Question

A radioactive source in the form of a metal sphere of diameter $3.2\times {10}^{-3}$ m emits $\beta$-particle at a constant rate of $6.25\times {10}^{10}$ particle/sec. The source is electrically insulated and all the $\beta$-particle are emitted from the surface. The potential of the sphere will rise to 1 V in time.

• A. $180\mu$ sec
• B. $90\mu$ sec
• C. $18\mu$ sec
• D. $9\mu$ sec

Answer 1

The correct option is C $18\mu$ sec

Radius $R=\frac{3.2}{2}\times {10}^{-3}meter$

so to make the potential $1Volt$ we need $\frac{Q}{4\pi {ϵ}_{0}R}=1$

so the charge needed is $Q=4\pi {ϵ}_{0}R=\frac{1.6\times {10}^{-3}}{9\times {10}^9}=\frac{1.6\times {10}^{-12}}{9}Coulomb$

So the number of electrons$(\beta -particle)$ needed to be ejected is $n=\frac{Q}{e}$

putting $e=1.6\times {10}^{-19}Coulomb$, we get $n=\frac{{10}^7}{9}$

time needed is $t=\frac{n}{rate}=\frac{{10}^7}{9\times 6.25\times {10}^{10}}=.01775\times {10}^{-3}sec=17.75\mu sec$ nearly $18\mu sec$