e6033809-8d58-4ca0-b705-a030c12883ec@
qu.18.6.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Magnetic Field Due to Current Carrying Wires;
  Difficulty=Hard;
  Features=Algebraic;
  Features=Algorithmic;
  Features=Diagram;
@

qu.18.7.question=<p>Four current carrying wires are arranged as seen in the diagram, where the arrows indicate the direction of current flow.<br />
The wires are each separated by a distance <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$w</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mstyle></math>. Calculate the magnetic field vector at the center point, given the following information:<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>B</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$B</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>C</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$C</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi mathvariant='normal'>D</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$D</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mstyle></math>.</p>
<p><img alt="" align="middle" width="753" height="500" src="__BASE_URI__img/MagneticFields/CurrentCarryingWires-1/Diagram2.png" /></p>
<p>&nbsp;<em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>xhat</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>yhat</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>z</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>zhat</mi></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.</p>@
qu.18.7.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uCT,mCT=0.5,uVS,mVS=0.5,uSM,mSM=0.8,subPackage=vectorNumericComponents);@
qu.18.7.allow2d=0@
qu.18.7.maple_answer=((-$A-$B-$C+$D)*(4*Pi*10^(-7))*zhat*T)/(Pi*($w/1000))@
qu.18.7.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.18.7.type=maple@
qu.18.7.mode=Maple@
qu.18.7.name=Magnetic Fields Due to Current Carrying Wires - 2 - Numeric@
qu.18.7.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT,mCT=0.5,uVS,mVS=0.5,uSM,mSM=0.8,subPackage=vectorNumericComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.18.7.editing=useHTML@
qu.18.7.solution=<p>The magnitude of the magnetic field due to an infinite current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>&pi;r</mi></mrow></mfrac></mrow></mstyle></math>.&nbsp; The direction of the magnetic field is found using the right-hand rule, where the thumb points in the direction of current flow and the curl of the fingers represents the direction of the magnetic field.&nbsp; When we consider the&nbsp;wires from the side, at any point on the screen, the magnetic field will either be directly into or out of the screen.&nbsp;</p>
<p>At the center point, for each wire, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mi>$w</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mstyle></math>.&nbsp; We are given the current in each wire and its direction.</p>
<p>For each wire, we need then to identify whether their resultant magnetic field is into or out of the screen, then add each together.</p>@
qu.18.7.algorithm=$A=rand(1.00,9.99,3);
$B=rand(1.00,9.99,3);
$C=rand(1.00,9.99,3);
$D=rand(1.00,9.99,3);
$w=rand(1.00,9.99,3);@
qu.18.7.uid=5b974ca0-8030-402a-85f6-1d6fce91bf08@
qu.18.7.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Magnetic Fields Due to Current Carrying Wires;
  Difficulty=Medium;
  Features=Algorithmic;
  Features=Diagram;
  Features=Partial Grading;
@

qu.18.8.question=<p>A charge&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$q</mi></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$qex</mi></mrow></msup><mi>C</mi></mrow></mstyle></math>&nbsp;is travelling with velocity&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>v</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$v1</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$v2</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow><mrow><mi></mi></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mi mathvariant='normal'></mi></mrow><mrow><mi mathvariant='normal'></mi></mrow><mrow><mi mathvariant='normal'></mi></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$vex</mi></mrow></msup><mfrac><mi>m</mi><mrow><mi>s</mi></mrow></mfrac></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>.&nbsp;<br />
What is the magnetic field at a point&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>r</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$r1</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$r2</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow><mrow><mi></mi></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow><mrow><mi>cm</mi></mrow><mrow><mi mathvariant='normal'></mi></mrow></mstyle></math>&nbsp;from the&nbsp;location of the charge?&nbsp;</p>
<p>&nbsp;</p>
<p>&nbsp;</p>
<p><em>Entry Notes</em>:&nbsp;<br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ihat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>jhat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>&nbsp; and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>khat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp;<br />
Ensure that you explicitly enter the multiplication symbol * between terms.</p>@
qu.18.8.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uVS=false,uUN);@
qu.18.8.allow2d=0@
qu.18.8.maple_answer=with(Physics[Vectors]);
v:=(($v1)*_i+($v2)*_j)*10^($vex);
rr:=(($r1)*_i+($r2)*_j)*10^(-2);
temp:=(10^(-7))*($q*10^($qex))*(v &x rr)/(Norm(rr)^3);
SigFigs[roundToSigFigs](Component(temp,1),3)*N*ihat+SigFigs[roundToSigFigs](Component(temp,2),3)*N*jhat+SigFigs[roundToSigFigs](Component(temp,3),3)*N*khat;@
qu.18.8.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.18.8.type=maple@
qu.18.8.mode=Maple@
qu.18.8.name=Magnetic Field Due to Moving Charge - 2D@
qu.18.8.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uVS=false,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.18.8.editing=useHTML@
qu.18.8.hint.1=Set up a determinant.&nbsp; The first row should be <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp; The second row should be the components of the first vector and the third row should be the components of the second vector.&nbsp; Evaluate the determinant.@
qu.18.8.solution=<p>The magnetic field due to a moving charged particle is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mrow><mn>4</mn><mrow><mi>&pi;</mi></mrow></mrow></mfrac><mfrac><mrow><mi>q</mi><mrow><mover><mrow><mi>v</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow><mrow><mover><mrow><mi></mi></mrow><mi></mi></mover></mrow><mrow><mover><mrow><mi>r</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow><mrow><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mstyle></math>.</p>
<p>We are given <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>q</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>v</mi></mrow><mi>&rarr;</mi></mover></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>r</mi></mrow><mi>&rarr;</mi></mover></mrow></mstyle></math>.&nbsp; Use the fact that <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>r</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mover><mrow><mi>r</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mfenced open='|' close='|' separators=','><mrow><mover><mrow><mi>r</mi></mrow><mi>&rarr;</mi></mover></mrow></mfenced></mrow></mfrac></mrow></mrow></mstyle></math>, and that <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='|' close='|' separators=','><mrow><mover><mrow><mi>r</mi></mrow><mi>&rarr;</mi></mover></mrow></mfenced></mrow></mstyle></math>.</p>@
qu.18.8.algorithm=$v1=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$v2=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$r1=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$r2=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$vex=range(2,7);
$q=switch(rint(2),rand(1.00,9.99,3),-rand(1.00,9.99,3));
$qex=range(-19,-5);@
qu.18.8.uid=263863f2-d119-4638-882a-4bafa2794bb7@
qu.18.8.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Magnetic Field Due to Moving Charges;
  Difficulty=Hard;
  Features=Algorithmic;
  Features=Partial Grading;
@

qu.18.9.question=<p>A charge&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$q</mi></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$qex</mi></mrow></msup><mi>C</mi></mrow></mstyle></math>&nbsp;is travelling with velocity&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>v</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$v1</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$v2</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$v3</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo></mrow><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow></mrow><mrow></mrow><mrow></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&times;</mo><msup><mn>10</mn><mrow><mi>$vex</mi></mrow></msup><mfrac><mi>m</mi><mrow><mi>s</mi></mrow></mfrac></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>.&nbsp;<br />
What is the magnetic field at a point&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>r</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$r1</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$r2</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$r3</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo></mrow><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow><mrow><mi>cm</mi></mrow><mrow></mrow></mstyle></math>&nbsp;from the&nbsp;location of the charge?&nbsp;</p>
<p>&nbsp;</p>
<p>&nbsp;</p>
<p><em>Entry Notes</em>:&nbsp;<br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ihat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>jhat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>&nbsp; and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>khat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp;<br />
Ensure that you explicitly enter the multiplication symbol * between terms.</p>@
qu.18.9.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uVS=false,uUN);@
qu.18.9.allow2d=0@
qu.18.9.maple_answer=with(Physics[Vectors]);
v:=(($v1)*_i+($v2)*_j+($v3)*_k)*10^($vex);
rr:=(($r1)*_i+($r2)*_j+($r3)*_k)*10^(-2);
temp:=(10^(-7))*($q*10^($qex))*(v &x rr)/(Norm(rr)^3);
SigFigs[roundToSigFigs](Component(temp,1),3)*N*ihat+SigFigs[roundToSigFigs](Component(temp,2),3)*N*jhat+SigFigs[roundToSigFigs](Component(temp,3),3)*N*khat;@
qu.18.9.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.18.9.type=maple@
qu.18.9.mode=Maple@
qu.18.9.name=Magnetic Field Due to Moving Charge - 3D@
qu.18.9.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uVS=false,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.18.9.editing=useHTML@
qu.18.9.hint.1=Set up a determinant.&nbsp; The first row should be <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp; The second row should be the components of the first vector and the third row should be the components of the second vector.&nbsp; Evaluate the determinant.@
qu.18.9.solution=<p>The magnetic field due to a moving charged particle is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mrow><mn>4</mn><mrow><mi>&pi;</mi></mrow></mrow></mfrac><mfrac><mrow><mi>q</mi><mrow><mover><mrow><mi>v</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow><mrow><mover><mrow><mi></mi></mrow><mi></mi></mover></mrow><mrow><mover><mrow><mi>r</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow><mrow><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mstyle></math>.</p>
<p>We are given <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>q</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>v</mi></mrow><mi>&rarr;</mi></mover></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>r</mi></mrow><mi>&rarr;</mi></mover></mrow></mstyle></math>.&nbsp; Use the fact that <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>r</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mover><mrow><mi>r</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mfenced open='|' close='|' separators=','><mrow><mover><mrow><mi>r</mi></mrow><mi>&rarr;</mi></mover></mrow></mfenced></mrow></mfrac></mrow></mrow></mstyle></math>, and that <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='|' close='|' separators=','><mrow><mover><mrow><mi>r</mi></mrow><mi>&rarr;</mi></mover></mrow></mfenced></mrow></mstyle></math>.</p>@
qu.18.9.algorithm=$v1=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$v2=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$v3=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$r1=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$r2=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$r3=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$vex=range(2,7);
$q=switch(rint(2),rand(1.00,9.99,3),-rand(1.00,9.99,3));
$qex=range(-19,-5);@
qu.18.9.uid=6ae8de61-d13b-4cec-b039-544a10064471@
qu.18.9.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Magnetic Field Due to Moving Charges;
  Difficulty=Hard;
  Features=Algorithmic;
  Features=Partial Grading;
@

qu.18.10.question=<p>Four current carrying wires are arranged as seen in the diagram, where the arrows indicate the direction of current flow.<br />
The wires are each separated by a distance <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>w</mi></mrow></mstyle></math>. Calculate the magnetic field vector at the center point, given the following information:<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>B</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$B</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>C</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$C</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi mathvariant='normal'>D</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$D</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>.</p>
<p><img alt="" align="middle" width="753" height="500" src="__BASE_URI__img/MagneticFields/CurrentCarryingWires-1/Diagram4.png" /></p>
<p>&nbsp;<em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>xhat</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>yhat</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>z</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>zhat</mi></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
Enter your response in terms of the given parameters and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>, entering <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi></mrow></mstyle></math>&nbsp;instead of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>.</p>@
qu.18.10.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT,mCT=0.5,uVS,mVS=0.5,uSM,mSM=0.8,subPackage=vectorAlgebraicComponents);@
qu.18.10.allow2d=0@
qu.18.10.maple_answer=(($A+$B-$C-$D)*I*u*zhat)/(Pi*w)@
qu.18.10.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.18.10.type=maple@
qu.18.10.mode=Maple@
qu.18.10.name=Magnetic Fields Due to Current Carrying Wires - 4@
qu.18.10.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT,mCT=0.5,uVS,mVS=0.5,uSM,mSM=0.8,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.18.10.editing=useHTML@
qu.18.10.solution=<p>The magnitude of the magnetic field due to an infinite current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>&pi;r</mi></mrow></mfrac></mrow></mstyle></math>.&nbsp; The direction of the magnetic field is found using the right-hand rule, where the thumb points in the direction of current flow and the curl of the fingers represents the direction of the magnetic field.&nbsp; When we consider the&nbsp;wires from the side, at any point on the screen, the magnetic field will either be directly into or out of the screen.&nbsp;</p>
<p>At the center point, for each wire, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>w</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mstyle></math>.&nbsp; We are given the current in each wire and its direction.</p>
<p>For each wire, we need then to identify whether their resultant magnetic field is into or out of the screen, then add each together.</p>@
qu.18.10.algorithm=$A=range(1,9);
$B=range(1,9);
$C=range(1,9);
$D=range(1,9);@
qu.18.10.uid=c7c96511-3def-4698-a392-90cbf3358dad@
qu.18.10.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Magnetic Fields Due to Current Carrying Wires;
  Difficulty=Medium;
  Features=Algorithmic;
  Features=Algebraic;
@

qu.18.11.question=<p>The wires <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>&nbsp;are both carrying a current <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>&nbsp;in the directions indicated in the diagram.&nbsp; In<br />
&nbsp;terms of the given parameters, provide an equation for the magnetic field at any point on the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></mstyle></math>axis.</p>
<p>&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="500" height="505">
<param name="image" value="__BASE_URI__img/MagneticFields/CurrentCarryingWiresVertical/Diagram2Wires1.png" />
<param name="size" value="3" />
<param name="label.1.x" value="232" />
<param name="label.1.y" value="215" />
<param name="label.1.text" value="$mag$var" />
<param name="label.2.x" value="232" />
<param name="label.2.y" value="315" />
<param name="label.2.text" value="$mag$var" />
<param name="label.3.x" value="315" />
<param name="label.3.y" value="257" />
<param name="label.3.text" value="x" /></applet></p>
<p>&nbsp;</p>
<p><em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>xhat</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>yhat</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>z</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>zhat</mi></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
Enter your response in terms of the given parameters and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>, entering&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi></mrow></mstyle></math> instead of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>.</p>@
qu.18.11.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents)@
qu.18.11.allow2d=0@
qu.18.11.maple_answer=(-xhat)*2*(u*$cur*I/(2*Pi*sqrt(x^2+($mag*$var)^2)))*(($mag*$var)/sqrt(x^2+($mag*$var)^2))@
qu.18.11.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.18.11.type=maple@
qu.18.11.mode=Maple@
qu.18.11.name=Current Carrying Wires - Vertical - 2 Wires - 1@
qu.18.11.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.18.11.editing=useHTML@
qu.18.11.solution=<p>The magnitude of the magnetic field due to an infinite current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>&pi;r</mi></mrow></mfrac></mrow></mstyle></math>.&nbsp; The direction of the magnetic field is found using the right-hand rule, where the thumb points in the direction of current flow and the curl of the fingers represents the direction of the magnetic field.&nbsp; If we draw a line from the wire to the point in question, then the magnetic field will always be perpendicular to that line.</p>
<p>&nbsp;</p>
<p>At point&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math> the magnitude of the magnetic field due to each of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math> and&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math> will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow></mfrac></mrow></mstyle></math>.</p>
<p>Based on the configuration, the vertical components will cancel.&nbsp; Thus, we need only calculate the horizontal components, which are:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>B</mi><mrow><mi>x</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>, and therefore the total magnetic field at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math>&nbsp;will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mn>2</mn><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac><mrow><mover><mrow><mi>x</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.</p>@
qu.18.11.algorithm=$var=switch($rint(5),'a','b','c','d');
$mag=range(1,9);
$cur=range(1,9);@
qu.18.11.uid=f3e52422-abc7-440b-b9a7-72cb76cb39ac@
qu.18.11.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Magnetic Field Due to Current Carrying Wires;
  Difficulty=Hard;
  Features=Algebraic;
  Features=Algorithmic;
  Features=Diagram;
@

qu.18.12.question=<p>The wires <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>&nbsp;are carrying a current <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;is carrying current <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$cur2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>, each in the&nbsp;<br />
directions indicated in the diagram.&nbsp; In&nbsp;terms of the given parameters, provide a vector&nbsp;expression for&nbsp;<br />
the&nbsp;force per unit length exerted by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>&nbsp;on <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>, for&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;placed at&nbsp;any point on the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></mstyle></math>axis.</p>
<p>&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="500" height="505">
<param name="image" value="__BASE_URI__img/MagneticFields/CurrentCarryingWiresVertical/Diagram3Wires2-1.png" />
<param name="size" value="3" />
<param name="label.1.x" value="232" />
<param name="label.1.y" value="215" />
<param name="label.1.text" value="$mag$var" />
<param name="label.2.x" value="232" />
<param name="label.2.y" value="315" />
<param name="label.2.text" value="$mag$var" />
<param name="label.3.x" value="315" />
<param name="label.3.y" value="257" />
<param name="label.3.text" value="x" /></applet></p>
<p>&nbsp;</p>
<p><em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>xhat</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>yhat</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>z</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>zhat</mi></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
Enter your response in terms of the given parameters and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>, entering&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi></mrow></mstyle></math> instead of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>.</p>@
qu.18.12.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents);@
qu.18.12.allow2d=0@
qu.18.12.maple_answer=+xhat*($cur2*I)*2*(u*$cur*I/(2*Pi*sqrt(x^2+($mag*$var)^2)))*((x)/sqrt(x^2+($mag*$var)^2));@
qu.18.12.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.18.12.type=maple@
qu.18.12.mode=Maple@
qu.18.12.name=Current Carrying Wires - Vertical - 3 Wires - 2-1@
qu.18.12.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.18.12.editing=useHTML@
qu.18.12.solution=<p>We will calculate the magnetic field due to wires <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>, and then the force due to this magnetic field on <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>.</p>
<p>The magnitude of the magnetic field due to an infinite current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>&pi;r</mi></mrow></mfrac></mrow></mstyle></math>.&nbsp; The direction of the magnetic field is found using the right-hand rule, where the thumb points in the direction of current flow and the curl of the fingers represents the direction of the magnetic field.&nbsp; If we draw a line from the wire to the point in question, then the magnetic field will always be perpendicular to that line.</p>
<p>&nbsp;</p>
<p>At point&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math> the magnitude of the magnetic field due to each of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math> and&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math> will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow></mfrac></mrow></mstyle></math>.</p>
<p>Based on the configuration, the horizontal components will cancel.&nbsp; Thus, we need only calculate the&nbsp;vertical components, which are:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>B</mi><mrow><mi>y</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>, and therefore the total magnetic field at&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mn>2</mn><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac><mover><mrow><mi>y</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>Finally, the force due to a magnetic field on a current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>I</mi><mrow><mover><mrow><mi>l</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math>.</p>
<p>The&nbsp;current&nbsp;flows perpendicular to the magnetic field, so:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>F</mi><mrow><mi>l</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>IB</mi></mrow></mstyle></math>.&nbsp; Using the right-hand rule, we get the direction of the force, thus:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mrow><mi>l</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mn>2</mn><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac><mover><mrow><mi>x</mi></mrow><mi>&#x005e;</mi></mover></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>.</p>@
qu.18.12.algorithm=$var=switch($rint(5),'a','b','c','d');
$mag=range(1,9);
$cur=range(1,9);
$cur2=range(1,9);@
qu.18.12.uid=051f6e21-6d12-41ff-a4c4-c8c295e0eb60@
qu.18.12.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Force Between Current Carrying Wires;
  Difficulty=Hard;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Partial Grading;
@

