Capacitance and charge density of an open conducting cylinder: Elliptic-kernel integral equation and asymptotics

Abstract

We study the electrostatics of a thin, finite-length conducting cylindrical shell held at constant potential $V_0$. 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 $a/L$, 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.