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

$\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 $=q=4\pi \epsilon _{0}VR$
$q=\frac{2 \times 10^{- 2}}{9 \times 10^{9}}=\frac{2}{9}\times 10^{- 11}C$
Now, $q=net$
Therefore, $t=\frac{q}{n e}$
$=\frac{\frac{2}{9} \times 10^{- 11}}{2 \times 10^{10} \times 1 . 6 \times 10^{- 19}}$
$=\frac{10^{- 2}}{9 \times 1 . 6}=\frac{100}{14 . 4}\times 10^{- 4}=6.94\times 10^{- 4}$
Hence, $\left[A\right]=\left[6 . 94\right]=6$