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”