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Homework Help: Permeable sphere placed in external magnetic field

  1. Jan 6, 2014 #1
    1. The problem statement, all variables and given/known data

    A permeable sphere with [tex]μ_r[/tex] is placed in a magnetic field of [tex]H[/tex]. Show that it satisfies Laplace's equation and the potentials are of the form:


    2. Relevant equations

    3. The attempt at a solution

    Part (a)

    Starting from maxwell's equations, I showed that ø satisfies laplace's equation. Followed by relating it to the electric case, where the general solution is similar.


    I've only managed to get 2 useful boundary conditions..

    Am I allowed to say that as [tex] r→ ∞, ø_2 → -H_0r cos θ [/tex] (Just as the electric case)

    Because this would immediately solve for [tex] C_2 = H_0 [/tex]

    Part (c)
    To find dipole moment m, we use the the formula for dipole scalar potential for r > r0:

    [tex] ø_2 = -C_2 r cos θ + C_3 r^{-2} cos θ = \frac {\vec {m} .\vec {r}}{4\pi r^3} [/tex]
    Last edited: Jan 6, 2014
  2. jcsd
  3. Jan 6, 2014 #2


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    Your equation (2) has a small error. Check the power of the ##a## in the denominator of the right side.

    Yes, I believe that's right.
  4. Jan 6, 2014 #3
    For part (c),

    [tex] Magnetization = NM_0 [/tex]

    How do we find the effective permeability?

    I know that:

    [tex] B = μM [/tex] where M is magnetization.

    Do I find an expression for the field of a dipole very far away?
  5. Jan 6, 2014 #4


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    If you have obtained a result for the magnetic dipole moment m0 of the paramagnetic sphere, then you should see that the moment is proportional to the applied field H.

    Then when you look at the composite material with N of these spheres per unit volume, the total magnetization of the material will be M = Nm0, as you say. Since m0 is proportional to H, so is M.

    Then consider how the proportionality constant between M and H is related to the effective permeability μeff.

    This might help: http://en.wikipedia.org/wiki/Magnetic_susceptibility.

    [There are some subtleties associated with the fact that in the composite material, the H-field felt by an individual paramagnetic sphere is not generally equal to the (averaged) macroscopic H field in the material. In the electric case, this leads to the Clausius-Mossotti equation. But in your problem, you are assuming that the spheres are "dilute" and so you can probably neglect this effect.]
  6. Jan 6, 2014 #5
    [tex] M = xH = Nm_0 = gH [/tex] where g is the proportionality constant I have found earlier

    [tex]μ_{eff} = μ_0(1+x) [/tex]

    Then solve for x in terms of g. Then plug it inside [tex] μ_{eff}[/tex]
  7. Jan 6, 2014 #6


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    Maybe a factor of N is missing in the last expression on the far right?

    I think that's right.
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