qu.18.13.question=<p>A region of space contains a magnetic field of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$B1</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$B2</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow><mrow><mi></mi></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow><mrow><mi>T</mi></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>.&nbsp; A current of&nbsp;<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mA</mi></mrow></mstyle></math>is travelling&nbsp;through a&nbsp;wire of displacement<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>l</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$l1</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$l2</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow><mrow><mi></mi></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow></mrow><mrow></mrow><mrow></mrow><mrow><mi>cm</mi></mrow></mstyle></math>&nbsp;in this region.&nbsp;<br />
What is the&nbsp;force on the wire due to the magnetic field?&nbsp;</p>
<p>&nbsp;</p>
<p><em>Entry Notes</em>:&nbsp;<br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ihat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>jhat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>&nbsp; and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>khat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp;<br />
Ensure that you explicitly enter the multiplication symbol * between terms.</p>@
qu.18.13.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uVS=false,uUN);@
qu.18.13.allow2d=0@
qu.18.13.maple_answer=with(Physics[Vectors]);
l:=(($l1)*_i+($l2)*_j)*10^(-2);
B:=(($B1)*_i+($B2)*_j);
temp:=($I*10^(-3))*(l &x B);
SigFigs[roundToSigFigs](Component(temp,1),3)*N*ihat+SigFigs[roundToSigFigs](Component(temp,2),3)*N*jhat+SigFigs[roundToSigFigs](Component(temp,3),3)*N*khat;@
qu.18.13.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.18.13.type=maple@
qu.18.13.mode=Maple@
qu.18.13.name=Magnetic Force On A Current Carrying Wire - 2D@
qu.18.13.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uVS=false,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.18.13.editing=useHTML@
qu.18.13.hint.1=Set up a determinant.&nbsp; The first row should be <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp; The second row should be the components of the first vector and the third row should be the components of the second vector.&nbsp; Evaluate the determinant.@
qu.18.13.solution=<p>The force due to a magnetic field on a current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>I</mi><mrow><mover><mrow><mi>l</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math>.</p>
<p align="left">We are given <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>l</mi></mrow><mi>&rarr;</mi></mover></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow></mstyle></math>.</p>@
qu.18.13.algorithm=$B1=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$B2=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$l1=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$l2=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$I=rand(1,9.99,3);@
qu.18.13.uid=76cdc495-8281-4b6f-ae5d-e914afcc5760@
qu.18.13.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Magnetic Force Due to Current Carrying Wire;
  Difficulty=Hard;
  Features=Algorithmic;
  Features=Partial Grading;
@

qu.18.14.question=<p>Four current carrying wires are arranged as seen in the diagram, where the arrows indicate the direction of current flow.<br />
The wires are each separated by a distance <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$w</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mstyle></math>. Calculate the magnetic field vector at the center point, given the following information:<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>B</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$B</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>C</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$C</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi mathvariant='normal'>D</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$D</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mstyle></math>.</p>
<p><img alt="" align="middle" width="753" height="500" src="__BASE_URI__img/MagneticFields/CurrentCarryingWires-1/Diagram4.png" /></p>
<p>&nbsp;<em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>xhat</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>yhat</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>z</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>zhat</mi></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.</p>@
qu.18.14.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uCT,mCT=0.5,uVS,mVS=0.5,uSM,mSM=0.8,subPackage=vectorNumericComponents);@
qu.18.14.allow2d=0@
qu.18.14.maple_answer=(($A+$B-$C-$D)*(4*Pi*10^(-7))*zhat*T)/(Pi*($w/1000))@
qu.18.14.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.18.14.type=maple@
qu.18.14.mode=Maple@
qu.18.14.name=Magnetic Fields Due to Current Carrying Wires - 4 - Numeric@
qu.18.14.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT,mCT=0.5,uVS,mVS=0.5,uSM,mSM=0.8,subPackage=vectorNumericComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.18.14.editing=useHTML@
qu.18.14.solution=<p>The magnitude of the magnetic field due to an infinite current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>&pi;r</mi></mrow></mfrac></mrow></mstyle></math>.&nbsp; The direction of the magnetic field is found using the right-hand rule, where the thumb points in the direction of current flow and the curl of the fingers represents the direction of the magnetic field.&nbsp; When we consider the&nbsp;wires from the side, at any point on the screen, the magnetic field will either be directly into or out of the screen.&nbsp;</p>
<p>At the center point, for each wire, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mi>$w</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mstyle></math>.&nbsp; We are given the current in each wire and its direction.</p>
<p>For each wire, we need then to identify whether their resultant magnetic field is into or out of the screen, then add each together.</p>@
qu.18.14.algorithm=$A=rand(1.00,9.99,3);
$B=rand(1.00,9.99,3);
$C=rand(1.00,9.99,3);
$D=rand(1.00,9.99,3);
$w=rand(1.00,9.99,3);@
qu.18.14.uid=c07c5ae1-1140-4025-9bbd-0acb4ea78c31@
qu.18.14.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Magnetic Fields Due to Current Carrying Wires;
  Difficulty=Medium;
  Features=Algorithmic;
  Features=Diagram;
  Features=Partial Grading;
@

qu.18.15.question=<p>The wires <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>&nbsp;are carrying a current <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;is carrying current <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$cur2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>, each in the&nbsp;<br />
directions indicated in the diagram.&nbsp; In&nbsp;terms of the given parameters, provide a vector&nbsp;expression for&nbsp;<br />
the&nbsp;force per unit length exerted by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>&nbsp;on <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>, for&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;placed at&nbsp;any point on the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></mstyle></math>axis.</p>
<p>&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="500" height="505">
<param name="image" value="__BASE_URI__img/MagneticFields/CurrentCarryingWiresVertical/Diagram3Wires3-2.png" />
<param name="size" value="3" />
<param name="label.1.x" value="232" />
<param name="label.1.y" value="215" />
<param name="label.1.text" value="$mag$var" />
<param name="label.2.x" value="232" />
<param name="label.2.y" value="315" />
<param name="label.2.text" value="$mag$var" />
<param name="label.3.x" value="315" />
<param name="label.3.y" value="257" />
<param name="label.3.text" value="x" /></applet></p>
<p>&nbsp;</p>
<p><em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>xhat</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>yhat</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>z</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>zhat</mi></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
Enter your response in terms of the given parameters and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>, entering&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi></mrow></mstyle></math> instead of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>.</p>@
qu.18.15.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents);@
qu.18.15.allow2d=0@
qu.18.15.maple_answer=+yhat*($cur2*I)*2*(u*$cur*I/(2*Pi*sqrt(x^2+($mag*$var)^2)))*(($mag*$var)/sqrt(x^2+($mag*$var)^2));@
qu.18.15.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.18.15.type=maple@
qu.18.15.mode=Maple@
qu.18.15.name=Current Carrying Wires - Vertical - 3 Wires - 3-2@
qu.18.15.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.18.15.editing=useHTML@
qu.18.15.solution=<p>We will calculate the magnetic field due to wires <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>, and then the force due to this magnetic field on <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>.</p>
<p>The magnitude of the magnetic field due to an infinite current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>&pi;r</mi></mrow></mfrac></mrow></mstyle></math>.&nbsp; The direction of the magnetic field is found using the right-hand rule, where the thumb points in the direction of current flow and the curl of the fingers represents the direction of the magnetic field.&nbsp; If we draw a line from the wire to the point in question, then the magnetic field will always be perpendicular to that line.</p>
<p>&nbsp;</p>
<p>At point&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math> the magnitude of the magnetic field due to each of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math> and&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math> will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow></mfrac></mrow></mstyle></math>.</p>
<p>Based on the configuration, the vertical components will cancel.&nbsp; Thus, we need only calculate the horizontal components, which are:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>B</mi><mrow><mi>x</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>, and therefore the total magnetic field at&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mn>2</mn><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac><mover><mrow><mi>x</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>Finally, the force due to a magnetic field on a current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>I</mi><mrow><mover><mrow><mi>l</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math>.</p>
<p>The&nbsp;current&nbsp;flows perpendicular to the magnetic field, so:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>F</mi><mrow><mi>l</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>IB</mi></mrow></mstyle></math>.&nbsp; Using the right-hand rule, we get the direction of the force, thus:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mrow><mi>l</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mn>2</mn><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac><mover><mrow><mi>y</mi></mrow><mi>&#x005e;</mi></mover></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>.</p>@
qu.18.15.algorithm=$var=switch($rint(5),'a','b','c','d');
$mag=range(1,9);
$cur=range(1,9);
$cur2=range(1,9);@
qu.18.15.uid=77d01e8d-0b3f-4e6c-b0a6-4a25ded4ab98@
qu.18.15.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Force Between Current Carrying Wires;
  Difficulty=Hard;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Partial Grading;
@

qu.18.16.question=<p>The wires <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>&nbsp;are both carrying a current <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>&nbsp;in the directions indicated in the diagram.&nbsp; In<br />
&nbsp;terms of the given parameters, provide an equation for the magnetic field at any point on the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></mstyle></math>axis.</p>
<p>&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="500" height="505">
<param name="image" value="__BASE_URI__img/MagneticFields/CurrentCarryingWiresVertical/Diagram2Wires4.png" />
<param name="size" value="3" />
<param name="label.1.x" value="232" />
<param name="label.1.y" value="215" />
<param name="label.1.text" value="$mag$var" />
<param name="label.2.x" value="232" />
<param name="label.2.y" value="315" />
<param name="label.2.text" value="$mag$var" />
<param name="label.3.x" value="315" />
<param name="label.3.y" value="257" />
<param name="label.3.text" value="x" /></applet></p>
<p>&nbsp;</p>
<p><em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>xhat</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>yhat</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>z</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>zhat</mi></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
Enter your response in terms of the given parameters and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>, entering&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi></mrow></mstyle></math> instead of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>.</p>@
qu.18.16.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents);@
qu.18.16.allow2d=0@
qu.18.16.maple_answer=(-yhat)*2*(u*$cur*I/(2*Pi*sqrt(x^2+($mag*$var)^2)))*(x/sqrt(x^2+($mag*$var)^2));@
qu.18.16.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.18.16.type=maple@
qu.18.16.mode=Maple@
qu.18.16.name=Current Carrying Wires - Vertical - 2 Wires - 4@
qu.18.16.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.18.16.editing=useHTML@
qu.18.16.solution=<p>The magnitude of the magnetic field due to an infinite current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>&pi;r</mi></mrow></mfrac></mrow></mstyle></math>.&nbsp; The direction of the magnetic field is found using the right-hand rule, where the thumb points in the direction of current flow and the curl of the fingers represents the direction of the magnetic field.&nbsp; If we draw a line from the wire to the point in question, then the magnetic field will always be perpendicular to that line.</p>
<p>&nbsp;</p>
<p>At point&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math> the magnitude of the magnetic field due to each of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math> and&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math> will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow></mfrac></mrow></mstyle></math>.</p>
<p>Based on the configuration, the horizontal components will cancel.&nbsp; Thus, we need only calculate the&nbsp;vertical&nbsp;components, which are:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>B</mi><mrow><mi>y</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mi>x</mi></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>, and therefore the total magnetic field at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math>&nbsp;will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mn>2</mn><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac><mrow><mover><mrow><mi>y</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.</p>@
qu.18.16.algorithm=$var=switch($rint(5),'a','b','c','d');
$mag=range(1,9);
$cur=range(1,9);@
qu.18.16.uid=2b879887-2cb8-4bdd-adb1-0afa6ababeb8@
qu.18.16.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Magnetic Field Due to Current Carrying Wires;
  Difficulty=Hard;
  Features=Algebraic;
  Features=Algorithmic;
  Features=Diagram;
@

qu.18.17.question=<p>The wires <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>&nbsp;are carrying a current <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;is carrying current <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$cur2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>, each in the&nbsp;<br />
directions indicated in the diagram.&nbsp; In&nbsp;terms of the given parameters, provide a vector&nbsp;expression for&nbsp;<br />
the&nbsp;force per unit length exerted by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>&nbsp;on <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>, for&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;placed at&nbsp;any point on the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></mstyle></math>axis.</p>
<p>&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="500" height="505">
<param name="image" value="__BASE_URI__img/MagneticFields/CurrentCarryingWiresVertical/Diagram3Wires1-1.png" />
<param name="size" value="3" />
<param name="label.1.x" value="232" />
<param name="label.1.y" value="215" />
<param name="label.1.text" value="$mag$var" />
<param name="label.2.x" value="232" />
<param name="label.2.y" value="315" />
<param name="label.2.text" value="$mag$var" />
<param name="label.3.x" value="315" />
<param name="label.3.y" value="257" />
<param name="label.3.text" value="x" /></applet></p>
<p>&nbsp;</p>
<p><em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>xhat</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>yhat</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>z</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>zhat</mi></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
Enter your response in terms of the given parameters and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>, entering&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi></mrow></mstyle></math> instead of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>.</p>@
qu.18.17.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents);@
qu.18.17.allow2d=0@
qu.18.17.maple_answer=+yhat*($cur2*I)*2*(u*$cur*I/(2*Pi*sqrt(x^2+($mag*$var)^2)))*(($mag*$var)/sqrt(x^2+($mag*$var)^2));@
qu.18.17.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.18.17.type=maple@
qu.18.17.mode=Maple@
qu.18.17.name=Current Carrying Wires - Vertical - 3 Wires - 1-1@
qu.18.17.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.18.17.editing=useHTML@
qu.18.17.solution=<p>We will calculate the magnetic field due to wires <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>, and then the force due to this magnetic field on <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>.</p>
<p>The magnitude of the magnetic field due to an infinite current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>&pi;r</mi></mrow></mfrac></mrow></mstyle></math>.&nbsp; The direction of the magnetic field is found using the right-hand rule, where the thumb points in the direction of current flow and the curl of the fingers represents the direction of the magnetic field.&nbsp; If we draw a line from the wire to the point in question, then the magnetic field will always be perpendicular to that line.</p>
<p>&nbsp;</p>
<p>At point&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math> the magnitude of the magnetic field due to each of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math> and&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math> will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow></mfrac></mrow></mstyle></math>.</p>
<p>Based on the configuration, the vertical components will cancel.&nbsp; Thus, we need only calculate the horizontal components, which are:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>B</mi><mrow><mi>x</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>, and therefore the total magnetic field at&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mn>2</mn><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac><mrow><mover><mrow><mi>x</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>Finally, the force due to a magnetic field on a current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>I</mi><mrow><mover><mrow><mi>l</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math>.</p>
<p>The&nbsp;current&nbsp;flows perpendicular to the magnetic field, so:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>F</mi><mrow><mi>l</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>IB</mi></mrow></mstyle></math>.&nbsp; Using the right-hand rule, we get the direction of the force, thus:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mrow><mi>l</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mn>2</mn><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac><mover><mrow><mi>y</mi></mrow><mi>&#x005e;</mi></mover></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>.</p>@
qu.18.17.algorithm=$var=switch($rint(5),'a','b','c','d');
$mag=range(1,9);
$cur=range(1,9);
$cur2=range(1,9);@
qu.18.17.uid=89bb2d5a-b698-47d1-ab83-9e52f288dacd@
qu.18.17.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Force Between Current Carrying Wires;
  Difficulty=Hard;
  Features=Algebraic;
  Features=Algorithmic;
  Features=Diagram;
@

qu.18.18.question=<p>Four current carrying wires are arranged as seen in the diagram, where the arrows indicate the direction of current flow.<br />
The wires are each separated by a distance <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$w</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mstyle></math>. Calculate the magnetic field vector at the center point, given the following information:<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>B</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$B</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>C</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$C</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi mathvariant='normal'>D</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$D</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mstyle></math>.</p>
<p><img alt="" align="middle" width="753" height="500" src="__BASE_URI__img/MagneticFields/CurrentCarryingWires-1/Diagram5.png" /></p>
<p>&nbsp;<em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>xhat</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>yhat</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>z</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>zhat</mi></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.</p>@
qu.18.18.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uCT,mCT=0.5,uVS,mVS=0.5,uSM,mSM=0.8,subPackage=vectorNumericComponents);@
qu.18.18.allow2d=0@
qu.18.18.maple_answer=(($A+$B+$C+$D)*(4*Pi*10^(-7))*zhat*T)/(Pi*($w/1000))@
qu.18.18.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.18.18.type=maple@
qu.18.18.mode=Maple@
qu.18.18.name=Magnetic Fields Due to Current Carrying Wires - 5 - Numeric@
qu.18.18.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT,mCT=0.5,uVS,mVS=0.5,uSM,mSM=0.8,subPackage=vectorNumericComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.18.18.editing=useHTML@
qu.18.18.solution=<p>The magnitude of the magnetic field due to an infinite current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>&pi;r</mi></mrow></mfrac></mrow></mstyle></math>.&nbsp; The direction of the magnetic field is found using the right-hand rule, where the thumb points in the direction of current flow and the curl of the fingers represents the direction of the magnetic field.&nbsp; When we consider the&nbsp;wires from the side, at any point on the screen, the magnetic field will either be directly into or out of the screen.&nbsp;</p>
<p>At the center point, for each wire, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mi>$w</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mstyle></math>.&nbsp; We are given the current in each wire and its direction.</p>
<p>For each wire, we need then to identify whether their resultant magnetic field is into or out of the screen, then add each together.</p>@
qu.18.18.algorithm=$A=rand(1.00,9.99,3);
$B=rand(1.00,9.99,3);
$C=rand(1.00,9.99,3);
$D=rand(1.00,9.99,3);
$w=rand(1.00,9.99,3);@
qu.18.18.uid=b1ec392c-417c-4d7a-bb26-defbdeb6f9ad@
qu.18.18.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Magnetic Fields Due to Current Carrying Wires;
  Difficulty=Medium;
  Features=Algorithmic;
  Features=Diagram;
  Features=Partial Grading;
@

qu.18.19.question=<p>Four current carrying wires are arranged as seen in the diagram, where the arrows indicate the direction of current flow.<br />
The wires are each separated by a distance <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>w</mi></mrow></mstyle></math>. Calculate the magnetic field vector at the center point, given the following information:<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>B</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$B</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>C</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$C</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi mathvariant='normal'>D</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$D</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>.</p>
<p><img alt="" align="middle" width="753" height="500" src="__BASE_URI__img/MagneticFields/CurrentCarryingWires-1/Diagram2.png" /></p>
<p>&nbsp;<em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>xhat</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>yhat</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>z</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>zhat</mi></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
Enter your response in terms of the given parameters and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>, entering <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi></mrow></mstyle></math>&nbsp;instead of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>.</p>@
qu.18.19.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT,mCT=0.5,uVS,mVS=0.5,uSM,mSM=0.8,subPackage=vectorAlgebraicComponents);@
qu.18.19.allow2d=0@
qu.18.19.maple_answer=((-$A-$B-$C+$D)*I*u*zhat)/(Pi*w)@
qu.18.19.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.18.19.type=maple@
qu.18.19.mode=Maple@
qu.18.19.name=Magnetic Fields Due to Current Carrying Wires - 2@
qu.18.19.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT,mCT=0.5,uVS,mVS=0.5,uSM,mSM=0.8,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.18.19.editing=useHTML@
qu.18.19.solution=<p>The magnitude of the magnetic field due to an infinite current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>&pi;r</mi></mrow></mfrac></mrow></mstyle></math>.&nbsp; The direction of the magnetic field is found using the right-hand rule, where the thumb points in the direction of current flow and the curl of the fingers represents the direction of the magnetic field.&nbsp; When we consider the&nbsp;wires from the side, at any point on the screen, the magnetic field will either be directly into or out of the screen.&nbsp;</p>
<p>At the center point, for each wire, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>w</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mstyle></math>.&nbsp; We are given the current in each wire and its direction.</p>
<p>For each wire, we need then to identify whether their resultant magnetic field is into or out of the screen, then add each together.</p>@
qu.18.19.algorithm=$A=range(1,9);
$B=range(1,9);
$C=range(1,9);
$D=range(1,9);@
qu.18.19.uid=8d1ef18a-0f89-476d-8396-9a20f1fbe897@
qu.18.19.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Magnetic Fields Due to Current Carrying Wires;
  Difficulty=Medium;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Diagram;
  Features=Partial Grading;
@

