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.
Zusammenfassung
Gravitationsfeld und Potential des homogenen Quaders sind von MacMillan (1930) in geschlossener Form dargestellt worden. Hier wird eine neue, konzise Herleitung dieser Resultate durch mehrfache Summationen gegeben. Die Komponenten des Kraftfeldes werden als Summen von 24 Artanh- und arctan-Termen geschrieben, während der Ausdruck für das Potential aus 48 Termen besteht. Als neue Anwendung wird die Selbstenergie des homogenen Quaders ebenfalls in geschlossener Form dargestellt.
This is a preview of subscription content, log in via an institution to check access.
References
W. D. MacMillan,The Theory of the Potential, McGraw-Hill, New York, 1930, Republished by Dover Publications, New York, 1958; pp. 72–80.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Waldvogel, J. The Newtonian potential of a homogeneous cube. Journal of Applied Mathematics and Physics (ZAMP) 27, 867–871 (1976). https://doi.org/10.1007/BF01595137
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01595137