  We consider the Johnson noise of a two-dimensional, two-terminal electrical
conductor for which the electron system obeys the Wiedemann-Franz law. We
derive two simple and generic relations between the Johnson Noise temperature
and the heat flux into the electron system. First, we consider the case where
the electron system is heated by Joule heating from a DC current, and we show
that there is a universal proportionality coefficient between the Joule power
and the increase in Johnson noise temperature. Second, we consider the case
where heat flows into the sample from an external source, and we derive a
simple relation between the Johnson noise temperature and the heat flux across
the boundary of the sample.
