Derivation of 1st london equation.

by Inertia
Tags: derivation, equation, london
 P: 26 Working the full derivation from Newton's second law we can say, $$m\frac{dv}{dt} = eE - \frac{mv}{τ}$$ The steady-state drift velocity implies we can write the Ohm's Law, $$J=nev=\frac{ne^2τ}{m}E=σE$$ If there is no scattering term, Ohm's law is replaced by an accelerative supercurrent. $$\frac{dJ_s}{dt}=\frac{n_se^2}{m}E=\frac{E}{\Lambda}=\frac{c^2}{4\piλ_l^ 2}E$$ This is the first London equation and you can see that factor of 1/4 that you are talking about. We can apply Maxwell's equations then, $$∇ X h = \frac{J4\pi}{c}\\ ∇ X E = -\frac{1}{c}\frac{∂h}{∂t}$$ From this we obtain, $$-∇ X ∇ X E = ∇^2E = \frac{E}{\lambda_l^2}$$