# What is the name of the paradox about (false) net electrical force in a system of two collinearly moving opposite charges?

Kirk McDonald said in his paper "Maxwell’s Objection to Lorenz’ Retarded Potentials" ( http://www.physics.princeton.edu/~mcdonald/examples/maxwell.pdf ) that (about Lorenz's concept of retarded potentials),

Maxwell made an objection [5] (p. 651) that if a pair of equal and opposite charges move collinearly, then the retarded potential experienced by the charge in front has smaller magnitude than that experienced by the charge in the rear because the former retarded distance is larger than the latter; hence, there must be a net electrical force on the system, which accelerates it without limit, providing an infinite source of free energy.

He also said Maxwell's objection was based on a misunderstanding, and the issue was avoided by using Lorentz transformation. I ask here whether this paradox has a name, and if so, what is it? I ask because I try to find papers about it. A name will greatly facilitate the search.

I have done some work in this area. A useful way in for you might be

A.M.Steane, The non-existence of the self-accelerating dipole, and related questions, Phys. Rev. D 89, 125006 (2014), doi 10.1103/PhysRevD.89.125006, arXiv:1311.5798

and

A.M.Steane, Self-force of a rigid ideal fluid, and a charged sphere in hyperbolic motion Phys. Rev. D 91, 065008 (2015) arXiv:1407.5914

These will both clarify the issue and provide a way in to other literature. I hope it is ok to cite my own stuff in this case; I don't know of any recent review to use instead and these papers are directly on what you are asking about.

I don't know of a name for the paradox you mention, but it has come up repeatedly in various guises. Gleaning from your question, I would guess that what Maxwell overlooked was that if the proper distance between the two charges is to remain constant as they accelerate, then the back charge should have a higher proper acceleration than the front charge.

• Hi, Prof Steane, it is a nice surprise to see you answering the question, because I have just discovered your book "Relativity Made Relatively Easy" (see physics.stackexchange.com/questions/525650 ). Actually I am studying Prof. Jefimenko's book "Electromagnetic Retardation and Theory of Relativity" (I know it is not well accepted by the mainstream) and want to clear any doubt I have met. I am writing a little book "Jefimenko made Easy" alongside the studying and have already quoted your aforementioned book. Feb 25, 2020 at 1:43
• For the sake of completeness, could you please include a summary of the relevant points of your paper here so your answer is self contained? Feb 25, 2020 at 2:04
• @KFGauss It is no longer a book but an article now, arxiv.org/abs/2306.14930 However, there is nothing there related to this question. Jun 28, 2023 at 1:48
• @verdelite, I was talking to A. Steane, not to the book you linked. Jun 28, 2023 at 3:33