Estimate the temperature profile of a wire of length L and radius r when an electric current I is driven through it, from the left end to the right end, causing it to heat up. The wire has an electric conductivity σ and a thermal conductivity κ. The heat propagates down the wire to its ends, i.e. no heat is dissipated at the surface of the wire. The Thomson heating is negligible compared to the Joule heating. The left end and right end are kept at a constant temperature T₀. Determine the temperature profile T (x) along the wire when it has reached a stationary state.