qu.18.20.question=<p>Four current carrying wires are arranged as seen in the diagram, where the arrows indicate the direction of current flow.<br />
The wires are each separated by a distance <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>w</mi></mrow></mstyle></math>. Calculate the magnetic field vector at the center point, given the following information:<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>B</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$B</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>C</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$C</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi mathvariant='normal'>D</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$D</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>.</p>
<p><img alt="" align="middle" width="753" height="500" src="__BASE_URI__img/MagneticFields/CurrentCarryingWires-1/Diagram1.png" /></p>
<p>&nbsp;<em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>xhat</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>yhat</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>z</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>zhat</mi></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
Enter your response in terms of the given parameters and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>, entering <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi></mrow></mstyle></math>&nbsp;instead of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>.</p>@
qu.18.20.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT,mCT=0.5,uVS,mVS=0.5,uSM,mSM=0.8,subPackage=vectorAlgebraicComponents);@
qu.18.20.allow2d=0@
qu.18.20.maple_answer=((-$A+$B-$C+$D)*I*u*zhat)/(Pi*w)@
qu.18.20.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.18.20.type=maple@
qu.18.20.mode=Maple@
qu.18.20.name=Magnetic Fields Due to Current Carrying Wires - 1@
qu.18.20.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT,mCT=0.5,uVS,mVS=0.5,uSM,mSM=0.8,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.18.20.editing=useHTML@
qu.18.20.solution=<p>The magnitude of the magnetic field due to an infinite current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>&pi;r</mi></mrow></mfrac></mrow></mstyle></math>.&nbsp; The direction of the magnetic field is found using the right-hand rule, where the thumb points in the direction of current flow and the curl of the fingers represents the direction of the magnetic field.&nbsp; When we consider the&nbsp;wires from the side, at any point on the screen, the magnetic field will either be directly into or out of the screen.&nbsp;</p>
<p>At the center point, for each wire, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>w</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mstyle></math>.&nbsp; We are given the current in each wire and its direction.</p>
<p>For each wire, we need then to identify whether their resultant magnetic field is into or out of the screen, then add each together.</p>@
qu.18.20.algorithm=$A=range(1,9);
$B=range(1,9);
$C=range(1,9);
$D=range(1,9);@
qu.18.20.uid=d8c0fcd7-1a5d-4ce1-a058-5c50c2d3d1f8@
qu.18.20.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Magnetic Fields Due to Current Carrying Wires;
  Difficulty=Medium;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Diagram;
  Features=Partial Grading;
@

qu.18.21.question=<p>Four current carrying wires are arranged as seen in the diagram, where the arrows indicate the direction of current flow.<br />
The wires are each separated by a distance <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>w</mi></mrow></mstyle></math>. Calculate the magnetic field vector at the center point, given the following information:<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>B</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$B</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>C</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$C</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi mathvariant='normal'>D</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$D</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>.</p>
<p><img alt="" align="middle" width="753" height="500" src="__BASE_URI__img/MagneticFields/CurrentCarryingWires-1/Diagram5.png" /></p>
<p>&nbsp;<em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>xhat</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>yhat</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>z</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>zhat</mi></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
Enter your response in terms of the given parameters and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>, entering <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi></mrow></mstyle></math>&nbsp;instead of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>.</p>@
qu.18.21.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT,mCT=0.5,uVS,mVS=0.5,uSM,mSM=0.8,subPackage=vectorAlgebraicComponents);@
qu.18.21.allow2d=0@
qu.18.21.maple_answer=((+$A+$B+$C+$D)*I*u*zhat)/(Pi*w)@
qu.18.21.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.18.21.type=maple@
qu.18.21.mode=Maple@
qu.18.21.name=Magnetic Fields Due to Current Carrying Wires - 5@
qu.18.21.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT,mCT=0.5,uVS,mVS=0.5,uSM,mSM=0.8,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.18.21.editing=useHTML@
qu.18.21.solution=<p>The magnitude of the magnetic field due to an infinite current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>&pi;r</mi></mrow></mfrac></mrow></mstyle></math>.&nbsp; The direction of the magnetic field is found using the right-hand rule, where the thumb points in the direction of current flow and the curl of the fingers represents the direction of the magnetic field.&nbsp; When we consider the&nbsp;wires from the side, at any point on the screen, the magnetic field will either be directly into or out of the screen.&nbsp;</p>
<p>At the center point, for each wire, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>w</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mstyle></math>.&nbsp; We are given the current in each wire and its direction.</p>
<p>For each wire, we need then to identify whether their resultant magnetic field is into or out of the screen, then add each together.</p>@
qu.18.21.algorithm=$A=range(1,9);
$B=range(1,9);
$C=range(1,9);
$D=range(1,9);@
qu.18.21.uid=1f636b43-0f2d-42cb-b49e-50699ab1d54c@
qu.18.21.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Magnetic Fields Due to Current Carrying Wires;
  Difficulty=Medium;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Diagram;
  Features=Partial Grading;
@

qu.18.22.question=<p>A current of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mA</mi></mrow></mstyle></math>&nbsp;is travelling with through a wire of displacement <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mover><mi>l</mi><mi>&rarr;</mi></mover></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$l1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>cm</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mi>$ldir</mi><mi>&#x005e;</mi></mover></mrow></mstyle></math>.&nbsp; If the region&nbsp;<br />
has a uniform magnetic field of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>B</mi><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$B1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>T</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mover><mrow><mi>$Bdir</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>, what is the force on the wires?&nbsp;</p>
<p>&nbsp;</p>
<p><em>Entry Notes</em>:&nbsp;<br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ihat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>jhat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>&nbsp; and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>khat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp;<br />
Ensure that you explicitly enter the multiplication symbol * between terms.</p>@
qu.18.22.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uVS=false,uUN);@
qu.18.22.allow2d=0@
qu.18.22.maple_answer=with(Physics[Vectors]);
B:=($B1)*subs({i=_i,j=_j,k=_k},$Bdir);
l:=($l1*10^(-2))*subs({i=_i,j=_j,k=_k},$ldir);
temp:=($I*10^(-3))*(l &x B);
SigFigs[roundToSigFigs](Component(temp,1),3)*N*ihat+SigFigs[roundToSigFigs](Component(temp,2),3)*N*jhat+SigFigs[roundToSigFigs](Component(temp,3),3)*N*khat;@
qu.18.22.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.18.22.type=maple@
qu.18.22.mode=Maple@
qu.18.22.name=Magnetic Force On A Current Carrying Wire - 1D@
qu.18.22.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uVS=false,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.18.22.editing=useHTML@
qu.18.22.hint.1=Set up a determinant.&nbsp; The first row should be <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp; The second row should be the components of the first vector and the third row should be the components of the second vector.&nbsp; Evaluate the determinant.@
qu.18.22.solution=<p>The force due to a magnetic field on a current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>I</mi><mrow><mover><mrow><mi>l</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math>.</p>
<p align="left">We are given <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>l</mi></mrow><mi>&rarr;</mi></mover></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow></mstyle></math>.</p>@
qu.18.22.algorithm=$B1=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$l1=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$I=rand(1.00,9.99,3);
$Bdir=switch(rint(0,3),'i','j','k');
$ldir=switch(rint(0,3),'i','j','k');
$BDisp='


<math><mrow><mn>$B1</mn><mover><mi>$Bdir</mi><mo>&Hat;</mo></mover></mrow></math>';
$lDisp='


<math><mrow><mn>$l1</mn><mover><mi>$ldir</mi><mo>&Hat;</mo></mover></mrow></math>';@
qu.18.22.uid=9c279d8e-3388-4e13-bdf5-5b01f37d4358@
qu.18.22.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Magnetic Force On Current Carrying Wire;
  Difficulty=Hard;
  Features=Algorithmic;
  Features=Partial Grading;
@

qu.18.23.question=<p>Four current carrying wires are arranged as seen in the diagram, where the arrows indicate the direction of current flow.<br />
The wires are each separated by a distance <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>w</mi></mrow></mstyle></math>. Calculate the magnetic field vector at the center point, given the following information:<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>B</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$B</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>C</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$C</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi mathvariant='normal'>D</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$D</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>.</p>
<p><img alt="" align="middle" width="753" height="500" src="__BASE_URI__img/MagneticFields/CurrentCarryingWires-1/Diagram3.png" /></p>
<p>&nbsp;<em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>xhat</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>yhat</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>z</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>zhat</mi></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
Enter your response in terms of the given parameters and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>, entering <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi></mrow></mstyle></math>&nbsp;instead of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>.</p>@
qu.18.23.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT,mCT=0.5,uVS,mVS=0.5,uSM,mSM=0.8,subPackage=vectorAlgebraicComponents);@
qu.18.23.allow2d=0@
qu.18.23.maple_answer=(($A-$B-$C+$D)*I*u*zhat)/(Pi*w)@
qu.18.23.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.18.23.type=maple@
qu.18.23.mode=Maple@
qu.18.23.name=Magnetic Fields Due to Current Carrying Wires - 3@
qu.18.23.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT,mCT=0.5,uVS,mVS=0.5,uSM,mSM=0.8,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.18.23.editing=useHTML@
qu.18.23.solution=<p>The magnitude of the magnetic field due to an infinite current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>&pi;r</mi></mrow></mfrac></mrow></mstyle></math>.&nbsp; The direction of the magnetic field is found using the right-hand rule, where the thumb points in the direction of current flow and the curl of the fingers represents the direction of the magnetic field.&nbsp; When we consider the&nbsp;wires from the side, at any point on the screen, the magnetic field will either be directly into or out of the screen.&nbsp;</p>
<p>At the center point, for each wire, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>w</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mstyle></math>.&nbsp; We are given the current in each wire and its direction.</p>
<p>For each wire, we need then to identify whether their resultant magnetic field is into or out of the screen, then add each together.</p>@
qu.18.23.algorithm=$A=range(1,9);
$B=range(1,9);
$C=range(1,9);
$D=range(1,9);@
qu.18.23.uid=cad43adf-31b4-4c84-a9eb-c67d140817d5@
qu.18.23.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Magnetic Fields Due to Current Carrying Wires;
  Difficulty=Medium;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Diagram;
  Features=Partial Grading;
@

qu.18.24.question=<p>The wires <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>&nbsp;are carrying a current <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;is carrying current <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$cur2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>, each in the&nbsp;<br />
directions indicated in the diagram.&nbsp; In&nbsp;terms of the given parameters, provide a vector&nbsp;expression for&nbsp;<br />
the&nbsp;force per unit length exerted by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>&nbsp;on <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>, for&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;placed at&nbsp;any point on the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></mstyle></math>axis.</p>
<p>&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="500" height="505">
<param name="image" value="__BASE_URI__img/MagneticFields/CurrentCarryingWiresVertical/Diagram3Wires1-2.png" />
<param name="size" value="3" />
<param name="label.1.x" value="232" />
<param name="label.1.y" value="215" />
<param name="label.1.text" value="$mag$var" />
<param name="label.2.x" value="232" />
<param name="label.2.y" value="315" />
<param name="label.2.text" value="$mag$var" />
<param name="label.3.x" value="315" />
<param name="label.3.y" value="257" />
<param name="label.3.text" value="x" /></applet></p>
<p>&nbsp;</p>
<p><em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>xhat</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>yhat</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>z</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>zhat</mi></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
Enter your response in terms of the given parameters and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>, entering&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi></mrow></mstyle></math> instead of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>.</p>@
qu.18.24.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents);@
qu.18.24.allow2d=0@
qu.18.24.maple_answer=(-yhat)*($cur2*I)*2*(u*$cur*I/(2*Pi*sqrt(x^2+($mag*$var)^2)))*(($mag*$var)/sqrt(x^2+($mag*$var)^2));@
qu.18.24.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.18.24.type=maple@
qu.18.24.mode=Maple@
qu.18.24.name=Current Carrying Wires - Vertical - 3 Wires - 1-2@
qu.18.24.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.18.24.editing=useHTML@
qu.18.24.solution=<p>We will calculate the magnetic field due to wires <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>, and then the force due to this magnetic field on <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>.</p>
<p>The magnitude of the magnetic field due to an infinite current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>&pi;r</mi></mrow></mfrac></mrow></mstyle></math>.&nbsp; The direction of the magnetic field is found using the right-hand rule, where the thumb points in the direction of current flow and the curl of the fingers represents the direction of the magnetic field.&nbsp; If we draw a line from the wire to the point in question, then the magnetic field will always be perpendicular to that line.</p>
<p>&nbsp;</p>
<p>At point&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math> the magnitude of the magnetic field due to each of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math> and&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math> will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow></mfrac></mrow></mstyle></math>.</p>
<p>Based on the configuration, the vertical components will cancel.&nbsp; Thus, we need only calculate the horizontal components, which are:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>B</mi><mrow><mi>x</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>, and therefore the total magnetic field at&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mn>2</mn><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac><mrow><mover><mrow><mi>x</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>Finally, the force due to a magnetic field on a current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>I</mi><mrow><mover><mrow><mi>l</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math>.</p>
<p>The&nbsp;current&nbsp;flows perpendicular to the magnetic field, so:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>F</mi><mrow><mi>l</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>IB</mi></mrow></mstyle></math>.&nbsp; Using the right-hand rule, we get the direction of the force, thus:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mrow><mi>l</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mn>2</mn><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac><mover><mrow><mi>y</mi></mrow><mi>&#x005e;</mi></mover></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>.</p>@
qu.18.24.algorithm=$var=switch($rint(5),'a','b','c','d');
$mag=range(1,9);
$cur=range(1,9);
$cur2=range(1,9);@
qu.18.24.uid=1ca78ee2-6622-4092-b527-4a6bce3d82d1@
qu.18.24.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Force Between Current Carrying Wires;
  Difficulty=Hard;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Partial Grading;
@

qu.18.25.question=<p>Four current carrying wires are arranged as seen in the diagram, where the arrows indicate the direction of current flow.<br />
The wires are each separated by a distance <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$w</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mstyle></math>. Calculate the magnetic field vector at the center point, given the following information:<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>B</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$B</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>C</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$C</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi mathvariant='normal'>D</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$D</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mstyle></math>.</p>
<p><img alt="" align="middle" width="753" height="500" src="__BASE_URI__img/MagneticFields/CurrentCarryingWires-1/Diagram1.png" /></p>
<p>&nbsp;<em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>xhat</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>yhat</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>z</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>zhat</mi></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.</p>@
qu.18.25.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uCT,mCT=0.5,uVS,mVS=0.5,uSM,mSM=0.8,subPackage=vectorNumericComponents);@
qu.18.25.allow2d=0@
qu.18.25.maple_answer=((-$A+$B-$C+$D)*(4*Pi*10^(-7))*zhat*T)/(Pi*($w/1000))@
qu.18.25.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.18.25.type=maple@
qu.18.25.mode=Maple@
qu.18.25.name=Magnetic Fields Due to Current Carrying Wires - 1 - Numeric@
qu.18.25.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT,mCT=0.5,uVS,mVS=0.5,uSM,mSM=0.8,subPackage=vectorNumericComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.18.25.editing=useHTML@
qu.18.25.solution=<p>The magnitude of the magnetic field due to an infinite current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>&pi;r</mi></mrow></mfrac></mrow></mstyle></math>.&nbsp; The direction of the magnetic field is found using the right-hand rule, where the thumb points in the direction of current flow and the curl of the fingers represents the direction of the magnetic field.&nbsp; When we consider the&nbsp;wires from the side, at any point on the screen, the magnetic field will either be directly into or out of the screen.&nbsp;</p>
<p>At the center point, for each wire, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mi>$w</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mstyle></math>.&nbsp; We are given the current in each wire and its direction.</p>
<p>For each wire, we need then to identify whether their resultant magnetic field is into or out of the screen, then add each together.</p>@
qu.18.25.algorithm=$A=rand(1.00,9.99,3);
$B=rand(1.00,9.99,3);
$C=rand(1.00,9.99,3);
$D=rand(1.00,9.99,3);
$w=rand(1.00,9.99,3);@
qu.18.25.uid=a1619aa3-e351-4b10-813d-f33b76373887@
qu.18.25.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Magnetic Fields Due to Current Carrying Wires;
  Difficulty=Medium;
  Features=Algorithmic;
  Features=Diagram;
  Features=Partial Grading;
@

qu.18.26.question=<p>The wires <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>&nbsp;are carrying a current <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;is carrying current <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$cur2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mstyle></math>, each in the&nbsp;<br />
directions indicated in the diagram.&nbsp; In&nbsp;terms of the given parameters, provide a vector&nbsp;expression for&nbsp;<br />
the&nbsp;force per unit length exerted by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>&nbsp;on <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>, for&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;placed at&nbsp;any point on the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></mstyle></math>axis.</p>
<p>&nbsp;</p>
<p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="500" height="505">
<param name="image" value="__BASE_URI__img/MagneticFields/CurrentCarryingWiresVertical/Diagram3Wires3-1.png" />
<param name="size" value="3" />
<param name="label.1.x" value="232" />
<param name="label.1.y" value="215" />
<param name="label.1.text" value="$mag$var" />
<param name="label.2.x" value="232" />
<param name="label.2.y" value="315" />
<param name="label.2.text" value="$mag$var" />
<param name="label.3.x" value="315" />
<param name="label.3.y" value="257" />
<param name="label.3.text" value="x" /></applet></p>
<p>&nbsp;</p>
<p><em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>xhat</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>yhat</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>z</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>zhat</mi></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.<br />
Enter your response in terms of the given parameters and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>, entering&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi></mrow></mstyle></math> instead of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub></mrow></mstyle></math>.</p>@
qu.18.26.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents);@
qu.18.26.allow2d=0@
qu.18.26.maple_answer=(-yhat)*($cur2*I)*2*(u*$cur*I/(2*Pi*sqrt(x^2+($mag*$var)^2)))*(($mag*$var)/sqrt(x^2+($mag*$var)^2));@
qu.18.26.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.18.26.type=maple@
qu.18.26.mode=Maple@
qu.18.26.name=Current Carrying Wires - Vertical - 3 Wires - 3-1@
qu.18.26.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uS12,mS12=0.8,uS23,mS23=0.5,subPackage=vectorAlgebraicComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.18.26.editing=useHTML@
qu.18.26.solution=<p>We will calculate the magnetic field due to wires <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math>, and then the force due to this magnetic field on <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>.</p>
<p>The magnitude of the magnetic field due to an infinite current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>&pi;r</mi></mrow></mfrac></mrow></mstyle></math>.&nbsp; The direction of the magnetic field is found using the right-hand rule, where the thumb points in the direction of current flow and the curl of the fingers represents the direction of the magnetic field.&nbsp; If we draw a line from the wire to the point in question, then the magnetic field will always be perpendicular to that line.</p>
<p>&nbsp;</p>
<p>At point&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math> the magnitude of the magnetic field due to each of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math> and&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi></mrow></mstyle></math> will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mrow><mi>&pi;</mi></mrow><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow></mfrac></mrow></mstyle></math>.</p>
<p>Based on the configuration, the vertical components will cancel.&nbsp; Thus, we need only calculate the horizontal components, which are:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>B</mi><mrow><mi>x</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>, and therefore the total magnetic field at&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi></mrow></mstyle></math>&nbsp;will be:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mn>2</mn><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac><mover><mrow><mi>x</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>Finally, the force due to a magnetic field on a current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>I</mi><mrow><mover><mrow><mi>l</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math>.</p>
<p>The&nbsp;current&nbsp;flows perpendicular to the magnetic field, so:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>F</mi><mrow><mi>l</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>IB</mi></mrow></mstyle></math>.&nbsp; Using the right-hand rule, we get the direction of the force, thus:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mrow><mi>l</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><mn>2</mn><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>$cur2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$cur</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mi>&pi;</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$mag$var</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></mfrac><mover><mrow><mi>y</mi></mrow><mi>&#x005e;</mi></mover></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>.</p>@
qu.18.26.algorithm=$var=switch($rint(5),'a','b','c','d');
$mag=range(1,9);
$cur=range(1,9);
$cur2=range(1,9);@
qu.18.26.uid=188d8388-a987-4ebe-b067-02769fbdc869@
qu.18.26.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Force Between Current Carrying Wires;
  Difficulty=Hard;
  Features=Algorithmic;
  Features=Algebraic;
  Features=Partial Grading;
@

