The scattering of electromagnetic waves in medium with randomly distributed discrete scatterers is studied. Analytical and numerical solutions to several problems with implications for the active and passive remote sensing of the Earth environment are obtained. The quasi-magnetostatic (QMS) solution for a conducting and permeable spheroid under arbitrary excitation is presented. The spheroid is surrounded by a weakly conducting background medium. The magnetic field inside the spheroid satisfies the vector wave equation, while the magnetic field outside can be expressed as the gradient of the Laplace solution. We solve this problem exactly using the separation of variables method in spheroidal coordinates by expanding the internal field in terms of vector spheroidal wavefunctions. The exact formulation works well for low to moderate frequencies; however, the solution breaks down at high frequency due to numerical difficulty in computing the spheroidal wavefunctions. To circumvent this difficulty, an approximate theory known as the small penetration-depth approximation (SPA) is developed. The SPA relates the internal field in terms of the external field by making use of the fact that at high frequency, the external field can only penetrate slightly into a thin skin layer below the surface of the spheroid. For spheroids with general permeability, the SPA works well at high frequency and complements the exact formulation. However, for high permeability, the SPA is found to give accurate broadband results. By neglecting mutual interactions, the QMS frequency response from a collection of conducting and permeable spheroids is also studied.
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