Question

A small metal plate of work function $\varphi$ is kept at a distance $d$ from a singly ionised, 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 electrons may go around the ion along a circle.

Answer 1

${\left(\frac{1}{2}m{v}^2\right)}_{max}=\frac{hc}{\lambda }-\varphi$
for the electron to move around the ion(fixed)

$\frac{1}{2}m{v}^2=PE+K$

$=\frac{-e^2}{4\pi {\epsilon }_{0}d}+\frac{e^2}{8\pi {\epsilon }_{0}d}$

Thus

$\frac{e^2}{8\pi {\epsilon }_{0}d}=\frac{hc}{\lambda }-\varphi$

or $e^2=\frac{hc8\pi {\epsilon }_{0}d}{\lambda }-8\pi {\epsilon }_0\varphi d$

or $\lambda =\frac{8\pi {\epsilon }_{0}hcd}{e^2+8\pi {\epsilon }_0\varphi d}=\frac{e^2}{8\pi {\epsilon }_{0}d}$