London equations: explanation of flux penetration

As we have already derived the London equations in last article. Now let us

explain the flux penetration (Meissner effect) from London equations:

To explain Meissner effect from London equations consider the differential form of Ampere’s circuital law:

del x B = µoJs

where B is magnetic flux density and Js is current density

Take curl on both sides of above equation

del x (del x B) = µo (del x Js)                                                     (5)

As del x (del  x B)= del(del.B) – del2B

Put above equation and London second equation (equation 4 is derived in last article) in equation (5), we get

del(del.B) – del2B = -[( µo nse2(B)/m]

But del.B = 0 (Maxwell’s second equation or Gauss law for magnetism)

Therefore above equation becomes

del2B = [( µo nse2(B)/m]                                                            (6)

del2B = B/λl2 (7)

where λl2 = m/ µo nse2

or λl = (m/ µo nse2)1/2

where λl is known as London’s penetration depth and it has units of length.

The solution of differential equation (7) is

B = B(0)e-x/ λl (8)

Where B(0) is the field at the surface and x is the depth inside the superconductor. Continue reading “London equations: explanation of flux penetration”

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: Continue reading “London equations in superconductors: derivation and discussion”