Helmholtz's theorem for two retarded fields and its application to Maxwell's equations

TL;DR

This paper extends Helmholtz's theorem to two retarded fields, providing a unique determination method that directly yields Maxwell's retarded electric and magnetic fields from their divergences and coupled curls.

Contribution

It introduces a new extension of Helmholtz's theorem for retarded fields, enabling direct derivation of Maxwell's fields from specified divergence and curl conditions.

Findings

The extended theorem guarantees unique solutions for two retarded vector fields.

Application to Maxwell's equations yields explicit formulas for electric and magnetic fields.

The proof includes a novel demonstration of the uniqueness of solutions to the vector wave equation.

Abstract

An extension of the Helmholtz theorem is proved, which states that two retarded vector fields ${\bf F}_1$ and ${\bf F}_2$ satisfying appropriate initial and boundary conditions are uniquely determined by specifying their divergences $\nabla\cdot{\bf F}_{1}$ and $\nabla\cdot{\bf F}_{2}$ and their coupled curls $-\nabla\times{\bf F}_{1}-\partial {\bf F}_{2}/\partial t$ and $\nabla\times{\bf F}_{2}-(1/c^2)\partial {\bf F}_{1}/\partial t$, where $c$ is the propagation speed of the fields. When a corollary of this theorem is applied to Maxwell's equations, the retarded electric and magnetic fields are directly obtained. The proof of the theorem relies on a novel demonstration of the uniqueness of the solutions of the vector wave equation.