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

Abstract

An extension of the Helmholtz theorem is proved, which states that two retarded vector fields ${\mathbf{F}}_1$ and ${\mathbf{F}}_2$ satisfying appropriate initial and boundary conditions are uniquely determined by specifying their divergences $\nabla \cdot {\mathbf{F}}_1$ and $\nabla \cdot {\mathbf{F}}_2$ and their coupled curls $-\nabla \times {\mathbf{F}}_1-\partial {\mathbf{F}}_2/\partial t$ and $\nabla \times {\mathbf{F}}_2-(1/c^2)\partial {\mathbf{F}}_1/\partial t$, 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.

References in corpus (2)

• Can Maxwell's equations be obtained from the continuity equation?
• Comment on 'A generalized Helmholtz theorem for time-varying vector fields by A. M. Davis, [Am. J. Phys. 74, 72-76 (2006)]'