# Divergences and Delta Functions

There’s an identity that shows up in electromagnetism which has been bugging me since college.

As soon as we start using Gauss’s Law

\[\del \cdot \b{E} = \rho\]in introductory E&M, we run into the problem that, in order to use it for a point charge — which is the most basic example in the subject! — we already don’t have the mathematical object we need to calculate the divergence on the left, or to represent the charge distribution on the right. The field of a point charge has to be

\[\b{E} = q \hat{\b{r}}/4 \pi r^2\]And its charge has to be concentrated at a point, i.e. it’s a delta function:

\[\del \cdot \frac{q \hat{\b{r}}}{4 \pi r^2} = q \delta(\b{x})\]In your multivariable-calculus-based E&M class you mention this briefly, maybe, but you don’t really use it. Yet it is… kinda weird? It feels like it should make sense inside of a larger framework.