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13.Nuclei
normal
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
A
$1$
B
$2$
C
$3$
D
$4$
(IIT-2015)
Solution
$\lambda_{ P }=\frac{1}{\tau} ; \lambda_{ Q }=\frac{1}{2 \tau}$
$\frac{ R _{ P }}{ R _{ Q }}=\frac{\left(A_0 \lambda_{ P }\right) e ^{-\lambda_2 t }}{A_0 \lambda_{ Q } e ^{-\lambda_{ Q } t}}$
$\text { At } t =2 \tau ; \frac{ R _{ P }}{ R _{ Q }}=\frac{2}{ e }$
Standard 12
Physics
