Find magnetic scalar potential for superconducting sphere

Question

In regions where $J = 0$, the curl of the magnetic field $B$ is necessarily zero (since $\nabla \times B = \mu_0 J$). Therefore $B$ can be written as $B = -\nabla V_m$, where $V_m$ is a scalar function of position.
A superconducting ball of radius $a$ is placed in an otherwise uniform magnetic field $B = B_0 \hat {z}$. Since the $B$ field is zero inside the ball, it must hold that $B_{\perp} = 0$ just outside the surface of the ball (by magnetic boundary conditions).
(a) Find the magnetic scalar potential $V_m (r, \theta)$ everywhere.
Far from the sphere the field is $B_0 \hat {z}$, and hence $V_m \rightarrow -B_0 z$. So in spherical coordinates, $V_m \rightarrow -B_0 r \cos \theta$ for $r \gg a$. Is the following true? $$ \frac {\partial V_{\text{above}}}{\partial n} = \mu_0 K $$ Since $V_{\text{below}} = 0$ due to the sphere being a superconductor. Using separation of variables, the first condition yields $$ \sum_{l=0}^{\infty} A_l r^l P_l (\cos \theta) = -B_0 r \cos \theta, $$ So, only one therm is present, $l = 1$. Then $A_1 = -B_0$ with all other $A_l$'s zero. From the second condition, $$ \frac {\partial V}{\partial n} = A_1 \cos \theta - \sum_{l =0}^{\infty} (l+1)\frac {B_l}{R^{l+2}} P_l (\cos \theta) = \mu_0 K. $$ Now I'm stuck as to how to proceed and remove some of the $B_l$ terms. Perhaps my boundary conditions are not correct.

Answer 1

The boundary conditions you should use are that as $r\rightarrow \infty$, $\mathbf{B}\rightarrow B_0\hat{\mathbf z}$; and that at $r=a$, $\mathbf B \cdot \hat{\mathbf r}=0$.