1.5　London equations16

Fritz London and Heinz London2assumed that the electrons move in a frictionless state and derived a phenomenological macroscopic theory of superconductivity.

In the normal conducting state, the current density Jn and the electric field E are connected by Ohm's law Jn =σn E, where σn is the normal conductivity.In the normal phase, the current density in the steady state is given by

whereτis the relaxation time andσn is the conductivity in the normal state.

According to the two-fluid model, the total density n of electrons is a sum of normal electrons, nn ，and superconducting electrons, ns . The superconducting electrons ns are 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

where vs is superconducting electrons velocity.

From Maxwell's equations, since the displacement current D inside the super conductor vanishes, one obtains

The superconducting current density Js =ns evs obeys the following equation（the first London equation）

Taking curl of both sides, we have

where∇is the curl operator, E is the electric field and B is the magnetic field.We have

Because

and∇·B=0，one obtains

Thus, B must satisfy this differential equation in superconductors, leading to a characteristic length scale known as the London penetration depth， λL ，

One obtain

or

This is the second London equation, and the first London equation can be written as

The second London equation leads to the Meissner-Ochsenfeld effect, and can also be derived from the BCS theory by assuming that the BCS state is spatially homogeneous.The London equation predicts that the magnetic field in a supercon ductor decays exponentially from the surface into the superconducting material.