GENERAL THEORY OF ELECTRICAL IMAGES IN SPHERE PAIRS

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Abstract
The classical method of images, used in solving electrostatics problems involving point charges near spherical and planar conductors, is extended to arbitrary charge distributions and fields, and more importantly to dielectric spheres. The charge distributions and fields are represented by differential operators and ‘imaging’ rules are found for these operators. Analytical results are obtained using an approximation of this method which becomes more accurate as the dielectric constant of the sphere increases. Exact results for conducting spheres are obtained as a limiting case. For the dielectric case exact results can be obtained numerically by iteration. An alternative formulation using continuous image-charge distributions and functional equations is also presented. The above methods are used to obtain explicit expressions for all the induced electrostatic multipole moments of a pair of unequal dielectric spheres in an arbitrary external field. Uniform asymptotic approximations are obtained which are valid for both large dielectric constants and small separations.

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