THE ELECTRICAL FORCES ON TWO TOUCHING SPHERES IN A UNIFORM FIELD

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Abstract
The boundary-value problem describing the electrostatics of two touching spherical conductors in an external electric field is solved and expressions derived for the capacitance, induced charge, and force on each sphere. Whereas uncharged separated spheres are known to attract one another, the force between uncharged touching spheres turns out to be repulsive. Furthermore, when charge is added the force is shown to be attractive or repulsive depending on the amount of charge, the relative sizes of the spheres, and the orientation of the field. Most of the differences between the behavior of touching and separated spheres are due to polarization of the spheres by the external field. Moreover, for a small particle resting on the surface of a much larger sphere, the repulsive force can be comparable in magnitude to the attractive forces of molecular origin. Extensive tabulations, covering a wide range of particle sizes enable the force or other relevant quantities to be calculated with ease. Several double series were summed in the course of the analysis and, in some instances, these series were non-convergent, a difficulty studied long ago by Bromwich and Hardy (6). Different summation methods gave different sums and to resolve the ambiguities reciprocal relations were derived based on the invariance properties of the physically acceptable solution.

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