An axiomatic approach to Maxwell's equations

TL;DR

This paper introduces an axiomatic framework for deriving Maxwell's equations, including cases with magnetic monopoles and media, emphasizing the fundamental roles of causality and charge conservation.

Contribution

It presents a novel axiomatic derivation of Maxwell's equations using a theorem based on localized functions and continuity equations, extending to media and magnetic monopoles.

Findings

Derived Maxwell's equations with magnetic monopoles from axioms.

Extended the axiomatic approach to material media.

Formulated covariant Maxwell's equations in Minkowski space-time.

Abstract

This paper suggests an axiomatic approach to Maxwell's equations. The basis of this approach is a theorem formulated for two sets of functions localized in space and time. If each set satisfies a continuity equation then the theorem provides an integral representation for each function. A corollary of this theorem yields Maxwell's equations with magnetic monopoles. It is pointed out that the causality principle and the conservation of electric and magnetic charges are the most fundamental physical axioms underlying these equations. Another application of the corollary yields Maxwell's equations in material media. The theorem is also formulated in the Minkowski space-time and applied to obtain the covariant form of Maxwell's equations with magnetic monopoles and the covariant form of Maxwell's equations in material media. The approach makes use of the infinite-space Green function of the wave equation and is therefore suitable for an advanced course in electrodynamics.