Dissipative Potentials and the Motion of a Classical Charge

The equation of motion of a charge is obtained as a second-order integro-differential equation, directly from Maxwell's field equations and Lorentz's force equation without the use of the Dirac equation of motion. The validity of the field equations is assumed everywhere, the position of the charge being treated as a point singularity. The force on the charge at the field point is given by the expression for the Lorentz force produced by the fields of a source charge in the limiting case where the field charge is identified with the source. The only fields which need to be considered are the retarded solutions of the field equations, in agreement with causality. In order to obtain the equation of motion, it is necessary to formulate the potentials of the field of a point source in dissipative form. In this form, the potentials satisfy the Lorentz gauge condition identically. They are Fourier integrals which contain ${\delta }_{\pm }$ functions in the Fourier coefficients. When the equation of motion is applied to the examples of relativistic motion of a free charge and to the nonrelativistic simple harmonic motion of a charge, it is found that the principal-part terms of the ${\delta }_{\pm }$ functions provide the electromagnetic kinetic reaction, while the $\delta$ function terms provide the dissipative effects. The only divergence stems from the principal-part integrals, and is the Coulomb self-energy of the charge. It can be removed by renormalization of mass. The results are consistent with the Abraham theory that the mass of an electron (and positron) is wholly electromagnetic.