qu.18.27.question=<p>Four current carrying wires are arranged as seen in the diagram, where the arrows indicate the direction of current flow.<br />
The wires are each separated by a distance <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$w</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mstyle></math>. Calculate the magnetic field vector at the center point, given the following information:<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>B</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$B</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi>C</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$C</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>I</mi><mrow><mi mathvariant='normal'>D</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$D</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi></mrow></mstyle></math>.</p>
<p><img alt="" align="middle" width="753" height="500" src="__BASE_URI__img/MagneticFields/CurrentCarryingWires-1/Diagram3.png" /></p>
<p>&nbsp;<em>Entry Notes</em>: <br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>x</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>xhat</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>y</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>yhat</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mi>z</mi><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>zhat</mi></mrow></mstyle></math>. <br />
Ensure that you explicitly enter the multiplication symbol * between terms.</p>@
qu.18.27.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uCT,mCT=0.5,uVS,mVS=0.5,uSM,mSM=0.8,subPackage=vectorNumericComponents);@
qu.18.27.allow2d=0@
qu.18.27.maple_answer=(($A-$B-$C+$D)*(4*Pi*10^(-7))*zhat*T)/(Pi*($w/1000))@
qu.18.27.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.18.27.type=maple@
qu.18.27.mode=Maple@
qu.18.27.name=Magnetic Fields Due to Current Carrying Wires - 3 - Numeric@
qu.18.27.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uCT,mCT=0.5,uVS,mVS=0.5,uSM,mSM=0.8,subPackage=vectorNumericComponents,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.18.27.editing=useHTML@
qu.18.27.solution=<p>The magnitude of the magnetic field due to an infinite current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>&mu;</mi><mrow><mi>o</mi></mrow></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>&pi;r</mi></mrow></mfrac></mrow></mstyle></math>.&nbsp; The direction of the magnetic field is found using the right-hand rule, where the thumb points in the direction of current flow and the curl of the fingers represents the direction of the magnetic field.&nbsp; When we consider the&nbsp;wires from the side, at any point on the screen, the magnetic field will either be directly into or out of the screen.&nbsp;</p>
<p>At the center point, for each wire, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mi>$w</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mstyle></math>.&nbsp; We are given the current in each wire and its direction.</p>
<p>For each wire, we need then to identify whether their resultant magnetic field is into or out of the screen, then add each together.</p>@
qu.18.27.algorithm=$A=rand(1.00,9.99,3);
$B=rand(1.00,9.99,3);
$C=rand(1.00,9.99,3);
$D=rand(1.00,9.99,3);
$w=rand(1.00,9.99,3);@
qu.18.27.uid=a77d3544-9fa6-44fa-8e21-1bcedef8ee8a@
qu.18.27.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Magnetic Fields Due to Current Carrying Wires;
  Difficulty=Medium;
  Features=Algorithmic;
  Features=Diagram;
  Features=Partial Grading;
@

qu.18.28.question=<p>A region of space contains a magnetic field of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$B1</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$B2</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$B3</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo></mrow><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow><mrow><mi>T</mi></mrow></mstyle></math>.&nbsp; A current of&nbsp;<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mA</mi></mrow></mstyle></math>is travelling&nbsp;through a&nbsp;wire of displacement<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>l</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$l1</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>i</mi><mi>&#x005e;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$l2</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mi>j</mi><mi>&#x005e;</mi></mover></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$l3</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo></mrow><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mi mathvariant='normal'></mi></mrow><mrow><mi mathvariant='normal'></mi></mrow><mrow><mi mathvariant='normal'></mi></mrow><mrow><mi>cm</mi></mrow></mstyle></math>&nbsp;in this region.&nbsp;<br />
What is the&nbsp;force on the wire due to the magnetic field?&nbsp;</p>
<p>&nbsp;</p>
<p><em>Entry Notes</em>:&nbsp;<br />
Vector components can be entered using <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ihat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>jhat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>&nbsp; and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>khat</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mrow><mover><mrow><mi>k</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp;<br />
Ensure that you explicitly enter the multiplication symbol * between terms.</p>@
qu.18.28.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uVS=false,uUN);@
qu.18.28.allow2d=0@
qu.18.28.maple_answer=with(Physics[Vectors]);
l:=(($l1)*_i+($l2)*_j+($l3)*_k)*10^(-2);
B:=(($B1)*_i+($B2)*_j+($B3)*_k);
temp:=($I*10^(-3))*(l &x B);
SigFigs[roundToSigFigs](Component(temp,1),3)*N*ihat+SigFigs[roundToSigFigs](Component(temp,2),3)*N*jhat+SigFigs[roundToSigFigs](Component(temp,3),3)*N*khat;@
qu.18.28.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.18.28.type=maple@
qu.18.28.mode=Maple@
qu.18.28.name=Magnetic Force On A Current Carrying Wire - 3D@
qu.18.28.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uVS=false,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.18.28.editing=useHTML@
qu.18.28.hint.1=Set up a determinant.&nbsp; The first row should be <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>i</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mrow><mi>j</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mover><mi>k</mi><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>.&nbsp; The second row should be the components of the first vector and the third row should be the components of the second vector.&nbsp; Evaluate the determinant.@
qu.18.28.solution=<p>The force due to a magnetic field on a current carrying wire is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>F</mi></mrow><mi>&rarr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>I</mi><mrow><mover><mrow><mi>l</mi></mrow><mi>&rarr;</mi></mover></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow></mrow></mstyle></math>.</p>
<p align="left">We are given <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>l</mi></mrow><mi>&rarr;</mi></mover></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>B</mi></mrow><mi>&rarr;</mi></mover></mrow></mstyle></math>.</p>@
qu.18.28.algorithm=$B1=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$B2=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$B3=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$l1=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$l2=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$l3=switch(rint(0,2),rand(1,9,3),-rand(1,9,3));
$I=rand(1,9.99,3);@
qu.18.28.uid=f4e9837c-810f-4a0c-93ed-87f117ef53e0@
qu.18.28.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Magnetic Force Due to Current Carrying Wire;
  Difficulty=Hard;
  Features=Algorithmic;
  Features=Partial Grading;
@

