A discontinuity in the electromagnetic field of a uniformly accelerated charge

TL;DR

This paper derives a delta-function electromagnetic field component for a uniformly accelerated charge, resolving a discontinuity issue and linking it to radiation reaction energy, using a simplified approach consistent with Maxwell's equations.

Contribution

It provides a straightforward derivation of the delta-field component for an accelerated charge, clarifying its physical significance and energy content.

Findings

The delta-field component ensures Maxwell's equations are satisfied everywhere.

The energy in the delta-field matches the energy lost due to radiation reaction.

A simple derivation method is presented, avoiding complex approximations.

Abstract

The electric field of a uniformly accelerated charge shows a plane of discontinuity, where the field extending only on one side of the plane, terminates abruptly on the plane with a finite value. This indicates a non-zero divergence of the electric field in a source-free region, implying a violation of Gauss law. In order to make the field compliant with Maxwell's equations everywhere, an additional field component, proportional to a $\delta$-function at the plane of discontinuity, is required. Such a "$\delta$-field" might be the electromagnetic field of the charge, moving with a uniform velocity approaching $c$, the speed of light, prior to the imposition of acceleration at infinity. However, some attempts to derive this $\delta$-field for such a case, have not been entirely successful. Some of the claims of the derivation involve elaborate calculations with some not-so-obvious mathematical approximations. Since the result to be derived is already known from the constraint of its compliance with Maxwell's equations, and the derivation involves the familiar text-book expressions for the field of a uniformly moving charge, one would expect an easy, simple approach, to lead to the correct result. Here, starting from the electromagnetic field of a uniformly accelerated charge in the instantaneous rest frame, in terms of the position and motion of the charge at the retarded time, we derive this $\delta$-field, consistent with Maxwell's equations, in a fairly simple manner. This is followed by a calculation of the energy in the $\delta$-field, in an analytical manner without making any approximation, where we show that this energy is exactly the one that would be lost by the charge because of the radiation reaction on the charge, proportional to its rate of change of acceleration, that was imposed on it at a distant past.