# Surprisingly Uniform Magnetic Field inside a Rotating Charged Sphere

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.

• The electric field inside a charged sphere is uniform. (It’s zero.) Coulomb’s Law and the Biot-Savart Law are both inverse square, so it shouldn’t be too surprising that the magnetic field is uniform. – G. Smith Oct 2 '19 at 16:18