Electrostatic Patch Effect in Cylindrical Geometry. I. Potential and Energy between Slightly Non-Coaxial Cylinders

TL;DR

This paper analyzes how uneven voltage distributions, or patch effects, influence the potential and energy between slightly non-coaxial cylindrical conductors, with implications for precision measurements like STEP.

Contribution

It provides a detailed solution for the potential distribution and energy in the gap, considering small transverse shifts and patch effects, extending understanding of electrostatic interactions in cylindrical geometries.

Findings

Energy includes capacitor, patch interaction, and uniform voltage terms.

Patch effects induce forces and torques analyzed in subsequent work.

Potential and energy are calculated to second order in small shifts.

Abstract

We study the effect of any uneven voltage distribution on two close cylindrical conductors with parallel axes that are slightly shifted in the radial and by any length in the axial direction. The investigation is especially motivated by certain precision measurements, such as the Satellite Test of the Equivalence Principle (STEP). By energy conservation, the force can be found as the energy gradient in the vector of the shift, which requires determining potential distribution and energy in the gap. The boundary value problem for the potential is solved, and energy is thus found to the second order in the small transverse shift, and to lowest order in the gap to cylinder radius ratio. The energy consists of three parts: the usual capacitor part due to the uniform potential difference, the one coming from the interaction between the voltage patches and the uniform voltage difference, and the energy of patch interaction, entirely independent of the uniform voltage. Patch effect forces and torques in the cylindrical configuration are derived and analyzed in the next two parts of this work.