Magnetoquasistatic response of conducting and permeable prolate spheroid under axial excitation

TLDR: An analytical solution is presented for the problem of magnetic diffusion into and scattering from a permeable, highly but not perfectly conducting prolate spheroid under axial excitation, expressed in terms of an infinite matrix equation.

Abstract: An analytical solution is presented for the problem of magnetic diffusion into and scattering from a permeable, highly but not perfectly conducting prolate spheroid under axial excitation, expressed in terms of an infinite matrix equation. The spheroid is assumed to be embedded in a homogeneous nonconducting medium as appropriate for low-frequency, high-contrast scattering governed by magnetoquasistatics. The solution is based on separation of variables and matching boundary conditions where the prolate spheroidal wavefunctions with complex wavenumber parameter are expanded in terms of spherical harmonics. For small skin depths, an approximate solution is developed that avoids any reference to the spheroidal wavefunctions. The problem of long spheroids and long circular cylinders is solved by using an infinite cylinder approximation. In some cases, our ability to evaluate the spheroidal wavefunctions breaks down at intermediate frequencies. To deal with this, a general broadband rational function approximation technique is developed and demonstrated. We treat special cases and provide numerical reference data for the induced magnetic dipole moment or, equivalently, the magnetic polarizability factor.