Completing the Li\'enard-Wiechert potentials: The origin of the delta function fields for a charged particle in hyperbolic motion

TL;DR

This paper clarifies the origin of delta function fields in the electromagnetic potentials of a uniformly accelerated charged particle by revisiting and correcting the derivation of Lie9nard-Wiechert potentials specifically for hyperbolic motion.

Contribution

It provides a new derivation that identifies and corrects missing terms in the standard Lie9nard-Wiechert potentials for hyperbolic motion.

Findings

Derived the complete electromagnetic field for hyperbolic motion.

Identified the source of delta function fields in the potentials.

Amended the standard Lie9nard-Wiechert derivation.

Abstract

Calculating the electromagnetic field of a uniformly accelerated charged particle is a surprisingly subtle problem that has been long discussed in the literature. While the correct field has been obtained many times and through various means, it remains somewhat unclear why the (supposedly general) field expression derived from the Li\'enard-Wiechert potentials misses some terms. We present new derivations of the missing field and potential terms by considering only pure hyperbolic motion, and we amend the offending step in the standard Li\'enard-Wiechert construction.