Penetration of fields through a circular hole in a wall of finite thickness

Consideration is given to a uniform, static electric field on one side of a plane metallic wall of finite thickness with a circular hole. The field induces a charge distribution on the metallic surface which behaves, at large distances from the hole, as a dipole moment, with different values for the polarizability on the inside (same side as the driving field) and outside surfaces of the hole. Two integral equations are derived for the potential in the hole and variational forms are constructed for the symmetric and asymmetric polarizabilities. Trial functions with adjustable parameters lead to accurate numerical values for the inside and outside polarizabilities. A similar approach yields corresponding values for the inside and outside magnetic susceptibilities