A thermal conductor with constant thermal and electrical conductivities, \(k\) and \(\lambda\) respectively, connects two reservoirs at different temperatures and also carries an electrical current of density, \(J_{\mathrm{I}}\). Show that the temperature distribution for one-dimensional flows is given by

\[\frac{\mathrm{d}^{2} T}{\mathrm{~d} x^{2}}-\frac{J_{\mathrm{I}} \sigma}{k} \frac{\mathrm{d} T}{\mathrm{~d} x}+\frac{J_{\mathrm{I}}^{2}}{\lambda}=0\]

where \(\sigma\) is the Thomson coefficient of the wire.