Q 12-13-097JEE MainJEE Main 2021 (27 Jul, Shift 1)Medium
A radioactive sample has an average life of $30$ ms and is decaying. A capacitor of capacitance $200\ \mu\text{F}$ is first charged and later connected with resistor $R$. If the ratio of the charge on the capacitor to the activity of the radioactive sample is fixed with respect to time then the value of $R$ should be ______ $\Omega$.
Numerical value type. Enter your answer.
Answer: 150
Charge: $q = q_0e^{-t/RC}$. Activity: $A = A_0e^{-t/\tau}$ where $\tau$ is the mean life.
The ratio stays constant only if $RC = \tau$:
$$R = \frac{30\times10^{-3}}{200\times10^{-6}} = 150\ \Omega$$
Solution by Sreeraj P, M.Sc Physics