Electrical leads carrying currents into cryogenic apparatus also introduce heat. Even with an ideal Carnot cycle, the mechanical power needed to remove this heat can be one hundred or more times the heat flow itself. If the currents and hence the input leads are heavy, a very sizeable refrigerator may be required. In this article the configuration of the leads which minimizes the influx of heat is derived theoretically, taking variations in thermal and electrical conductivity into account. Graphs are given for the minimum heat flow and optimum cross section of a copper input lead carrying arbitrary currentI. The optimum is found to be fairly sharp. If the diameter of the lead differs by a factor of two from the optimum, the influx of heat is increased by over 100%.