First slide

Radio activity

Question

An isolated radioactive source in the form a metal sphere of radius 1 mm undergoes $β$ –ray decay at a constant rate of 6.25 × 10¹⁰ /sec. Assuming 70% of the emitted $β$ –particles escape from the source, time after which potential of the sphere rises by 1 V is

Moderate

A

15.5 $μ$ S
B

20 $μ$ S
C

25 $μ$ S
D

10 $μ$ S

Solution

$V = \frac{1}{4 {πε}_{0}} \frac{q}{r} \Rightarrow 1 = \frac{9 x 10^{9} x n^{1} e}{10^{- 3}} \Rightarrow n^{1} = \frac{10^{- 3}}{9 x 10^{9} x 1.6 x 10^{- 19}} = \frac{10^{7}}{14.4} Since n^{1} = \frac{70}{100} (n) [n \to no . of β particle s emitted] \Rightarrow n = \frac{100}{70} x \frac{10^{7}}{14.4} \Rightarrow n = \frac{10^{8}}{7 x 14.4}$
For 6.25 x10¹⁰ particles → 1sec
$For \frac{10^{8}}{7 x 14.4} particles \to' t' \sec ∴ t = \frac{(\frac{10^{8}}{14.4 x 7})}{6.25 x 10^{10}} = 15.5 x 10^{- 6} \sec = 15.5 μ \sec$

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