Retarded electric and magnetic fields of a moving charge: Feynman's derivation of Li\'enard-Wiechert potentials revisited

TL;DR

This paper revisits Feynman's derivation of Lie9nard-Wiechert potentials, clarifies differences with conventional potentials, and compares various formulations of retarded fields, highlighting a mathematical error in Jefimenko's derivation.

Contribution

It provides a detailed analysis of retarded electromagnetic potentials, explaining discrepancies and correcting a derivation error in Jefimenko's formulas.

Findings

Retarded potentials differ from conventional ones due to charge density considerations.

Feynman's fields match Lie9nard-Wiechert but not Jefimenko's formulas.

A derivation error in Jefimenko's equations is identified and corrected.

Abstract

Retarded electromagnetic potentials are derived from Maxwell's equations and the Lorenz condition. The difference found between these potentials and the conventional Li\'{e}nard-Wiechert ones is explained by neglect, for the latter, of the motion-dependence of the effective charge density. The corresponding retarded fields of a point-like charge in arbitary motion are compared with those given by the formulae of Heaviside, Feynman, Jefimenko and other authors. The fields of an accelerated charge given by the Feynman are the same as those derived from the Li\'{e}nard-Wiechert potentials but not those given by the Jefimenko formulae. A mathematical error concerning partial space and time derivatives in the derivation of the Jefimenko equations is pointed out.