Derivation and Physical Interpretation of the General Solutions to the Wave Equations for Electromagnetic Potentials

Abstract

The inhomogeneous wave equations for the scalar, vector, and Hertz potentials are derived starting from retarded charge, current, and polarization densities and then solved in the reciprocal (or k-) space to obtain general solutions, which are formulated as nested integrals of such densities over the source volume, k-space, and time. The solutions provide real-space forms of the potentials and fields that are inherently free of singularities and do not require introduction by fiat of combinations of advanced and retarded terms as done previously to cure such singularities for the point-charge model. Physical implications of these general solutions are discussed through specific examples involving either the real or reciprocal space forms of the different potentials. This approach allows for real- and reciprocal-space expansions of potentials and fields for arbitrary distributions of charges and may lead to applications in condensed matter research and fluorescence-based imaging.