Question

A small metal plate (work function φ) is kept at a distance d from a singly-ionised, fixed 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.

Answer 1

From Einstein's photoelectric equation,
$$\begin{gathered} e{V}_0=\frac{hc}{\lambda }-\mathrm{\varphi } \\ \Rightarrow V_0=\left(\frac{hc}{\lambda }-\mathrm{\varphi }\right)\frac{1}{e} \end{gathered}$$
Here, V₀ = stopping potential
h = Planck's constant
c = speed of light
$\varphi$ = work function

The particle will move in a circle when the stopping potential is equal to the potential due to the singly charged ion at that point so that the particle gets the required centripetal force for its circular motion.
$$\begin{gathered} \Rightarrow \frac{\mathrm{K}e}{2d}=\left(\frac{hc}{\lambda }-\mathrm{\varphi }\right)\frac{1}{e} \\ \Rightarrow \frac{\mathrm{K}{e}^2}{2d}=\frac{hc}{\lambda }-\mathrm{\varphi } \\ \Rightarrow \frac{hc}{\lambda }=\frac{{\mathrm{Ke}}^2}{2d}+\mathrm{\varphi }=\frac{\mathrm{K}{e}^2+2d\mathrm{\varphi }}{2d} \\ \Rightarrow \lambda =\frac{\left(hc\right)\left(2d\right)}{k{e}^2+2d\mathrm{\varphi }} \\ \Rightarrow \lambda =\frac{2hdc}{{\displaystyle \frac{1}{4\mathrm{\pi }{\in }_0}}{e}^2+2d\mathrm{\varphi }} \\ \Rightarrow \lambda =\frac{8\mathrm{\pi }{\in }_{0}hcd}{e^2+8\mathrm{\pi }{\in }_{0}d\mathrm{\varphi }} \end{gathered}$$