Exploring the electric field around a loop of static charge: Rectangles, stadiums, ellipses, and knots

TL;DR

This paper investigates the electric field around various charged loops and knots, revealing bifurcations, numerical bounds, and visualizations of equipotential surfaces, advancing understanding of electrostatic configurations in complex geometries.

Contribution

It introduces numerical analysis of electric fields around charged knots, challenges existing bounds, and visualizes equipotential surfaces, providing new insights into electrostatic topology.

Findings

Symmetry-breaking bifurcation in rectangular loops

Previous bounds on equilibrium points are not sharp

First visualizations of equipotential surfaces around charged knots

Abstract

We study the electric field around a continuous one-dimensional loop of static charge, under the assumption that the charge is distributed uniformly along the loop. For rectangular or stadium-shaped loops in the plane, we find that the electric field can undergo a symmetry-breaking pitchfork bifurcation as the loop is elongated; the field can have either one or three zeros, depending on the loop's aspect ratio. For knotted charge distributions in three-dimensional space, we compute the electric field numerically and compare our results to previously published theoretical bounds on the number of equilibrium points around charged knots. Our computations reveal that the previous bounds are far from sharp. The numerics also suggest conjectures for the actual minimum number of equilibrium points for all charged knots with five or fewer crossings. In addition, we provide the first images of the equipotential surfaces around charged knots, and visualize their topological transitions as the level of the potential is varied.