The influence of surface roughness on the capacitance between a sphere and a plane

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Abstract
The problem of capacitance between a rough sphere and a rough plane very close to each other is studied. The problem is simplified by gathering the roughness of both members onto the plane. We begin by considering the case where the sphere and the plane are both ideally smooth. Starting from a known exact formula of the capacitance involving an infinite sum, we derive a simple and easy to use approximate expression. Then we consider the case where the sphere is smooth and the plane rough. The new formula for the capacitance involves an integral in which the weighting function is the height distribution. To validate this formula, the results of two experiments, during which the derivative of the capacitance with respect to the separation has been recorded, are considered. Comparison of the measurements with theoretical calculations indicates the approach is valid.