Electric polarizability and magnetic susceptibility of small holes in a thin screen

It is shown that the fields inside a cavity with a hole can be written in terms of a coupling integral involving the tangential electric field in the plane of the hole. For a hole whose dimensions are small compared to the wavelength, this coupling integral was separated into an electric term proportional to the (scalar) polarizability of the hole and a magnetic term proportional to the (vector) susceptibility of the hole. Integral equations for the determination of these static geometrical parameters are developed. These equations can be solved exactly for holes of circular and elliptical shape, but only approximately for other shapes. The main result is the derivation of what appears to be a new relation between the electric polarizability and the (diagonalized) magnetic susceptibility for a hole of general shape. This relation can be used to test the accuracy of various numerical methods which are used to calculate these parameters