Energy conservation for a radiating charge in classical electrodynamics

TL;DR

This paper clarifies the discrepancy in classical electrodynamics between radiated power and radiation reaction power by distinguishing between real time and retarded time, leading to a modified expression for power loss.

Contribution

It introduces a proper distinction between real time and retarded time in calculating power loss, resolving a longstanding disparity in classical electrodynamics.

Findings

Real-time radiative power loss differs from Larmor's formula.

The disparity is resolved by distinguishing real time from retarded time.

Modified power loss expression aligns with physical observations.

Abstract

It is shown that the well-known disparity in classical electrodynamics between the power radiated in electromagnetic fields and the power-loss, as calculated from the radiation reaction on a charge undergoing a non-uniform motion, is successfully resolved when a proper distinction is made between quantities expressed in terms of a "real time" and those expressed in terms of a retarded time. It is shown that the expression for the real-time radiative power loss from a charged particle is somewhat different from the familiar Larmor's formula, or in a relativistic case, from Li\'{e}nard's formula.