Two-fluid model of a superconductor on the two-fluid model of a super conductor we assume that at temperatures 0 < T < T₀, the current density may be written as the sum of the contributions of normal and superconducting electrons: j = j_N, + j_S, where j_N, = σ₀E and js is given by the London equation. Here σ₀ is an ordinary normal conductivity, decreased by the reduction in the number of normal electrons at temperature T as compared to the normal state. Neglect inertial effects on both j_N, and j_S. 

(a) Show from the Maxwell equations that the dispersion relation connecting wave vector k and frequency w for electromagnetic waves in the superconductor is (CGS) k²c² = 4πσ₀2wt – c²λL^–2 + w²; or (SI) k²c² = (σ_0/ε₀)wt c²λL^–2 + w² where A; is given by (148) with n replaced by ns. Recall that curl B = – Δ²B.

(b) If T is the relaxation time of the normal electrons and n_N is their concentration, show by use of the expression σ₀ = n_Ne²τ/m that at frequencies w << 1/τ the dispersion relation does not involve the normal electrons in an important way, so that the motion of the electrons is described by the London equation alone. The super current short-circuits the normal electrons. The London equation itself only holds true if hw is small in comparison with the energy gap. Note: The frequencies of interest are such that w << w_p, where w_p is the plasma frequency.