Superconductor in an external magnetic field

[Raihan amin]

Homework Statement

A superconducting spherical shell of radius R is placed in a uniform magnetic field $\vec{B_0}$
1)Find the magnetic field everywhere outside the shell
2)the sutface current density

Homework Equations

Inside the shell the net magnetic field is 0, and at the surface also.
The magnetic field of a magnetic diople of moment $\vec{m}$ is
$$\vec{B_m}=\frac{μ_0}{4\pi}[\frac{3(\vec{m}.\vec{r})\vec{r}}{r^5} - \frac{\vec{m}}{r^3}]$$

The Attempt at a Solution

The boundary condition at the surface which is at an angle $\theta$ with the vertical is
$$\vec{B_{0,\hat{n}}}+\vec{B_{m,\hat{n}}}=0$$
So,$$B_0\cos{\theta}+\frac{μ_0}{4\pi}(\frac{2m\cos{\theta}}{R^3} )=0$$
Therefore at $$\vec{m}=-(\frac{2\pi}{μ_0})R^3 \vec{B_0} $$,the boundary condition are satisfied on the surface of the shell.Hence,$$\vec{B}=\vec{B_0}-\frac{(3R^3)(\vec{B_0}.\vec{r})\vec{r}}{2r^5} + \vec{B_0}(\frac{R^3}{2r^3})$$
But i can't find the surface current density in this way. In my textbook,the author has written that we can find that using tangential B's continuity though i didn't get that.

 

[mfb]

You calculated m as function of B and then plugged in m as function of B again (and I'm a bit surprised the terms don't cancel). You need m as function of the rotation of the surface current.

 

[Raihan amin]

You calculated m as function of B and then plugged in m as function of B again (and I'm a bit surprised the terms don't cancel). You need m as function of the rotation of the surface current.

I know that way,but it is also a valid process i think.
You can see page 305 of "A Guide to Physics Problems" by Cahn and Nadgorny

 

[Raihan amin]

Here it is