The activity of a radioactive sample is measured as $N_0$ counts per minute at $t = 0$ and $N_0/e$ counts per minute at $t = 5\, \text{minutes}$. The time (in minutes) at which the activity reduces to half of its initial value is

If a radioactive element having a half-life of $30 \ min$ is undergoing beta decay,the fraction of the radioactive element that remains undecayed after $90 \ min$ will be:

$A$ radioactive sample decays by $\beta$-emission. In the first $2 \ s$,$n$ $\beta$-particles are emitted,and in the next $2 \ s$,$0.25n$ $\beta$-particles are emitted. The half-life of the radioactive nuclei is ...... $s$.

The half-life of a stream of radioactive particles moving along a straight path with a constant kinetic energy of $4 \text{ eV}$ is $1 \text{ minute}$. The percentage of particles which decay before travelling a distance of $3.6 \text{ km}$ is (Mass of the radioactive particles $= 3.2 \times 10^{-21} \text{ kg}$ and charge of the electron $= 1.6 \times 10^{-19} \text{ C}$).

An element used for radioactive carbon dating for more than $5600$ years is

At some instant,a radioactive sample $S_1$ having an activity $5\,\mu Ci$ has twice the number of nuclei as another sample $S_2$ which has an activity of $10\,\mu Ci$. The half-lives of $S_1$ and $S_2$ are: