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

摘要

An extension of the Helmholtz theorem is proved, which states that two retarded vector fields F-1 and F-2 satisfying appropriate initial and boundary conditions are uniquely determined by specifying their divergences del.F-1 and del.F-2 and their coupled curls -del x F-1 - partial derivative F-2/partial derivative t and del x F-2 - (1/c(2))partial derivative F-1/partial derivative 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.