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 = µ_oJ_s

where B is magnetic flux density and J_s is current density

Take curl on both sides of above equation

del x (del x B) = µ_o (del x J_s)                                                     (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 = -[( µ_o n_se²(B)/m]

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

Therefore above equation becomes

del²B = [( µ_o n_se²(B)/m]                                                            (6)

del²B = B/λ_l² (7)

where λ_l² = m/ µ_o n_se²

or λ_l = (m/ µ_o n_se²)^1/2

where λ_l 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/ λ}_l (8)

Where B(0) is the field at the surface and x is the depth inside the superconductor.

The equation (8) shows that a uniform magnetic field equal to zero can not exist in a superconductor, which is Meissner effect. In the pure superconducting state the only field allowed in the exponentially decreasing field as the flux penetrated from external surface and it is given by equation (8) (Refer figure).

Suppose x = λ_l

Then equation (8) becomes

B = B(0)/e

Definition of London penetration depth: The London penetration depth is the distance inside the surface of a superconductor at which the magnetic field reduces to 1/e times its value at the surface.

The London penetration depth depends strongly on the temperature and becomes much larger as T approaches critical temperature Tc. The relation is

λ_l(T)/ λ_l(0)= [1 – T/T_c)⁴]^-1/2

where λ_l(T) and λ_l(0) are the London penetration depths at temperature T kelvin and 0 k respectively.