Find magnetic scalar potencial of a spherical shell

Question

I don't know how to calculate magnetic scalar potential due to spherical shell with inner radius $R_1$ and outer radius $R_2$. The magentization is uniform and in $z$ direction ($\vec{M} = M_0\hat{k}$). My first attempt was the following: $$ \varphi_{ext}^* = \sum_{n=0}^{\infty}\frac{D_n}{r^{n+1}}P_n(\cos{\theta}) \\ \varphi_{int}^* = \sum_{n=0}^{\infty}C_nr^nP_n(\cos{\theta}) $$ Of course, the tangent component of $\vec{H}$ must be continuous at $r=R_2$ and the normal component of $\vec{B}$ must be continuous too.

But in the region $ R1 \le r \le R_2$ I don't know how is the expression for the scalar potential. Any comment or sugestion will be very helpful

Answer 1

For the inner region you could try a general solution, e.g. $$ \varphi_{inner}^* = \sum_{n=0}^{\infty}\frac{D_n}{r^{n+1}}P_n(\cos{\theta}) +\sum_{n=0}^{\infty}C_nr^nP_n(\cos{\theta}) $$ and use $r = R_1$ and $r = R_2$ to find $D_n$ and $C_n$, i.e. $$ \varphi_{inner}^*(R_1,\theta) = \sum_{n=0}^{\infty}\frac{D_n}{R_1^{n+1}}P_n(\cos{\theta}) +\sum_{n=0}^{\infty}C_n R_1^nP_n(\cos{\theta}) $$ $$ \varphi_{inner}^*(R_2,\theta) = \sum_{n=0}^{\infty}\frac{D_n}{R_2^{n+1}}P_n(\cos{\theta}) +\sum_{n=0}^{\infty}C_n R_2^nP_n(\cos{\theta}) $$ by matching boundary conditions. To be clear, the coefficients $C_n$ and $D_n$ I wrote above are different from the ones you wrote in your question for the inside and outside fields, I was just trying to be consistent with your notation.