Question #43330

An air-filled parallel plate capacitor is constructed using two identical flat metal plates, each of initial area $A_0$ and coefficient of linear thermal expansion $\alpha$. The plates are held parallel to each other at an initial separation $d_0$ by a composite insulating spacer that keeps the plates perfectly parallel. The spacer consists of Material 1 (Young's modulus $2Y$, total cross-sectional area $S$, coefficient of linear expansion $\alpha$) and Material 2 (Young's modulus $Y$, total cross-sectional area $S$, coefficient of linear expansion $3\alpha$). Both materials are rigidly fastened to the plates at their ends, forcing them to have the same length at all temperatures. The capacitor is connected to an AC voltage source of constant angular frequency $\omega$. If the temperature of the entire assembly is increased by $\Delta T$, what is the fractional change in the capacitive reactance ($\frac{\Delta X_C}{X_{C0}}$) of the capacitor? (Assume $\Delta T$ is small enough that higher-order terms are negligible).