When a metallic surface is illuminated with a radiation of wavelength ' \(\lambda\) ', the stopping potential is ' \(V\) '. If' the same surface is illuminated with radiation of wavelength ' \(3 \lambda\) ', the stopping potential is " \(\left(\frac{\mathrm{V}}{6}\right)\) '. The threshold wavelength for the surface is

From Einstein's equation, \(\mathrm{h} \nu=\mathrm{e} \mathrm{V}_0+\mathrm{h} \nu_0\)
\(\therefore \quad \frac{\mathrm{hc}}{\lambda}-\frac{\mathrm{hc}}{\lambda_0}=\mathrm{eV}_0\)
case (i) \(\lambda=\lambda ; \mathrm{V}_0=\mathrm{V}\)
\(\frac{\mathrm{hc}}{\lambda}-\frac{\mathrm{hc}}{\lambda_0}=\mathrm{eV}\)
case (ii) \(\lambda=3 \lambda ; \mathrm{V}_0=\frac{\mathrm{V}}{6}\)
\(\frac{\mathrm{hc}}{3 \lambda}-\frac{\mathrm{hc}}{\lambda_0}=\frac{\mathrm{eV}}{6}\)
\(\begin{aligned}
& \text {dividing equation (i) by equation (ii) } \\
& \therefore \quad \frac{\left(\frac{\mathrm{hc}}{\lambda}-\frac{\mathrm{hc}}{\lambda_0}\right)}{\left(\frac{\mathrm{hc}}{3 \lambda}-\frac{\mathrm{hc}}{\lambda_0}\right)}=6 \\
& \therefore \quad \frac{1}{\lambda}-\frac{1}{\lambda_0}=6\left(\frac{1}{3 \lambda}-\frac{1}{\lambda_0}\right) \\
& \therefore \quad \frac{1}{\lambda}-\frac{1}{\lambda_0}=\frac{2}{\lambda}-\frac{6}{\lambda_0} \\
& \therefore \quad \frac{-1}{\lambda_0}+\frac{6}{\lambda_0}=\frac{2}{\lambda}-\frac{1}{\dot{\lambda}}
\end{aligned}\)
\(\therefore \quad \frac{5}{\lambda_0}=\frac{1}{\lambda} \quad \Rightarrow \lambda_0=5 \lambda\)