The Maxwell Conjecture is False
arXiv:2607.27197
summary
The authors present a configuration of five point charges that yields at least 24 non‑degenerate critical points of the electrostatic potential, disproving Maxwell's conjecture about the maximum number of such points.
Abstract
We exhibit a configuration of five point charges in Euclidean space whose electrostatic potential admits at least 24 critical points all of which are non-degenerate. Maxwell's conjecture that the field of \(n\) point charges has at most \((n-1)^2\) critical points which are all non-degenerate is therefore false.
Topics & keywords
#electrostatics#critical points#maxwell conjecture#point charges#potential theoryelectrostatic potentialnon-degenerate critical points(n-1)^2 boundcounterexamplefive point charges