Electrostatics of a finite-thickness conducting cylindrical shell: coupled elliptic-kernel integral equations

TL;DR

This paper presents an exact electrostatic formulation for finite-thickness conducting cylindrical shells, combining numerical solutions with asymptotic analysis to understand the effects of geometry, thickness, and dielectric contrast.

Contribution

It introduces a coupled integral equation approach for finite-thickness shells, providing high-accuracy solutions and insights into classical limiting regimes and charge distribution behaviors.

Findings

Finite thickness regularizes the short-cylinder behavior, preventing singularities.

The coupled equations explain charge redistribution between inner and outer surfaces.

Results serve as benchmarks for validating electrostatic solvers.

Abstract

We develop an exact electrostatic formulation for a finite-length conducting cylindrical shell of finite thickness separating two dielectric media with arbitrary permittivity contrast. The boundary-value problem is reduced to a coupled system of singular integral equations with elliptic kernels governing the induced surface-charge densities on the inner and outer faces. High-accuracy numerical solutions are combined with a systematic asymptotic analysis that elucidates the interplay between geometry, thickness, and dielectric contrast. All classical limiting regimes are recovered, including the slender-body limit, the short-cylinder (ring-like) asymptote, and the thick-shell regime dominated by the outer surface. We demonstrate that the logarithmic short-cylinder behavior of zero-thickness models is a singular feature, which is regularized for any finite thickness, giving rise instead to a finite capacitance plateau. The asymptotic structure of the coupled equations explains both the electrostatic decoupling of the inner cavity in the thick-shell limit and the redistribution of charge between the two surfaces. The results provide exact benchmarks for finite cylindrical conductors, bridging classical analytical treatments and modern numerical approaches, and furnish a high-accuracy reference solution for the validation of axisymmetric electrostatic solvers.