Current density of a filamentary loop

[fadecomic]

Hi,

I'm trying to wrap my head around the derivation of the current density of a filamentary loop. On the face of it, the result seems obvious since it involves Dirac delta functions, but it's the rest of the formulation I don't quite follow.

The derivation begins by considering a loop of radius a at z=h carrying a current I. Considering a small portion of the loop dS, a quantity $$Id\vec{S}=Iad\phi\hat{\phi}$$. Already I'm not sure why we're considering the current times the length of the element of the loop. In any case, the current density of that small element can be written (apparently) as $$\vec{j}=Iad\phi\delta(\vec{r}-\vec{r_0})$$ where $$r_0=a\hat{\rho}+\phi\hat{\phi}+h\hat{z}$$. This is where I'm hung so far. Can someone explain what's going on here to me? I know what a current density is, and I know what the purpose of the delta function is. I don't see this as an expression of "current per unit area perpendicular to flow", though.

Thanks.

 

[fadecomic]

I should probably clarify that the loop is at x=y=0, and that $$\rho$$, $$\phi$$, and z are cylindrical coordinates.