London equations in superconductors: derivation and discussion

London Equations:

As discussed in the Meissner effect that one of the conditions of the superconducting state is that Magnetic flux density (B) = 0 inside the superconductors that is the magnetic flux cannot penetrate inside the superconductor. But experimentally it is not so. The magnetic flux does not suddenly drop to zero inside the surface. The phenomenon of flux penetration inside the superconductors was explained by H. London and F. London.

Derivation of London first equation:

Let ns and vs be the number density (number/volume) and velocity of superconducting electrons respectively. The equation of motion or acceleration of electrons in the superconducting state is given by

m(dvs/dt) = -eE

or dvs/dt = -eE/m                                              (1)

where m is the mass of electrons and e is the charge on the electrons.

Also the current density is given by

Js = -nsevs

Differentiate it with respect to time,

dJs/dt = -nse(dvs/dt)

Put equation (1) in above equation, we get

dJs/dt = (nse2 E)/m                                            (2)

Equation (2) is known as London’s first equation

Derivation of London second equation:

Take curl (that is cross or vector product of del operator with a vector) of London’s first equation, we get

del operator x dJs/dt =  [(nse2 )del operator x E]/m                                (3)

By differential form of Faraday’s law of electromagnetic induction (or Maxwell’s third equation)

del x E = -dB/dt

Put this in equation (3), we get

del x dJs/dt =  -[(nse2(dB/dt)/m)

Integrate both sides with respect to time, we get

del x Js = -[(nse2(B)/m]                                                             (4)

This is known as London’s second equation.

Note: Read the importance of London’s equations in the article:

London equations: explanation of flux penetration

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