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Homework Help: A couple of simple magnetization problems

  1. Jan 31, 2015 #1
    1. The problem statement, all variables and given/known data
    These are actually two problems that I'm merging into one because each of them seem to have conflicting solutions, and I want to clear this up.

    Consider a long cylinder (very long) extending in the z direction with a constant magnetization ##\vec{M}=M\hat{z}##. What are the H and B fields everywhere? Now drill a very small hole through it in the z direction, what are the H and B fields everywhere now?

    2. Relevant equations
    ##\vec{j}=\nabla \times \vec{M}##
    ##\vec{K}=\vec{M}\times \hat{n}##
    There are more, of course, but let me know if something crucial is missing.

    3. The attempt at a solution
    Without hole: there is no current density inside (curl of that constant magnetization is zero) but we do have surface current ##\vec{K}=M\hat{\phi}## because the walls are perpendicular to M. So with that it's easy to calculate B (it's zero outside and ##\mu_0 M## inside, like a solenoid). And to find H is also simple, using ##\vec{H}=\vec{B}/\mu_0-\vec{M}## we see that H is zero everywhere, inside and out.

    Now with the drilled hole for some reason we have H and B fields not zero everywhere. Shouldn't it be the at least similar? Sure, there's a current in the central surface going in opposite direction as the outer current, but I woud still expect that outside the whole thing the fields to be zero. What am I missing?
  2. jcsd
  3. Feb 1, 2015 #2
    I don't see anything wrong with your reasoning: the only changes should be the inner surface current and B-field within the cavity created.
    Where did you get that there should be non-zero fields outside? (in the "very long" cylinder approximation, of course)
  4. Feb 1, 2015 #3
    Oh!! I feel dumb. I misread the second problem statement. It's not infinitely long, it's infinitely wide. That changes everything (we could really use a facepalm 'smiey')
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