(a) Referring to Prob. 5.52(a) and Eq. 7.18, show that
E=−∂t∂A,(7.66)
for Faraday-induced electric fields. Check this result by taking the divergence and curl of both sides.
(b) A spherical shell of radius R carries a uniform surface charge σ. It spins about a fixed axis at an angular velocity ω(t) that changes slowly with time. Find the electric field inside and outside the sphere. [Hint: There are two contributions here: the Coulomb field due to the charge, and the Faraday field due to the changing B. Refer to Ex. 5.11.]