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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.

Additional recommended knowledge

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,


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

is the London penetration depth.


  • Tinkham, Michael, Introduction to Superconductivity McGraw-Hill, Inc, 1996
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "London_equations". A list of authors is available in Wikipedia.
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