We show that the differential contribution, at energy E, to the persistent currents of independent electrons in infinitely extended quantum systems is given by (2\ensuremath{\pi}i${)}^{\mathrm{\ensuremath{-}}1}$${\mathrm{\ensuremath{\partial}}}_{\mathrm{\ensuremath{\varphi}}}$ [ln detS(E,\ensuremath{\varphi})]dE, where S(E,\ensuremath{\varphi}) 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.