A photoelectric surface is illuminated successively by monochromatic light of wavelength ' \((\lambda)\) ' and ' \(\left(\frac{\lambda}{2}\right)\). If the maximum kinetic energy of the emitted photoelectrons in the first case is one-third that in the second case, the work function of the surface of the material is ( \(\mathrm{c}=\) speed of light, \(\mathrm{h}=\) Planck's constant.)

The correct option is (B).
Concept: Einstien's Photoelectric effect theory
The maximum kinetic energy \(\mathrm{K}\) of the photoelectric electrons is related to the incident wavelength \(\lambda\) and work function \(\phi\) of the metal as follows:
\(\mathrm{K}=\frac{\mathrm{hc}}{\lambda}-\phi\)
For the incident wavelength \(\lambda\), the equation is \(\mathrm{K}=\frac{\mathrm{hc}}{\lambda}-\phi\)
And for the incident wavelength \(\frac{\lambda}{2}\), the equation is \(3 K=\frac{\mathrm{hc}}{\frac{\lambda}{2}}-\phi\)
Taking the ratio of two equations:
\(\frac{1}{3}=\frac{\left(\frac{\mathrm{hc}}{\lambda}-\phi\right)}{\left(\frac{\mathrm{hc}}{\frac{\lambda}{2}}-\phi\right)}\)
On solving, we get \(\phi=\frac{\mathrm{hc}}{2 \lambda}\)