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We extend a theorem of Bloch, which concerns the net orbital current carried by an interacting electron system in equilibrium, to include mesoscopic effects. We obtain a rigorous upper bound to the allowed ground-state current in a ring or disc, for an interacting electron system in the presence of static but otherwise arbitrary electric and magnetic fields. We also investigate the effects of spin-orbit and current-current interactions on the upper bound. Current-current interactions, caused bydoi:10.1103/physrevb.53.9550 pmid:9982497 fatcat:nq2ey4srhfaxjkwqhpuug4v46a