Interaction between two charged dielectric balls with strongly different radii

The interaction between two charged dielectric balls that have significantly different radii and whose surfaces are spaced by a small distance is considered. Analytical solutions are found for the first time for the expansion coefficients of the potential for high multipole moments. Exact analytical expressions for the interaction strength are obtained in the method of expansion of the potential in the bispherical coordinate system in the case where the dielectric constant of one of the balls coincides with the dielectric constant of the medium where balls are located. The passage to the infinite radius of the ball with the dielectric constant different from the dielectric constant of the medium is performed. It is shown that these solutions coincide with known solutions of problems of the interaction of the point charge with the dielectric ball and with the charged plane interface between dielectrics. The passage to the infinite radius of one of the balls is performed in the case where the dielectric constants of both balls differ from the dielectric constant of the medium. An analytical solution is found for the first time for the problem of the interaction of the charged dielectric ball with the charged plane interface between homogeneous dielectrics.