Derivation and physical interpretation of the general solutions to the wave equations for electromagnetic potentials

TL;DR

This paper derives general, singularity-free solutions to wave equations for electromagnetic potentials from retarded source densities, providing a basis for advanced applications in condensed matter physics and imaging.

Contribution

It introduces a novel derivation of wave equation solutions that are free of singularities and applicable to arbitrary charge distributions, without relying on advanced-retarded combinations.

Findings

Solutions are expressed as nested integrals over source distributions.

The approach avoids spatial singularities present in previous models.

Potential applications include condensed matter physics and fluorescence imaging.

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 thus obtained are inherently free of spatial 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 in the context of specific examples involving either the real or reciprocal space forms of the different potentials. The present approach allows for k-space expansions of the potentials for arbitrary distributions of charges and may lead to applications in condensed matter and fluorescence-based imaging.