In a superconductor, the charge carriers have a charge $q$, mass $m$ and number density $n$. Describe how to construct the superconducting current in terms of the vector potential A and the wavefunction of the charge carriers.

Show that the current is gauge invariant.

Derive the Helmholtz equation

$$\nabla^{2} \mathbf{B}=\mathbf{B} / \ell^{2}$$

for a time-independent magnetic field and evaluate the length scale $\ell$ in terms of $n, q$ and $m$.

Why does this imply that magnetic flux cannot exist in a superconductor?