Coulomb-gauge vector potential of a long solenoid

For an ideal infinitely long solenoid with interior magnetic field $B_0\hat{\mathbf{z}}$ and radius $R$, an axisymmetric Coulomb gauge representative of the magnetic vector potential isThis follows from $2\pi r{A}_{\varphi }=\Phi (r)$ by Stokes theorem. The exterior has zero curl but nonzero circulation around the enclosed magnetic flux. The Coulomb gauge condition alone does not eliminate every residual gauge transformation; symmetry and regularity select this customary representative.