What is the current density of a circular current? Lets consider a wire loop in the $xy$-plane of radius $R$ which carries a current $I$. I want to find the current density $\vec{j}(x)$ of that configuration expressed with delta functions.
Intuitively, I would say that in spherical coordinates:
$$
\vec{j}(x) = \delta(r - R)\delta(\theta - \pi/2) I \hat\phi
$$
so that in the integration of $\vec{j}(x)$, $r$ is set to $R$ and $\theta$ to $\pi/2$.
But based on one of my exercise sheets, I am missing a factor $1/R$ on the rhs of the above equation.
Is that true? If yes, where does this factor come from?
 A: If you compute the integral $\int_A\vec j \cdot \mathrm d\vec S$ where $A$ is a surface oriented in the $\hat\phi$ direction as illustrated below, then you would obtain
$$\int _A  \big(\delta(r-R)\delta(\theta-\pi/2) I\big)\color{red}{r} \mathrm dr \mathrm d\theta  = I\color{red}{R}$$
where the extra factor of $R$ arises from the area element $r\mathrm dr \mathrm d\theta$.  As a result, your $\vec j$ should be multiplied by a factor of $1/r$ to cancel this out.

One might be able to guess this by noting that current density should have units of current per area, but since $\delta(\theta-\pi/2)$ is dimensionless and $\delta(r-R)$ has dimensions of inverse length, your proposed $\vec j$ has dimensions of current per length.
As a general rule, delta functions in non-Cartesian coordinates are tricky to work with. You should always integrate them as a test to make sure you get what you want.
A: I think the flaw here is that you're equating a current density to a current. The line density of the current in the loop would be expressed as:
$$\lambda = \frac{I}{2\pi R}$$
So that you can integrate:
$$I=\int_0^\infty\int_0^\pi\int_0^{2\pi}drd\theta d\phi \mathbf{j(r,\theta,\phi)}=\int_0^\infty\int_0^\pi\int_0^{2\pi}drd\theta d\phi \delta(r-R)\delta(\theta-\frac{\pi}{2})\frac{I}{2\pi R}$$
A: I agree with the accepted answer.
As an extra insight, you can see from the equation you proposed that it doesn't have the right dimensions.
The current density has units of current/length^2, yours has current/length.
