The Dipole Moments of Chains of Equally Spaced Dielectric Spheres

A self-similar fractal model is presented for obtaining the dipole moments of chains of equally spaced dielectric spheres immersed in a uniform electric field. The sphere pair is selected as the generator. The dipole moments are calculated iteratively in terms of those of this generator. At each stage, the pair is replaced by an equivalent sphere with the same dipole moment which is used to construct a new larger pair for the next stage. The effective radii of the equivalent spheres, used to determine the dipole moments of the chains, are obtained from a fractal generating process, and the effect of multipole interactions of the spheres is accounted for in this way. Approximate fractal characters of the dipole moments are found and simple expressions for the dipole moments are given.