Can Maxwell's equations be obtained from the continuity equation?

TL;DR

This paper proves that Maxwell's equations can be derived from the continuity equation, highlighting charge conservation as the fundamental principle behind electromagnetic field equations.

Contribution

It presents an existence theorem linking localized sources satisfying the continuity equation to Maxwell's equations, emphasizing charge conservation as fundamental.

Findings

Maxwell's equations can be derived from the continuity equation.

Retarded fields satisfying Maxwell's equations exist given sources obeying charge conservation.

Charge conservation underpins the structure of electromagnetic field equations.

Abstract

We formulate an existence theorem that states that given localized scalar and vector time-dependent sources satisfying the continuity equation, there exist two retarded fields that satisfy a set of four field equations. If the theorem is applied to the usual electromagnetic charge and current densities, the retarded fields are identified with the electric and magnetic fields and the associated field equations with Maxwell's equations. This application of the theorem suggests that charge conservation can be considered to be the fundamental assumption underlying Maxwell's equations.