qu.19.topic=Diffraction@

qu.19.1.question=<p>The narrow slits of a diffraction grating are&nbsp;separated by a distance of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$d</mi></mrow><mrow><mi></mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&mu;m</mi></mrow></mstyle></math>,&nbsp;with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mrow><mo lspace='0.0em' rspace='0.0em'>&Gt;</mo></mrow><mi>d</mi></mrow></mstyle></math>.&nbsp; A&nbsp;<br />
beam of&nbsp;monochromatic light&nbsp;of wavelength <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mstyle></math>passes through the grating at normal&nbsp;<br />
incidence.&nbsp; What order of&nbsp;dark fringes&nbsp;are observed at an angle <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>$theta</mi><mrow><mi>o</mi></mrow></msup></mrow></mstyle></math>&nbsp;above and below <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math>?</p>
<p>&nbsp;</p>
<p><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="593" height="500">
<param name="image" value="__BASE_URI__img/Interference/DiagramGrating.png" />
<param name="size" value="1" />
<param name="label.1.x" value="230" />
<param name="label.1.y" value="10" />
<param name="label.1.text" value="X" /></applet></p>
<p>&nbsp;</p>
<p>&nbsp;</p>@
qu.19.1.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uEM,uSF=false);@
qu.19.1.allow2d=0@
qu.19.1.maple_answer=SigFigs[roundToSigFigs]($m,3)@
qu.19.1.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.19.1.type=maple@
qu.19.1.mode=Maple@
qu.19.1.name=Diffraction Grating - Find Fringe Order - Dark ~ PGc@
qu.19.1.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uEM,uSF=false,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.19.1.editing=useHTML@
qu.19.1.solution=<p>For a diffraction grating, destructive interference occurs when the parameters satisfy:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mi>m</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced><mi>&lambda;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>d</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>.</p>
<p>Thus, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>m</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mi>&mu;</mi></mrow><mi>m</mi></mrow></mfenced><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mfenced></mrow></mfrac><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>$theta</mi><mrow><mi>o</mi></mrow></msup></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>&nbsp;</p>@
qu.19.1.algorithm=$m=range(1,6);
$theta=rand(0,45,3);
$lambda=range(100,999);
$d=sig(3,($m/2)*$lambda*10^(-9)*10^(6)/sin($theta*2*Pi/360));@
qu.19.1.uid=1b6ce0e0-1c10-4064-93cd-622792bf145a@
qu.19.1.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Diffraction;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.19.2.question=<p>The slit in the diagram&nbsp;is of width&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow><mrow><mi></mi></mrow></mstyle></math>.&nbsp; If the&nbsp;light&nbsp;incident&nbsp;on the slit is of wavelength <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mstyle></math>,<br />
&nbsp;at what distance above and below <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math> would&nbsp;one&nbsp;find&nbsp;the&nbsp;order-$m bright fringes on the screen?</p>
<p>&nbsp;</p>
<p><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="593" height="500">
<param name="image" value="__BASE_URI__img/Interference/DiagramSingleSlit.png" />
<param name="size" value="1" />
<param name="label.1.x" value="230" />
<param name="label.1.y" value="10" />
<param name="label.1.text" value="$X m" /></applet></p>
<p>&nbsp;</p>
<p>&nbsp;</p>@
qu.19.2.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25);@
qu.19.2.allow2d=0@
qu.19.2.maple_answer=SigFigs[roundToSigFigs]($ans,3)*cm@
qu.19.2.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.19.2.type=maple@
qu.19.2.mode=Maple@
qu.19.2.name=Single Slit Diffraction - Find Screen Height - Bright ~ PGc@
qu.19.2.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.19.2.editing=useHTML@
qu.19.2.solution=<p>For a single slit, constructive interference occurs when the parameters satisfy:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mi>m</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced><mi>&lambda;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>d</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>.</p>
<p>Thus, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&theta;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msup><mi>sin</mi><mn>-1</mn></msup><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$m</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mfenced></mrow><mrow><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mfrac></mrow></mfenced></mrow></mrow></mstyle></math>.</p>
<p>&nbsp;Given the distance to the screen, we have:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>y</mi><mrow><mi>$m</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>m</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>.</p>@
qu.19.2.algorithm=$d=rand(0.5,0.9,3);
$m=range(1,4);
$lambda=range(400,800);
$X=rand(1.00,9.00,3);
$ans=tan(arcsin((($m/2)*$lambda*10^(-9))/($d*10^(-3))))*$X*10^(2);@
qu.19.2.uid=fba95af7-3fa8-4213-a690-6ead9f503a1e@
qu.19.2.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Diffraction;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.19.3.question=<p>The single slit in the diagram&nbsp;is of width&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow><mrow><mi></mi></mrow></mstyle></math>.&nbsp; If&nbsp;light&nbsp;incident&nbsp;on&nbsp;the slit&nbsp;of wavelength <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mstyle></math><br />
&nbsp;produces&nbsp;order-$m dark fringes&nbsp;on the screen&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$y</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>cm</mi></mrow></mstyle></math>above&nbsp;and below <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math>, what is the distance <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math>&nbsp;to&nbsp;<br />
the screen?</p>
<p>&nbsp;</p>
<p><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="593" height="500">
<param name="image" value="__BASE_URI__img/Interference/DiagramSingleSlit.png" />
<param name="size" value="1" />
<param name="label.1.x" value="230" />
<param name="label.1.y" value="10" />
<param name="label.1.text" value="X" /></applet></p>
<p>&nbsp;</p>
<p>&nbsp;</p>@
qu.19.3.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25);@
qu.19.3.allow2d=0@
qu.19.3.maple_answer=SigFigs[roundToSigFigs]($ans,3)*m@
qu.19.3.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.19.3.type=maple@
qu.19.3.mode=Maple@
qu.19.3.name=Single Slit Diffraction - Find Distance to Screen - Dark ~ PGc@
qu.19.3.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.19.3.editing=useHTML@
qu.19.3.solution=<p>For a single slit, destructive interference occurs when the parameters satisfy:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>m&lambda;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>d</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>.</p>
<p>Thus, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&theta;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msup><mi>sin</mi><mn>-1</mn></msup><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$m</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mfenced></mrow><mrow><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mfrac></mrow></mfenced></mrow></mrow></mstyle></math>.</p>
<p>Given the height of the fringe on the screen, we have:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mi>$y</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>cm</mi></mrow></mfenced><mrow><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>.</p>
<p>&nbsp;</p>@
qu.19.3.algorithm=$d=rand(0.5,0.9,3);
$m=range(1,4);
$lambda=range(400,800);
$X=rand(1.00,9.00,3);
$y=rand(0.1,1.5,3);
$ans=($y*10^(-2))/tan(arcsin(($m*$lambda*10^(-9))/($d*10^(-3))));@
qu.19.3.uid=0162448e-8651-4812-ac0d-e4ff252dfe34@
qu.19.3.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Diffraction;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.19.4.question=<p>The slit in the diagram&nbsp;is of width&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$d</mi></mrow><mrow><mi></mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&mu;m</mi></mrow></mstyle></math>,&nbsp;with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mrow><mo lspace='0.0em' rspace='0.0em'>&Gt;</mo></mrow><mi>d</mi></mrow></mstyle></math>.&nbsp; A&nbsp;beam of&nbsp;monochromatic light&nbsp;of&nbsp;<br />
wavelength <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mstyle></math>passes through the slit at normal&nbsp;incidence.&nbsp; What order of&nbsp;bright fringes&nbsp;are&nbsp;<br />
observed at an angle <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>$theta</mi><mrow><mi>o</mi></mrow></msup></mrow></mstyle></math>&nbsp;above and below <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math>?</p>
<p>&nbsp;</p>
<p><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="593" height="500">
<param name="image" value="__BASE_URI__img/Interference/DiagramSingleSlit.png" />
<param name="size" value="1" />
<param name="label.1.x" value="230" />
<param name="label.1.y" value="10" />
<param name="label.1.text" value="X" /></applet></p>
<p>&nbsp;</p>
<p>&nbsp;</p>@
qu.19.4.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uEM,uSF=false);@
qu.19.4.allow2d=0@
qu.19.4.maple_answer=SigFigs[roundToSigFigs]($m,3)@
qu.19.4.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.19.4.type=maple@
qu.19.4.mode=Maple@
qu.19.4.name=Single Slit Diffraction - Find Fringe Order - Bright ~ PGc@
qu.19.4.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uEM,uSF=false,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.19.4.editing=useHTML@
qu.19.4.solution=<p>For a single slit, constructive interference occurs when the parameters satisfy:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mi>m</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced><mi>&lambda;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>d</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>.</p>
<p>Thus, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>m</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&mu;m</mi></mrow></mfenced><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mfenced></mrow></mfrac></mrow><mrow><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>$theta</mi><mrow><mi>o</mi></mrow></msup></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>&nbsp;</p>@
qu.19.4.algorithm=$m=range(1,6);
$theta=rand(0,45,3);
$lambda=range(100,999);
$d=sig(3,($m/2)*$lambda*10^(-9)*10^(6)/sin($theta*2*Pi/360));@
qu.19.4.uid=d092470f-9051-426b-8c62-1128359bf661@
qu.19.4.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Diffraction;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.19.5.question=<p>The slit in the diagram&nbsp;is of&nbsp;width&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mi>&mu;</mi></mrow><mi>m</mi></mrow><mrow></mrow></mstyle></math>, and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mrow><mo lspace='0.0em' rspace='0.0em'>&Gt;</mo></mrow><mi>d</mi></mrow></mstyle></math>.&nbsp; A beam of monochromatic light&nbsp;passes&nbsp;<br />
through the slit at normal incidence.&nbsp; If the&nbsp;order-$m bright fringes are visible at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>$theta</mi><mrow><mi>o</mi></mrow></msup></mrow></mstyle></math>, what is the&nbsp;<br />
wavelength of the light?</p>
<p>&nbsp;</p>
<p><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="593" height="500">
<param name="image" value="__BASE_URI__img/Interference/DiagramSingleSlit.png" />
<param name="size" value="1" />
<param name="label.1.x" value="230" />
<param name="label.1.y" value="10" />
<param name="label.1.text" value="X" /></applet></p>
<p>&nbsp;</p>
<p>&nbsp;</p>
<p><span>&nbsp;</span></p>@
qu.19.5.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25);@
qu.19.5.allow2d=0@
qu.19.5.maple_answer=SigFigs[roundToSigFigs]($ans,3)*nm@
qu.19.5.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.19.5.type=maple@
qu.19.5.mode=Maple@
qu.19.5.name=Single Slit Diffraction - Find Wavelength - Bright ~ PGc@
qu.19.5.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.19.5.editing=useHTML@
qu.19.5.solution=<p>For a single slit, constructive interference occurs when the parameters satisfy:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mi>m</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced><mi>&lambda;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>d</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>.</p>
<p>Thus, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&lambda;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$d</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&mu;m</mi></mrow></mfenced><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>$theta</mi><mrow><mi>o</mi></mrow></msup></mrow></mfenced></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$m</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>.</p>
<p>&nbsp;</p>@
qu.19.5.algorithm=$d=rand(1.0,9.99,3);
$m=range(1,4);
$theta=rand(0,20,3);
$ans=($d*10^(-6))*sin($theta*2*Pi/360)*10^(9)/($m/2);@
qu.19.5.uid=f6493de5-cd6c-420e-9f30-83cfccec9dc0@
qu.19.5.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Diffraction;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.19.6.question=<p>The slit in the diagram&nbsp;is of width&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi></mrow></mstyle></math>, and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mrow><mo lspace='0.0em' rspace='0.0em'>&Gt;</mo></mrow><mi>d</mi></mrow></mstyle></math>.&nbsp; A beam of&nbsp;monochromatic light&nbsp;passes through the&nbsp;<br />
slit at normal incidence.&nbsp; If the&nbsp;order-$m&nbsp;bright fringes&nbsp;are&nbsp;visible at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>$theta</mi><mrow><mi>o</mi></mrow></msup></mrow></mstyle></math>&nbsp;for light of wavelength&nbsp;<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mstyle></math>,&nbsp;what is the slit width <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi></mrow></mstyle></math>?</p>
<p>&nbsp;</p>
<p><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="593" height="500">
<param name="image" value="__BASE_URI__img/Interference/DiagramSingleSlit.png" />
<param name="size" value="1" />
<param name="label.1.x" value="230" />
<param name="label.1.y" value="10" />
<param name="label.1.text" value="X" /></applet></p>
<p>&nbsp;</p>
<p>&nbsp;</p>@
qu.19.6.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25);@
qu.19.6.allow2d=0@
qu.19.6.maple_answer=SigFigs[roundToSigFigs]($ans,3)*um@
qu.19.6.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.19.6.type=maple@
qu.19.6.mode=Maple@
qu.19.6.name=Single Slit Diffraction - Find Slit Spacing - Bright ~ PGc@
qu.19.6.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.19.6.editing=useHTML@
qu.19.6.solution=<p>For a single slit, constructive interference occurs when the parameters satisfy:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mi>m</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced><mi>&lambda;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>d</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>.</p>
<p>Thus, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$m</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mfenced></mrow><mrow><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>$theta</mi><mrow><mi>o</mi></mrow></msup></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>.</p>
<p>&nbsp;</p>@
qu.19.6.algorithm=$m=range(1,4);
$theta=rand(0,20,3);
$lambda=range(100,999);
$ans=($m/2)*$lambda*10^(-9)*10^(6)/sin($theta*2*Pi/360);@
qu.19.6.uid=ee01051e-7f9a-442e-bb1d-3971ddcefdce@
qu.19.6.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Diffraction;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.19.7.question=<p>The narrow slits of a diffraction grating are&nbsp;separated by a distance of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mi>&mu;</mi></mrow><mi>m</mi></mrow><mrow></mrow></mstyle></math>, and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mrow><mo lspace='0.0em' rspace='0.0em'>&Gt;</mo></mrow><mi>d</mi></mrow></mstyle></math>.&nbsp; A<br />
beam of monochromatic light&nbsp;passes through the grating at normal incidence.&nbsp; If the&nbsp;order-$m dark<br />
fringes are visible at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>$theta</mi><mrow><mi>o</mi></mrow></msup></mrow></mstyle></math>, what is the wavelength of the light?</p>
<p>&nbsp;</p>
<p><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="593" height="500">
<param name="image" value="__BASE_URI__img/Interference/DiagramGrating.png" />
<param name="size" value="1" />
<param name="label.1.x" value="230" />
<param name="label.1.y" value="10" />
<param name="label.1.text" value="X" /></applet></p>
<p>&nbsp;</p>
<p>&nbsp;</p>@
qu.19.7.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25);@
qu.19.7.allow2d=0@
qu.19.7.maple_answer=SigFigs[roundToSigFigs]($ans,3)*nm@
qu.19.7.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.19.7.type=maple@
qu.19.7.mode=Maple@
qu.19.7.name=Diffraction Grating - Find Wavelength - Dark ~ PGc@
qu.19.7.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.19.7.editing=useHTML@
qu.19.7.solution=<p>For a diffraction grating, destructive interference occurs when the parameters satisfy:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mi>m</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced><mi>&lambda;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>d</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>.</p>
<p>Thus, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&lambda;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$d</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mi>&mu;</mi></mrow><mi>m</mi></mrow></mfenced><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>$theta</mi><mrow><mi>o</mi></mrow></msup></mrow></mfenced></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$m</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>.</p>
<p>&nbsp;</p>@
qu.19.7.algorithm=$d=rand(1.0,9.99,3);
$m=range(1,4);
$theta=rand(0,20,3);
$ans=($d*10^(-6))*sin($theta*2*Pi/360)*10^(9)/($m/2);@
qu.19.7.uid=476310bc-a585-481e-ad23-7a71be2b1044@
qu.19.7.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Diffraction;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.19.8.question=<p>The&nbsp;slit in the diagram&nbsp;is of width&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi></mrow></mstyle></math>, and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mrow><mo lspace='0.0em' rspace='0.0em'>&Gt;</mo></mrow><mi>d</mi></mrow></mstyle></math>.&nbsp; A beam of&nbsp;monochromatic light&nbsp;passes through the&nbsp;<br />
slit at normal incidence.&nbsp; If the&nbsp;order-$m dark fringes&nbsp;are&nbsp;visible at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>$theta</mi><mrow><mi>o</mi></mrow></msup></mrow></mstyle></math>&nbsp;for light of wavelength&nbsp;<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mstyle></math>,&nbsp;what is the width of the slit <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi></mrow></mstyle></math>?</p>
<p>&nbsp;</p>
<p><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="593" height="500">
<param name="image" value="__BASE_URI__img/Interference/DiagramSingleSlit.png" />
<param name="size" value="1" />
<param name="label.1.x" value="230" />
<param name="label.1.y" value="10" />
<param name="label.1.text" value="X" /></applet></p>
<p>&nbsp;</p>
<p>&nbsp;</p>@
qu.19.8.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25);@
qu.19.8.allow2d=0@
qu.19.8.maple_answer=SigFigs[roundToSigFigs]($ans,3)*um@
qu.19.8.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.19.8.type=maple@
qu.19.8.mode=Maple@
qu.19.8.name=Single Slit Diffraction - Find Slit Spacing - Dark ~ PGc@
qu.19.8.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.19.8.editing=useHTML@
qu.19.8.solution=<p>For a single slit, destructive interference occurs when the parameters satisfy:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>m&lambda;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>d</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>.</p>
<p>Thus, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$m</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mfenced></mrow><mrow><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>$theta</mi><mrow><mi>o</mi></mrow></msup></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>.</p>
<p>&nbsp;</p>@
qu.19.8.algorithm=$m=range(1,4);
$theta=rand(0,20,3);
$lambda=range(100,999);
$ans=$m*$lambda*10^(-9)*10^(6)/sin($theta*2*Pi/360);@
qu.19.8.uid=f22bfe0a-077c-4d94-9a9f-b33617c05e86@
qu.19.8.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Diffraction;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.19.9.question=<p>The slit in the diagram&nbsp;is of width&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow><mrow><mi></mi></mrow></mstyle></math>, and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mrow><mo lspace='0.0em' rspace='0.0em'>&Gt;</mo></mrow><mi>d</mi></mrow></mstyle></math>. If&nbsp;the light&nbsp;incident on the slit is of&nbsp;<br />
wavelength <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mstyle></math>, at what angle would one find&nbsp;the&nbsp;order-$m bright fringes?</p>
<p>&nbsp;</p>
<p><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="593" height="500">
<param name="image" value="__BASE_URI__img/Interference/DiagramSingleSlit.png" />
<param name="size" value="1" />
<param name="label.1.x" value="230" />
<param name="label.1.y" value="10" />
<param name="label.1.text" value="X" /></applet></p>
<p>&nbsp;</p>
<p>&nbsp;</p>
<p>
<p>&nbsp;</p>
</p>
<p><em>Note: Enter </em>deg<em> for </em><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi></mi><mrow><mi>o</mi></mrow></msup></mrow></mstyle></math>&nbsp;<em>or </em>rad<em> for </em>radians.</p>@
qu.19.9.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25);@
qu.19.9.allow2d=0@
qu.19.9.maple_answer=SigFigs[roundToSigFigs]($ans,3)*deg@
qu.19.9.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.19.9.type=maple@
qu.19.9.mode=Maple@
qu.19.9.name=Single Slit Diffraction - Find Angle - Bright ~ PGc@
qu.19.9.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.19.9.editing=useHTML@
qu.19.9.solution=<p>For a single slit, constructive interference occurs when the parameters satisfy:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mi>m</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced><mi>&lambda;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>d</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>.</p>
<p>Thus, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&theta;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msup><mi>sin</mi><mn>-1</mn></msup><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$m</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mfenced></mrow><mrow><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mfrac></mrow></mfenced></mrow></mrow></mstyle></math>.</p>
<p>&nbsp;</p>@
qu.19.9.algorithm=$d=rand(0.5,0.9,3);
$m=range(1,4);
$lambda=range(400,800);
$ans=arcsin((($m/2)*$lambda*10^(-9))/($d*10^(-3)))*(360/(2*Pi));@
qu.19.9.uid=51252171-f11d-48f7-92f1-d2cdcc75783f@
qu.19.9.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Diffraction;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.19.10.question=<p>The slit in the diagram&nbsp;is of width <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mi>&mu;</mi></mrow><mi>m</mi></mrow><mrow></mrow></mstyle></math>, and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mrow><mo lspace='0.0em' rspace='0.0em'>&Gt;</mo></mrow><mi>d</mi></mrow></mstyle></math>.&nbsp; A beam of monochromatic light&nbsp;passes&nbsp;<br />
through the slit at normal incidence.&nbsp; If the&nbsp;order-$m dark fringes are visible at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>$theta</mi><mrow><mi>o</mi></mrow></msup></mrow></mstyle></math>, what is the&nbsp;<br />
wavelength of the light?</p>
<p>&nbsp;</p>
<p><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="593" height="500">
<param name="image" value="__BASE_URI__img/Interference/DiagramSingleSlit.png" />
<param name="size" value="1" />
<param name="label.1.x" value="230" />
<param name="label.1.y" value="10" />
<param name="label.1.text" value="X" /></applet></p>
<p>&nbsp;</p>
<p>&nbsp;</p>
<p><span>&nbsp;</span></p>@
qu.19.10.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25);@
qu.19.10.allow2d=0@
qu.19.10.maple_answer=SigFigs[roundToSigFigs]($ans,3)*nm@
qu.19.10.libname=__BASE_URI__repositories/PartialGrading.12-09-01.14-02.lib@
qu.19.10.type=maple@
qu.19.10.mode=Maple@
qu.19.10.name=Single Slit Diffraction - Find Wavelength - Dark ~ PGc@
qu.19.10.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25,giveComments);,libname=__BASE_URI__repositories/PartialGrading.12-09-01.14-02.lib"))}</p>@
qu.19.10.editing=useHTML@
qu.19.10.solution=<p>For a single slit, destructive interference occurs when the parameters satisfy:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>m&lambda;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>d</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>.</p>
<p>Thus, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&lambda;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&mu;m</mi></mrow></mfenced><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$m</mi></mrow></mfenced></mrow></mfrac></mrow><mrow><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>$theta</mi><mrow><mi>o</mi></mrow></msup></mrow></mfenced></mrow></mstyle></math>.</p>
<p>&nbsp;</p>@
qu.19.10.algorithm=$d=rand(1.0,9.99,3);
$m=range(1,4);
$theta=rand(0,20,3);
$ans=($d*10^(-6))*sin($theta*2*Pi/360)*10^(9)/$m;@
qu.19.10.uid=e8dcd74c-d315-498d-819d-4c4214acab39@
qu.19.10.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Diffraction;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.19.11.question=<p>The narrow slits of a diffraction grating are&nbsp;separated by a distance of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi></mrow></mstyle></math>, and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mrow><mo lspace='0.0em' rspace='0.0em'>&Gt;</mo></mrow><mi>d</mi></mrow></mstyle></math>.&nbsp; A beam of&nbsp;<br />
monochromatic light&nbsp;passes through the grating at normal incidence.&nbsp; If the&nbsp;order-$m bright&nbsp;fringes&nbsp;are&nbsp;<br />
visible at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>$theta</mi><mrow><mi>o</mi></mrow></msup></mrow></mstyle></math>&nbsp;for light of wavelength <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mstyle></math>,&nbsp;what is the slit spacing <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi></mrow></mstyle></math>?</p>
<p>&nbsp;</p>
<p><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="593" height="500">
<param name="image" value="__BASE_URI__img/Interference/DiagramGrating.png" />
<param name="size" value="1" />
<param name="label.1.x" value="230" />
<param name="label.1.y" value="10" />
<param name="label.1.text" value="X" /></applet></p>
<p>&nbsp;</p>
<p>&nbsp;</p>@
qu.19.11.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25);@
qu.19.11.allow2d=0@
qu.19.11.maple_answer=SigFigs[roundToSigFigs]($ans,3)*um@
qu.19.11.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.19.11.type=maple@
qu.19.11.mode=Maple@
qu.19.11.name=Diffraction Grating - Find Slit Spacing - Bright ~ PGc@
qu.19.11.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.19.11.editing=useHTML@
qu.19.11.solution=<p>For a diffraction grating, constructive interference occurs when the parameters satisfy:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>m&lambda;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>d</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mi>sin</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mrow></mstyle></math>.</p>
<p>Thus, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi mathvariant='normal'>$</mi><mi>m</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mfenced></mrow><mrow><mi>sin</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><msup><mi>$theta</mi><mrow><mi>o</mi></mrow></msup></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>.</p>
<p>&nbsp;</p>@
qu.19.11.algorithm=$m=range(1,4);
$theta=rand(0,20,3);
$lambda=range(100,999);
$ans=$m*$lambda*10^(-9)*10^(6)/sin($theta*2*Pi/360);@
qu.19.11.uid=6644d5d6-0247-4403-8f02-ebd1245b02f3@
qu.19.11.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Diffraction;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.19.12.question=<p>The single slit&nbsp;in the diagram is of width&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow><mrow><mi></mi></mrow></mstyle></math>, and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mrow><mo lspace='0.0em' rspace='0.0em'>&Gt;</mo></mrow><mi>d</mi></mrow></mstyle></math>. &nbsp;If&nbsp;the&nbsp;light&nbsp;incident on the slit is of&nbsp;<br />
wavelength <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mstyle></math>, at what angle would one find&nbsp;the&nbsp;order-$m dark fringes?</p>
<p>&nbsp;</p>
<p><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="593" height="500">
<param name="image" value="__BASE_URI__img/Interference/DiagramSingleSlit.png" />
<param name="size" value="1" />
<param name="label.1.x" value="230" />
<param name="label.1.y" value="10" />
<param name="label.1.text" value="X" /></applet></p>
<p>&nbsp;</p>
<p>&nbsp;</p>
<p>&nbsp;</p>
<p>&nbsp;</p>
<p>&nbsp;</p>
<p><em>Note: Enter </em>deg<em> for </em><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi></mi><mrow><mi>o</mi></mrow></msup></mrow></mstyle></math>&nbsp;<em>or </em>rad<em> for </em>radians.</p>@
qu.19.12.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25);@
qu.19.12.allow2d=0@
qu.19.12.maple_answer=SigFigs[roundToSigFigs]($ans,3)*deg@
qu.19.12.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.19.12.type=maple@
qu.19.12.mode=Maple@
qu.19.12.name=Single Slit Diffraction - Find Angle - Dark ~ PGc@
qu.19.12.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.19.12.editing=useHTML@
qu.19.12.solution=<p>For a single slit, destructive interference occurs when the parameters satisfy:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>m&lambda;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>d</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>.</p>
<p>Thus, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&theta;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msup><mi>sin</mi><mn>-1</mn></msup><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$m</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mfenced></mrow><mrow><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mfrac></mrow></mfenced></mrow></mrow></mstyle></math>.</p>
<p>&nbsp;</p>@
qu.19.12.algorithm=$d=rand(0.5,0.9,3);
$m=range(1,4);
$lambda=range(400,800);
$ans=arcsin(($m*$lambda*10^(-9))/($d*10^(-3)))*(360/(2*Pi));@
qu.19.12.uid=365ea7d3-f73e-46b1-9b55-06ebb7bb3554@
qu.19.12.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Diffraction;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.19.13.question=<p>The narrow slits of a diffraction grating are&nbsp;separated by a distance of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow><mrow><mi></mi></mrow></mstyle></math>.&nbsp; If the light&nbsp;incident&nbsp;<br />
on the slits is of wavelength <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mstyle></math>, at what distance above and below <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math> would one&nbsp;find&nbsp;the&nbsp;<br />
order-$m bright fringes on the screen?</p>
<p>&nbsp;</p>
<p><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="593" height="500">
<param name="image" value="__BASE_URI__img/Interference/DiagramGrating.png" />
<param name="size" value="1" />
<param name="label.1.x" value="230" />
<param name="label.1.y" value="10" />
<param name="label.1.text" value="$X m" /></applet></p>
<p>&nbsp;</p>
<p>&nbsp;</p>
<p><span>&nbsp;</span></p>@
qu.19.13.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25);@
qu.19.13.allow2d=0@
qu.19.13.maple_answer=SigFigs[roundToSigFigs]($ans,3)*cm@
qu.19.13.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.19.13.type=maple@
qu.19.13.mode=Maple@
qu.19.13.name=Diffraction Grating - Find Screen Height - Bright ~ PGc@
qu.19.13.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.19.13.editing=useHTML@
qu.19.13.solution=<p>For a diffraction grating, constructive interference occurs when the parameters satisfy:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>m&lambda;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>d</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mi>sin</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mrow></mstyle></math>.</p>
<p>Thus, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&theta;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msup><mi>sin</mi><mn>-1</mn></msup><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$m</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mfenced></mrow><mrow><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mfrac></mrow></mfenced></mrow></mrow></mstyle></math>.</p>
<p>Given that the distance to the screen is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>m</mi></mrow></mstyle></math>, we can find the height of the fringe by:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>m</mi></mrow></mfenced><mrow><mi>tan</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mrow></mstyle></math>.</p>
<p>&nbsp;</p>@
qu.19.13.algorithm=$d=rand(0.5,0.9,3);
$m=range(1,4);
$lambda=range(400,800);
$X=rand(1.00,9.00,3);
$ans=tan(arcsin(($m*$lambda*10^(-9))/($d*10^(-3))))*$X*10^(2);@
qu.19.13.uid=f4beac60-50e5-4288-aa57-d83205bf4290@
qu.19.13.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Diffraction;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.19.14.question=<p>The narrow slits of a diffraction grating are&nbsp;separated by a distance of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi></mrow></mstyle></math>, and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mrow><mo lspace='0.0em' rspace='0.0em'>&Gt;</mo></mrow><mi>d</mi></mrow></mstyle></math>.&nbsp; A beam of&nbsp;<br />
monochromatic light&nbsp;passes through the grating at normal incidence.&nbsp; If the&nbsp;order-$m dark fringes&nbsp;are&nbsp;<br />
visible at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>$theta</mi><mrow><mi>o</mi></mrow></msup></mrow></mstyle></math>&nbsp;for light of wavelength <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mstyle></math>,&nbsp;what is the slit spacing <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi></mrow></mstyle></math>?</p>
<p>&nbsp;</p>
<p><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="593" height="500">
<param name="image" value="__BASE_URI__img/Interference/DiagramGrating.png" />
<param name="size" value="1" />
<param name="label.1.x" value="230" />
<param name="label.1.y" value="10" />
<param name="label.1.text" value="X" /></applet></p>
<p>&nbsp;</p>
<p>&nbsp;</p>@
qu.19.14.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25);@
qu.19.14.allow2d=0@
qu.19.14.maple_answer=SigFigs[roundToSigFigs]($ans,3)*um@
qu.19.14.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.19.14.type=maple@
qu.19.14.mode=Maple@
qu.19.14.name=Diffraction Grating - Find Slit Spacing - Dark ~ PGc@
qu.19.14.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.19.14.editing=useHTML@
qu.19.14.solution=<p>For a diffraction grating, destructive interference occurs when the parameters satisfy:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mi>m</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced><mi>&lambda;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>d</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>.</p>
<p>Thus, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$m</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mfenced></mrow><mrow><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>$theta</mi><mrow><mi>o</mi></mrow></msup></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>.</p>
<p>&nbsp;</p>@
qu.19.14.algorithm=$m=range(1,4);
$theta=rand(0,20,3);
$lambda=range(100,999);
$ans=($m/2)*$lambda*10^(-9)*10^(6)/sin($theta*2*Pi/360);@
qu.19.14.uid=4b84eaca-cc0b-445d-a4b2-7da9497aa119@
qu.19.14.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Diffraction;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.19.15.question=<p>The narrow slits of a diffraction grating are&nbsp;separated by a distance of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$d</mi></mrow><mrow><mi></mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&mu;m</mi></mrow></mstyle></math>,&nbsp;with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mrow><mo lspace='0.0em' rspace='0.0em'>&Gt;</mo></mrow><mi>d</mi></mrow></mstyle></math>.&nbsp; A&nbsp;<br />
beam of&nbsp;monochromatic light&nbsp;of wavelength <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mstyle></math>passes through the grating at normal&nbsp;<br />
incidence.&nbsp; What order of&nbsp;bright fringes&nbsp;are observed at an angle <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>$theta</mi><mrow><mi>o</mi></mrow></msup></mrow></mstyle></math>&nbsp;above and below <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math>?</p>
<p>&nbsp;</p>
<p><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="593" height="500">
<param name="image" value="__BASE_URI__img/Interference/DiagramGrating.png" />
<param name="size" value="1" />
<param name="label.1.x" value="230" />
<param name="label.1.y" value="10" />
<param name="label.1.text" value="X" /></applet></p>
<p>&nbsp;</p>
<p>&nbsp;</p>@
qu.19.15.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uEM,uSF=false);@
qu.19.15.allow2d=0@
qu.19.15.maple_answer=SigFigs[roundToSigFigs]($m,3)@
qu.19.15.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.19.15.type=maple@
qu.19.15.mode=Maple@
qu.19.15.name=Diffraction Grating - Find Fringe Order - Bright ~ PGc@
qu.19.15.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uEM,uSF=false,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.19.15.editing=useHTML@
qu.19.15.solution=<p>For a diffraction grating, constructive interference occurs when the parameters satisfy:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>m&lambda;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>d</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mi>sin</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mrow></mstyle></math>.</p>
<p>Thus, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>m</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mfrac><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>$theta</mi><mrow><mi>o</mi></mrow></msup></mrow></mfenced></mrow></mstyle></math>.</p>
<p>&nbsp;</p>@
qu.19.15.algorithm=$m=range(1,6);
$theta=rand(0,45,3);
$lambda=range(100,999);
$d=sig(3,$m*$lambda*10^(-9)*10^(6)/sin($theta*2*Pi/360));@
qu.19.15.uid=2dabdd25-8856-49cd-9d3f-1a7ae90d1291@
qu.19.15.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Diffraction;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.19.16.question=<p>The narrow slits of a diffraction grating are&nbsp;separated by a distance of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow><mrow><mi></mi></mrow></mstyle></math>.&nbsp; If the light&nbsp;incident&nbsp;<br />
on the slits is of wavelength <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mstyle></math>, at what distance above and below <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math> would one&nbsp;find&nbsp;the&nbsp;<br />
order-$m dark fringes on the screen?</p>
<p>&nbsp;</p>
<p><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="593" height="500">
<param name="image" value="__BASE_URI__img/Interference/DiagramGrating.png" />
<param name="size" value="1" />
<param name="label.1.x" value="230" />
<param name="label.1.y" value="10" />
<param name="label.1.text" value="$X m" /></applet></p>
<p>&nbsp;</p>
<p>&nbsp;</p>@
qu.19.16.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25);@
qu.19.16.allow2d=0@
qu.19.16.maple_answer=SigFigs[roundToSigFigs]($ans,3)*cm@
qu.19.16.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.19.16.type=maple@
qu.19.16.mode=Maple@
qu.19.16.name=Diffraction Grating - Find Screen Height - Dark ~ PGc@
qu.19.16.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.19.16.editing=useHTML@
qu.19.16.solution=<p>For a diffraction grating, destructive interference occurs when the parameters satisfy:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mi>m</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced><mi>&lambda;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>d</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>.</p>
<p>Thus, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&theta;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msup><mi>sin</mi><mn>-1</mn></msup><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$m</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mfenced></mrow><mrow><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mfrac></mrow></mfenced></mrow></mrow></mstyle></math>.</p>
<p>Given the distance from the screen to the grating, we have:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi mathvariant='normal'>y</mi><mrow><mi>$m</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>m</mi></mrow></mfenced><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>.</p>
<p>&nbsp;</p>@
qu.19.16.algorithm=$d=rand(0.5,0.9,3);
$m=range(1,4);
$lambda=range(400,800);
$X=rand(1.00,9.00,3);
$ans=tan(arcsin((($m/2)*$lambda*10^(-9))/($d*10^(-3))))*$X*10^(2);@
qu.19.16.uid=01ba7a0e-d7b1-409a-b8e5-10de569f24d1@
qu.19.16.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Diffraction;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.19.17.question=<p>The narrow slits of a diffraction grating are&nbsp;separated by a distance of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow><mrow><mi></mi></mrow></mstyle></math>, and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mrow><mo lspace='0.0em' rspace='0.0em'>&Gt;</mo></mrow><mi>d</mi></mrow></mstyle></math>. If the<br />
light&nbsp;incident on the slits is of wavelength <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mstyle></math>, at what angle would one find&nbsp;the&nbsp;order-$m&nbsp;bright fringes?</p>
<p>&nbsp;</p>
<p><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="593" height="500">
<param name="image" value="__BASE_URI__img/Interference/DiagramGrating.png" />
<param name="size" value="1" />
<param name="label.1.x" value="230" />
<param name="label.1.y" value="10" />
<param name="label.1.text" value="X" /></applet></p>
<p>&nbsp;<em>Note:&nbsp; Enter </em>deg<em> for</em>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi></mi><mrow><mi>o</mi></mrow></msup></mrow></mstyle></math> <em>or </em>rad<em> for </em>radians.</p>@
qu.19.17.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uUN);@
qu.19.17.allow2d=0@
qu.19.17.maple_answer=SigFigs[roundToSigFigs]($ans,3)*deg@
qu.19.17.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.19.17.type=maple@
qu.19.17.mode=Maple@
qu.19.17.name=Diffraction Grating - Find Angle - Bright ~ PGc@
qu.19.17.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.19.17.editing=useHTML@
qu.19.17.solution=<p>For a diffraction grating, constructive interference occurs when the parameters satisfy:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>m&lambda;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>d</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mi>sin</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mrow></mstyle></math>.</p>
<p>Thus, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&theta;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msup><mi>sin</mi><mn>-1</mn></msup><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$m</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mfenced></mrow><mrow><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mfrac></mrow></mfenced></mrow></mrow></mstyle></math>.</p>
<p>&nbsp;</p>@
qu.19.17.algorithm=$d=rand(0.5,0.9,3);
$m=range(1,4);
$lambda=range(400,800);
$ans=arcsin(($m*$lambda*10^(-9))/($d*10^(-3)))*(360/(2*Pi));@
qu.19.17.uid=abed2435-1db1-4572-a490-34cf10956b23@
qu.19.17.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Diffraction;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.19.18.question=<p>The narrow slits of a diffraction grating are&nbsp;separated by a distance of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow><mrow><mi></mi></mrow></mstyle></math>, and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mrow><mo lspace='0.0em' rspace='0.0em'>&Gt;</mo></mrow><mi>d</mi></mrow></mstyle></math>. If the<br />
light&nbsp;incident on the slits is of wavelength <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mstyle></math>, at what angle would one find&nbsp;the&nbsp;order-$m dark fringes?</p>
<p>&nbsp;</p>
<p><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="593" height="500">
<param name="image" value="__BASE_URI__img/Interference/DiagramGrating.png" />
<param name="size" value="1" />
<param name="label.1.x" value="230" />
<param name="label.1.y" value="10" />
<param name="label.1.text" value="X" /></applet></p>
<p>&nbsp;</p>
<p>&nbsp;</p>
<p>
<p>&nbsp;</p>
</p>
<p><em>Note: Enter </em>deg<em> for </em><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi></mi><mrow><mi>o</mi></mrow></msup></mrow></mstyle></math><em>or </em>rad<em> for </em>radians.</p>@
qu.19.18.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25);@
qu.19.18.allow2d=0@
qu.19.18.maple_answer=SigFigs[roundToSigFigs]($ans,3)*deg@
qu.19.18.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.19.18.type=maple@
qu.19.18.mode=Maple@
qu.19.18.name=Diffraction Grating - Find Angle - Dark ~ PGc@
qu.19.18.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.19.18.editing=useHTML@
qu.19.18.solution=<p>For a diffraction grating, destructive interference occurs when the parameters satisfy:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mi>m</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced><mi>&lambda;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>d</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>.</p>
<p>Thus, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&theta;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msup><mi>sin</mi><mn>-1</mn></msup><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$m</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mfenced></mrow><mrow><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mfrac></mrow></mfenced></mrow></mrow></mstyle></math>.</p>
<p>&nbsp;</p>@
qu.19.18.algorithm=$d=rand(0.5,0.9,3);
$m=range(1,4);
$lambda=range(400,800);
$ans=arcsin((($m/2)*$lambda*10^(-9))/($d*10^(-3)))*(360/(2*Pi));@
qu.19.18.uid=47e0eb27-6a4d-4180-ae15-c30137780083@
qu.19.18.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Diffraction;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.19.19.question=<p>The narrow slits of a diffraction grating are&nbsp;separated by a distance of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mi>&mu;</mi></mrow><mi>m</mi></mrow><mrow></mrow></mstyle></math>, and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mrow><mo lspace='0.0em' rspace='0.0em'>&Gt;</mo></mrow><mi>d</mi></mrow></mstyle></math>.&nbsp; A<br />
beam of monochromatic light&nbsp;passes through the grating at normal incidence.&nbsp; If the&nbsp;order-$m bright&nbsp;<br />
fringes are visible at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>$theta</mi><mrow><mi>o</mi></mrow></msup></mrow></mstyle></math>, what is the wavelength of the light?</p>
<p>&nbsp;</p>
<p><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="593" height="500">
<param name="image" value="__BASE_URI__img/Interference/DiagramGrating.png" />
<param name="size" value="1" />
<param name="label.1.x" value="230" />
<param name="label.1.y" value="10" />
<param name="label.1.text" value="X" /></applet></p>
<p>&nbsp;</p>
<p>&nbsp;</p>
<p><span>&nbsp;</span></p>@
qu.19.19.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25);@
qu.19.19.allow2d=0@
qu.19.19.maple_answer=SigFigs[roundToSigFigs]($ans,3)*nm@
qu.19.19.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.19.19.type=maple@
qu.19.19.mode=Maple@
qu.19.19.name=Diffraction Grating - Find Wavelength - Bright ~ PGc@
qu.19.19.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.19.19.editing=useHTML@
qu.19.19.solution=<p>For a diffraction grating, constructive interference occurs when the parameters satisfy:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>m&lambda;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>d</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mi>sin</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mrow></mstyle></math>.</p>
<p>Thus, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&lambda;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mi>$d</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mi>&mu;</mi></mrow><mi>m</mi></mrow></mfenced><mrow><mi>$m</mi></mrow></mfrac><mrow><mi>sin</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><msup><mi>$theta</mi><mrow><mi>o</mi></mrow></msup></mrow></mfenced></mrow></mrow></mstyle></math>.</p>
<p>&nbsp;</p>@
qu.19.19.algorithm=$d=rand(1.0,9.99,3);
$m=range(1,4);
$theta=rand(0,20,3);
$ans=($d*10^(-6))*sin($theta*2*Pi/360)*10^(9)/$m;@
qu.19.19.uid=32fdd8a3-8554-420a-bfe8-d1b470eb9b29@
qu.19.19.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Diffraction;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.19.20.question=<p>The narrow slits of a diffraction grating are&nbsp;separated by a distance of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow><mrow><mi></mi></mrow></mstyle></math>.&nbsp; If light&nbsp;incident&nbsp;on&nbsp;<br />
the slits&nbsp;of wavelength <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mstyle></math> produces&nbsp;order-$m dark fringes&nbsp;on the screen <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$y</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>cm</mi></mrow></mstyle></math>above&nbsp;<br />
and below <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math>, what is the distance <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math>&nbsp;to the screen?</p>
<p>&nbsp;</p>
<p><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="593" height="500">
<param name="image" value="__BASE_URI__img/Interference/DiagramGrating.png" />
<param name="size" value="1" />
<param name="label.1.x" value="230" />
<param name="label.1.y" value="10" />
<param name="label.1.text" value="X" /></applet></p>
<p>&nbsp;</p>
<p>&nbsp;</p>@
qu.19.20.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25);@
qu.19.20.allow2d=0@
qu.19.20.maple_answer=SigFigs[roundToSigFigs]($ans,3)*m@
qu.19.20.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.19.20.type=maple@
qu.19.20.mode=Maple@
qu.19.20.name=Diffraction Grating - Find Distance to Screen - Dark ~ PGc@
qu.19.20.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.19.20.editing=useHTML@
qu.19.20.solution=<p>For a diffraction grating, destructive interference occurs when the parameters satisfy:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mi>m</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced><mi>&lambda;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>d</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>.</p>
<p>Thus, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&theta;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msup><mi>sin</mi><mn>-1</mn></msup><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$m</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mfenced></mrow><mrow><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mfrac></mrow></mfenced></mrow></mrow></mstyle></math>.</p>
<p>Given that the fringes occur at a height <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$y</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>cm</mi></mrow></mstyle></math>,</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mi>$y</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>cm</mi></mrow></mfenced><mrow><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>.</p>
<p>&nbsp;</p>@
qu.19.20.algorithm=$d=rand(0.5,0.9,3);
$m=range(1,4);
$lambda=range(400,800);
$X=rand(1.00,9.00,3);
$y=rand(0.1,1.5,3);
$ans=($y*10^(-2))/tan(arcsin((($m/2)*$lambda*10^(-9))/($d*10^(-3))));@
qu.19.20.uid=e7c4a0b3-49b1-4df6-b1e8-de0b960cc144@
qu.19.20.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Diffraction;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.19.21.question=<p>The slit in the diagram is of width&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow><mrow><mi></mi></mrow></mstyle></math>.&nbsp; If&nbsp;light&nbsp;incident&nbsp;on&nbsp;the slit&nbsp;of wavelength <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mstyle></math>&nbsp;<br />
produces&nbsp;order-$m bright fringes&nbsp;on the screen&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$y</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>cm</mi></mrow></mstyle></math>above&nbsp;and below <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math>, what is the distance <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math>&nbsp;to&nbsp;<br />
the screen?</p>
<p>&nbsp;</p>
<p><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="593" height="500">
<param name="image" value="__BASE_URI__img/Interference/DiagramSingleSlit.png" />
<param name="size" value="1" />
<param name="label.1.x" value="230" />
<param name="label.1.y" value="10" />
<param name="label.1.text" value="X" /></applet></p>
<p>&nbsp;</p>
<p>&nbsp;</p>
<p><span>&nbsp;</span></p>@
qu.19.21.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25);@
qu.19.21.allow2d=0@
qu.19.21.maple_answer=SigFigs[roundToSigFigs]($ans,3)*m@
qu.19.21.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.19.21.type=maple@
qu.19.21.mode=Maple@
qu.19.21.name=Single Slit Diffraction - Find Distance to Screen - Bright ~ PGc@
qu.19.21.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.19.21.editing=useHTML@
qu.19.21.solution=<p>For a single slit, constructive interference occurs when the parameters satisfy:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mi>m</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced><mi>&lambda;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>d</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>.</p>
<p>Thus, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&theta;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msup><mi>sin</mi><mn>-1</mn></msup><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$m</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mfenced></mrow><mrow><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mfrac></mrow></mfenced></mrow></mrow></mstyle></math>.</p>
<p>&nbsp;Given the height of the fringe on the screen, we can find the distance to the screen:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mi>$y</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>cm</mi></mrow></mfenced><mrow><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>.</p>@
qu.19.21.algorithm=$d=rand(0.5,0.9,3);
$m=range(1,4);
$lambda=range(400,800);
$X=rand(1.00,9.00,3);
$y=rand(0.1,1.5,3);
$ans=($y*10^(-2))/tan(arcsin((($m/2)*$lambda*10^(-9))/($d*10^(-3))));@
qu.19.21.uid=ab6bb4a8-23cc-442a-ba33-ab5da82223b9@
qu.19.21.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Diffraction;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.19.22.question=<p>The slit in the diagram&nbsp;is of width&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow><mrow><mi></mi></mrow></mstyle></math>.&nbsp; If the&nbsp;light&nbsp;incident&nbsp;on the slit is of wavelength <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mstyle></math>,<br />
&nbsp;at what distance above and below <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math> would&nbsp;one&nbsp;find&nbsp;the&nbsp;order-$m dark fringes on the screen?</p>
<p>&nbsp;</p>
<p><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="593" height="500">
<param name="image" value="__BASE_URI__img/Interference/DiagramSingleSlit.png" />
<param name="size" value="1" />
<param name="label.1.x" value="230" />
<param name="label.1.y" value="10" />
<param name="label.1.text" value="$X m" /></applet></p>
<p>&nbsp;</p>
<p>&nbsp;</p>@
qu.19.22.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25);@
qu.19.22.allow2d=0@
qu.19.22.maple_answer=SigFigs[roundToSigFigs]($ans,3)*cm@
qu.19.22.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.19.22.type=maple@
qu.19.22.mode=Maple@
qu.19.22.name=Single Slit Diffraction - Find Screen Height - Dark ~ PGc@
qu.19.22.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.19.22.editing=useHTML@
qu.19.22.solution=<p>For a single slit, destructive interference occurs when the parameters satisfy:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>m&lambda;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>d</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>.</p>
<p>Thus, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&theta;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msup><mi>sin</mi><mn>-1</mn></msup><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$m</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mfenced></mrow><mrow><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mfrac></mrow></mfenced></mrow></mrow></mstyle></math>.</p>
<p>Given the distance to the screen, we have:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>y</mi><mrow><mi>$m</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>m</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>.</p>
<p>&nbsp;</p>@
qu.19.22.algorithm=$d=rand(0.5,0.9,3);
$m=range(1,4);
$lambda=range(400,800);
$X=rand(1.00,9.00,3);
$ans=tan(arcsin(($m*$lambda*10^(-9))/($d*10^(-3))))*$X*10^(2);@
qu.19.22.uid=caef3959-2471-4693-b421-17512445c7ba@
qu.19.22.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Diffraction;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.19.23.question=<p>The narrow slits of a diffraction grating are&nbsp;separated by a distance of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow><mrow><mi></mi></mrow></mstyle></math>.&nbsp; If light&nbsp;incident&nbsp;on&nbsp;<br />
the slits&nbsp;of wavelength <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mstyle></math> produces&nbsp;order-$m bright fringes&nbsp;on the screen <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$y</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>cm</mi></mrow></mstyle></math>above&nbsp;<br />
and below <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math>, what is the distance <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math>&nbsp;to the screen?</p>
<p>&nbsp;</p>
<p><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="593" height="500">
<param name="image" value="__BASE_URI__img/Interference/DiagramGrating.png" />
<param name="size" value="1" />
<param name="label.1.x" value="230" />
<param name="label.1.y" value="10" />
<param name="label.1.text" value="X" /></applet></p>
<p>&nbsp;</p>
<p>&nbsp;</p>
<p><span>&nbsp;</span></p>@
qu.19.23.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25);@
qu.19.23.allow2d=0@
qu.19.23.maple_answer=SigFigs[roundToSigFigs]($ans,3)*m@
qu.19.23.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.19.23.type=maple@
qu.19.23.mode=Maple@
qu.19.23.name=Diffraction Grating - Find Distance to Screen - Bright ~ PGc@
qu.19.23.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.19.23.editing=useHTML@
qu.19.23.solution=<p>For a diffraction grating, constructive interference occurs when the parameters satisfy:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>m&lambda;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>d</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mi>sin</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mrow></mstyle></math>.</p>
<p>Thus, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&theta;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msup><mi>sin</mi><mn>-1</mn></msup><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$m</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mfenced></mrow><mrow><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mfrac></mrow></mfenced></mrow></mrow></mstyle></math>.</p>
<p>Given <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$y</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>cm</mi></mrow></mstyle></math>, we can then solve for:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>y</mi><mrow><mi>tan</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>&nbsp;</p>@
qu.19.23.algorithm=$d=rand(0.5,0.9,3);
$m=range(1,4);
$lambda=range(400,800);
$X=rand(1.00,9.00,3);
$y=rand(0.1,1.5,3);
$ans=($y*10^(-2))/tan(arcsin(($m*$lambda*10^(-9))/($d*10^(-3))));@
qu.19.23.uid=e7394a77-7942-489b-bef5-4e97641c3073@
qu.19.23.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Diffraction;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.19.24.question=<p>The slit in the diagram is of width&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$d</mi></mrow><mrow><mi></mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&mu;m</mi></mrow></mstyle></math>,&nbsp;with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mrow><mo lspace='0.0em' rspace='0.0em'>&Gt;</mo></mrow><mi>d</mi></mrow></mstyle></math>.&nbsp; A beam of&nbsp;monochromatic light&nbsp;of&nbsp;<br />
wavelength <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mstyle></math>passes through the slit at normal&nbsp;incidence.&nbsp; What order of&nbsp;dark fringes&nbsp;are&nbsp;<br />
observed at an angle <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>$theta</mi><mrow><mi>o</mi></mrow></msup></mrow></mstyle></math>&nbsp;above and below <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math>?</p>
<p>&nbsp;</p>
<p><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="593" height="500">
<param name="image" value="__BASE_URI__img/Interference/DiagramSingleSlit.png" />
<param name="size" value="1" />
<param name="label.1.x" value="230" />
<param name="label.1.y" value="10" />
<param name="label.1.text" value="X" /></applet></p>
<p>&nbsp;</p>
<p>&nbsp;</p>
<p><span>&nbsp;</span></p>@
qu.19.24.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uEM,uSF=false);@
qu.19.24.allow2d=0@
qu.19.24.maple_answer=SigFigs[roundToSigFigs]($m,3)@
qu.19.24.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.19.24.type=maple@
qu.19.24.mode=Maple@
qu.19.24.name=Single Slit Diffraction - Find Fringe Order - Dark ~ PGc@
qu.19.24.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uEM,uSF=false,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.19.24.editing=useHTML@
qu.19.24.solution=<p>For a single slit, destructive interference occurs when the parameters satisfy:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>m&lambda;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>d</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>.</p>
<p>Thus, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>m</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&mu;m</mi></mrow></mfenced><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mfenced></mrow></mfrac><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>$theta</mi><mrow><mi>o</mi></mrow></msup></mrow></mfenced></mrow></mstyle></math>.</p>
<p>&nbsp;</p>@
qu.19.24.algorithm=$m=range(1,6);
$theta=rand(0,45,3);
$lambda=range(100,999);
$d=sig(3,$m*$lambda*10^(-9)*10^(6)/sin($theta*2*Pi/360));@
qu.19.24.uid=ba21e65f-6acc-4582-aead-2cf8b4f538c5@
qu.19.24.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Diffraction;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.20.topic=Interference@

