I was assigned this question related to Dirac strings:
Given a vector potential $\vec{A}= \frac{1-\cos(\theta)}{r\sin(\theta)}\hat{\phi}$, show that there is a singularity in the $B$ field proportional to $\Theta(-z)\delta(x)\delta(y)$ on the $z$ axis. ($\Theta(x-x_0)$ is the step function with its jump at $x_0$) Find the proportionality constant.
My attempt at a solution:
So showing there is a singularity simply results from the fact that at $\theta=0$, the vector potential explodes because the denominator goes to zero. The same holds true for $r$. My question becomes quantifying the magnitude of this singularity via this proportionality constant. I'm assuming it's essentially a measure of how quickly the field increases close the the $z$ axis, but I'm not sure how to procede from here. I've looked at quite a bit of literature on the matter, including Dirac's original paper, but they all simply state there is a singularity, and make no statement about the size of it. Any insight that can be offered about the nature of this singularity would be deeply appreciated!
