1
$\begingroup$

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

$\endgroup$
1
  • $\begingroup$ Was my answer helpful? $\endgroup$ Dec 4, 2020 at 14:29

1 Answer 1

3
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.