Surface current of superconducting sphere in magnetic field

[Jan05]

Homework Statement

Suppose a sphere of superconductiong material is placed in a uniform magnetic field $\mathbf{B} = B \, \hat{\mathbf{z}}$. What is the induced surface current distribution?

Relevant Equations

$\mathbf{B}_{above} - \mathbf{B}_{below} = \mu_0 (\mathbf{K} \times \hat{\mathbf{n}} )$

My idea was to use the continuity of parallel components of the magnetic field and the spherical coordinate system. Because the magnetic field in a superconducting material is 0 and the current is completely confined to the surface, there only is a $\mathbf{B}_{above}$ component. The equation then reduces to $\mathbf{K} \times \hat{\mathbf{r}} = B / \mu_0 \, \hat{\mathbf{z}}$. Then evaluating components and using $\hat{\mathbf{z}} = \cos \theta \, \hat{\mathbf{r}} - \sin \theta \,\hat{\boldsymbol{\theta}}$ we obtain $K_\varphi \, \hat{\boldsymbol{\theta}} = B / \mu_0 \sin \theta \, \hat{\boldsymbol{\theta}}$. So $\mathbf{K} = B / \mu_0 \sin \theta \, \hat{\boldsymbol{\varphi}}$.

 

[Charles Link]

See https://www.physicsforums.com/threa...uniform-external-b-field.923406/#post-5828157

Without looking at it very carefully, I think you will get a factor of 1/3 or 3 in there for the spherical geometry.

Edit: $H_m=-M_i/(3 \mu_o)$ and $H_i=H_m+H_o =0$, so that $H_o=M_i/(3 \mu_o )$, and $M_i=3 \mu_o H _o$.

Meanwhile $\vec{K}=\vec{M} \times \hat{n}/\mu_o$.