Three-dimensional approximation of the total force on uncharged spheres in electric fields

Droplets on outdoor high-voltage equipment suffer a total force which is non-vanishing in general. We consider a model problem of an uncharged and conductive sphere. We show that the total force can be given as a series of inhomogeneity indicators of the undisturbed electric field in the absence of the droplet. The proof of the series involves several aspects of the spherical harmonics in the three-dimensional Fourier technique. The series expansion establishes a relation between the solutions of two Poisson's equations on different domains. It is found that the expansion converges fast. The results are applied for droplets on a realistically shaped insulator