A New Type of Expansion in Radiation Problems

A new type of expansion of $\frac{\mathrm{i}(1){{e}^{\mathrm{ikr}}}_{12}}{{r}_{12}}$ is developed. Here i is a vector function of the spherical coordinates denoted by 1 and ${r}_{12}$ is the distance between two points denoted by 1 and 2. This expansion is used in the solution of Maxwell's equations and a simple general expression is found for the energy radiated from a known current distribution. A brief application to Dirac's theory of radiation is given. An expansion for $\frac{\mathrm{i}(1)}{{r}_{12}}$ is developed which can be used to find the vector potential due to a steady current distribution.