# 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 = µ_{o}J_{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) – del^{2}B

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

del(del.B) – del^{2}B = -[( µ_{o} n_{s}e^{2}(B)/m]

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

Therefore above equation becomes

del^{2}B = [( µ_{o} n_{s}e^{2}(B)/m] (6)

del^{2}B = B/λ_{l}^{2 }(7)

where λ_{l}^{2} = m/ µ_{o} n_{s}e^{2}

or λ_{l} = (m/ µ_{o} n_{s}e^{2})^{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).

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.

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