Question

A silver sphere of radius $1\, cm$ and work function $4.7 \,eV$ is suspended from an insulating thread in free-space. It is under continuous illumination of $200 \,nm$ 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 $R=1\,cm=1\times10^{-2}m$
Work function, $W = 4.7 \,eV$
Energy of incident radiation
$=\frac{hc}{\lambda}=\frac{1240\,eV\,nm}{200\,nm}=6.2\,eV$
According to Einsteins photoelectric equation
$\frac{hc}{/lambda}=\phi+eV_{s}$
$6.2\,eV=4.7\,eV+eV_{s}$
$V_{s}=1.5\,V$

The sphere will stop emitting photoelectrons, when the potential on its surface becomes equal to $1.5\, V$
$\therefore \frac{1}{4\pi\varepsilon_{0}} \frac{Q}{R}=1.5$
$\Rightarrow \frac{1}{4\pi\varepsilon_{0}} \frac{Ne}{R}=1.5$
where $N$ = Number of photoelectrons emitted
$e$ = charge of each electron
$N=\frac{1.5\times R}{\frac{1}{4\pi\varepsilon_{0}}\times e}=\frac{1.5\times1\times10^{-2}}{9\times10^{9}\times1.6\times10^{-19}}$
$N=\frac{15}{16}\times\frac{1}{9}\times10^{8}=\frac{5}{48}\times10^{8}$
$N=\frac{50}{48}\times10^{7}=1.04\times10^{7}$
$\therefore Z=7$