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Sphere with permanent radial magnetization

  1. Apr 9, 2014 #1
    1. The problem statement, all variables and given/known data

    We have a sphere of radius [itex]a [/itex] with permanent magnetization [itex]\mathbf{M}=M\hat{e}_{\mathbf{r}}[/itex].
    Find the magnetic scalar potential.



    2. Relevant equations

    $$\Phi_M(\mathbf{x})=-\frac{1}{4\pi}\int_V \frac{\mathbf{\nabla}'\cdot\mathbf{M}(\mathbf{x}')}{|\mathbf{x}-\mathbf{x}'|}d^3 x' +\frac{1}{4\pi}\int_S \frac{\mathbf{n}'\cdot\mathbf{M}(\mathbf{x}')}{|\mathbf{x}-\mathbf{x}'|}da'$$



    3. The attempt at a solution

    $$\mathbf{\nabla}'\cdot\mathbf{M}(\mathbf{x}')=\frac{2M}{r'}$$

    $$\mathbf{n}'\cdot\mathbf{M}(\mathbf{x}')=M$$

    $$\Phi_M(\mathbf{x})=-\frac{M}{2\pi}\int_{0}^{a} r'dr'\int\int\frac{1}{|\mathbf{x}-\mathbf{x}'|}d\Omega' +\frac{Ma^2}{4\pi}\int \int\frac{1}{|\mathbf{x}-\mathbf{x}'|}d\Omega'$$

    I expanded the [itex]1/|\mathbf{x}-\mathbf{x}'| [/itex] in terms of spherical harmonics (and because of the spherical symmetry we have [itex]m=0,\ell=0 [/itex]) and solved the integrals. What I got is:
    $$\Phi_M(\mathbf{x})=-2M\int_{0}^{a}\frac{r'}{r_{>}}dr'+\frac{Ma^2}{r_{>}}$$

    where [itex]r_{>}=max(r,a)[/itex]

    Inside the sphere we have [itex]r_{>}=a[/itex], therefore:
    $$\Phi_M(r)=\frac{-2M}{a}\int_{0}^{a}r'dr'+Ma=0$$

    This is constant. However the [itex]\Phi_M(r)[/itex] inside the sphere has to satisfy the Poisson equation:
    $$\nabla^2 \Phi_M(r)=\nabla\cdot \mathbf{M}=\frac{2M}{r}$$

    This is not true for the potential that I found..
     
    Last edited: Apr 9, 2014
  2. jcsd
  3. Apr 10, 2014 #2
    We could also compute the vector potential [itex]\mathbf{A}[/itex]

    $$\mathbf{A}(\mathbf{x})=\frac{\mu_0}{4\pi}\int_V \frac{\mathbf{\nabla}'\times\mathbf{M}(\mathbf{x}')}{|\mathbf{x}-\mathbf{x}'|}d^3 x' +\frac{\mu_0}{4\pi}\int_S \frac{\mathbf{M}(\mathbf{x}')\times\mathbf{n}'}{|\mathbf{x}-\mathbf{x}'|}da'$$


    $$\mathbf{\nabla}'\times\mathbf{M}(\mathbf{x}')=M\mathbf{\nabla}'\times \hat{e}_{\mathbf{r}'}=0$$

    $$\mathbf{M}(\mathbf{x}')\times\mathbf{n}'=M\hat{e}_{\mathbf{r}'}\times \hat{e}_{\mathbf{r}'}=0$$

    So [itex]\mathbf{A}=0[/itex].

    Both methods give zeros. Where is the problem?
     
  4. Apr 10, 2014 #3

    TSny

    User Avatar
    Homework Helper
    Gold Member

    Interesting. I think such a sphere is impossible.

    Note that the divergence of M is undefined at the center of the sphere (r = 0). So, you have a singularity there that would need to be handled when integrating over the volume of the sphere to find ##\Phi_M##.

    It is easy to see that ##\vec{B} = 0## everywhere. If ##\vec{B} \neq 0## at some point located a distance r from the center of the magnetized sphere, then by spherical symmetry ##\vec{B}## is radial at that point and there would exist a radial ##\vec{B}## at every point on the surface of a sphere of radius r. Thus, there would be a nonzero magnetic flux through a closed surface which would violate the law ##\vec{\nabla} \cdot \vec{B} = 0## everywhere.
     
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