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Dual Nature of Radiation and Matter — Physics STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 SciencePhysicsDual Nature of Radiation and Matter3 Marks
Question
Write Einstein's Photoelectric equation and mention which important features in photoelectric effect can be explained with the help of this equation.
The maximum kinetic energy of the photoelectrons gets doubled when the wavelength of light incident on the surface changes from $\lambda_{1}$to $\lambda_{2}$. Derive the expressions for the threshold wavelength $\lambda_{\circ}$ and work function for the metal surface.
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Answer

$\text{hv} = \varphi_{o} + \text{k}_{max}$
or $\text{hv} =\text{hv}_{0} + \frac{1}{2}\text{mv}^{2}_{max}$
Important features:
  1. $k_{max}$ depends linearly on frequency v.
  2. Existence of threshold frequency for the metal surface.
$\text{hv} =\varphi_{o} +\text{k}_{max}$
$\frac{\text{hc}}{\lambda}_{1} = \frac{\text{hc}}{\lambda_{0}} + \text{k}_{max}$.............................(i)
$\frac{\text{hc}}{\lambda_{2}} =\frac{\text{hc}}{\lambda_{0}} +2\text{k}_{max}$...........................(ii)
From (i) and (ii)
$\frac{2\text{hc}}{\lambda_{1}} - \frac{\text{hc}}{\lambda_{2}} = \frac{\text{hc}}{\lambda}_{0}$
$\frac{1}{\lambda_{0}} =\bigg(\frac{2}{\lambda_{1}} -\frac{1}{\lambda_{2}}\bigg)$
$\lambda_{0} = \frac{\lambda_{1}\lambda_{2}}{2\lambda_{2} - \lambda_{1}}$
Work function $\varphi_{o} =\frac{\text{hc}}{\lambda_{0}} = \frac{\text{hc}(2\lambda_{2} - \lambda_{1})}{\lambda_{1}\lambda_{2}}$.

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