High-Discretization Method of Moments for Capacitance Calculation: A Cube and a Hollow Cylinder

TL;DR

This work tackles the long-standing problem of computing capacitances for a unit cube and a hollow cylinder, where analytic solutions are unavailable. It advances the method of moments by discretizing surface elements into up to 600×600 sub-areas per cube face and exploiting geometric symmetry and parallel computing to achieve substantial efficiency. The cube exhibits a nonmonotonic convergence, peaking at $C_{ ext{max}} = 73.519014\, \mathrm{pF}$ for a 1 m edge at 90×90 discretization, a behavior attributed to discretization errors and an overestimation of local potentials, as explained via Thomson's theorem. For hollow cylinders, the MOM results agree well with Lekner's theoretical formulations and Cavendish's experiments across various aspect ratios, validating the approach, while demonstrating efficient computation times and scalable performance. Overall, the paper provides a practical, symmetry-exploiting, high-resolution MOM framework for capacitance calculations with clear insights into discretization effects and numerical validation against established benchmarks.

Abstract

This paper employs the method of moments (MOM) to calculate the capacitances of a cube and a hollow cylinder. For the cube, each face was divided into a maximum of 600 x 600 sub-areas. By fully exploiting the geometric symmetry between sub-areas and incorporating parallel computing, computational resources were significantly conserved. Our results show that the calculated capacitance of the cube first increases and then decreases as the number of sub-areas increases. When each face was divided into 90 x 90 sub-areas, the capacitance of the unit cube (with an edge length of 1 m) reached a maximum reference value of 73.519014 pF. This indicates that higher accuracy cannot be achieved merely by indefinitely increasing the number of discretized sub-areas. Subsequently, the method was applied to compute the capacitance of a hollow cylinder. The results were compared with numerical solutions based on Lekner's theoretical formula and Cavendish's experimental values, showing good agreement among the three.

Figures (7)

• Figure 1: Cube with Each Face Divided into $7\times7$ Sub-areas (1 m Side Length).
• Figure 2: Capacitance of a Unit Cube (edge length: 1 m) with Each Face Divided into N$\times$ N Sub-areas (N: X-Axis).
• Figure 3: Surface Charge Density Distribution on a Cube (edge length: 1 m) with Each Face Divided into $30\times30$ Sub-areas.
• Figure 4: The surface of a hollow cylinder is divided into $L$ annular rings of 1 m width, each subdivided into $K$ square sub-areas with 1 m side length.
• Figure 5: The Charge Density Distribution Curve of the Hollow Cylinder with Length $L= 1250$ m and Base Circumference $K = 3927$ m ( Diameter $D = 1250$ m ).