Vector Spherical Harmonic Expansion in the Time Domain of the Retarded Hertz Vector for a Distributed, Transient Source-Current Configuration

Abstract

An equation giving the retarded Hertz vector II(r, t) in terms of a source-current distribution J(r1, t1) is derived. Both vector functions are written as expansions in vector spherical harmonics, with the expansion coefficients containing the dependence upon radial distance and time, while the angular dependence is kept within the harmonic functions. After integration over two angles, expressions are obtained giving the expansion coefficients for the Hertz vector, which depend upon (r, t), in terms of the corresponding coefficients for the current, which depend upon (r1, t1). The original four-dimensional problem is thus reduced to two dimensions, but with the four-dimensional causality requirements satisfied at each step of the analysis.