Fractal model of the dipole moments of conducting particle chains

A self-similar fractal model is developed for calculating the dipole moments of linear chains of equally spaced conducting spheres immersed in uniform electric field. We assume that chains may be treated as larger equivalent spheres with same dipole moment and that two such imaginary spheres behave like two real ones. The effective radii of the equivalent spheres are determined from a fractal generating process. In this process, only the knowledge of particle pair is required. The fractal dimension is determined, which is only a function of the spacing of the spheres. The longitudinal and transverse dipole moments are expressed as simple functions of the fractal dimension and the number of the spheres in a chain. The results for touching spheres are in excellent agreement with the known numerical calculations of the image method. Comparison is made with the nearest-neighbor approximation, which is extended to the transverse case here. The fractal model is acceptable.