Equivalent antenna radius for narrow slot apertures having depth

The problem of electromagnetic penetration of a narrow slot aperture in a thick perfectly conducting plane is reduced to the solution of Hallen´s integral equation with an equivalent antenna radius. The depth and width of the slot are assumed to be small compared to both the length of the slot and the wavelength. The equivalent radius is evaluated in terms of the solution to the transverse static problem. Simple approximations for the equivalent radius is also given. Hallen´s integral equation is solved by the Galerkin method with piecewise sinusoidal basis functions. Large slot depth-to-width ratios give rise to vanishingly small equivalent radii and thus large antenna resonance quality factors. A simple correction to the static field distribution in the slot is also given. This correction allows the depth to become somewhat larger relative to the wavelength