Concise presentation of the Coulomb electrostatic potential of a uniformly charged cube

Abstract

a b s t r a c tWe use a novel method to calculate in closed form the Coulomb electrostatic potential created by a uniformly charged cube at an arbitrary point in space. We apply a suitable transformation of variables that allows us to obtain a simple presentation of the electrostatic potential in one-dimensional integral form. The final concise closed form expression of the Coulomb electrostatic potential of the uniformly charged cube is obtained after completing the calculation of the resulting one-dimensional integrals. Such integrals consist of combinations of products of error functions and power functions that can be solved exactly despite their intimidating appearance. The exact analytic formula for the Coulomb electrostatic potential that we derive reflects the symmetry of the cube and is easy to implement. We illustrate its use by calculating the exact values of the electrostatic potential at some points of symmetry such as the center of cube, center of face of cube, center of edge of cube and corner of cube.

The Challenge of Sixfold Integrals: The Closed-Form Evaluation of Newton Potentials between Two Cubes

“…By rewriting the inverse hyperbolic tangent in terms of logarithms, this structure is also present in the concrete cases (2), (4), and (6). 5 Recall the notation ρ δ = δ 2 1 + δ 2 2 + δ 2 3 that we introduced in Waldvogel's formula (5).…”

“…He sketches the modifications of his method but does not offer any specific formulae. The Laplace transform technique was previously used by Ciftja [5,Eq. (32)] in 2015 for expressing the potential V (y 1 , y 2 , y 3 ) in form of the single integral (29) with an integrand similar to (30).…”

“…(32)] in 2015 for expressing the potential V (y 1 , y 2 , y 3 ) in form of the single integral (29) with an integrand similar to (30). However, the subsequent explicit evaluation, ultimately yielding Waldvogel's formula (5) in the form [5, Eq. ( 14)], extends over 4 additional pages in his paper.…”

“…As a bonus, we show in Section 6 that the method presented here allows us to straightforwardly reproduce the lovely symmetric formula (5). The main objective of this paper is to give these three spotlights a common frame that allows an understanding of their structure as well as a comparatively simple algorithm to obtain, among other things, all previous results and formulae.…”

The electric field of a uniformly charged cubic shell

“…They could then investigate qualitatively and then quantitatively (say by approximating the density by a finite regular grid of point charges) the electric field and potential. (An analytical solution for V is known in this case20 , and a qualitative discussion for a uniform cubic mass has been given by Sanny and Smith 21 .) How do the potential and electrical field differ from those for a sphere of constant charge density ρ?…”