Small Angle Approximation for Non-parallel Plate Capacitors with Applications in Experimental Gravitation

Abstract

An approximate analytical formula for the capacitance of a non-parallel plate capacitor with small values of inclination angles and distance separations of the plates is presented. Most applications involving position sensing that are employed in precision experiments and micromechanical devices often use a parallel plate approximation for modeling the capacitance. The analytical approximation presented here is based on the more general, but hard to implement formalism for small angles, and that may be used directly to compute the forces acting on the proof masses by taking the derivative of the capacitance. Effects of varying the inclination angle, plate dimension and separation on the value of the capacitance per unit length are analyzed in detail. We use the formula derived in this paper to compute the acceleration of the proof mass due to the presence of sensing electrodes (used for positioning the proof masses) with an assumed tilt of $\sim 10^{-6}$ rad as applicable to SR-POEM: an experiment aiming to test the weak equivalence principle - a fundamental postulate of general relativity.