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

The maximum wavelength of the light beam, so that some of the photo electrons may go round the ion along a circle is  $\frac{8 \pi \in_{0} h_{c} d}{e^{2}+8 \pi \in_{0} d \phi}$

Explanation:

Step 1:

From the Photoelectric Equation of Einstein,

$e V_{0}=\frac{h c}{\lambda}-\phi$

$V_{0}=\left(\frac{h c}{\lambda}-\phi\right) \frac{1}{e}$

where  

$V_0$ = stopping potential  

h = Planck's constant  

c = speed of light  

ϕ = work function  

Step 2:

Due to the single charged ion at that point, When the stopping potential is equal to the potential, the particle must pass in a circle. So that the particle receives the centripetal force it requires for circular motion.

$\frac{K e}{2 d}=\left(\frac{h c}{\lambda}-\phi\right) \frac{1}{e}$

$\frac{K e^{2}}{2 d}=\frac{h c}{\lambda}-\phi$

$\frac{h c}{\lambda}=\frac{K e^{2}}{2 d}+\phi=\frac{K e^{2}+\phi}{2 d}$

$\lambda=\frac{(h c)(2 d)}{K e^{2}+\phi}$

$\lambda=\frac{2 h d c}{\frac{1}{4 \pi \epsilon_{0}} e^{2}+2 d \phi}$

$\lambda=\frac{8 \pi \in_{0} h c d}{e^{2}+8 \pi \in_{0} d \phi}$