Question #86141

A parallel-plate capacitor is completely filled with a solid isotropic dielectric and has an initial capacitance $C_0$ and initial volume $V_0$ at temperature $T_0$. The capacitor is charged to a constant charge $Q$ and then electrically isolated. The dielectric has a heat capacity at constant pressure $C_p$ (measured when uncharged), a volume thermal expansion coefficient $\gamma$, and its permittivity varies with temperature as $\epsilon(T) = \epsilon_i [1 - \beta (T - T_0)]$, where $\beta > 0$. The capacitor is kept in an environment at a constant atmospheric pressure $P_0$. The conducting plates are thin, perfectly flexible, and rigidly attached to the dielectric, expanding isotropically with it. If the temperature of the system is increased by a small amount $\Delta T$, which of the following expressions gives the total heat $\Delta Q_{heat}$ supplied to the system? (Assume linear approximations hold for small $\Delta T$ and ignore the heat capacity of the thin plates).