qu.20.1.mode=Inline@
qu.20.1.name=Two Source Interference - Constructive@
qu.20.1.comment=@
qu.20.1.editing=useHTML@
qu.20.1.hint.1=<p>Look at path-length differences.</p>@
qu.20.1.solution=<p>For two sources that are in phase, constructive interference will be observed whenever the path-length difference from the two sources to a point is an integer multiple of the wavelength:</p>
<p align="left"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>r</mi><mrow><mn>2</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mi>r</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>m</mi><mrow><mi>&lambda;</mi></mrow></mrow></mstyle></math>.</p>
<p align="left">&nbsp;</p>
<p align="left">Here, we are given the wavelength, so we need to calculate the distance from each source to the candidate points.&nbsp; If the difference in those two distances, for a given point, is an integer multiple of the wavelength, then constructive interference will be observed.</p>@
qu.20.1.algorithm=$x1i=range(1,10);
$x1=50*$x1i;
$y1i=range(1,10);
$y1=50*$y1i;
$m=maple("
randomize():
a:={}:
idx:=0:
for i from 0 to 10 do
for j from 0 to 10 do
idx:=idx+1:
a:={a[],[i,j]}:
end do:
end do:
S1:=RandomTools[Generate](choose(a)):
a:=a minus {S1};
S2:=RandomTools[Generate](choose(a)):
a:=a minus {S2};
a:=combinat[randperm](a):
for i from 1 to 98 while not(type(student[distance](a[i],S2)-student[distance](a[i],S1),integer)) do end do:
CI:=a[i]:
a:={a[]} minus {CI};
a:=combinat[randperm](a):
for i from 1 to 97 while (type(student[distance](a[i],S2)-student[distance](a[i],S1),integer)) do end do:
P1:=a[i]:
a:={a[]} minus {P1}:
a:=combinat[randperm](a):
for i from 1 to 96 while (type(student[distance](a[i],S2)-student[distance](a[i],S1),integer)) do end do:
P2:=a[i]:
a:={a[]} minus {P2}:
a:=combinat[randperm](a):
for i from 1 to 95 while (type(student[distance](a[i],S2)-student[distance](a[i],S1),integer)) do end do:
P3:=a[i]:
a:={a[]} minus {P3}:
n1:=RandomTools[Generate](choose({A,B,C,D})):
n2:=RandomTools[Generate](choose({A,B,C,D} minus {n1})):
n3:=RandomTools[Generate](choose({A,B,C,D} minus {n1,n2})):
n4:=RandomTools[Generate](choose({A,B,C,D} minus {n1,n2,n3})):
S1,S2,CI,P1,P2,P3,n1,n2,n3,n4;
");
$S1x=switch(0,$m)*50+10;
$S1y=switch(1,$m)*50+15;
$S2x=switch(2,$m)*50+10;
$S2y=switch(3,$m)*50+15;
$CIx=switch(4,$m)*50+10;
$CIy=switch(5,$m)*50+15;
$P1x=switch(6,$m)*50+10;
$P1y=switch(7,$m)*50+15;
$P2x=switch(8,$m)*50+10;
$P2y=switch(9,$m)*50+15;
$P3x=switch(10,$m)*50+10;
$P3y=switch(11,$m)*50+15;
$CI=switch(12,$m);
$P1=switch(13,$m);
$P2=switch(14,$m);
$P3=switch(15,$m);
$lambda=range(100,900);
$n=range(1,9);
$d=$n*$lambda*10^(-2);@
qu.20.1.uid=672f246d-89a9-413e-86c9-a5fc2199ca4f@
qu.20.1.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Two Source Interference;
  Difficulty=Hard;
  Features=Algorithmic;
  Features=Algorithmic Diagram;
@
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qu.20.1.question=<p align="left">Two sources, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>S1</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>S2</mi></mrow></mstyle></math>&nbsp;output monochromatic electromagnetic waves,&nbsp;in phase, at wavelength<br /><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>cm</mi></mrow></mstyle></math>.&nbsp; If the major grid divisions are <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>m</mi></mrow></mstyle></math>&nbsp;apart, at which point will constructive interference&nbsp;<br />be observed?&nbsp;</p><p align="center">&nbsp;</p><p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="520" height="520"><param name="image" value="__BASE_URI__img/Interference/TwoSource/Diagram.png" /><param name="size" value="6" /><param name="label.1.x" value="$S1x" /><param name="label.1.y" value="$S1y" /><param name="label.1.text" value="S1" /><param name="label.2.x" value="$S2x" /><param name="label.2.y" value="$S2y" /><param name="label.2.text" value="S2" /><param name="label.3.x" value="$CIx" /><param name="label.3.y" value="$CIy" /><param name="label.3.text" value="$CI" /><param name="label.4.x" value="$P1x" /><param name="label.4.y" value="$P1y" /><param name="label.4.text" value="$P1" /><param name="label.5.x" value="$P2x" /><param name="label.5.y" value="$P2y" /><param name="label.5.text" value="$P2" /><param name="label.6.x" value="$P3x" /><param name="label.6.y" value="$P3y" /><param name="label.6.text" value="$P3" /></applet></p><p>&nbsp;</p><p>&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.20.2.question=<p>The centres of the two slits in the diagram are&nbsp;separated by a distance of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow><mrow><mi></mi></mrow></mstyle></math>, and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mrow><mo lspace='0.0em' rspace='0.0em'>&Gt;</mo></mrow><mi>d</mi></mrow></mstyle></math>. If&nbsp;<br />
the coherent light&nbsp;incident on the slits is of wavelength <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mstyle></math>, at what angle, in <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>degrees</mi></mrow></mstyle></math>,&nbsp;<br />
would one find&nbsp;the&nbsp;order-$m dark fringes?</p>
<p>&nbsp;</p>
<p><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="593" height="500">
<param name="image" value="__BASE_URI__img/Interference/DiagramDoubleSlit.png" />
<param name="size" value="1" />
<param name="label.1.x" value="230" />
<param name="label.1.y" value="10" />
<param name="label.1.text" value="X" /></applet></p>
<p>&nbsp;</p>
<p>&nbsp;</p>
<p><span><em>Note: Enter </em>deg<em> for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi></mi><mrow><mi>o</mi></mrow></msup></mrow></mstyle></math>&nbsp;or </em>rad<em> for </em>radians.</span></p>@
qu.20.2.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25);@
qu.20.2.allow2d=0@
qu.20.2.maple_answer=SigFigs[roundToSigFigs]($ans,3)*deg@
qu.20.2.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.20.2.type=maple@
qu.20.2.mode=Maple@
qu.20.2.name=Double Slit Interference - Find Angle - Dark ~ PGc@
qu.20.2.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.20.2.editing=useHTML@
qu.20.2.solution=<p>For a double slit, destructive interference occurs when the parameters satisfy:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mi>m</mi><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn mathvariant='italic'>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&lambda;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>d</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>.</p>
<p>Thus, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&theta;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msup><mi>sin</mi><mn>-1</mn></msup><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$m</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mfenced></mrow><mrow><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mfrac></mrow></mfenced></mrow></mrow></mstyle></math>.</p>
<p>&nbsp;</p>@
qu.20.2.algorithm=$d=rand(0.5,0.9,3);
$m=range(1,4);
$lambda=range(400,800);
$ans=arcsin((($m/2)*$lambda*10^(-9))/($d*10^(-3)))*(360/(2*Pi));@
qu.20.2.uid=6533ca28-aeca-4db2-ba45-b12ded34eb8f@
qu.20.2.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Two Source Interference;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.20.3.question=<p>The centres of the two slits in the diagram&nbsp;are&nbsp;separated by a distance of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi></mrow></mstyle></math>, and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mrow><mo lspace='0.0em' rspace='0.0em'>&Gt;</mo></mrow><mi>d</mi></mrow></mstyle></math>.&nbsp; A beam of&nbsp;<br />
monochromatic, coherent&nbsp;light&nbsp;passes through the slits at normal incidence.&nbsp; If the&nbsp;order-$m&nbsp;<br />
bright&nbsp;fringes&nbsp;are&nbsp;visible at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>$theta</mi><mrow><mi>o</mi></mrow></msup></mrow></mstyle></math>&nbsp;for light of wavelength <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mstyle></math>,&nbsp;what is the slit spacing <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi></mrow></mstyle></math>?</p>
<p>&nbsp;</p>
<p><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="593" height="500">
<param name="image" value="__BASE_URI__img/Interference/DiagramDoubleSlit.png" />
<param name="size" value="1" />
<param name="label.1.x" value="230" />
<param name="label.1.y" value="10" />
<param name="label.1.text" value="X" /></applet></p>
<p>&nbsp;</p>
<p>&nbsp;</p>@
qu.20.3.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25);@
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qu.20.3.maple_answer=SigFigs[roundToSigFigs]($ans,3)*um@
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qu.20.3.mode=Maple@
qu.20.3.name=Double Slit Interference - Find Slit Spacing - Bright ~ PGc@
qu.20.3.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.20.3.editing=useHTML@
qu.20.3.solution=<p>For a double slit, constructive interference occurs when the parameters satisfy:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>m&lambda;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>d</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mi>sin</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mrow></mstyle></math>.</p>
<p>Thus, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$m</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mfenced></mrow><mrow><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>$theta</mi><mrow><mi>o</mi></mrow></msup></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>.</p>
<p>&nbsp;</p>@
qu.20.3.algorithm=$m=range(1,4);
$theta=rand(0,20,3);
$lambda=range(100,999);
$ans=$m*$lambda*10^(-9)*10^(6)/sin($theta*2*Pi/360);@
qu.20.3.uid=91a588ad-7637-4e9d-88ee-ff25001b7ffa@
qu.20.3.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Two Source Interference;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.20.4.question=<p>The centres of the two slits in the diagram are&nbsp;separated by a distance of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow><mrow><mi></mi></mrow></mstyle></math>.&nbsp; If the&nbsp;coherent<br />
light&nbsp;incident&nbsp;on the slits is of wavelength <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mstyle></math>, at what distance above and below <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math> would&nbsp;<br />
one&nbsp;find&nbsp;the&nbsp;order-$m dark fringes on the screen?</p>
<p>&nbsp;</p>
<p><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="593" height="500">
<param name="image" value="__BASE_URI__img/Interference/DiagramDoubleSlit.png" />
<param name="size" value="1" />
<param name="label.1.x" value="230" />
<param name="label.1.y" value="10" />
<param name="label.1.text" value="$X m" /></applet></p>
<p>&nbsp;</p>
<p>&nbsp;</p>@
qu.20.4.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25);@
qu.20.4.allow2d=0@
qu.20.4.maple_answer=SigFigs[roundToSigFigs]($ans,3)*cm@
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qu.20.4.type=maple@
qu.20.4.mode=Maple@
qu.20.4.name=Double Slit Interference - Find Screen Height - Dark ~ PGc@
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qu.20.4.editing=useHTML@
qu.20.4.solution=<p>For a double slit, destructive interference occurs when the parameters satisfy:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mi>m</mi><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn mathvariant='italic'>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&lambda;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>d</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>.</p>
<p>Thus, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&theta;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msup><mi>sin</mi><mn>-1</mn></msup><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$m</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mfenced></mrow><mrow><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mfrac></mrow></mfenced></mrow></mrow></mstyle></math>.</p>
<p>&nbsp;Given the distance to the screen, we have:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>m</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>.</p>@
qu.20.4.algorithm=$d=rand(0.5,0.9,3);
$m=range(1,4);
$lambda=range(400,800);
$X=rand(1.00,9.00,3);
$ans=tan(arcsin((($m/2)*$lambda*10^(-9))/($d*10^(-3))))*$X*10^(2);@
qu.20.4.uid=e9457242-a4f0-484c-82de-e06602534417@
qu.20.4.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Two Source Interference;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.20.5.question=<p>The centres of the two slits in the diagram are&nbsp;separated by a distance of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow><mrow><mi></mi></mrow></mstyle></math>, and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mrow><mo lspace='0.0em' rspace='0.0em'>&Gt;</mo></mrow><mi>d</mi></mrow></mstyle></math>. If&nbsp;<br />
the coherent&nbsp;light&nbsp;incident on the slits is of wavelength <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mstyle></math>, at what angle would one&nbsp;<br />
find&nbsp;the&nbsp;order-$m bright fringes?</p>
<p>&nbsp;</p>
<p><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="593" height="500">
<param name="image" value="__BASE_URI__img/Interference/DiagramDoubleSlit.png" />
<param name="size" value="1" />
<param name="label.1.x" value="230" />
<param name="label.1.y" value="10" />
<param name="label.1.text" value="X" /></applet></p>
<p>&nbsp;</p>
<p>&nbsp;<em>Note:&nbsp; Enter </em>deg<em> for </em><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi></mi><mrow><mi>o</mi></mrow></msup></mrow></mstyle></math>&nbsp;<em>or </em>rad<em> for </em>radians.</p>@
qu.20.5.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN);@
qu.20.5.allow2d=0@
qu.20.5.maple_answer=SigFigs[roundToSigFigs]($ans,3)*deg@
qu.20.5.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.20.5.type=maple@
qu.20.5.mode=Maple@
qu.20.5.name=Double Slit Interference - Find Angle - Bright ~ PGc@
qu.20.5.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.20.5.editing=useHTML@
qu.20.5.solution=<p>For a double slit, constructive interference occurs when the parameters satisfy:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>m&lambda;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>d</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mi>sin</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mrow></mstyle></math>.</p>
<p>Thus, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&theta;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msup><mi>sin</mi><mn>-1</mn></msup><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$m</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mfenced></mrow><mrow><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mfrac></mrow></mfenced></mrow></mrow></mstyle></math>.</p>
<p>&nbsp;</p>@
qu.20.5.algorithm=$d=rand(0.5,0.9,3);
$m=range(1,4);
$lambda=range(400,800);
$ans=arcsin(($m*$lambda*10^(-9))/($d*10^(-3)))*(360/(2*Pi));@
qu.20.5.uid=ef2082ea-c163-4b1b-8e6f-e49b08fe4b75@
qu.20.5.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Two Source Interference;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.20.6.question=<p>The centres of the two slits in the diagram&nbsp;are&nbsp;separated by a distance of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$d</mi></mrow><mrow><mi></mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&mu;m</mi></mrow></mstyle></math>,&nbsp;with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mrow><mo lspace='0.0em' rspace='0.0em'>&Gt;</mo></mrow><mi>d</mi></mrow></mstyle></math>.&nbsp; A&nbsp;<br />
beam of&nbsp;monochromatic, coherent&nbsp;light&nbsp;of wavelength <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mstyle></math>passes through the slits at normal&nbsp;<br />
incidence.&nbsp; What order of&nbsp;dark fringes&nbsp;are observed at an angle <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>$theta</mi><mrow><mi>o</mi></mrow></msup></mrow></mstyle></math>&nbsp;above and below <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math>?</p>
<p>&nbsp;</p>
<p><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="593" height="500">
<param name="image" value="__BASE_URI__img/Interference/DiagramDoubleSlit.png" />
<param name="size" value="1" />
<param name="label.1.x" value="230" />
<param name="label.1.y" value="10" />
<param name="label.1.text" value="X" /></applet></p>
<p>&nbsp;</p>
<p>&nbsp;</p>@
qu.20.6.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uEM,uSF=false);@
qu.20.6.allow2d=0@
qu.20.6.maple_answer=SigFigs[roundToSigFigs]($m,3)@
qu.20.6.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.20.6.type=maple@
qu.20.6.mode=Maple@
qu.20.6.name=Double Slit Interference - Find Fringe Order - Dark ~ PGc@
qu.20.6.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uEM,uSF=false,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.20.6.editing=useHTML@
qu.20.6.solution=<p>For a double slit, destructive interference occurs when the parameters satisfy:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mi>m</mi><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn mathvariant='italic'>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&lambda;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>d</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>.</p>
<p>Thus, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>m</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&mu;m</mi></mrow></mfenced><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mfenced></mrow></mfrac><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>$theta</mi><mrow><mi>o</mi></mrow></msup></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>&nbsp;</p>@
qu.20.6.algorithm=$m=range(1,6);
$theta=rand(0,45,3);
$lambda=range(100,999);
$d=sig(3,($m/2)*$lambda*10^(-9)*10^(6)/sin($theta*2*Pi/360));@
qu.20.6.uid=b7745b5d-49a1-4610-b22f-c5e65073bebf@
qu.20.6.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Two Source Interference;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.20.7.mode=Inline@
qu.20.7.name=Two Source Interference - Destructive@
qu.20.7.comment=@
qu.20.7.editing=useHTML@
qu.20.7.hint.1=Look at path-length differences.@
qu.20.7.solution=<p>For two sources that are in phase, destructive interference will be observed whenever the path-length difference from the two sources to a point is a half-integer multiple of the wavelength:</p>
<p align="left"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>r</mi><mrow><mn>2</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mi>r</mi><mrow><mn>1</mn></mrow></msub><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>m</mi><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn mathvariant='italic'>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&lambda;</mi></mrow></mstyle></math>.</p>
<p align="left">&nbsp;</p>
<p align="left">Here, we are given the wavelength, so we need to calculate the distance from each source to the candidate points.&nbsp; If the difference in those two distances, for a given point, is a half-integer multiple of the wavelength, then destructive interference will be observed.</p>@
qu.20.7.algorithm=$x1i=range(1,10);
$x1=50*$x1i;
$y1i=range(1,10);
$y1=50*$y1i;
$m=maple("
randomize():
a:={}:
idx:=0:
for i from 0 to 10 do
for j from 0 to 10 do
idx:=idx+1:
a:={a[],[i,j]}:
end do:
end do:
S1:=RandomTools[Generate](choose(a)):
a:=a minus {S1};
S2:=RandomTools[Generate](choose(a)):
a:=a minus {S2};
a:=combinat[randperm](a):
for i from 1 to 98 while not(type(student[distance](a[i],S2)-student[distance](a[i],S1),integer)) do end do:
CI:=a[i]:
a:={a[]} minus {CI};
a:=combinat[randperm](a):
for i from 1 to 97 while (type(student[distance](a[i],S2)-student[distance](a[i],S1),integer)) do end do:
P1:=a[i]:
a:={a[]} minus {P1}:
a:=combinat[randperm](a):
for i from 1 to 96 while (type(student[distance](a[i],S2)-student[distance](a[i],S1),integer)) do end do:
P2:=a[i]:
a:={a[]} minus {P2}:
a:=combinat[randperm](a):
for i from 1 to 95 while (type(student[distance](a[i],S2)-student[distance](a[i],S1),integer)) do end do:
P3:=a[i]:
a:={a[]} minus {P3}:
n1:=RandomTools[Generate](choose({A,B,C,D})):
n2:=RandomTools[Generate](choose({A,B,C,D} minus {n1})):
n3:=RandomTools[Generate](choose({A,B,C,D} minus {n1,n2})):
n4:=RandomTools[Generate](choose({A,B,C,D} minus {n1,n2,n3})):
S1,S2,CI,P1,P2,P3,n1,n2,n3,n4;
");
$S1x=switch(0,$m)*50+10;
$S1y=switch(1,$m)*50+15;
$S2x=switch(2,$m)*50+10;
$S2y=switch(3,$m)*50+15;
$CIx=switch(4,$m)*50+10;
$CIy=switch(5,$m)*50+15;
$P1x=switch(6,$m)*50+10;
$P1y=switch(7,$m)*50+15;
$P2x=switch(8,$m)*50+10;
$P2y=switch(9,$m)*50+15;
$P3x=switch(10,$m)*50+10;
$P3y=switch(11,$m)*50+15;
$CI=switch(12,$m);
$P1=switch(13,$m);
$P2=switch(14,$m);
$P3=switch(15,$m);
$lambda=range(100,900);
$n=rint(1,11,2);
$d=($n/2)*$lambda*10^(-2);@
qu.20.7.uid=e06ff3f1-d847-4fe6-aff4-22c9106ddef7@
qu.20.7.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Two Source Interference;
  Difficulty=Hard;
  Features=Algorithmic;
  Features=Algorithmic Diagram;
@
qu.20.7.weighting=1@
qu.20.7.numbering=alpha@
qu.20.7.part.1.grader=exact@
qu.20.7.part.1.name=sro_id_1@
qu.20.7.part.1.editing=useHTML@
qu.20.7.part.1.display.permute=true@
qu.20.7.part.1.answer.4=$P3@
qu.20.7.part.1.answer.3=$P2@
qu.20.7.part.1.question=(Unset)@
qu.20.7.part.1.answer.2=$P1@
qu.20.7.part.1.answer.1=$CI@
qu.20.7.part.1.mode=List@
qu.20.7.part.1.display=menu@
qu.20.7.part.1.credit.4=0.0@
qu.20.7.part.1.credit.3=0.0@
qu.20.7.part.1.credit.2=0.0@
qu.20.7.part.1.credit.1=1.0@
qu.20.7.question=<p align="left">Two sources, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>S1</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>S2</mi></mrow></mstyle></math>&nbsp;output monochromatic electromagnetic waves,&nbsp;in phase, at wavelength<br /><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>cm</mi></mrow></mstyle></math>.&nbsp; If the major grid divisions are <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>m</mi></mrow></mstyle></math>&nbsp;apart, at which point will destructive interference&nbsp;<br />be observed?&nbsp;</p><p align="center">&nbsp;</p><p align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="520" height="520"><param name="image" value="__BASE_URI__img/Interference/TwoSource/Diagram.png" /><param name="size" value="6" /><param name="label.1.x" value="$S1x" /><param name="label.1.y" value="$S1y" /><param name="label.1.text" value="S1" /><param name="label.2.x" value="$S2x" /><param name="label.2.y" value="$S2y" /><param name="label.2.text" value="S2" /><param name="label.3.x" value="$CIx" /><param name="label.3.y" value="$CIy" /><param name="label.3.text" value="$CI" /><param name="label.4.x" value="$P1x" /><param name="label.4.y" value="$P1y" /><param name="label.4.text" value="$P1" /><param name="label.5.x" value="$P2x" /><param name="label.5.y" value="$P2y" /><param name="label.5.text" value="$P2" /><param name="label.6.x" value="$P3x" /><param name="label.6.y" value="$P3y" /><param name="label.6.text" value="$P3" /></applet></p><p>&nbsp;</p><p>&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.20.8.question=<p>The centres of the two slits in the diagram&nbsp;are&nbsp;separated by a distance of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$d</mi></mrow><mrow><mi></mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&mu;m</mi></mrow></mstyle></math>,&nbsp;with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mrow><mo lspace='0.0em' rspace='0.0em'>&Gt;</mo></mrow><mi>d</mi></mrow></mstyle></math>.&nbsp; A&nbsp;<br />
beam of&nbsp;monochromatic, coherent light&nbsp;of wavelength <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mstyle></math>passes through the slits at normal&nbsp;<br />
incidence.&nbsp; What order of&nbsp;bright fringes&nbsp;are observed at an angle <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>$theta</mi><mrow><mi>o</mi></mrow></msup></mrow></mstyle></math>&nbsp;above and below <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math>?</p>
<p>&nbsp;</p>
<p><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="593" height="500">
<param name="image" value="__BASE_URI__img/Interference/DiagramDoubleSlit.png" />
<param name="size" value="1" />
<param name="label.1.x" value="230" />
<param name="label.1.y" value="10" />
<param name="label.1.text" value="X" /></applet></p>
<p>&nbsp;</p>
<p>&nbsp;</p>@
qu.20.8.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uEM,uSF=false);@
qu.20.8.allow2d=0@
qu.20.8.maple_answer=SigFigs[roundToSigFigs]($m,3)@
qu.20.8.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.20.8.type=maple@
qu.20.8.mode=Maple@
qu.20.8.name=Double Slit Interference - Find Fringe Order - Bright ~ PGc@
qu.20.8.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uEM,uSF=false,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.20.8.editing=useHTML@
qu.20.8.solution=<p>For a double slit, constructive interference occurs when the parameters satisfy:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>m&lambda;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>d</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mi>sin</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mrow></mstyle></math>.</p>
<p>Thus, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>m</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&mu;m</mi></mrow></mfenced><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mfenced></mrow></mfrac></mrow><mrow><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>$theta</mi><mrow><mi>o</mi></mrow></msup></mrow></mfenced></mrow></mstyle></math>.</p>
<p>&nbsp;</p>@
qu.20.8.algorithm=$m=range(1,6);
$theta=rand(0,45,3);
$lambda=range(100,999);
$d=sig(3,$m*$lambda*10^(-9)*10^(6)/sin($theta*2*Pi/360));@
qu.20.8.uid=25cab014-b02f-4aa7-b7fc-fb2023a20e06@
qu.20.8.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Two Source Interference;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.20.9.question=<p>The centres of the two slits in the diagram are&nbsp;separated by a distance of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow><mrow><mi></mi></mrow></mstyle></math>.&nbsp; If coherent&nbsp;<br />
light&nbsp;incident&nbsp;on&nbsp;the slits&nbsp;of wavelength <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mstyle></math> produces&nbsp;order-$m bright fringes&nbsp;on the screen&nbsp;<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$y</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>cm</mi></mrow></mstyle></math>above&nbsp;and below <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math>, what is the distance <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math>&nbsp;to the screen?</p>
<p>&nbsp;</p>
<p><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="593" height="500">
<param name="image" value="__BASE_URI__img/Interference/DiagramDoubleSlit.png" />
<param name="size" value="1" />
<param name="label.1.x" value="230" />
<param name="label.1.y" value="10" />
<param name="label.1.text" value="X" /></applet></p>
<p>&nbsp;</p>
<p>&nbsp;</p>@
qu.20.9.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25);@
qu.20.9.allow2d=0@
qu.20.9.maple_answer=SigFigs[roundToSigFigs]($ans,3)*m@
qu.20.9.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.20.9.type=maple@
qu.20.9.mode=Maple@
qu.20.9.name=Double Slit Interference - Find Distance to Screen - Bright ~ PGc@
qu.20.9.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.20.9.editing=useHTML@
qu.20.9.solution=<p>For a double slit, constructive interference occurs when the parameters satisfy:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>m&lambda;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>d</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mi>sin</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mrow></mstyle></math>.</p>
<p>Thus, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&theta;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msup><mi>sin</mi><mn>-1</mn></msup><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$m</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mfenced></mrow><mrow><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mfrac></mrow></mfenced></mrow></mrow></mstyle></math>.</p>
<p>Given the height of the fringes, we have:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mi>$y</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>cm</mi></mrow></mfenced><mrow><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>.</p>
<p>&nbsp;</p>@
qu.20.9.algorithm=$d=rand(0.5,0.9,3);
$m=range(1,4);
$lambda=range(400,800);
$X=rand(1.00,9.00,3);
$y=rand(0.1,1.5,3);
$ans=($y*10^(-2))/tan(arcsin(($m*$lambda*10^(-9))/($d*10^(-3))));@
qu.20.9.uid=5f2bd6c3-17b3-424d-9ab7-777c6c27e1e8@
qu.20.9.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Two Source Interference;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
@

