Critical field of thin films

(a) Using the result of Problem lb, show that the free energy density at T = 0 K within a superconducting film of thickness δ in an external magnetic field B_a is given by, for δ << A, (CGS) F_S (x, B_a) = U_S (0) + (δ² – 4x²) B²_a/64πλ². In SI the factor π is replaced by ¼μ₀. We neglect a kinetic energy contribution to the problem. 

(b) Show that the magnetic contribution to F_S when averaged over the thickness of the film is B²_a (δ /λ)²/96π.

(c) S how that the critical field of the thin film is proportional to (λ/δ) H_c, where H_c, is the bulk critical field, if we consider only the magnetic contribution to U_S.