Question #42162

A long cylindrical metal rod of initial radius $r_0$ and length $L_0$ is placed between two perfectly rigid, unyielding walls such that its length cannot change. Assume the rod is initially unstressed at $0 \text{ K}$ and its material properties (emissivity $\epsilon$, coefficient of linear thermal expansion $\alpha$, Young's modulus $Y$, and Poisson's ratio $\nu$) are temperature-independent. It is heated by a constant internal power source $P$ and radiates heat to the surrounding vacuum at $0 \text{ K}$. Neglecting heat loss from the flat ends and assuming $\alpha T \ll 1$, what is the steady-state temperature $T$ of the rod to first order in $\alpha$?

(Let $T_0 = \left( \frac{P}{\epsilon \sigma 2\pi r_0 L_0} \right)^{1/4}$, where $\sigma$ is the Stefan-Boltzmann constant).