Electrostatic Patch Effect in Cylindrical Geometry. III. Torques

TL;DR

This paper investigates electrostatic torques caused by uneven voltage patches on cylindrical conductors, deriving formulas for axial and slanting torques, analyzing patch interactions, and applying findings to the STEP mission conditions.

Contribution

It extends previous work by explicitly calculating electrostatic torques in cylindrical geometries, including patch interactions, using energy and surface integration methods.

Findings

Derived formulas for axial and slanting torques in cylindrical conductors.

Analyzed the dependence of torques on patch parameters and positions.

Applied results to the experimental conditions of the STEP mission.

Abstract

We continue to study the effect of uneven voltage distribution on two close cylindrical conductors with parallel axes started in our papers [1] and [2], now to find the electrostatic torques. We calculate the electrostatic potential and energy to lowest order in the gap to cylinder radius ratio for an arbitrary relative rotation of the cylinders about their symmetry axis. By energy conservation, the axial torque, independent of the uniform voltage difference, is found as a derivative of the energy in the rotation angle. We also derive both the axial and slanting torques by the surface integration method: the torque vector is the integral over the cylinder surface of the cross product of the electrostatic force on a surface element and its position vector. The slanting torque consists of two parts: one coming from the interaction between the patch and the uniform voltages, and the other due to the patch interaction. General properties of the torques are described. A convenient model of a localized patch suggested in [2] is used to calculate the torques explicitly in terms of elementary functions. Based on this, we analyze in detail patch interaction for one pair of patches, namely, the torque dependence on the patch parameters (width and strength) and their mutual positions. The effect of the axial torque is then studied for the experimental conditions of the STEP mission.