The pedagogical value of the four-dimensional picture III: Solutions to Maxwell's equations

TL;DR

This paper presents a systematic approach to solving Maxwell's equations using vector potentials, revealing fundamental solutions that encompass all classical electromagnetic configurations.

Contribution

It introduces a regular method for solving Maxwell's equations based on vector potentials and identifies two fundamental solutions that generate all classical electromagnetic fields.

Findings

Existence of two fundamental solutions to Maxwell's equations.

Linear combinations of these solutions cover all classical electromagnetic configurations.

Vector potentials provide a consistent framework for solving Maxwell's equations.

Abstract

We outline a regular way for solving Maxwell's equations. We take, as the starting point, the notion of vector potentials. The rationale for introducing this notion in electrodynamics is that the set of Maxwell's equations is seemingly overdetermined. We demonstrate the existence of two fundamental solutions to Maxwell's equations whose linear combinations comprise the whole variety of classical electromagnetic field configurations.