Question #43062

A uniform, purely resistive metallic rod of Young's modulus $Y$ (in $\text{N/m}^2$) and coefficient of linear thermal expansion $\alpha$ (in $\text{K}^{-1}$) is rigidly clamped between two fixed, perfectly insulating walls at an initial temperature $T_0$. The rod has a temperature-dependent electrical resistance $R(T) = R_0[1 + \beta(T - T_0)]$, where $R_0$ is the resistance at $T_0$ (in $\Omega$) and $\beta$ is the temperature coefficient of resistivity (in $\text{K}^{-1}$). The rod is connected in parallel with a pure inductor, and this parallel combination is driven by an AC source.

In the steady state, the rod reaches a constant temperature $T > T_0$. At this steady state, the total RMS current drawn from the AC source is $I_0$ (in $\text{A}$), and the overall power factor of the parallel circuit is $\cos\phi$. The rod loses heat to the surroundings (which remain at $T_0$) at a rate $K(T - T_0)$, where $K$ is a positive constant (in $\text{W/K}$).

Assuming the walls do not yield and the rod does not buckle, which of the following expressions gives the compressive thermal stress $\sigma$ (in $\text{N/m}^2$) developed in the rod in the steady state?