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B Field Inside of Sphere using Sep. Variables

  1. Oct 29, 2013 #1
    Done editing I hope.

    1. The problem statement, all variables and given/known data
    If Jf = 0 everywhere, then (as we showed in class), one can express H as the gradient of a scalar potential, W. W satisfies Poisson’s equation with ∇⋅M as the source. Use this fact to find the field inside a uniformly magnetized sphere. (Griffiths has some additional verbiage
    intended to help, but I think you already know what he says.) Compare your answer with
    example 6.1 (p. 264-5), which is this problem solved by another method.



    2. Relevant equations

    ##H^\perp_{above} - H^\perp_{below}=-(M^\perp_{above}- M^\perp_{below})##

    3. The attempt at a solution

    My question is with the constraints and in particular the one I have in the equations section. I had this as ##∇W^\perp_{in} - ∇W^\perp_{out}=M^\perp## Since at r=R they are equivalent. I know that ##M^\perp## must writable as some version of M but I cannot determine what. I know the solutions manual has ##M\hat{z}\hat{r}= Mcos(θ)## but I don't understand how they have ##\hat{z}## in spherical coordinates...

    Thanks for any clarification.
     
    Last edited: Oct 29, 2013
  2. jcsd
  3. Oct 29, 2013 #2

    marcusl

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    Can you be more precise about your confusion regarding [itex]\hat{z}[/itex]?
     
  4. Oct 29, 2013 #3
    More than the z vector I'm just confused how they got from M orthogonal to Mcos(θ). What is the reasoning or the process?
     
  5. Oct 29, 2013 #4

    marcusl

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    I don't own a copy of Griffiths and you haven't shown the whole problem, so I'm going to make a guess that the magnetization is along the z axis? The component of M normal to the boundary (at r=R) is proportional to cos(theta). You can see this intuitively: at theta=0, M is normal to the boundary so Mnormal=M. At 90 deg, M is tangential to the circle so Mnormal=0.
     
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