My watch list

London equations

The London equations relate the current to electromagnetic fields in and around a superconductor. Their purpose is to describe the magnetic field exclusion that is characteristic of a superconductor, and known as the Meissner effect.

Additional recommended knowledge

Daily Sensitivity Test

Don't let static charges disrupt your weighing accuracy

Safe Weighing Range Ensures Accurate Results

The first London equation relates the superconducting current to the electric field:

$\frac{\partial \mathbf{j}_s}{\partial t} = \frac{n_se_s^2}{m}\mathbf{E}$ ,

where $n s$ is the density of Cooper pairs, and $e s$ and m are the charge and mass of a Cooper pair, respectively, which is simply twice the charge and mass of an electron.

The second London equation relates the supercurrent to the magnetic field:

$\mathbf{\nabla}\times\mathbf{j}_s =-\frac{n_se_s^2}{m}\mathbf{B}$ .

Writing the magnetic field in terms of the vector potential $\mathbf{B} = \mathbf{\nabla}\times\mathbf{A}$ , we find that the current is simply,

$\mathbf{j}_s = -\frac{n_se_s^2}{m}\mathbf{A} - \frac{n_s\hbar}{m}\mathbf{\nabla}\phi$ ,

where $φ$ is an arbitrary phase. Substituting this equation into the fourth of Maxwell's equations, $\mathbf{\nabla}\times\mathbf{B} = 4\pi\mathbf{j}_s$ , and making use of Maxwell's third equation, $\mathbf{\nabla}\cdot\mathbf{B}=0$ , we have