General expressions for vector and scalar potentials

Abstract

This paper derives general expressions for the vector and scalar potentials in a bounded region. The necessary conditions for a vector function to be, respectively, the curl of another vector function and the gradient of a scalar function are first presented. Then with the help of Helmholtz's theorem, general expressions for the vector and scalar potentials are derived. The results are summarized in two theorems. Three corollaries are presented. The first one indicates that a vector function in a simply bounded space is the curl of another vector function, if and only if, its divergence is equal to zero. The second one indicates that a vector function in a simply connected space is the gradient of a scalar function, if and only if, its curl is equal to zero. The last one indicates that the magnetic induction is the curl of a vector potential. The classical textbook by Jeans (1915) presents the same conclusion about the magnetic induction using an erroneous discussion. The point where the discussion breaks down is indicated. Applications to antenna theory as well as four simple examples are shown to facilitate the understanding of the present discussion.