# Liénard-Wiechert potential of a charged infinite sheet uniformly translating parallel to itself

This question is motivated by the model used in The Feynman Lectures on Physics Vol II Section 18-4. In the model there are two plane sheets extending infinitely in both their dimensions. Each has a uniform electric charge equal in magnitude and opposite in sign to the other. They are place in immediate mutual proximity. One is accelerated abruptly to a constant velocity parallel to itself.

The purpose of the model is to illustrate some basic features of electromagnetic waves. But my question regards the actual consequences of that models, taking to its logical conclusions. In particular, it is evident to me that the Liénard-Wiechert potential of such a construct would overwhelm any other effect such as the electric and magnetic fields produced by the relative motion.

NB:This is a non-relativistic result.

The Liénard-Wiechert potential is discussed in The Feynman Lectures on Physics Vol II 21-5 The potentials of a moving charge; the general solution of Liénard and Wiechert As well as in this PDF document of lecture notes UIUC Physics 436 EM Fields & Sources II

The gist of the Liénard-Wiechert potential theory is that a charge distribution moving toward a filed point will have an apparent volume greater than the actual volume. In the case of a receding charge distribution, it will appear to have less volume than it actually has. Thus an approaching charge will appear greater than it would were it stationary or receding. A receding charge will appear to be less than if it were stationary or approaching.

I didn't set out to destroy Feynman's example, but this appears to the the final nail in the coffin.

Is it not the case that the moving charged sheet of infinite extent would produce an electric field approaching an infinite magnitude due to its Liénard-Wiechert potential?

• The Lienard Wiechert potentials are for point charges, not infinite sheets. Using them heuriatically for continuous charge distributions can lead to erroneous conclusions. – Dale Dec 15 '18 at 14:37
• That seems to contradict both sources I provided. Can you explain why a Liénard-Wiechert potential is not applicable to the case at hand? Can you explain why it is not applicable to a finite charge distribution? – Steven Thomas Hatton Dec 15 '18 at 14:43
• Not really, comments and answers are not well suited to that sort of extended conversation. Your best bet would be to ask on a more discussion-oriented forum. Also, both references specify point charges – Dale Dec 15 '18 at 15:00
• In the case of Feynman he writes "point" charge and actually discusses a finite charge distribution. In the case of Errede, he is clearly discussing finite charge distributions and then argues that his result is also applicable to so-called point charges. As a matter of fact, both discussions strongly indicate that "point" charges have some finite volume of "charge" distribution. Or at least some extended structure. – Steven Thomas Hatton Dec 15 '18 at 16:07
• (1/2) It seems your question is whether the charged sheet will produce a potential approaching an infinite magnitude. It is known that charged sheet produces constant electric field anywhere (using Gauss's Law). If you integrate the electric field from infinitely far far away to the point of interest near the sheet, you surely get an infinite potential. Thus when treating such problems people do not define potential at far far away as 0. They may define potential at the sheet as 0. Your Lienard-Weichert potential defines potential at far far away as 0 so it is not compatible with this problem. – verdelite Apr 25 at 14:26