MHT CET · Physics · Dual Nature of Matter
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-fourth 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 \()\)
- A \(\frac{2 \mathrm{hc}}{\lambda}\)
- B \(\frac{\mathrm{hc}}{\lambda}\)
- C \(\frac{2 \mathrm{hc}}{3 \lambda}\)
- D \(\frac{\mathrm{hc}}{3 \lambda}\)
Answer & Solution
Correct Answer
(C) \(\frac{2 \mathrm{hc}}{3 \lambda}\)
Step-by-step Solution
Detailed explanation
\(\begin{aligned}
& E_1=\frac{h c}{\lambda}-\phi ...(i)\\
& E_2=\frac{h c}{\lambda / 2}-\phi=\frac{2 h c}{\lambda}-\phi \ldots \ldots \text { (ii) } \\
& \text { Given, } E_1=\frac{E_2}{4} \\
& 4 E_1=E_2 ...(iii)\\
& \text { From (ii) and (iii), } \\
& 4 E_1=\frac{2 h c}{\lambda}-\phi \\
& 4\left(\frac{h c}{\lambda}-\phi\right)=\frac{2 h c}{\lambda}-\phi
\end{aligned}\)
[Substituting from (i)]
\(\begin{array}{ll}\therefore & \frac{4 h c}{\lambda}-4 \phi=\frac{2 h c}{\lambda}-\phi \\ \therefore & 3 \phi=\frac{2 h c}{\lambda} \\ \therefore & \phi=\frac{2 h c}{3 \lambda}\end{array}\)
& E_1=\frac{h c}{\lambda}-\phi ...(i)\\
& E_2=\frac{h c}{\lambda / 2}-\phi=\frac{2 h c}{\lambda}-\phi \ldots \ldots \text { (ii) } \\
& \text { Given, } E_1=\frac{E_2}{4} \\
& 4 E_1=E_2 ...(iii)\\
& \text { From (ii) and (iii), } \\
& 4 E_1=\frac{2 h c}{\lambda}-\phi \\
& 4\left(\frac{h c}{\lambda}-\phi\right)=\frac{2 h c}{\lambda}-\phi
\end{aligned}\)
[Substituting from (i)]
\(\begin{array}{ll}\therefore & \frac{4 h c}{\lambda}-4 \phi=\frac{2 h c}{\lambda}-\phi \\ \therefore & 3 \phi=\frac{2 h c}{\lambda} \\ \therefore & \phi=\frac{2 h c}{3 \lambda}\end{array}\)
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