  Recent radioastronomical observations of Faraday rotation in the solar corona
can be interpreted as evidence for coronal currents, with values as large as
$2.5 \times 10^9$ Amperes (Spangler 2007). These estimates of currents are used
to develop a model for Joule heating in the corona. It is assumed that the
currents are concentrated in thin current sheets, as suggested by theories of
two dimensional magnetohydrodynamic turbulence. The Spitzer result for the
resistivity is adopted as a lower limit to the true resistivity. The calculated
volumetric heating rate is compared with an independent theoretical estimate by
Cranmer et al (2007). This latter estimate accounts for the dynamic and
thermodynamic properties of the corona at a heliocentric distance of several
solar radii. Our calculated Joule heating rate is less than the Cranmer et al
estimate by at least a factor of $3 \times 10^5$. The currents inferred from
the observations of Spangler (2007) are not relevant to coronal heating unless
the true resistivity is enormously increased relative to the Spitzer value.
However, the same model for turbulent current sheets used to calculate the
heating rate also gives an electron drift speed which can be comparable to the
electron thermal speed, and larger than the ion acoustic speed. It is therefore
possible that the coronal current sheets are unstable to current-driven
instabilities which produce high levels of waves, enhance the resistivity and
thus the heating rate.
