Question #84223

A uniform wire of initial length $L_0$, cross-sectional area $A$, and perimeter $p$ is suspended in a vacuum chamber maintained at a constant ambient temperature $T_0$. A constant mechanical tension $F$ is applied to the wire. Simultaneously, a constant direct current $I$ is passed through it. The resistivity of the wire varies with temperature as $\rho(T) = \rho_0 [1 + \beta(T - T_0)]$, where $\rho_0$ is the resistivity at $T_0$. Due to lattice anharmonicity, the Young's modulus of the wire decreases with temperature as $Y(T) = Y_0 [1 - \alpha(T - T_0)]$. The wire loses heat to the surroundings purely by radiation. The emissivity of the wire surface is $\epsilon$, and the Stefan-Boltzmann constant is $\sigma$. Assuming the steady-state temperature rise $\Delta T = T - T_0$ is small ($\Delta T \ll T_0$), what is the steady-state elastic elongation $\Delta L$ of the wire? (Ignore the thermal expansion of the wire and assume the cross-sectional area remains approximately constant).