Sphere with permanent radial magnetization

[maxverywell]

Homework Statement

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.

Homework Equations

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

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..

 

[maxverywell]

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?

 

[TSny]

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.