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. Continue reading “London equations: explanation of flux penetration”