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A monochromatic light of wavelength $λ$ is incident on an isolated metallic sphere of radius a. The threshold wavelength is $λ_{0}$ which is larger than $λ .$ Find the number of photoelectrons emitted before the emission of photo electrons stops.

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$\frac{a h c}{4 π ε_{0} e^{2}} (\frac{1}{λ} - \frac{1}{λ_{0}})$

$\frac{4 π ε_{0} a h c}{e^{2}} (\frac{1}{λ} - \frac{1}{λ_{0}})$

$\frac{e^{2} a h c}{4 π ε_{0}} (\frac{1}{λ} - \frac{1}{λ_{0}})$

$\frac{4 π ε_{0} a V}{e^{2}} (\frac{1}{λ} - \frac{1}{λ_{0}})$

answer is C.

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Detailed Solution

The emission of photoelectrons will stop when

$\frac{h c}{λ} = w_{0} + e V = \frac{h c}{λ_{0}} + e V or$

$V = \frac{h c}{e} (\frac{1}{λ} - \frac{1}{λ_{0}})$

The charge on the sphere needed to take its potential to V is $Q = (4 π ε_{0} a) V$

The number of electrons emitted is, therefore,

$n = \frac{Q}{e} = \frac{4 π ε_{0} a V}{e} = \frac{4 π ε_{0} a h c}{e^{2}} (\frac{1}{λ} - \frac{1}{λ_{0}})$

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