Derivation of 1st london equation.

[Inertia]

This is my first time posting here, I apologize if this is the wrong place to ask such a question. In my book I have the following London equation written (1st) for a superconductor:


E=μ₀λ²_L∂J/∂t

where: λ²_L is the london penetration depth.

My understanding is that it can be derived from Newtons 2nd law, by simply assuming the electron is accelerated indefinitely, and writing in terms of current densities. My issue with this is that there is an ernous factor of 1/4 that turns up in the final answer which is sometimes included in λ²_L that I cannot resolve. The wikipedia article here: (http://en.wikipedia.org/wiki/London_penetration_depth) does not include the factor of 1/4 in λ²_L.

I can't find anywhere to help with this inconsistency, I can only think that the mass is half of that in the drude model (after the scattering term is removed) or the charge is a factor of √2 greater.

 

[Greg Bernhardt]

I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?

 

[Bashkir]

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,

$$-∇ X ∇ X E = ∇^2E = \frac{E}{\lambda_l^2}$$