Question #43347

An $LC$ circuit consists of a capacitor of capacitance $C$ initially charged to a voltage $V_0$, an ideal inductor of inductance $L$, and a thermally insulated resistor. The initial resistance of the resistor is $R_0$. As the circuit oscillates, the resistor heats up due to Joule heating. The resistance varies with the total heat energy $H$ dissipated in it according to the relation $R(H) = R_0 (1 + \alpha H)$, where $\alpha$ is a positive constant. It is given that if the entire initial stored energy of the capacitor were to be dissipated in the resistor, its resistance would become $3 R_0$. Assuming the circuit is highly underdamped ($R \ll \sqrt{L/C}$) such that the fractional energy loss per cycle is negligible, find the time $t$ taken for the total electromagnetic energy of the circuit to reduce to half of its initial value.