Question #43882

A cylindrical metal rod is rigidly clamped between two immovable, perfectly insulating walls. The rod acts as the only resistor in a series LCR circuit driven by an AC voltage source $V(t) = V_0 \sin(\omega t)$. Initially, at temperature $T_0$, the circuit has a phase angle $\phi_0 = \tan^{-1}(2)$, with the voltage leading the current.

The temperature of the rod is then uniformly increased by $\Delta T$. Due to the elastoresistance effect, the resistivity of the rod material varies with both temperature and compressive mechanical stress $\sigma_c$ according to the relation $\rho = \rho_0 (1 + \beta \Delta T + \gamma \sigma_c)$, where $\rho_0$ is the resistivity at $T_0$.

Given the following properties for the rod material: $\bullet$ Temperature coefficient of resistivity, $\beta = 5.0 \times 10^{-3} \text{ K}^{-1}$ $\bullet$ Elastoresistance coefficient, $\gamma = 1.0 \times 10^{-9} \text{ m}^2/\text{N}$ $\bullet$ Young's modulus, $Y = 2.0 \times 10^{11} \text{ N/m}^2$ $\bullet$ Coefficient of linear thermal expansion, $\alpha = 1.5 \times 10^{-5} \text{ K}^{-1}$

Assume the cross-sectional area of the rod remains practically constant and the walls do not yield. If the new phase angle of the circuit becomes $\phi = \frac{\pi}{4}$ rad, what is the value of the temperature increase $\Delta T$?