Consider this:
A spherical shell of radius R, carrying a uniform surface charge $σ$, is set spinning at an angular velocity $ω$. What can be said about the magnetic field inside the sphere?
I found the magnetic vector potential* at an arbitrary point in space, and used $B = curl(A)$ to find the magnetic field. To my surprise, the magnetic field inside the sphere is uniform!
$$\mathbf{B}=\mathbf{\nabla}\times\mathbf{A}
=\frac{2}{3}\mu_0\sigma R\mathbf{\omega}.$$
Why is this so? Is there a way to predict the same (uniformity of B inside the sphere) without going through the mathematical derivation? I'd love to get some more insight into this problem, after all, there is more to physics than just mathematics!
P.S.
*The calculation, if you want to go through it, can be found in the Electrodynamics text by Griffiths.
Best Answer
The electric field inside a charged sphere is uniform. (It’s zero.) Coulomb’s Law and the Biot-Savart Law both have inverse-square spatial dependence, so it shouldn’t be too surprising that the magnetic field is uniform.