Exterior electromagnetic boundary value problems for spheres and cones

The problem of determining a harmonic time-varying electromagnetic field where the electric vector assumes prescribed values for its tangential components over given spherical or conical boundaries and which has proper radiation characteristics at infinity is considered by a procedure very much like that used in the theory of slots in waveguide walls. The technique used in solving this type of boundary value problem is to establish, by an application of the Lorentz Reciprocity Theorem, a Green's function which represents the electric and magnetic fields of a point generator (infinitesimal dipole) applied at an arbitrary position on the conducting surface where the fields satisfy homogeneous boundary conditions. The total fields for an arbitrary source are then obtained by superposition; i.e., direct integration over the aperture. Since detailed results for the case of a sphere have been obtained by many authors, we confine the details of the technique to the infinite cone. It is assumed that in each case the tangential components of the electric vector are given functions over the entire boundary surface. The results apply directly to the theory of radiating apertures in a perfectly conducting spherical wall or a cone, since the tangential components of the electric vector are different from zero only in the area of the aperture, where it is presumed they are known. The results are also applicable to scattering by conducting spheres and cones, since the tangential electric field components over the boundary surfaces are the negative of those of the incident field. To illustrate the applicability and the limitations of the results, we shall present the formal solutions for arbitrarily shaped apertures on cones and apply them to the several types of delta slots which are usually discussed in connection with other radiating structures.