A monochromatic light is incident on a metallic plate having work function $\phi$. An electron, emitted…

A monochromatic light is incident on a metallic plate having work function $\phi$. An electron, emitted normally to the plate from a point $A$ with maximum kinetic energy, enters a constant magnetic field, perpendicular to the initial velocity of electron. The electron passes through a curve and hits back the plate at a point $B$. The distance between A and B is :
(Given : The magnitude of charge of an electron is e and mass is $\mathrm{m}, \mathrm{h}$ is Planck's constant and c is velocity of light. Take the magnetic field exists throughout the path of electron)
  1. $\sqrt{2 \mathrm{~m}\left(\frac{\mathrm{hc}}{\lambda}-\phi\right)} / \mathrm{eB}$
  2. $\sqrt{\mathrm{m}\left(\frac{\mathrm{hc}}{\lambda}-\phi\right)} / \mathrm{eB}$
  3. $\sqrt{8 \mathrm{~m}\left(\frac{\mathrm{hc}}{\lambda}-\phi\right)} / \mathrm{eB}$
  4. $2 \sqrt{\mathrm{~m}\left(\frac{\mathrm{hc}}{\lambda}-\phi\right)} / \mathrm{eB}$

Solution

$\begin{aligned} & \mathrm{KE}_{\max }=\frac{\mathrm{hc}}{\lambda}-\phi \\ & \mathrm{p}=\sqrt{2 \mathrm{mK}_{\max }} \\ & \mathrm{p}=\sqrt{2 \mathrm{~m}\left(\frac{\mathrm{hc}}{\lambda}-\phi\right)} \\ & \mathrm{d}_{\mathrm{A}-\mathrm{B}}=2 \mathrm{R} \\ & =2\left[\frac{\mathrm{p}}{\mathrm{qB}}\right]\end{aligned}$
$\mathrm{d}_{\mathrm{AB}}=\frac{2 \sqrt{2 \mathrm{~m}\left(\frac{\mathrm{hc}}{\lambda}-\phi\right)}}{\mathrm{eB}}=\frac{\sqrt{8 \mathrm{~m}\left(\frac{\mathrm{hc}}{\lambda}-\phi\right)}}{\mathrm{eB}}$

Asked in: JEE Main 2025 (02 Apr Shift 1)

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