Joule-heated cylindrical wire ($T(r)-T(a)=I^2(a^2-r^2)/(4{\pi }^2{a}^{4}k\sigma )$)

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.

The surface flux of a uniformly heated cylinder is $Ha/2$. Equating it to the Newton-cooling flux $\alpha [T(a)-T_{\mathrm{air}}]$ gives the displayed surface temperature. The full temperature profile adds the internal parabolic excess to this surface value. Electrical power per length equals the surface heat loss per length in steady state.