High Temperature Superconducting Magnetic Levitation

1.5 London equations Ë 7

The above discussion deals with the phase transition between the normal state
and the superconducting state atT<Tcin a magnetic field. It is a first-order transition
and which requires a latent heat for the phase change. AtT=Tcthe superconducting
phase transition has no latent heat, but it has a specific heat jump. This shows that
the superconducting phase transition atT=Tcis a second-order phase transition.

1.5 London equations [16]

London and London [2] assumed that the electrons move in a frictionless state and
derived a phenomenological macroscopic theory of superconductivity.
In the normal conducting state, the current densityJnand the electric fieldEare
connected by Ohm’s lawJn=휎nE, where휎nis the normal conductivity. In the normal
phase, the current density in the steady state is given by

Jn=휎nE=ne

(^2) 휏
m E, (1.9)
where휏is the relaxation time and휎nis the conductivity in the normal state.
According to the two-fluid model, the total densitynof electrons is a sum of nor-
mal electrons,nn, and superconducting electrons,ns. The superconducting electrons
nsare not scattered by either impurities or lattice vibrations, i.e., phonons do not
contribute to the resistivity. These electrons are freely accelerated by an electric field.
The equation of the superconducting electrons motion is
m휕vs
휕t
=eE, (1.10)
wherevsis superconducting electrons velocity.
From Maxwell’s equations, since the displacement currentDinside the supercon-
ductor vanishes, one obtains
∇×E= −휕B
휕t

, (1.11)

∇×B=휇 0 J. (1.12)

The superconducting current densityJs=nsevsobeys the following equation (the first
London equation)

휕

휕t

Js=nse

2

m

E. (1.13)

Adding curl to both sides, we have

휕

휕t

∇×Js=nse

2

m

∇×E, (1.14)