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A small metal plate, having work function $ϕ_{0}$ , kept at a distance $d$ from a singly ionized, fixes ion. A monochromatic light beam is incident on the metal plate and photoelectrons are emitted. Find the maximum wavelength of the light beam so that some of the photoelectrons may go round the ion along a circle of radius d.

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$\frac{π ε_{0} h c d}{e^{2} + 8 π ε_{0} ϕ_{0} d}$

$\frac{8 π ε_{0} h c d}{e^{2} + 8 π ε_{0} ϕ_{0} d}$

$\frac{3 π ε_{0} h c d}{e^{2} + 8 π ε_{0} ϕ_{0} d}$

$\frac{8 π ε_{0} h c d}{e^{2} + 4 π ε_{0} ϕ_{0} d}$

answer is B.

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Detailed Solution

For circular motion of electrons around the ion, th electrostatic force between the electron and the posi tively charged ion must provide the necessary cen tripetal force to the electron to revolve in a circle a radius $d$ .

$\Rightarrow \frac{1}{4 π ε_{0}} \frac{e^{2}}{d^{2}} = \frac{m v^{2}}{d}$

$\Rightarrow \frac{e^{2}}{8 π ε_{0} d} = \frac{1}{2} m v^{2} = \frac{h c}{λ} - ϕ_{0}$

$\Rightarrow λ = \frac{8 π ε_{0} h c d}{e^{2} + 8 π ε_{0} ϕ_{0} d}$

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