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# 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.

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 ns is the density of Cooper pairs, and es 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

$\nabla^2\mathbf{B} - \frac{1}{\lambda_L^2}\mathbf{B}=0$,

where

$\lambda_L = \sqrt{\frac{m}{4\pi n_se_s^2}}$

is the London penetration depth.

## References

• Tinkham, Michael, Introduction to Superconductivity McGraw-Hill, Inc, 1996