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:
∂ j s ∂ t = n s e 2 m E , ∇ × j s = − n s e 2 m B . {\displaystyle {\frac {\partial \mathbf {j} _{\rm {s}}}{\partial t}}={\frac {n_{\rm {s}}e^{2}}{m}}\mathbf {E} ,\qquad \mathbf {\nabla } \times \mathbf {j} _{\rm {s}}=-{\frac {n_{\rm {s}}e^{2}}{m}}\mathbf {B} .}
Here j s {\displaystyle {\mathbf {j} }_{\rm {s}}} is the (superconducting) current density, E and B are respectively the electric and magnetic fields within the superconductor,
e {\displaystyle e\,} is the charge of an electron or proton,
m {\displaystyle m\,} is electron mass, and
n s {\displaystyle n_{\rm {s}}\,} 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 A s {\displaystyle \mathbf {A} _{\rm {s}}} which has been gauge fixed to the "London gauge", giving:
j s = − n s e 2 m A s . {\displaystyle \mathbf {j} _{\rm {s}}=-{\frac {n_{\rm {s}}e^{2}}{m}}\mathbf {A} _{\rm {s}}.}
In the London gauge, the vector potential obeys the following requirements, ensuring that it can be interpreted as a current density:
∇ ⋅ A s = 0 , {\displaystyle \nabla \cdot \mathbf {A} _{\rm {s}}=0,}
A s = 0 {\displaystyle \mathbf {A} _{\rm {s}}=0} in the superconductor bulk,
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