London equations

The London equations, developed by brothers Fritz and Heinz London in 1935, are constitutive relations for a superconductor relating its superconducting current to electromagnetic fields in and around it. Whereas Ohm's law is the simplest constitutive relation for an ordinary conductor, the London equations are the simplest meaningful description of superconducting phenomena, and form the genesis of almost any modern introductory text on the subject. A major triumph of the equations is their ability to explain the Meissner effect, wherein a material exponentially expels all internal magnetic fields as it crosses the superconducting threshold.
Description.
There are two London equations when expressed in terms of measurable fields: 
Here formula_2 is the (superconducting) current density, E and B are respectively the electric and magnetic fields within the superconductor, 
formula_3 
is the charge of an electron or proton, 
formula_4 
is electron mass, and 
formula_5 
is a phenomenological constant loosely associated with a number density of superconducting carriers.
The two equations can be combined into a single "London Equation"
in terms of a specific vector potential formula_6 which has been gauge fixed to the "London gauge", giving:
In the London gauge, the vector potential obeys the following requirements, ensuring that it can be interpreted as a current density:
These requirements do away with all gauge freedom and uniquely determine the vector potential. One can also write the London equation in terms of an arbitrary gauge formula_12 by simply defining formula_13, where formula_14 is a scalar function and formula_15 is the change in gauge which shifts the arbitrary gauge to the London gauge.
The vector potential expression holds for magnetic fields that vary slowly in space.
London penetration depth.
If the second of London's equations is manipulated by applying Ampere's law,
then it can be turned into the Helmholtz equation for magnetic field:
where the inverse of the laplacian eigenvalue:
is the characteristic length scale, formula_19, over which external magnetic fields are exponentially suppressed: it is called the London penetration depth: typical values are from 50 to 500 nm.
For example, consider a superconductor within free space where the magnetic field outside the superconductor is a constant value pointed parallel to the superconducting boundary plane in the "z" direction. If "x" leads perpendicular to the boundary then the solution inside the superconductor may be shown to be
From here the physical meaning of the London penetration depth can perhaps most easily be discerned.
Rationale for the London equations.
Original arguments.
While it is important to note that the above equations cannot be formally derived,
the Londons did follow a certain intuitive logic in the formulation of their theory. Substances across a stunningly wide range of composition behave roughly according to Ohm's law, which states that current is proportional to electric field. However, such a linear relationship is impossible in a superconductor for, almost by definition, the electrons in a superconductor flow with no resistance whatsoever. To this end, the London brothers imagined electrons as if they were free electrons under the influence of a uniform external electric field. According to the Lorentz force law 
these electrons should encounter a uniform force, and thus they should in fact accelerate uniformly. This is precisely what the first London equation states.
To obtain the second equation, take the curl of the first London equation and apply Faraday's law,
to obtain
As it currently stands, this equation permits both constant and exponentially decaying solutions. The Londons recognized from the Meissner effect that constant nonzero solutions were nonphysical, and thus postulated that not only was the time derivative of the above expression equal to zero, but also that the expression in the parentheses must be identically zero. This results in the second London equation.
Canonical momentum arguments.
It is also possible to justify the London equations by other means.
Current density is defined according to the equation
Taking this expression from a classical description to a quantum mechanical one, we must replace values j and v by the expectation values of their operators. The velocity operator
is defined by dividing the gauge-invariant, kinematic momentum operator by the particle mass "m". Note we are using formula_26 as the electron charge.
We may then make this replacement in the equation above. However, an important assumption from the microscopic theory of superconductivity is that the superconducting state of a system is the ground state, and according to a theorem of Bloch's,
in such a state the canonical momentum p is zero. This leaves
which is the London equation according to the second formulation above.