Maxwellian Fields and Geometry, Field Behaviour Near Region Boundaries and Sharp Discontinuities – a Review

Abstract

The solution of Maxwell’s equation is normally determined subject to boundary conditions, and conducting boundaries have the most influence on the field behaviors, especially near sharp discontinuities, where the transverse field components become singular. It is shown that if the region outside the boundary with curvature discontinuities is mapped onto a region with a smooth boundary the reciprocal of the metric coefficient of the transformation describes exactly the singular behavior of the field components near the surface discontinuities. This is shown first mathematically for a conducting wedge. Then, the problem of a TM scattering by the wedge and a conducting prism is investigated, and shown again the same relationship between the surface currents and the reciprocal of the transformation metric coefficients. This property is used to develop a new integral equation by eliminating the singularity of the surface current in the integral equation. This study also showed that the product of the prism surface current and the metric coefficient of the transformation behaved nearly the same as the surface current on a corresponding circular cylinder, an interesting geometrical relationship.