Advanced Solid State Physics

16.3.2 TheLondonEquations

It is the aim of the following considerations to derive theMeissner - Ochsenfeld effect from a
particular ansatz for the electrodynamic laws in the superconducting state. We write the equation of
motion of a charged particle in an electrical fieldE

mv ̇=−qE. (275)

We now insert the current densityjs=−qnsv, wherensis the density of superconducting electrons,
in order to obtain the firstLondonequation

j ̇s=nsq

2

m

E. (276)

From∇×E=−B ̇ we obtain

∂

∂t

(

m

nsq^2

∇×js+B

)

= 0. (277)

This equation describes an ideal conductor, i.e. ρ= 0. However, the expulsion of the magnetic field
in theMeissner - Ochsenfeldeffect is not yet included. Integrating this equation with respect
to time and neglecting the integration constant yields the secondLondonequation, which already
includes to correct physical description:

∇×js=−

nsq^2

m

B. (278)

Note that sinceB=∇×Awe can conclude that

js=−

nsq^2

m

A≡−

1

μ 0 λ^2 L

A, (279)

where we definedλL=

√ m
nsμ 0 q^2 with the dimension of length. Here,Ahas to be understood in the
Londongauge. For a simply connected superconductor it may be expressed by∇·A= 0andAn= 0 ,
whereAnis the normal component ofAon an external surface, through which no current flows. The
secondLondonequation can be rewritten in the following way

∇×js=−

1

μ 0 λ^2 L

B. (280)

Further, fromMaxwell’s equations

∇×B=μ 0 js, (281)

we obtain

∇×∇×B=−∇^2 B=μ 0 ∇×j. (282)

Combining Eqs. (280) and (282) yields

∇^2 B−

1

λ^2 L

B= 0. (283)