Steve Roper

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Steve The three vectors _r, _theta and _...

Answered: Steve Roper 180

May 03 2020

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@ecterrab Hi Edgardo,

That's interesting. I was expecting k_.k_ to be the square of the length of the vector, which would be a^2. I was expecting the angles theta and phi to have no impact.

Steve

Study Maxwell’s Equations...

Answered: Steve Roper 180

March 15 2020

0 0

@ecterrab Thank you Edgardo, you have given me a great deal to think about :-)

I completely agree with your comments on Mathematica. The syntax is quite difficult to read, whereas the Maple document looks pretty much the same as a text book or paper.

Thanks to your help, I now have this:

Maxwell's Equations

Initialise

(1.1)

(1.2)

Maxwell's Equations

(2.1)

(2.2)

(2.3)

(2.4)

Constitutive Relations

(3.1)

(3.2)

Solution

We need an expression for the Curl of H, so we start by taking the Curl of both sides of Maxwell_1:

(4.1)

Now substitute B for μH:

(4.2)

Now substitute the Curl of H with an expression in D:

(4.3)

Now substitute E for D:

Physics:-Vectors:-Curl(Physics:-Vectors:-Curl(D__flux_(x, y, z, t)/varepsilon)) = -mu*(diff(diff(D__flux_(x, y, z, t), t), t))

(4.4)

Let's see what happens if we expand this:

(4.5)

This is impressive! Maple knows that V×(V×D) = V(V·D) - V²D

Now we can use Maxwell_3 to arrive at an expression in V²D because we know that V·D=0:

$simplify(Physics[Vectors][Curl](Physics[Vectors][Curl](D__flux_(x, y, z, t)/varepsilon)) = -mu*(diff(diff(D__flux_(x, y, z, t), t), t)), {Maxwell_3})$

(-(diff(diff(D__flux_(x, y, z, t), x), x))-(diff(diff(D__flux_(x, y, z, t), y), y))-(diff(diff(D__flux_(x, y, z, t), z), z)))/varepsilon = -mu*(diff(diff(D__flux_(x, y, z, t), t), t))

(4.6)

Now we have an expression that looks like a wave, we can substitute D for E:

(-(diff(diff(varepsilon*E__field_(x, y, z, t), x), x))-(diff(diff(varepsilon*E__field_(x, y, z, t), y), y))-(diff(diff(varepsilon*E__field_(x, y, z, t), z), z)))/varepsilon = -mu*(diff(diff(varepsilon*E__field_(x, y, z, t), t), t))

(4.7)

(4.8)

Loose the "-" sign:

(4.9)

Finaly, we can use the inert operator "%" to make expression look more concise like this:

(4.10)

Now we need to use pdsolve to find a solution!

You are quite right about equation lables. It makes it possible to insert and delete and re-execute individual statements. Using %, I often found that it was necessary to re-execute the whole workbook with !!!.

And the intert "%" operator helped with the final substitution in (4.10) :-)

I haven't had time to experiment with pdsolve yet. Presumably it will be necessary to solve for waves propagating allong the x,y and z axies seperately before re-combining?

Download Maxwells_Equations.mw

Study Maxwell’s Equations...

Answered: Steve Roper 180

March 12 2020

0 0

@ecterrab Thank you Edgardo, you make it look so easy :-)

Can I ask how that works please? Why does subs need a little help in this case in order to substitute the LHS of Maxwell_2 with the RHS?

I have attached the updated worksheet and now have some new questions. The first is how to substitute using the left-hand side of the equation rather than the right and the second is how to invoke vector identities such as ∇×(∇×F)=∇(∇·F)-∇2F?

Maxwell's Equations