Steady state Joule heating with temperature dependent conductivities

A general solution is given to the nonlinear steady state heat conduction equation for the case in which a metal is heated by electrical conduction currents. The solution is valid for any temperature variation in the thermal and electrical conductivities and is illustrated by application to a typical Joule heating situation in one dimension. Comparison of the case of temperature dependent conductivities for a metal with the solution which assumes both of these conductivities to remain constant with varying temperature reveals some interesting differences between the two cases.