(a) Referring to Prob. 5.52(a) and Eq. 7.18, show that for Faraday-induced elec...

(a) Referring to Prob. 5.52(a) and Eq. 7.18, show that

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.]

Reference prob 5.52

(a) One way to fill in the “missing link” in Fig. 5.48 is to exploit the analogy between the defining equations for A (viz. ∇ · A = 0, ∇ × A = B) and Maxwell’s equations for B (viz. ∇ · B = 0, ∇ × B = μ₀J). Evidently A depends on B in exactly the same way that B depends on μ₀J (to wit: the Biot-Savart law). Use this observation to write down the formula for A in terms of B.

(b) The electrical analog to your result in (a) is

Derive it, by exploiting the appropriate analogy.

Figure 5.48

Refer Ex. 5.11

A spherical shell of radius R, carrying a uniform surface charge σ, is set spinning at angular velocity ω. Find the vector potential it produces at point r (Fig. 5.45).

Equation 7.18