Charged line segments and ellipsoidal equipotentials

TL;DR

This survey explores electrostatic potentials from charged line segments across various dimensions, highlighting unique charge distributions that produce ellipsoidal equipotentials, with findings differing notably between two and higher dimensions.

Contribution

It systematically compares uniform and non-uniform charge distributions on line segments across dimensions, revealing dimension-dependent properties of ellipsoidal equipotentials.

Findings

Uniform charge yields ellipsoidal equipotentials only in 3D.

In higher dimensions, charge peaks at the center for ellipsoidal equipotentials.

In 2D, charge peaks at the ends, contrasting with higher dimensions.

Abstract

This is a survey of the electrostatic potentials produced by charged straight-line segments, in various numbers of spatial dimensions, with comparisons between uniformly charged segments and those having non-uniform linear charge distributions that give rise to ellipsoidal equipotentials surrounding the segments. A uniform linear distribution of charge is compatible with ellipsoidal equipotentials only for three dimensions. In higher dimensions, the linear charge density giving rise to ellipsoidal equipotentials is counter-intuitive --- the charge distribution has a maximum at the center of the segment and vanishes at the ends of the segment. Only in two dimensions is the continuous charge distribution intuitive --- for that one case of ellipsoidal equipotentials, the charge is peaked at the ends of the segment and minimized at the center.