Broad-band electromagnetic induction (EMI) methods are promising in the detection and discrimination of subsurface metallic targets. In this paper, the quasi-magnetostatic solution for a conducting and permeable prolate spheroid under arbitrary excitation by a time-harmonic primary field is obtained by using the separation of variables method with vector spheroidal wave functions. Numerical results for the induced dipole moments are presented for uniform axial and transverse excitations, where the primary field is oriented along the major and minor axis of the prolate spheroid, respectively. We show that the EMI frequency responses are sensitive to the orientation and permeability of the spheroid. An approximation is also developed that aims to extend the exact solution to higher frequencies by assuming slight penetration of the primary field into the spheroid. Under this approximation, a system of equations that refers only to the external field expansions is derived. It is shown that, for spheroids with high relative permeability, this approximation is in fact capable of yielding an accurate broad-band response even for highly elongated spheroids.

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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, highcontrast 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.