The current-voltage relation within narrow auroral current sheets is examined through the use of high-resolution data from the high-altitude Dynamics Explorer 1 satellite. The north-south perpendicular electric field and the east-west magnetic field are shown for three cases in which there are large amplitude, oppositely directed paired electric fields which are confined to a region less than 20 km wide. The magnetic field variations are found to be proportional to the second integral of the high-altitude perpendicular electric field. It is shown that at the small-scale limit, this relationship between ΔB and E is consistent with a linear ''Ohm's law'' relationship had previously been verified for large-scale auroral formations greater than 20 km wide at the ionosphere. The evidence shown here extends our knowledge down to the scale size of discrete auroral arcs.