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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 $A \times 10^2$ (where $1 < A < 10$ ). The value of $'Z '$ is
$7$
$8$
$9$
$1$
Solution
Let $x$ be the maximum number of photo electrons emitted.
The potential $V$ of the sphere can be calculated as : $U =\frac{ Kq }{ R }$
$V =\frac{9 \times 10^9 \times x \times 1.6 \times 10^{-1} 19}{10^{-2}}= x \times 14.4 \times 10^{-8} $
$ U = eV = x \times 14.4 \times 10^{-8} \times 1.6 \times 10^{-19}= x \times 23.04 \times 10^{-27} J$
From conservation of energy :
$U =\frac{ hC }{\lambda}-\phi $
$x \times 23.04 \times 10^{-27}=\frac{2 \times 10^{-25}}{200 \times 10^{-9}}-4.7 \times 1.6 \times 10^{-19} $
$x =\frac{10^{-18}-7.52 \times 10^{-19}}{23.04 \times 10^{-27}}=1.076 \times 10^7$
$\therefore Z =7$
