Asymptotic Directions for the Zero Sets of the Components of an Electrical Field from a Finite Number of Point Charges on the Plane Part II

TL;DR

This paper investigates the asymptotic directions of zero sets of the components of a finite point charge electrical field in the plane, establishing equations for these directions and showing their finiteness.

Contribution

It derives equations governing the asymptotic directions of zero sets and proves there are finitely many such directions for each component.

Findings

Finitely many asymptotic directions for zero sets of each component.

Equations satisfied by the asymptotic directions.

Suspected distinctness of asymptotic directions for different components.

Abstract

We study the structure of the zero set of a nontrivial finite point charge electrical field $F = (X,Y)$ in the plane $\mathbb R^2$. We establish equations satisfied by the possible directions for the zero sets \{X = 0\} and $\{Y = 0\}$ separately, and we show that there are only finitely many possible asymptotic directions for both of these zero sets. We suspect that the set of asymptotic directions for \{X = 0\} and the set of asymptotic directions for $\{Y = 0\}$ are (essentially) distinct.