Question #43065

An isolated LCR circuit consists of an inductor of inductance $L$, a capacitor of capacitance $C$, and a resistor of mass $m$, specific heat capacity $s$, and initial resistance $R_0$ at ambient temperature $T_0$. The resistance varies with temperature $T$ as $R(T) = R_0 [1 + \alpha (T - T_0)]$, where $\alpha$ is the temperature coefficient of resistance. At $t=0$, the capacitor is charged to a voltage $V_0$ (giving an initial total energy $E_0 = \frac{1}{2} C V_0^2$) and the circuit is closed. The resistor is perfectly thermally insulated from its surroundings. Assume the circuit is highly underdamped ($Q \gg 1$) so that the fractional energy loss per cycle is very small, allowing the cycle-averaged power dissipation to be approximated as $\langle P \rangle = \frac{E}{L} R(T)$, where $E$ is the instantaneous total electrical energy of the circuit. Which of the following expressions gives the time $t$ at which the total electrical energy of the circuit reduces to $E_0 / 2$? (Let $\gamma = \frac{\alpha E_0}{ms}$)