Write Einstein's Photoelectric equation and mention which importa… - Vidyadip

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:

k_max depends linearly on frequency v.
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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