A model of the scrape-off layer outside the separatrix of a diverted tokamak plasma is used to calculate the current flowing into the divertor plates. The model extends previous work by P.J. Harbour. This current can be due to thermoelectric effects or to an externally applied voltage. It is found that when the target plates are grounded, as in present-day experiments, asymmetries in the power deposition to the target plates can drive current along magnetic field lines in the scrape-off layer. These currents can modify the collisional transport across the separatrix.