The figure shows $\displaystyle \mathrm{v}_{\mathrm{m}}^{2}$ versus $\displaystyle \frac{1}{\lambda}$ graph for photoelectrons emitted from a surface where $\displaystyle \mathrm{v}_{\mathrm{m}}$ is the maximum speed of electrons and $\displaystyle \lambda$ is the wavelength of incident radiation. Using this graph and Einstein's photoelectric equation, obtain the expression for Planck's constant and work function of the surface.

\[\begin{aligned} & \frac{1}{2} \mathrm{m} \mathrm{v}_{\mathrm{m}}^{2}=\frac{\mathrm{h} c}{\lambda}-\phi_{0} \\ & \mathrm{v}_{\mathrm{m}}^{2}=\left(\frac{2 \mathrm{h} c}{\mathrm{m}}\right) \frac{1}{\lambda}-\frac{2}{\mathrm{m}} \phi_{0} \end{aligned} \] According to this equation a plot of \(\displaystyle \mathrm{v}_{\mathrm{m}}^{2}\) versus \(\displaystyle (1 / \lambda)\) is a straight line. Slope of the graph \(\displaystyle =\frac{2 \mathrm{h} c}{\mathrm{m}}\) \[\text { Intercept }=\frac{2}{\mathrm{m}} \phi_{0} \] Slope and intercept can be found from the graph \[\begin{aligned} & \mathrm{h}=\frac{\mathrm{m}}{2 c} \times \text { slope } \\ & \phi_{0}=\frac{\mathrm{m}}{2} \times \text { intercept } \end{aligned} \]