Maxwell's Equations

TL;DR

This paper presents a unified matrix-based formulation of Maxwell's equations, simplifying their structure and revealing dualities, while connecting to stress-energy and Lorentz law, and comparing to algebraic approaches.

Contribution

It introduces a novel matrix divergence approach to express Maxwell's equations as a single, elegant equation, unifying the electromagnetic field and current representations.

Findings

Simplifies Maxwell's equations into a single matrix divergence equation

Derives the wave equation and Lorentz law naturally from the matrix formulation

Establishes connections to biquaternion and Clifford algebra descriptions

Abstract

We express Maxwell's equations as a single equation, first using the divergence of a special type of matrix field to obtain the four current, and then the divergence of a special matrix to obtain the Electromagnetic field. These two equations give rise to a remarkable dual set of equations in which the operators become the matrices and the vectors become the fields. The decoupling of the equations into the wave equation is very simple and natural. The divergence of the stress energy tensor gives the Lorentz Law in a very natural way. We compare this approach to related descriptions of Maxwell's equations by biquaternions and Clifford algebras.