The Electromagnetic Field of Constant-Density Distributions in Finite Regions

Abstract

The analytical and continuity properties of the electromagnetic field both inside and outside finite regions with constant current-density distributions are examined in detail. Exact results in closed form are obtained for spherical regions and combinations thereof. The approach here is based on a direct evaluation of the vector potential Ā and the electromagnetic fields E, H, which are found to be continuous and analytic vector functions everywhere, experiencing only a finite step discontinuity in the normal E -field component and in certain second derivatives of Ā at the boundary of the constant-density source region, as required by the inhomogeneous partial differential equations. These general results are further extended to finite regions with arbitrary shape; also to infinitesimal regions, establishing the fact that the electromagnetic field and the vector potential never become singular. The implications of these field properties in the solution of integral equations are discussed in detail. Finally, certain related remarks for regions with varying current density are also included.