There are a collection $C$ of charges in $\mathbb{R}^2$ which cause an electric vector field $V$ to form. Each charge's contribution to $V$ follows the inverse-square law. Let $\gamma$ be a curve encolsing all charges and critical points (not at infinity is there is a critical point there).
Let the index of $\gamma$ on $V$ be $n$.
In the cases of $n=1$ and $n=2$, I have noticed the following, and am curious to see a proof or disproof of the general case
The magnitude of the vectors in $V$ is asymptotically proportional to $r^{-1-n}$, where $r$ is the distance from the origin.