Electrostatics of a Finite Conducting Cylinder: Elliptic-Kernel Integral Equation and Capacitance Asymptotics

TL;DR

This paper develops a precise numerical method using elliptic integrals and Chebyshev collocation to analyze the electrostatics and capacitance of finite conducting cylinders across all aspect ratios.

Contribution

It introduces a novel integral equation approach with a specialized numerical scheme for accurate electrostatics of finite cylinders, including intermediate aspect ratios.

Findings

Accurate capacitance values for various aspect ratios.

Rapid convergence of the numerical scheme.

Recovery of known asymptotic limits.

Abstract

We study the electrostatics of a thin, finite-length conducting cylindrical shell held at constant potential V0. Exploiting axial symmetry, we recast the problem as a one-dimensional singular integral equation for the axial surface-charge density, with a kernel written in terms of complete elliptic integrals. A Chebyshev-weighted collocation scheme that incorporates the square-root edge singularity yields rapidly convergent charge profiles and dimensionless capacitances for arbitrary aspect ratios L/a, recovering known long- and short-cylinder limits and providing accurate benchmark values in the intermediate regime. The method offers a compact, numerically robust reference formulation for the electrostatics of finite cylindrical conductors.