The Zero Set of an Electrical Field from a Finite Number of Point Charges: One, Two, and Three Dimensions

TL;DR

This paper investigates the structure of zero sets of electrical fields generated by finite point charges, proving finiteness in a special case and providing bounds on the number of zeros.

Contribution

It establishes the finiteness of the zero set when charges lie on a line and offers detailed structural insights and bounds in this special case.

Findings

Zero set is finite when charges are collinear.

Zero set contains at most 9M^2 4^M points, with M charges.

Provides structural information about zero sets in the special case.

Abstract

We study the structure of the zero set of a finite point charge electrical field $F = (X,Y,Z)$ in $\mathbb R^3$. Indeed, mostly we focus on a finite point charge electrical field $F =(X,Y)$ in $\mathbb R^2$. The well-known conjecture is that the zero set of $F = (X,Y)$ is finite. We show that this is true in a Special Case: when the point charges for $F = (X,Y)$ lie on a line. In addition, we give fairly complete structural information about the zero sets of $X$ and $Y$ for $F = (X,Y)$ in the Special Case. A highlight of the paper states that in the Special Case the zero set of $F = (X,Y)$ contains at most $9M^24^M$ points, where $M$ is the number of point charges.