EM Field Induced in Inhomogeneous Dielectric Spheres by External Sources

Abstract-The electromagnetic field induced in the interior of inhomogeneous dielectric bodies by external sources can be evaluated by solving the well-known electric field integrodifferential equation (EFIDE). For spheres with constant magnetic permeability µ, but variable dielectric constant ε(r, θ, ϕ) a direct, mainly analytical solution can be used even in case when the inhomogeneity in ε renders separation of variables inapplicable. This approach constitutes a generalization of the hybrid (analytical-numerical) scalar method developed by the authors in two recent papers, for the corresponding acoustic (scalar) field induced in spheres with variable density and/or compressibility. This extension, by no means trivial, owing to the vector and integrodifferential nature of the equation, is based on field-vector expansions using the set of three harmonic surface vectors, orthogonal and complete over the surface of the sphere, for their angular (θ, ϕ) dependence, and Dini's expansions of a general type for their radial functions. The use of the latter has been shown to be superior to other possible sets of orthogonal expansions and as far as its convergence is concerned it may further be improved by properly choosing a crucial parameter in their eigenvalue equation. The restriction to the spherical shape is imposed here to allow use of the well-known expansion of Green's dyadic in spherical eigenvectors of the vector wave equation

EM Field Induced in Inhomogeneous Dielectric Spheres by External Sources

Abstract-The electromagnetic field induced in the interior of inhomogeneous dielectric bodies by external sources can be evaluated by solving the well-known electric field integrodifferential equation (EFIDE). For spheres with constant magnetic permeability µ, but variable dielectric constant ε(r, θ, ϕ) a direct, mainly analytical solution can be used even in case when the inhomogeneity in ε renders separation of variables inapplicable. This approach constitutes a generalization of the hybrid (analytical-numerical) scalar method developed by the authors in two recent papers, for the corresponding acoustic (scalar) field induced in spheres with variable density and/or compressibility. This extension, by no means trivial, owing to the vector and integrodifferential nature of the equation, is based on field-vector expansions using the set of three harmonic surface vectors, orthogonal and complete over the surface of the sphere, for their angular (θ, ϕ) dependence, and Dini's expansions of a general type for their radial functions. The use of the latter has been shown to be superior to other possible sets of orthogonal expansions and as far as its convergence is concerned it may further be improved by properly choosing a crucial parameter in their eigenvalue equation. The restriction to the spherical shape is imposed here to allow use of the well-known expansion of Green's dyadic in spherical eigenvectors of the vector wave equation

Mellin-transform method for integral evaluation

This book introduces the Mellin-transform method for the exact calculation of one-dimensional definite integrals, and illustrates the application if this method to electromagnetics problems. Once the basics have been mastered, one quickly realizes that the method is extremely powerful, often yielding closed-form expressions very difficult to come up with other methods or to deduce from the usual tables of integrals. Yet, as opposed to other methods, the present method is very straightforward to apply; it usually requires laborious calculations, but little ingenuity. Two functions, the general