Einar Snäll: Maxwell's Conjecture - An Overview

Abstract: Maxwell's Conjecture is a claim made by James Clerk Maxwell regarding the number of critical points of the electric potential generated by point charges. It is also an open problem in mathematics. This paper aims to give the reader an introduction to and overview of Maxwell's Conjecture, and to provide statistics for the distributions of critical points through computer experiments. A basic introduction to the electrostatics necessary to understand the conjecture is given, and the upper bound of (N-1)^2 critical points proposed by Maxwell is presented. The paper then outlines and summarises some previous research from other authors, presenting the many inferior upper bounds that have been found, alongside some other relevant results like specific configurations that reach the bound. We then dedicate the rest of the paper to computer experiments that test the conjecture. Random configurations of N point charges are generated for N ranging from 3 to 7, and the number of critical points of each configuration is investigated using repeated attempts at root finding. We then present the data on the number and distribution of critical points. The data clearly indicates that, at the very least, for random configurations generated 'haphazardly', configurations that approach the bound are rare, especially as the number of charges increases. This supports the conjecture's claim, though it does nothing to further any attempts at proving it. These results are approximate given the numerical approach, and there is room for error and undercounting given the method used. Finally, we discuss possible expansions or improvements that can be made to obtain more relevant or reliable results.