In the auroral zone of the earth currents flow along magnetic field lines. In the downward current region currents are mainly carried by upflowing electrons from the ionosphere. Because of the low plasma density along auroral field lines, substantial currents in the range of microamps per square meter require substantial potential drops parallel to the magnetic field in the range of a few hundred to a few thousand volts. The current-voltage relation along such magnetic field lines can be determined for simple profiles of the background ion density by invoking the condition of charge neutrality. For typical parameters, the current density is found to be a few times larger in the downward current region compared to currents in the upward current region for similar potential drops. Thus potential drops up to a few thousand volts and the consequent acceleration of ionospheric electrons up to keV energies, such as has been observed by the FAST satellite, are a necessary consequence of the observed current densities in the downward auroral current region. ¿ 1998 American Geophysical Union