  Under mesoscopic conditions, the transport potential on a thin film with
current is theoretically expected to bear spatial variation due to quantum
interference. Scanning tunneling potentiometry is the ideal tool to investigate
such variation, by virtue of its high spatial resolution. We report in this
{\it Letter} the first detailed measurement of transport potential under
mesoscopic conditions. Epitaxial graphene at a temperature of 17K was chosen as
the initial system for study because the characteristic transport length scales
in this material are relatively large. Tip jumping artifacts are a major
possible contribution to systematic errors; and we mitigate such problems by
using custom-made slender and sharp tips manufactured by focussed ion beam. In
our data, we observe residual resistivity dipoles associated with
topoographical defects, and local peaks and dips in the potential that are not
associated with topographical defects.
