On the fields due to line segments

TL;DR

This paper explores the electrostatic fields and equipotential geometries generated by uniformly charged line segments, providing an elementary discussion with historical context and connections to classical physics problems.

Contribution

It offers an accessible analysis of the electrostatic fields due to line segments, linking modern explanations with historical solutions and geometric properties.

Findings

Ellipsoidal equipotential surfaces are produced by line charges.

Gradient fields exhibit characteristic geometries related to the charge distribution.

Historical methods by Green and others are connected to modern understanding.

Abstract

The remarkable geometries of ellipsoidal equipotentials and their associated gradient fields, as produced by uniformly charged straight-line segments, are discussed at an elementary level, motivated by recent treatments intended for introductory physics classes. Some effort is made to put the results into a broader conceptual and historical context. The equipotentials and vector fields were first obtained for the electrostatic problem by George Green in his famous 1828 essay. Related problems were commonly found on the Mathematical Tripos examinations given at the University of Cambridge, and their solutions were widely disseminated by William Thomson (Lord Kelvin), Peter Guthrie Tait, and Edward Routh during the last half of the 19th century.