Question

A radioactive source, in the form of a metallic sphere of radius $10^{-2}m$ emits $\beta$ -particles at the rate of $5\times 10^{10}$ particles per second. The source is electrically insulated. The time required for its potential to be raised by $2V$ , assuming that $40\%$ of the emitted $\beta$ -particles escape the source is $A\times 10^{-4}s$ . Find $\left[A\right],$ where $\left[\right]$ is the greatest integer function.

Answer 1

Answer: 6

$\beta$ -particle emitted per second
$=\frac{40}{100}\times 5\times 10^{10}=2\times 10^{10}$
Charge on a $\beta$ -particle $=1.6\times 10^{-19}C$
$\therefore$ Charge acquired by sphere in one second
$=\left(2\times 10^{10}\right)\left(1.6\times 10^{-19}\right)$
$=3.2\times 10^{-9}C$
Total charge $=q=4\pi {ϵ}_{0}VR$
$q=\frac{2\times 10^{-2}}{9\times 10^9}=\frac{2}{9}\times 10^{-11}C$
Now, $q=net$
Therefore, $t=\frac{q}{ne}$
$=\frac{\frac{2}{9}\times 10^{-11}}{2\times 10^{10}\times 1.6\times 10^{-19}}$
$=\frac{10^{-2}}{9\times 1.6}=\frac{100}{14.4}\times 10^{-4}=6.94\times 10^{-4}$
Hence, $\left[A\right]=\left[6.94\right]=6$