qu.20.10.question=<p>The centres of the two slits in the diagram are&nbsp;separated by a distance of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow><mrow><mi></mi></mrow></mstyle></math>.&nbsp; If&nbsp;coherent<br />
light&nbsp;incident&nbsp;on&nbsp;the slits&nbsp;of wavelength <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mstyle></math> produces&nbsp;order-$m dark fringes&nbsp;on the screen&nbsp;<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$y</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>cm</mi></mrow></mstyle></math>above&nbsp;and below <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math>, what is the distance <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math>&nbsp;to the screen?</p>
<p>&nbsp;</p>
<p><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="593" height="500">
<param name="image" value="__BASE_URI__img/Interference/DiagramDoubleSlit.png" />
<param name="size" value="1" />
<param name="label.1.x" value="230" />
<param name="label.1.y" value="10" />
<param name="label.1.text" value="X" /></applet></p>
<p>&nbsp;</p>
<p>&nbsp;</p>@
qu.20.10.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25);@
qu.20.10.allow2d=0@
qu.20.10.maple_answer=SigFigs[roundToSigFigs]($ans,3)*m@
qu.20.10.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.20.10.type=maple@
qu.20.10.mode=Maple@
qu.20.10.name=Double Slit Interference - Find Distance to Screen - Dark ~ PGc@
qu.20.10.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.20.10.editing=useHTML@
qu.20.10.solution=<p>For a double slit, destructive interference occurs when the parameters satisfy:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mi>m</mi><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn mathvariant='italic'>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&lambda;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>d</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>.</p>
<p>Thus, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&theta;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msup><mi>sin</mi><mn>-1</mn></msup><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$m</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mfenced></mrow><mrow><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mfrac></mrow></mfenced></mrow></mrow></mstyle></math>.</p>
<p>&nbsp;Given the fringe height, we have:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mi>$y</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>cm</mi></mrow></mfenced><mrow><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>.</p>@
qu.20.10.algorithm=$d=rand(0.5,0.9,3);
$m=range(1,4);
$lambda=range(400,800);
$X=rand(1.00,9.00,3);
$y=rand(0.1,1.5,3);
$ans=($y*10^(-2))/tan(arcsin((($m/2)*$lambda*10^(-9))/($d*10^(-3))));@
qu.20.10.uid=b487427e-86a1-475a-8f2e-042fc71331ca@
qu.20.10.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Two Source Interference;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.20.11.question=<p>The centres of the two slits in the diagram&nbsp;are&nbsp;separated by a distance of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mi>&mu;</mi></mrow><mi>m</mi></mrow><mrow></mrow></mstyle></math>, and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mrow><mo lspace='0.0em' rspace='0.0em'>&Gt;</mo></mrow><mi>d</mi></mrow></mstyle></math>.&nbsp; A<br />
beam of monochromatic, coherent&nbsp;light&nbsp;passes through the slits at normal incidence.&nbsp; If the&nbsp;order-$m&nbsp;<br />
bright&nbsp;fringes are visible at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>$theta</mi><mrow><mi>o</mi></mrow></msup></mrow></mstyle></math>, what is the wavelength of the light?</p>
<p>&nbsp;</p>
<p><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="593" height="500">
<param name="image" value="__BASE_URI__img/Interference/DiagramDoubleSlit.png" />
<param name="size" value="1" />
<param name="label.1.x" value="230" />
<param name="label.1.y" value="10" />
<param name="label.1.text" value="X" /></applet></p>
<p>&nbsp;</p>
<p>&nbsp;</p>@
qu.20.11.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25);@
qu.20.11.allow2d=0@
qu.20.11.maple_answer=SigFigs[roundToSigFigs]($ans,3)*nm@
qu.20.11.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.20.11.type=maple@
qu.20.11.mode=Maple@
qu.20.11.name=Double Slit Interference - Find Wavelength - Bright ~ PGc@
qu.20.11.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.20.11.editing=useHTML@
qu.20.11.solution=<p>For a double slit, constructive interference occurs when the parameters satisfy:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>m&lambda;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>d</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mi>sin</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mrow></mstyle></math>.</p>
<p>Thus, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&lambda;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&mu;m</mi></mrow></mfenced><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$m</mi></mrow></mfenced></mrow></mfrac></mrow><mrow><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>$theta</mi><mrow><mi>o</mi></mrow></msup></mrow></mfenced></mrow></mstyle></math>.</p>
<p>&nbsp;</p>@
qu.20.11.algorithm=$d=rand(1.0,9.99,3);
$m=range(1,4);
$theta=rand(0,20,3);
$ans=($d*10^(-6))*sin($theta*2*Pi/360)*10^(9)/$m;@
qu.20.11.uid=adfb3520-ac2f-4534-a9c3-ab244aa0ef5b@
qu.20.11.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Two Source Interference;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.20.12.question=<p>The centres of the two slits in the diagram are&nbsp;separated by a distance of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow><mrow><mi></mi></mrow></mstyle></math>.&nbsp; If the&nbsp;coherent<br />
light&nbsp;incident&nbsp;on the slits is of wavelength <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mstyle></math>, at what distance above and below <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi></mrow></mstyle></math> would&nbsp;<br />
one&nbsp;find&nbsp;the&nbsp;order-$m bright fringes on the screen?</p>
<p>&nbsp;</p>
<p><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="593" height="500">
<param name="image" value="__BASE_URI__img/Interference/DiagramDoubleSlit.png" />
<param name="size" value="1" />
<param name="label.1.x" value="230" />
<param name="label.1.y" value="10" />
<param name="label.1.text" value="$X m" /></applet></p>
<p>&nbsp;</p>
<p>&nbsp;</p>
<p><span>&nbsp;</span></p>@
qu.20.12.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25);@
qu.20.12.allow2d=0@
qu.20.12.maple_answer=SigFigs[roundToSigFigs]($ans,3)*cm@
qu.20.12.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.20.12.type=maple@
qu.20.12.mode=Maple@
qu.20.12.name=Double Slit Interference - Find Screen Height - Bright ~ PGc@
qu.20.12.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.20.12.editing=useHTML@
qu.20.12.solution=<p>For a double slit, constructive interference occurs when the parameters satisfy:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>m&lambda;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>d</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mi>sin</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mrow></mstyle></math>.</p>
<p>Thus, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&theta;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msup><mi>sin</mi><mn>-1</mn></msup><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>$m</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mfenced></mrow><mrow><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mm</mi></mrow></mfrac></mrow></mfenced></mrow></mrow></mstyle></math>.</p>
<p>&nbsp;Given the distance to the screen, we have:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>m</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>.</p>@
qu.20.12.algorithm=$d=rand(0.5,0.9,3);
$m=range(1,4);
$lambda=range(400,800);
$X=rand(1.00,9.00,3);
$ans=tan(arcsin(($m*$lambda*10^(-9))/($d*10^(-3))))*$X*10^(2);@
qu.20.12.uid=af13d381-e5ac-4059-991b-850f5cb2e380@
qu.20.12.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Two Source Interference;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.20.13.question=<p>The centres of the two slits in the diagram&nbsp;are&nbsp;separated by a distance of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mi>&mu;</mi></mrow><mi>m</mi></mrow><mrow></mrow></mstyle></math>, and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mrow><mo lspace='0.0em' rspace='0.0em'>&Gt;</mo></mrow><mi>d</mi></mrow></mstyle></math>.&nbsp; A<br />
beam of monochromatic, coherent&nbsp;light&nbsp;passes through the slits at normal incidence.&nbsp; If the&nbsp;order-$m&nbsp;<br />
dark fringes are visible at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>$theta</mi><mrow><mi>o</mi></mrow></msup></mrow></mstyle></math>, what is the wavelength of the light?</p>
<p>&nbsp;</p>
<p><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="593" height="500">
<param name="image" value="__BASE_URI__img/Interference/DiagramDoubleSlit.png" />
<param name="size" value="1" />
<param name="label.1.x" value="230" />
<param name="label.1.y" value="10" />
<param name="label.1.text" value="X" /></applet></p>
<p>&nbsp;</p>
<p>&nbsp;</p>
<p><span>&nbsp;</span></p>@
qu.20.13.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25);@
qu.20.13.allow2d=0@
qu.20.13.maple_answer=SigFigs[roundToSigFigs]($ans,3)*nm@
qu.20.13.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.20.13.type=maple@
qu.20.13.mode=Maple@
qu.20.13.name=Double Slit Interference - Find Wavelength - Dark ~ PGc@
qu.20.13.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.20.13.editing=useHTML@
qu.20.13.solution=<p>For a double slit, destructive interference occurs when the parameters satisfy:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mi>m</mi><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn mathvariant='italic'>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&lambda;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>d</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>.</p>
<p>Thus, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&lambda;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mi>$d</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&mu;m</mi></mrow></mfenced><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$m</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced></mrow></mfrac><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>$theta</mi><mrow><mi>o</mi></mrow></msup></mrow></mfenced></mrow></mstyle></math>.</p>
<p>&nbsp;</p>@
qu.20.13.algorithm=$d=rand(1.0,9.99,3);
$m=range(1,4);
$theta=rand(0,20,3);
$ans=($d*10^(-6))*sin($theta*2*Pi/360)*10^(9)/($m/2);@
qu.20.13.uid=f0bd930b-dc95-43df-9e07-37537f839b75@
qu.20.13.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Two Source Interference;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

