In a superconductor the number density of charge carriers of charge $q$ is $n_{s}$. Suppose that there is a time-independent magnetic field described by the three-vector potential $\mathbf{A} .$

Derive an expression for the superconducting current.

Explain how your answer is gauge invariant.

Suppose that for $z<0$ there is a constant magnetic field $\mathbf{B}_{0}$ in a vacuum and, for $z>0$, there is a uniform superconductor. Derive the magnetic field for $z>0$.