A Vector Wave Identity and Its Application to the Electromagnetic Field

Abstract

In this paper, we derive a new and very useful vector identity based on the vector wave operator. It relates certain retarded volume and surface integrals of a vector function which, along with its first and second partial derivatives, must be continuous within and at a closed surface, but which may otherwise be arbitrary. This identity applied to the inhomogeneous vector wave equations (obtained from Maxwell's equations by elimination of E and of H in the usual way) yields immediately a formulation of Huygens' principle for the fields E and H. Although, these formulas for E and H are solutions of their respective vector wave equations, it is shown that they are not necessarily a solution of Maxwell's equations. We prove a theorem giving the necessary and sufficient conditions that they be a solution of those equations. Further tranformations on these formulae by use of Maxwell's equations on the surface yield the formulations of Larmor, Tedone, Baker and Copson, and Jones. It is also shown that these latter formulations are not necessarily solutions of either the vector wave equations or of Maxwell's equations, but they will be, if they are subject to the conditions of the above mentioned theorem.