Question

A small metal plate (work function $\phi$)is kept at a distance d from a singly ionized, fixed ion. A monochromatic light beam is incident on the metal plate and photo electrons are emitted. Find the maximum wavelength of the light beam so that some of the photo electrons may go round the ion along a circle.

Answer 1

The particle will move in a circle when the stopping potential is equal to the potential due the singly charged on at that point.

$e{V}_0=\frac{hc}{\lambda }-\varphi$

$\Rightarrow V_0=(\frac{hc}{\lambda }-\varphi )\frac{1}{e}$

$\Rightarrow \frac{Ke}{2d}=(\frac{hc}{\lambda }-\varphi )\frac{1}{e}$

$\Rightarrow \frac{K{e}^2}{2d}=\frac{hc}{\lambda }-\varphi$

$\Rightarrow \frac{hc}{\lambda }=\frac{K{e}^2}{2d}+\varphi =\frac{K{e}^2+2d\varphi }{2d}$

$\lambda =\frac{\left(hc\right)\left(2d\right)}{k{e}^2+2d\varphi }$

$=\frac{2hdc}{\frac{1}{4\pi {ϵ}_0}{e}^2+2d\varphi }$

$=\frac{8\pi {ϵ}_{0}hcd}{e^2+8\pi {ϵ}_{0}d\varphi }$