On the limit for β→0 of the Toraldo di Francia expansion

Abstract

In a very interesting paper Toraldo di Francia has shown that the fields generated by a charged particle moving uniformly in vacuum can be exactly expanded in a set of transversal electric (TE) and transversal magnetic (TM) evanescent waves. This expansion is valid in a whole half-space having no points in common with the line of motion of the particle, and the evanescent waves are all attenuated in the direction perpendicular to the boundary of the half-space. Since this expansion is exact, we have taken the limit when the particle's velocity goes to zero, and what we have obtained is a representation of the Coulomb field in TM evanescent waves of zero frequency. This expansion enables us to analize boundary-value problems in electrostatic in a novel way, by studying the behaviour of each wave of the series independently. As an example we solve the classical problem of a point charge in vacuum near a semi-infinite dielectric medium.