Just as ∇ · B = 0 allows us to express B as the curl of a vector potential (B = ∇ × A),...

Just as ∇ · B = 0 allows us to express B as the curl of a vector potential (B = ∇ × A), so ∇ · A = 0 permits us to write A itself as the curl of a “higher” potential: A = ∇ ×W. (And this hierarchy can be extended ad infinitum.) (a) Find the general formula for W (as an integral over B), which holds when

B → 0 at∞.

(b) Determine W for the case of a uniform magnetic field B. [Hint: see Prob. 5.25.]

(c) Find W inside and outside an infinite solenoid. [Hint: see Ex. 5.12.]

Prob. 5.25

Ex. 5.12

Find the vector potential of an infinite solenoid with n turns per unit length, radius R, and current I .