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.

There are two London equations when expressed in terms of measurable fields:

Here ${\displaystyle {\mathbf {j} }_{\rm {s}}}$ is the (superconducting) current density, E and B are respectively the electric and magnetic fields within the superconductor, ${\displaystyle e\,}$ is the charge of an electron or proton, ${\displaystyle m\,}$ is electron mass, and ${\displaystyle n_{\rm {s}}\,}$ 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 ${\displaystyle \mathbf {A} _{\rm {s}}}$ 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:

The first requirement, also known as Coulomb gauge condition, leads to the constant superconducting electron density ${\displaystyle {\dot {\rho }}_{\rm {s}}=0}$ as expected from the continuity equation. The second requirement is consistent with the fact that supercurrent flows near the surface. The third requirement ensures no accumulation of superconducting electrons on the surface. 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 ${\displaystyle \mathbf {A} }$ by simply defining ${\displaystyle \mathbf {A} _{\rm {s}}=(\mathbf {A} +\nabla \phi )}$, where ${\displaystyle \phi }$ is a scalar function and ${\displaystyle \nabla \phi }$ 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.

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: