Electrostatic boundary problems and T-matrix for the dielectric half-spheroid

We solve the electrostatic boundary problems of a dielectric or conducting hemispheroid (half-spheroid) under arbitrary excitation. The solutions are obtained by expanding the potentials as series of spheroidal harmonics, and integrating over the boundary to obtain matrix equations which can be used to solve for the coefficients. The solutions are used to derive the capacity, polarizability and spatial fields. We simplify the results to that for a hemisphere, which for specific excitation fields agrees with the literature. We make a link to the T-matrix method, and present analytic expressions for the T-matrix and auxiliary Q and P matrices in the electrostatic limit. We show that the standard T-matrix approach of the extended boundary condition method (EBCM) cannot be used for this geometry, and that the P and Q-matrices do not match the EBCM form.