Enlarged Lorentz–Dirac equations

Abstract

Dirac's approach to incorporate the radiation into the equation of motion for a point charge in classical electrodynamics is based on three structural components: the point model for the electron, the Maxwell equations and the principle of relativity. These fundamental components lead to an equation of motion that involves an undetermined 4-vector Bµ. The Lorentz–Dirac equation corresponds to the case in which Bµ = 0, but in general there is a large family of 4-vectors Bµ consistent with the above three basic components. This paper deals with the study of these equations of motion in the case of the three simplest permissible choices for Bµ. We show that these equations admit as exact solutions the motion of an arbitrary number of identical charges that are equally spaced in a circumference and that rotate at constant angular velocity. These solutions show that the rate of radiation emitted by the system of charges is completely independent of the 4-vector Bµ. We also study the restrictions over the dimensionless parameters that appear in the four-vectors Bµ, in order that the trajectories of the corresponding equations cannot be discriminated from the trajectory determined by the Lorentz–Dirac equation in a practical case, as for instance the design and operation of a synchrotron.