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Magnetic Field of a cylinder

  1. May 31, 2007 #1
    1. The problem statement, all variables and given/known data
    An infinitely long cylinder, of radius R, carries a "frozen-in" magnetization. parallel to the axis, [itex] M = ks\hat{z} [/itex] where k is a constant and s is the distance from the axis; there is no free current anywhere. Find the magnetic field inside and outside the cylinder

    2. Relevant equations
    [tex] J_{b} = \nabla \times M [/tex]
    [tex] K_{b} = M\times \hat{n} [/tex]
    [tex] \oint B \cdot dl = \mu_{0} I_{enc} [/tex]

    3. The attempt at a solution

    Here [tex] J_{b} = -k\hat{\phi} [/tex]
    and [tex] K_{b} = kR \hat{\phi} [/tex]

    so the field inside s<R
    [tex] B \cdot 2\pi s = \mu_{0} \int J_{b} da = \mu_{0} \int -k s'ds' d\phi' [/tex]
    so i get [tex] B = -\mu_{0} ks \hat{z} [/tex]
    but the answer is supposed to be positive...
    why is that? Am i supposed to include the surface current density to find the field? But for a question in the past (for a cylinder with magnetization [itex] M = ks^2 \hat{\phi} [/itex].. however that time the enclosed current in the enitre (s>R) cylinder was zero - there was symmetry between the two surface currents. The amperian loop was a circlular loop within the cylinder...

    Is this question to be solved differently because there is no symmtery between the surface and volume current densities?
     
  2. jcsd
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