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Let us first consider the magnetic penetration depth
in the context of the conventional
London theory [7] at .
If a magnetic field is applied to a superconductor
which is initially in zero field, the magnetic field is a function
of time. According to the Maxwell equation
,the time-varying magnetic field gives rise to an electric field. In a
normal metal this will induce eddy currents, but in a superconductor the
-field will give rise to persistent currents (*i.e.*
supercurrents).
The induced supercurrents will in turn generate a magnetic field of
their own which opposes the applied magnetic field. If the applied
magnetic field is weak, the flux is totally screened from
the bulk of the superconductor.
This phenomenon is often described as ``perfect diamagnetism''.
From Newton's law, the equation of motion
for a superconducting carrier with mass *m* and charge -*e*
in the presence of an electric field is

| |
(6) |

where is the velocity of the superconducting carrier.
The field-induced supercurrent density is given by
| |
(7) |

where *n*_{s} is the local density of superconducting carriers.
Substituting Eq. (2.7) into Eq. (2.6) gives
| |
(8) |

which is known as the ``first London equation''.
Taking the *curl* of both sides of Eq. (2.8) gives
| |
(9) |

which can be rewritten using the Maxwell equation
to give
| |
(10) |

In order to obtain the Meissner effect (*i.e.* in the
bulk of the superconductor) the London brothers removed the time
derivative in Eq. (2.10) and postulated the new equation
| |
(11) |

Equation (2.11) is commonly
referred to as the ``second London equation''.
Since the supercurrent density is related to the
field by another Maxwell equation
| |
(12) |

substitution of Eq. (2.12) into Eq. (2.11) gives
| |
(13) |

where
| |
(14) |

The variable is called the London penetration depth.
Note that is proportional to the superfluid
density *n*_{s}.
The relationship between and
is best realized by a simple example. Consider the
vacuum-superconductor interface illustrated in Fig. 2.1.
If a field
is applied parallel to the surface of the superconductor,
the solution of Eq. (2.13) is

| |
(15) |

Thus, both the magnetic field *B*(*x*) and the
supercurrent density *J*_{s}(*x*) decay exponentially with distance inside
the superconductor over the length scale .

Equation (2.14) was derived for .At nonzero temperature the behaviour of can
be approximated by incorporating the two-fluid model of Gorter
and Casimir [8]. In this model the electron system
is assumed to contain a superconducting component with electron
density *n*_{s}, and a normal component with an electron density
*n*_{n}. The total electron density becomes
at , and
for , where *T*_{c} is
the superconducting transition temperature.
For arbitrary temperature, Gorter and Casimir found that good
agreement with early experiments could be obtained if one
assumes that .When this expression is combined with Eq. (2.14)
the penetration depth is given by

| |
(16) |

As the temperature *T* increases, *n*_{s} is reduced and the magnetic
penetration depth increases with *T*,
such that as .Although this relation gives a good first-order description
of many of the experimental
results for conventional superconductors, the temperature dependence
is not a prediction from the two-fluid model. Better agreement
is obtained with the behaviour predicted from the
microscopic theory developed by
Bardeen, Cooper and Schrieffer (BCS) [9].
Before considering the BCS theory, let us work backwards to determine
the kernel in Eq. (2.2) for the London theory.
The second London equation [*i.e.* Eq. (2.13)] can be rewritten
using the Maxwell
relation in Eq. (2.12) and the relation
to give

| |
(17) |

which is easily solved for to give
| |
(18) |

The th Fourier component of is thus
| |
(19) |

Comparing Eq. (2.19)
with Eq. (2.2) gives the following simple result for
the London kernel
| |
(20) |

Clearly, *Q*_{L} is independent of .
This implies that the London theory
is a local theory, where the value of at a point only depends on the value of at that same point .Generally, the value of in the
vicinity of the point where is being measured is
also important. In a more general nonlocal theory, depends
on .
The current density due to the normal electrons is

| |
(21) |

where is the finite conductivity of the normal fluid.
Thus, the total current density at arbitrary temperature
below *T*_{c} is simply

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