3.) Flux Quantization: For a superconducting ring with wall thickness >> 1/λ, unlike a single homogeneous slab, the superconductor wave function's phase (θ) can vary with position along the ring because of the hole. However, deep inside the body of the ring (well away from inner and outer walls), in equilibrium Jsc=0, even though n(r)>0.
a) If we take a simple closed loop path around the ring that is inside the body of the ring and encloses the hole in the ring, use this and the result of Prob. 2 to show that the path integral: ∮ V0(r)·ds = q s A, where ds is a small element of the path length and the integral is taken once around the loop.
From Prob. 2:
b) From Maxwell's equations, the path integral of A around any closed loop = magnetic flux through the area of that loop, i.e. ∮ A·ds = B Area enclosed by loop. If we require that, upon going exactly once around the loop, the phase (θ) can only change by a multiple of 2πn, where n = 0, 1, 2, 3, ..., then show that the magnetic flux enclosed inside the loop is quantized, that is Φ = n(2πh/q). Experiments show that Φ = n(2πh/2e) = h/2e, where e here being the fundamental charge of a single electron. Φ is called the flux quantum, which is a simple combination of fundamental constants.
It may be useful to recall from vector calculus that for any differentiable scalar function f(r), the path integral between any two points r1 and r2:
∫ Vf(r)·ds = f(r2) - f(r1)
holds for any continuous path connecting the points.