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Homework Help: Ampere's law in a nonuniform conducting cylinder.

  1. Feb 29, 2012 #1
    1. The problem statement, all variables and given/known data
    A long, cylindrical conductor of radius R carries a current I. The current density, J, is not uniform and is given by the equation J = br, where b is a constant. Find an expression for the magnetic field magnitude B at distance r < R and at distance r > R.

    2. Relevant equations
    I = ∫JxdA
    B= μI/(2pi*r)

    3. The attempt at a solution
    So I think that I have this figured out, but it's an even numbered problem so I don't know if I'm correct.
    a) First finding I inside r
    I = ∫JxdA = ∫br2(pi)rdr = 2(pi)b∫r^2dr = 2/3(pi)br^3 when integrating from 0 to r

    So B = μI/(2(pi)r) = 2μ(pi)br^3/(6(pi)r) = μbr^2/3

    b) First finding total I
    I = ∫JxdA = ∫br2(pi)rdr = 2(pi)b∫r^2dr = 2/3(pi)bR^3 when integrating from 0 to R

    so B = μI/(2(pi)r) = 2μ(pi)bR^3/(6(pi)r) = μbR^3/(3r)

    I'm pretty sure it's correct, but not entirely sure.
  2. jcsd
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