On the force of electrostatic interaction between two conducting spheres

A compact solution is obtained to the problem on the force of interaction between two conducting spheres with preset charges on their surfaces in zero external field. The derivation is based on exact solution of the problem of the potential distribution in the bispherical coordinate system. The expression for the force was derived by differentiating the potential energy of interaction between the spheres with respect to the distance between their centers. It is shown using numerical calculations that with decreasing distance between the spheres, the ratio of their charges for which the forces of interaction between the charges are zero tends to the ratio of the charges of contacting spheres. It follows hence that for any ratio of charges of the same polarity, which differs from the ratio of charges of the contacting spheres, there always exists a small distance between the spheres, at which they attract each other.