(a) Construct the scalar potential U(r) for a “pure” magnetic dipole m .
(b) Construct a scalar potential for the spinning spherical shell (Ex. 5.11). [Hint: for r > R this is a pure dipole field, as you can see by comparing Eqs. 5.69 and 5.87.]
A (r, \theta, \phi)= \begin{cases}\frac{\mu_{0} R \omega \sigma}{3} r \sin \theta \hat{\phi}, & (r \leq R), \\ \frac{\mu_{0} R^{4} \omega \sigma}{3} \frac{\sin \theta}{r^{2}} \hat{\phi}, & (r \geq R).\end{cases} (5.69)
A _{ dip }( r )=\frac{\mu_{0}}{4 \pi} \frac{m \sin \theta}{r^{2}} \hat{ \phi } (5.87)
(c) Try doing the same for the interior of a solid spinning sphere. [Hint: If you solved Prob. 5.30, you already know the field; set it equal to –∇U, and solve for U. What’s the trouble?]