Question

Radiation of wavelength 200 nm is incident on a photosensitive surface of work function 4.2 eV. The kinetic energy of fastest photoelectrons emitted from this surface will be close to :

• A. 3.5 eV
• B. 3.0 eV
• C. 2.5 eV
• D. 2.0 eV

Hint:

Using \( E = \frac{1240 \text{ eV}\cdot\text{nm}}{\lambda (\text{nm})} \) is an essential shortcut for modern physics problems.
It completely bypasses the tedious process of calculating \( \frac{(6.63 \times 10^{-34}) \times (3 \times 10^8)}{\lambda \times 10^{-9}} \) and then dividing by \( 1.6 \times 10^{-19} \) to convert from Joules to eV.

Answer 1

The Correct Option is D

Concept:
• The photoelectric effect is governed by Einstein's photoelectric equation, which is essentially an energy conservation statement.

• The equation states that the maximum kinetic energy (\( K_{\text{max}} \)) of an emitted photoelectron equals the energy of the incident photon (\( E \)) minus the work function (\( \Phi \)) of the metal surface.

• Mathematically: \( K_{\text{max}} = E - \Phi \).

• The energy of an incident photon is inversely proportional to its wavelength, given by \( E = \frac{hc}{\lambda} \).

Step 1: Calculate the energy of the incident photon
The wavelength of the incident radiation is provided as \( \lambda = 200 \text{ nm} \).
To compute the photon energy directly in electron-volts (eV), we use the highly practical approximation formula:
\[ E (\text{in eV}) = \frac{1240}{\lambda (\text{in nm})} \quad (\text{or } 1242 \text{ for slightly higher precision}) \]
Substituting the given wavelength:
\[ E = \frac{1240}{200} \]
\[ E = 6.2 \text{ eV} \]

Step 2: Calculate the maximum kinetic energy
The work function of the photosensitive surface is provided as \( \Phi = 4.2 \text{ eV} \).
Apply Einstein's photoelectric equation:
\[ K_{\text{max}} = E - \Phi \]
Substitute the known energy values:
\[ K_{\text{max}} = 6.2 \text{ eV} - 4.2 \text{ eV} \]
\[ K_{\text{max}} = 2.0 \text{ eV} \]

Step 3: Conclusion
The kinetic energy of the fastest emitted photoelectrons is calculated to be 2.0 eV.
This perfectly matches option (D).