 Joule-heated cylindrical wire

A long wire with uniform current density $I/(\pi a^2)$ generates uniform heat per volume $H=I^2/({\pi }^2{a}^4\sigma )$. Radial steady conduction obeys $(r{T}_r{)}_r=-Hr/k$. Regularity at the centre removes a logarithmic solution, leaving the displayed parabolic temperature excess relative to the surface. Conductivity, electrical conductivity and current density are assumed constant.