The Newtonian Potential and the Demagnetizing Factors of the General Ellipsoid

TL;DR

This paper introduces a modern, concise derivation of the Newtonian potential for homogeneous ellipsoids in higher dimensions, and explores the demagnetizing factors relevant to ferromagnetism.

Contribution

It provides a new, simplified derivation method for the Newtonian potential of ellipsoids and details the calculation of demagnetizing factors in a unified framework.

Findings

Explicit formulas for interior and exterior potentials

Simplified derivation using Reynolds transport and Green-Stokes formulas

Calculation of demagnetizing factors for general ellipsoids

Abstract

The objective of this paper is to present a modern and concise new derivation for the explicit expression of the interior and exterior Newtonian potential generated by homogeneous ellipsoidal domains in $\mathbb{R}^N$ (with $N \geqslant 3$). The very short argument is essentially based on the application of Reynolds transport theorem in connection with Green-Stokes integral representation formula for smooth functions on bounded domains of $\mathbb{R}^N$, which permits to reduce the N-dimensional problem to a 1-dimensional one. Due to its physical relevance, a separate section is devoted to the the derivation of the demagnetizing factors of the general ellipsoid which are one of the most fundamental quantities in ferromagnetism.