Relation between persistent currents and the scattering matrix

We show that the differential contribution, at energy E, to the persistent currents of independent electrons in infinitely extended quantum systems is given by (2πi${)}^{\mathrm{-}1}$ ${\mathrm{\partial }}_{\mathrm{\phi }}$ [ln detS(E,φ)]dE, where S(E,φ) is the (on-shell) scatteirng matrix. We apply this result to the calculation of the persistent currents in two examples: a mesoscopic loop connected to one infinitely long lead, and a plane pierced by a flux line. In the last example spin plays a remarkable role.