Circular ring in xy-plane with current, find current density

[jajay504]

Homework Statement

Consider a circular ring of wire of radius a that resides in the x-y plane through the origin. The center of the ring coincides with the origin and you can regard the thickness of the wire to be infinitesimal.

a. Given that a current I flows in the ring, find an expression for the current density J.

Homework Equations

∫B(r).dl = μ∫J(r).da

The Attempt at a Solution

ok, I know J(r)=Io * δ(x)*δ(y) z
How do I get this in cylindrical coordinates (ρ,θ,z)?

 

[Steely Dan]

Well, the current density is only non-zero at z = 0, that corresponds to it being non-zero only at θ = π / 2, or cos(θ) = 0. So you should have a delta function for that. Additionally, it's non-zero only at ρ = R, so you should have a delta function for that as well. If I remember Jackson correctly, you'll also need a normalization factor of 1/R because θ (and therefore the first delta function) is a dimensionless quantity, so you divide by 1/R to maintain the proper units for current density of current / area (I'm not really sure why this works, I just took Jackson at his word on this one). The original delta functions are functions of position coordinates and therefore already have 1/L dependence.