In a superconductor, there are superconducting charge carriers with number density $n$, mass $m$ and charge $q$. Starting from the quantum mechanical wavefunction $\Psi=R e^{i \Phi}$ (with real $R$ and $\Phi$ ), construct a formula for the electric current and explain carefully why your result is gauge invariant.

Now show that inside a superconductor a static magnetic field obeys the equation

$$\nabla^{2} \mathbf{B}=\frac{\mu_{0} n q^{2}}{m} \mathbf{B}$$

A superconductor occupies the region $z>0$, while for $z<0$ there is a vacuum with a constant magnetic field in the $x$ direction. Show that the magnetic field cannot penetrate deep into the superconductor.