Question

A silver sphere of radius $1cm$ and work function $4.7eV$ is suspended from an insulating thread in free-space. It is under continuous illumination of $200nm$ wavelength light. As photoelectrons are emitted, the sphere gets charged and acquires a potential. The maximum number of photoelectrons emitted from the sphere is $A\times 10^z$ (where $1<A<10)$. The value of $Z$ is

• A. $6$
• B. $7$
• C. $8$
• D. $9$

Answer 1

Answer: (B)

Here, radius of sphere $R=1cm=1\times 10^{-2}m$
Work function, $W=4.7eV$
Energy of incident radiation
$=\frac{hc}{\lambda }=\frac{1240eVnm}{200nm}=6.2eV$
According to Einsteins photoelectric equation
$\frac{hc}{/lambda}=\varphi +e{V}_s$
$6.2eV=4.7eV+e{V}_s$
$V_s=1.5V$

The sphere will stop emitting photoelectrons, when the potential on its surface becomes equal to $1.5V$
$\therefore \frac{1}{4\pi {\epsilon }_0}\frac{Q}{R}=1.5$
$\Rightarrow \frac{1}{4\pi {\epsilon }_0}\frac{Ne}{R}=1.5$
where $N$ = Number of photoelectrons emitted
$e$ = charge of each electron
$N=\frac{1.5\times R}{\frac{1}{4\pi {\epsilon }_0}\times e}=\frac{1.5\times 1\times 10^{-2}}{9\times 10^9\times 1.6\times 10^{-19}}$
$N=\frac{15}{16}\times \frac{1}{9}\times 10^8=\frac{5}{48}\times 10^8$
$N=\frac{50}{48}\times 10^7=1.04\times 10^7$
$\therefore Z=7$