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”

Integral Theorems

INTEGRAL THEOREMS

Gauss’s Divergence Theorem:

Statement. It states that the volume integral of the divergence of a vector  field A, taken over  any volume, V is equal to the surface integral of A taken over the closed surface surrounding  the volume V and vice versa.

Stoke’s Theorem:

Statement. It states  that the surface integral of curl of a vector field over an open surface equals the line integral of the vector field over the closed curve bounding the surface area and vice versa.

Helmholtz’s Theorem: Continue reading “Integral Theorems”