Q. Surface of certain metal is first illuminated with light of wavelength $\lambda_1 =350 \; nm$ and then, by light of wavelength $\lambda_2=54D \; nm$. It is found that the maximum speed of the photo electrons in the two cases differ by a factor of 2. The work function of the metal (in eV) is close to :
(Energjr of photon = $\frac{1240}{\lambda (in \; nm)} eV$)

Solution:

$\frac{hc}{\lambda_{1}} =\phi + \frac{1}{2} m\left(2v\right)^{2} $
$\frac{hc}{\lambda _{2}} =\phi + \frac{1}{2} mv^{2} $
$\Rightarrow \frac{\frac{hc}{\lambda _{1}}-\phi}{\frac{hc}{\lambda _{2}} -\phi} = 4 $
$\Rightarrow \frac{hc}{\lambda _{1}} - \phi = \frac{4hc}{\lambda _{2}} -4 \phi $
$ \Rightarrow \frac{4hc}{\lambda _{2}} - \frac{hc}{\lambda _{1}} = 3\phi $
$ \Rightarrow \phi = \frac{1}{3} hc \left(\frac{4}{\lambda _{2}} -\frac{1}{\lambda _{1}}\right) $
$= \frac{1}{3} \times1240 \left(\frac{4 \times350 -540}{350\times540} \right) $
$= 1.8 eV $

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