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Homework Help: Circular ring in xy-plane with current, find current density

  1. Nov 28, 2011 #1
    1. The problem statement, all variables and given/known data
    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.


    2. Relevant equations

    ∫B(r).dl = μ∫J(r).da

    3. 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)?
     
  2. jcsd
  3. Nov 28, 2011 #2
    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.
     
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