I Shape and value of the Liénard–Wiechert potential

[TCO]

TL;DR Summary

symmetry of the Liénard–Wiechert potential in the lab frame

1) What is the shape of equipotential surfaces of the Liénard–Wiechert potential for a uniformly moving charge in the lab frame in the x-direction, like behind(x=-r) and in front(x=r) of the charge? Is the value the same as in the static case, Coulomb potential at x=r?

2) Is there some kind of asymmetry in the x-axis or is it symmetric like shown at fig. b?


Fig. a:

Fig. b:

Any pictures of the moving potential would be appreciated...

 

[Ibix]

Can you state the expressions for the Liénard-Wiechert potentials?

 

[renormalize]

Any pictures of the moving potential would be appreciated...

Would you settle for moving field-lines instead of moving equipotentials?
From https://physics.weber.edu/schroeder/mrr/MRRtalk.html:

 

[TCO]

I was a specifically interested in the potential shape (as I'm aware of the pancake field, that has a reduction on the x-axis compared to the static case):

(Griffiths 4th ed (10.46))
- where v is the velocity of the charge at the retarded time, and r is the vector from the retarded position to the field point r.

It seems that (10.46) can be written as:

(Griffiths 4th ed (10.51))
with

and

as maybe I should just actually read the book...so on the x-axis at θ = 0 and θ = pi, the potential is actually the same as the static one.

So, I guess the video is actually correct in showing this here:

right after this, this picture is shown:

but this picture is wrong at this time (@17:39), because we here see a moving spherical symmetric field equal to the static one both in the lab frame (indicated by the velocity vector)...

 

[Ibix]

The potential isn't uniquely defined - it depends on your choice of gauge. So the first picture from the video may or may not be correct. It's inconsistent with Griffiths' 10.51, which is manifestly mirror symmetric about a plane perpendicular to the x axis and passing through the charge, but that may not necessarily be wrong.

The electric field, however, is uniquely defined. It is spherically symmetric in both frames, but should differ in strength by a factor of $\gamma$. It's not clear from the diagram if the arrows are different lengths, or whether they are intended to be, but to my eye neither looks less spherical than the other.

 

[TCO]

okay, thanks...