For a radioactive material, its activity $A$ and rate of change of its activity $R$ are defined as $A=-\frac{d N}{d t}$ and $R=-\frac{d A}{d t}$, where $N(t)$ is the number of nuclei at time $t$. Two radioactive sources $P$ (mean life $\tau$ ) and $Q$ (mean life $2 \tau$ ) have the same activity at $t=0$. Their rates of change of activities at $t=2 \tau$ are $R_p$ and $R_Q$, respectively. If $\frac{R_p}{R_Q}=\frac{n}{e}$, then the value of $n$ is

Half-life is measured by

A radioactive sample has an average life of $30\, {ms}$ and is decaying. A capacitor of capacitance $200\, \mu\, {F}$ is first charged and later connected with resistor $^{\prime}{R}^{\prime}$. If the ratio of charge on capacitor to the activity of radioactive sample is fixed with respect to time then the value of $^{\prime}R^{\prime}$ should be $....\,\Omega$

The half life of radioactive Radon is $3.8$ days. The time at the end of which $1/{20^{th}}$ of the Radon sample will remain undecayed is ........... $day$ (Given ${\log _{10}}e = 0.4343$)

$37$ Rutherford equals

Draw a graph of the time $t$ versus the number of undecay nucleus in a radioactive sample and write its characteristics.