This is an interesting exercise, the computation of the Liénard–Wiechert potentials describing the classical electromagnetic field of a moving electric point charge, a problem of a 3^{rd} year undergrad course in Electrodynamics. The calculation is nontrivial and is performed below using the Physics package, following the presentation in [1] (Landau & Lifshitz "The classical theory of fields"). I have not seen this calculation performed on a computer algebra worksheet before. Thus, this also showcases the more advanced level of symbolic problems that can currently be tackled on a Maple worksheet. At the end, the corresponding document is linked and with it the computation below can be reproduced. There is also a link to a corresponding PDF file with all the sections open.
Moving charges:
The retarded and LiénardWiechert potentials, and the fields and
Freddy Baudine^{(1)}, Edgardo S. ChebTerrab^{(2)}
(1) Retired, passionate about Mathematics and Physics
(2) Physics, Differential Equations and Mathematical Functions, Maplesoft
Generally speaking, determining the electric and magnetic fields of a distribution of charges involves determining the potentials and , followed by determining the fields and from
,
In turn, the formulation of the equations for and is simple: they follow from the 4D second pair of Maxwell equations, in tensor notation
where is the electromagnetic field tensor and is the 4D current. After imposing the Lorentz condition
, i.e.
we get
which in 3D form results in
where is the current and is the charge density.
Following the presentation shown in [1] (Landau and Lifshitz, "The classical theory of fields", sec. 62 and 63), below we solve these equations for and resulting in the socalled retarded potentials, then recompute these fields as produced by a charge moving along a given trajectory  the socalled LiénardWiechert potentials  finally computing an explicit form for the corresponding and .
While the computation of the generic retarded potentials is, in principle, simple, obtaining their form for a charge moving along a given trajectory , and from there the form of the fields and shown in Landau's book, involves nontrivial algebraic manipulations. The presentation below thus also shows a technique to map onto the computer the manipulations typically done with paper and pencil for these problems. To reproduce the contents below, the Maplesoft Physics Updates v.1252 or newer is required.

(1) 

The retarded potentials and


The equations which determine the scalar and vector potentials of an arbitrary electromagnetic field are input as

(2) 

(3) 

(4) 
The solutions to these inhomogeneous equations are computed as the sum of the solutions for the equations without righthand side plus a particular solution to the equation with righthand side.


The LiénardWiechert potentials of a charge moving along


From (13), the potential at the point is determined by the charge , i.e. by the position of the charge e at the earlier time
The quantityis the 3D distance from the position of the charge at the time to the 3D point of observation. In the previous section, the charge was located at the origin and at rest, so , the radial coordinate. If the charge is moving, say on a path , we have
From (13)≡ and the definition of above, the potential of a moving charge can be written as
When the charge is at rest, in the Lorentz gauge we are working, the vector potential is . When the charge is moving, the form of can be found searching for a solution to that gives when . Following [1], this solution can be written as
where is the four velocity of the charge, .
Without showing the intermediate steps, [1] presents the three dimensional vectorial form of these potentials and as
,


The electric and magnetic fields and of a charge moving along


The electric and magnetic fields at a point are calculated from the potentials and through the formulas
,
where, for the case of a charge moving on a path , these potentials were calculated in the previous section as (24) and (18)
These two expressions, however, depend on the time only through the retarded time . This dependence is within and through the velocity of the charge . So, before performing the differentiations, this dependence on must be taken into account.

(29) 

(30) 

The Electric field


Collecting the results of the two previous subsections, we have for the electric field

(60) 

(61) 
The book, presents this result as equation (63.8):
where and . To rewrite (61) as in the above, introduce the two triple vector products

(62) 

(63) 

(64) 

(65) 
Split now this result into two terms, one of them involving the acceleration . For that purpose first expand the expression without expanding the cross products

(66) 
Introduce the notation used in the textbook, and and proceed with the splitting

(67) 
Rearrange only the first term using simplify; that can be done in different ways, perhaps the simplest is using subsop

(68) 
By eye this result is mathematically equal to equation (63.8) of the textbook, shown here above before (62) .


The magnetic field


The book does not show an explicit form for , it only indicates that it is related to the electric field by the formula
Thus in this section we compute the explicit form of and show that this relationship mentioned in the book holds. To compute we proceed as done in the previous sections, the righthand side should be taken at the previous (retarded) time . For clarity, turn OFF the compact display of functions.
We need to calculate

(75) 
So applying to (75) the chain rule derived in the previous subsection we have

(87) 
where is taken as a function of only in . Now that the functionality is understood, turning ON the compact display of functions and displaying the fields by their names,

(88) 
The value of is computed lines above as (48)

(89) 
The expression for with no dependency is computed lines above, as (28),

(90) 
The expressions for and the velocity in terms of with no dependency are

(91) 

(92) 

(93) 
Introducing this into ,

(94) 
Before computing the first term , for readability, reintroduce the velocity , the acceleration , then remove the dependency of these functions on , not relevant anymore since there are no more derivatives with respect to . Performing these substitutions in sequence,

(95) 

(96) 
Activate now the inert curl

(97) 
From (34)≡, and reintroducing

(98) 

(99) 
To conclude, rearrange this expression as done with the one for the electric field at (65), so first expand (99) without expanding the cross products

(100) 
Then introduce the notation used in the textbook, and and split into two terms, one of which contains the acceleration

(101) 


References


[1] Landau, L.D., and Lifshitz, E.M. Course of Theoretical Physics Vol 2, The Classical Theory of Fields. Elsevier, 1975.


