# Find magnetic scalar potencial of a spherical shell

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

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.