qu.20.14.question=<p>The centres of the two slits in the diagram&nbsp;are&nbsp;separated by a distance of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi></mrow></mstyle></math>, and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mrow><mo lspace='0.0em' rspace='0.0em'>&Gt;</mo></mrow><mi>d</mi></mrow></mstyle></math>.&nbsp; A beam of&nbsp;<br />
monochromatic, coherent&nbsp;light&nbsp;passes through the slits at normal incidence.&nbsp; If the&nbsp;order-$m dark&nbsp;<br />
fringes&nbsp;are&nbsp;visible at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>$theta</mi><mrow><mi>o</mi></mrow></msup></mrow></mstyle></math>&nbsp;for light of wavelength <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mstyle></math>,&nbsp;what is the slit spacing <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi></mrow></mstyle></math>?</p>
<p>&nbsp;</p>
<p><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="593" height="500">
<param name="image" value="__BASE_URI__img/Interference/DiagramDoubleSlit.png" />
<param name="size" value="1" />
<param name="label.1.x" value="230" />
<param name="label.1.y" value="10" />
<param name="label.1.text" value="X" /></applet></p>
<p>&nbsp;</p>
<p>&nbsp;</p>
<p><span>&nbsp;</span></p>@
qu.20.14.maple=PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25);@
qu.20.14.allow2d=0@
qu.20.14.maple_answer=SigFigs[roundToSigFigs]($ans,3)*um@
qu.20.14.libname=__BASE_URI__repositories/PartialGrading.lib@
qu.20.14.type=maple@
qu.20.14.mode=Maple@
qu.20.14.name=Double Slit Interference - Find Slit Spacing - Dark ~ PGc@
qu.20.14.comment=<p>${if(eq($RESPONSE,No answer),"No answer entered.",maple("PartialGrading[Grade]($ANSWER,`$RESPONSE`,uSF,nSF=3,uUN,nUN=2,dUN=3,mUN=0.66,uDM,mDM=0.25,giveComments);,libname=__BASE_URI__repositories/PartialGrading.lib"))}</p>@
qu.20.14.editing=useHTML@
qu.20.14.solution=<p>For a double slit, destructive interference occurs when the parameters satisfy:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mi>m</mi><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn mathvariant='italic'>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&lambda;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>d</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>.</p>
<p>Thus, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>d</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$m</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi>$lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>nm</mi></mrow></mfenced></mrow><mrow><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>$theta</mi><mrow><mi>o</mi></mrow></msup></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>.</p>
<p>&nbsp;</p>@
qu.20.14.algorithm=$m=range(1,4);
$theta=rand(0,20,3);
$lambda=range(100,999);
$ans=($m/2)*$lambda*10^(-9)*10^(6)/sin($theta*2*Pi/360);@
qu.20.14.uid=a65159fe-d271-4c4f-8d54-c33e094f0761@
qu.20.14.info=  Course=Introductory Electricity and Magnetism;
  Author=Aron Pasieka;
  Topic=Two Source Interference;
  Difficulty=Medium;
  Features=Partial Grading;
  Features=Algorithmic;
  Features=Diagram;
@

