  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.
