Electrical capacitance of the unit cube

Abstract

It is well known that there is no analytical expression for the electrical capacitance of a cube, even though it has been claimed that one can compute this capacitance numerically to high precision. However, there have been some disparities between reported numerical results of the capacitance of the unit cube. In this article, the ''walk on planes'' ͑WOP͒ algorithm ͓M. L. Mansfield, J. F. Douglas, and E. J. Garboczi, Phys. Rev. E 64, 061401 ͑2001͔͒ is used to compute the capacitance of the unit cube. With WOP, we remove the error from the ⑀-absorption layer commonly used in ''walk on spheres'' computations so that there is no inherent error introduced in these WOP computations except the intrinsic Monte Carlo sampling error of size O(N 1/2). This WOP technique comes from the isomorphism, provided by probabilistic potential theory, between the electrostatic Dirichlet problem on a conducting surface, and the corresponding Brownian motion first-passage expectation. The numerical result we obtain with WOP, 0.660 678 2Ϯ1ϫ10 Ϫ7 , supports the deterministic calculations by Read ͓F.

The Capacitance of Pristine Ice Crystals and Aggregate Snowflakes

“…For the nonsmooth shapes of realistic ice particles, this sampling approach turns out to be an accurate and efficient method of solving Laplace's equation for C. Also, since we are sampling steady-state diffusion onto a stationary crystal, statistics for C can be built up sampling one random walk trajectory at a time, and this means that very little computer memory is required. It is interesting to note that random walker sampling has recently been applied in the electrostatics community to calculate electrical capacitances for conductors with sharp edges (e.g., Hwang and Mascagni 2004) since it bypasses many of the artifacts introduced by boundary-element, finite-difference, and finiteelement methods that can cause systematic errors (Wintle 2004).…”

Electromagnetic Modeling of Metallic Elliptical Plates

“…For m = n, this expression for the matrix element is not valid. Thus the diagonal elements have to be treated separately [6]. The diagonal elements are calculated by considering each triangular section to be a circle of equal area.…”

“…They did not present any data on the capacitance for the case of two parallel elliptical plates of finite size. Thus it is worthwhile to carry out the analysis of elliptical structures using the method of moments with triangular subsections for the geometry under consideration unlike that reported in [1][2][3][4][5][6].…”

“…The analysis of three-dimensional spherical, paraboloidal and truncated conical surfaces and two-dimensional square, rectangular, circular and annular metallic disks have been examined using the method of moments [1][2][3][4][5][6]. In these works, the capacitance of the different geometrical structures was obtained by subdividing the structure into uniform rectangular planar subsections and computing the effect of the charge on each subsection on the potential of the others.…”

The random walk on the boundary method for calculating capacitance

“…This problem has no analytic solution, and has long been regarded as a benchmark in the electrostatic theory [3]. Different computational methods were used to solve it: boundary element [21,23,4], finite-difference [22], and stochastic algorithms [24,9,7] as well. The results (in units of 4π 0 ) and their published errors (in different senses) are given in Table 1.…”