The surprising subtlety of electrostatic field lines

TL;DR

This paper explores the limitations and subtleties of visualizing electric fields with field lines, examining whether field lines can reveal if a field is conservative and identifying special cases with straight lines.

Contribution

It provides a differential geometry framework to analyze electric field lines, addressing questions about their ability to indicate conservativeness and special configurations.

Findings

Field lines alone cannot always determine if an electric field is conservative.

Certain conservative fields can have straight field lines beyond common symmetric cases.

The paper introduces differential geometry tools to analyze electric field line properties.

Abstract

Electric fields are commonly visualized with field line diagrams, which only unambiguously specify the field's direction. We consider two simple questions. First, can one deduce if an electric field is conservative, as required e.g. in electrostatics, from its field lines alone? Second, are there conservative electric fields with straight field lines, besides the familiar textbook examples with spherical, cylindrical, or planar symmetry? We give a self-contained introduction to the differential geometry required to answer these questions, assuming only vector calculus background.