On the inversion problem in classical electrodynamics and the “Casimir theorem”

Abstract

The inversion problem in classical electrodynamics in investigated in great detail in connection with the “Casimir theorem” which states that given all multipoles (both electric and magnetic) of a given charge and current distribution localized in a finite region, the electromagnetic field outside the region will not be sufficient to determine uniquely such a distribution. We wish to determine whether supplementary conditions exist which allow a determination of such distribution. We show that if the system contains only currents and charges (no magnetization) the divergences of the currents will allow such a determination. A similar result holds if the system contains only magnetization (no current and charges). If currents, charges and magnetization are present, then not even the knowledge of the divergences is a sufficient condition for such a determination.