The maximum kinetic energies of photoelectrons emitted are \(K_1\) and \(K_2\) when lights of wavelengths \(\lambda_1\) and \(\lambda_2\) are incident on a metallic surface. If \(\lambda_1=3 \lambda_2\) then

Kinetic energy of the photoelectrons
\(
\mathrm{K}=\frac{\mathrm{hc}}{\lambda}-\phi
\)
\(\therefore \quad\) For wavelength \(\lambda_1\),
\(
\mathrm{K}_1=\frac{\mathrm{hc}}{\lambda_1}-\phi
\)
\(\therefore \quad\) For wavelength \(\lambda_1\),
\(
\mathrm{K}_2=\frac{\mathrm{hc}}{\lambda_2}-\phi
\)
Subtracting equation (i) from equation (ii),
\(
\begin{aligned}
& \mathrm{K}_2-\mathrm{K}_1=\frac{\mathrm{hc}}{\lambda_2}-\phi-\frac{\mathrm{hc}}{\lambda_1}+\phi \\
& \mathrm{K}_2-\mathrm{K}_1=\frac{\mathrm{hc}}{\lambda_2}-\frac{\mathrm{hc}}{3 \lambda_2} \\
& \mathrm{~K}_2-\mathrm{K}_1=\frac{2}{3} \frac{\mathrm{hc}}{\lambda_2} \\
& \frac{\mathrm{hc}}{\lambda_2}=\frac{3}{2}\left(\mathrm{~K}_2-\mathrm{K}_1\right)
\end{aligned}
\)
Substituting equation (iii) in equation (ii),
\(
\begin{aligned}
& \mathrm{K}_2=\frac{3}{2}\left(\mathrm{~K}_2-\mathrm{K}_1\right)-\phi \\
& 2 \mathrm{~K}_2=3 \mathrm{~K}_2-3 \mathrm{~K}_1-2 \phi \\
& \mathrm{K}_2-3 \mathrm{~K}_1=2 \phi \\
& \mathrm{As}, \phi>0 \\
& \mathrm{~K}_2-3 \mathrm{~K}_1>0 \\
& \mathrm{~K}_1 < \frac{\mathrm{K}_2}{3}
\end{aligned}
\)