A superconducting wire expells a magnetic field.The Proca-formulation of electrodynamics allows to account for a hypothetical massive photon. This formulation lead to astonishing experiments but is formally equivalent to the London theory of superconductivity. Learn in this problem how a magnetic field is expelled from a superconductor by "massive photons".

Problem Statement

A slab superconductor with a constant magnetic field.A superconducting half-plane is terminated at \(z=0\). Outside of the superconductor, there is a magnetostatic field \(\mathbf{B}\left(\mathbf{r}\right)=B_{0}\mathbf{e}_{y}\) parallel to the surface. Inside the superconductor, the Proca equation \[\left(\Delta-\mu_{\mathrm{L}}^{2}\right)\mathbf{A}\left(\mathbf{r}\right) = 0\] shall be applicable.
Calculate the magnetic induction with respect to the London penetration depth \(\lambda_{\mathrm{L}}=1/\mu_{\mathrm{L}}\). Now consider the case if the superconductor is a slab with a finite thickness \(d\), centered at \(z=0\). What does the induction look like now? Calculate an effective current \(\mathbf{j}\left(\mathbf{r}\right)\) inside the superconductor consistent with the calculated magnetic induction. Discuss qualitatively how the magnetic field and current should look like if one replaces the superconducting slab with a superconducting wire of some radius.

Background: Photon Mass and Superconductivity

If photons had a nonvanishing mass, the electromagnetic fields would show different characteristics than those we are used to: a wavelength dependence of the speed of light, modifications of Coulomb's and Ampères law and thus different fields for charges and dipoles and so on. Without a doubt, the question for a photon mass is very fundamental. Here, however, we can use it to explain superconductivity following the Londons. For much more information please have a look at The "Mass" of the Photon - Magnetic Fields in Superconductors.

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