Question

A radioactive source in the form of a metal sphere of radius \[{10^{ - 2}}\] emits beta particles (electrons) at the rate of \[5 \times {10^{10}}\] particles per second. The source is electrically insulated. How long will it take for its potential to be raised by 2 V assuming that 40% of the emitted beta particles escape from the source.

Answer 1

Hint:Before we proceed with the problem let’s see what is given. They have given the radius of the metal sphere, the rate at which the beta particle emits, and the potential. We need to find how much time it will take if the potential is raised by 2V. for this we need to find the charge first then calculate the time taken.

Formula Used:
The formula to find the potential of a sphere is given by,
\[V = \dfrac{1}{{4\pi {\varepsilon _0}}}\dfrac{q}{r}\]……… (1)
Where, \[q\] is charge on the sphere and \[r\] is radius of the sphere.

Complete step by step solution:
Consider equation (1) and rearrange the equation for q
\[q = 4\pi {\varepsilon _0}Vr\]……… (2)
Substitute the value of r and V in equation (2) we get,
\[q = 4 \times 3.142 \times 8.854 \times {10^{ - 12}} \times 2 \times {10^{ - 2}}\]
\[ \Rightarrow q = 0.22 \times {10^{ - 11}}C\]
At one second, the rate of emission of \[\beta \]-particles \[ = 5 \times {10^{10}}/s\]
Then, n is the number of \[\beta \]-particles emitted in t seconds then,
\[n = R \times t\]
\[ \Rightarrow n = 5 \times {10^{10}} \times t\]

But, we know that,
\[q = ne\]
\[ \Rightarrow q = 1.6 \times {10^{ - 19}} \times n\]
If 40% of \[\beta \]-particles are escaped then,
\[q = 1.6 \times {10^{ - 19}} \times 5 \times {10^{10}} \times t \times \dfrac{{40}}{{100}}\]
\[ \Rightarrow t = 0.0693 \times {10^{ - 2}}s\]
\[ \therefore t = 693\mu s\]

Therefore, the time taken to raise its potential is \[693\mu s\].

Note:Beta particles are basically electrons which arise from the radioactive decay of the unstable atoms. Its penetration power is higher than α-particles and can penetrate through a thin metal foil. These particles can be affected by both the electric field and magnetic field.