Capacitors in DC Circuits & RC question from JEE Main 2022 (29 Jun, Shift 2)

Answer: (d)

1. Discharging: $q = q_0e^{-t/\tau}$ and $U = U_0e^{-2t/\tau}$, with $\tau = RC$.
2. Energy halves: $e^{-2t_1/\tau} = \frac{1}{2} \Rightarrow t_1 = \frac{\tau\ln 2}{2}$
3. Charge to one eighth: $e^{-t_2/\tau} = \frac{1}{8} \Rightarrow t_2 = 3\tau\ln 2$
4. $\frac{t_1}{t_2} = \frac{1/2}{3} = \frac{1}{6}$
5. So the answer is (d) $1/6$.