The Newtonian potential of a homogeneous cube

Abstract

Closed form expressions for the Newtonian force field and the Newtonian potential of a homogeneous rectangular parallelepiped were given by MacMillan (1930). Here a new concise derivation by means of multiple summations is presented. Each component of the force is written as a sum of 24 terms involving Artanh and arctan functions, whereas the expression for the potential consists of 48 terms. As a new application, the self-energy is represented in closed form, too.