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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 but variable dielectric constant , a direct, mainly analytical solution can be used even in case when the inhomogeneity in 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 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.