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

TL;DR

This work addresses the axisymmetric electrostatics of a thin finite-length conducting cylinder held at a uniform potential by deriving a one-dimensional integral equation with an elliptic-kernel for the axial surface-charge density. A Chebyshev-weighted collocation scheme explicitly factors out the rim singularity, yielding stable, rapidly convergent solutions for $\sigma(z)$ and the dimensionless capacitance $\widetilde{C}(\alpha)=C/(2\pi\varepsilon_{0}a)$ across a wide range of aspect ratios $\alpha=a/L$. The paper provides detailed asymptotic analyses for long and short cylinders, delivering explicit formulas and benchmark values, and demonstrates an exact equivalence between the real-space elliptic-kernel formulation and a spectral Hankel–Bessel representation via dual integral equations. The resulting high-accuracy results serve as a reproducible reference for finite cylindrical conductors and offer a robust tool for validating numerical solvers and informing electrode-design applications, with potential extensions to nonuniform boundary conditions and multi-conductor geometries.

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.

Figures (4)

• Figure 1: Schematic representation of a thin conducting cylindrical shell of radius $a$ and finite length $L$, aligned with the $z$-axis and held at a uniform electrostatic potential $V_0$. The cylinder occupies the region $-L/2<z<L/2$ and has zero thickness. Owing to axial symmetry and the absence of external fields, the induced surface-charge density is independent of the azimuthal angle and depends only on the axial coordinate, $\sigma=\sigma(z)$, on the surface $\rho=a$.
• Figure 2: Dimensionless surface-charge density on the lateral wall of an open conducting cylinder held at a uniform potential $V_0$. The curves show $\tilde{\sigma}(z)=a\,\sigma(z)/(\varepsilon_0 V_0)$ as a function of the normalized axial coordinate $z/L$, for four aspect ratios $\alpha=1/3,\,1,\,2,$ and $5$ (different line styles). The vertical scale is restricted to emphasize the interior region; the rapid rise near $z/L=\pm 1/2$ reflects the edge (rim) charge crowding at the open ends.
• Figure 3: Finite-size scaling of the numerically computed cylinder-center surface-charge density. The dimensionless center value $\tilde{\sigma}(0)$ for $\alpha=1$ is plotted versus $1/N_c$, where $N_c$ is the number of Chebyshev panels used in the axial discretization of the integral equation, with within-panel quadrature fixed at $n_g=16$ Gauss--Chebyshev points. In the asymptotic regime ($N_c\ge 220$) the data fall on an approximately straight line, consistent with a leading discretization error $\tilde{\sigma}(0,N_c)=\tilde{\sigma}_\infty(0)+A/N_c+\mathcal{O}(N_c^{-2})$. The dashed line is the corresponding linear least-squares fit, whose intercept provides the $N_c\to\infty$ extrapolation $\tilde{\sigma}_\infty(0)$.
• Figure 4: Dimensionless total capacitance of an open conducting cylinder (lateral surface only) held at a uniform potential $V_0$, shown as a function of the aspect parameter $\alpha=a/L$. The plotted quantity is $C(\alpha)$ obtained from the numerically computed surface-charge distribution, Eq. (\ref{['c13']}). The vertical range is restricted to $0\le C(\alpha)\le 2.0$ to emphasize the rapid growth as $\alpha\to 0$ (long-cylinder limit) while preserving detail at moderate $\alpha$. Inset: $1/C(\alpha)$ versus $\ln\alpha$, emphasizing the slow logarithmic trend in the short-cylinder regime $\alpha\gg 1$, see Eq. (\ref{['c13n']}).