  Patch potentials arising from the polycrystalline structure of material
samples may contribute significantly to measured signals in Casimir force
experiments. Most of these experiments are performed in the sphere-plane
geometry, yet, up to now all analysis of patch effects has been taken into
account using the proximity force approximation which, in essence, treats the
sphere as a plane. In this paper we present the exact solution for the
electrostatic patch interaction energy in the sphere- plane geometry, and
derive exact analytical formulas for the electrostatic patch force and
minimizing potential. We perform numerical simulations to analyze the distance
dependence of the minimizing potential as a function of patch size, and
quantify the sphere-plane patch force for a particular patch layout. Once the
patch potentials on both surfaces are measured by dedicated experiments our
formulas can be used to exactly quantify the sphere-plane patch force in the
particular experimental situation.
