Counting Equilibria of the Electrostatic Potential

TL;DR

This paper improves the upper bound on the number of equilibria of electric fields generated by point charges, constructs counterexamples to existing conjectures, and explores configurations with high ratios of zeroes.

Contribution

It provides the best known upper bound on electric field zeroes, disproves a conjecture relating equilibria to distance function maxima, and investigates configurations with maximal ratios of zeroes.

Findings

Established a new upper bound on the number of electric field zeroes.

Constructed a counterexample to a conjecture about the maximum number of equilibria.

Identified configurations with high ratios of zeroes relative to charge arrangements.

Abstract

In 1873, James C. Maxwell conjectured that the electric field generated by $n$ point charges in generic position has at most $(n-1)^2$ isolated zeroes. The first (non-optimal) upper bound was only obtained in 2007 by Gabrielov, Novikov and Shapiro, who also posed two additional interesting conjectures. In this article, we give the best upper bound known to date on the number of zeroes of the electric field, and construct a counterexample to a conjecture of Gabrielov, Novikov and Shapiro that the number of equilibria cannot exceed those of the distance function defined by the unit point charges. Finally, we note that it is quite possible that Maxwell's quadratic upper bound is not tight, so it is prudent to find smaller bounds. Hence, we also explore examples and construct configurations of charges achieving the highest ratios of the number of electric field zeroes by point charges found to this day.