Question

Write down Einstein's photoelectric equation. Explain with the help of this equation that what is the effect on the maximum kinetic energy of the emitted electrons, if frequency of the incident light (photons) is increased by \( n \) times. What is the relationship between the work function of the metal surface and threshold wavelength?

Hint:

When frequency increases, the energy of the incident light increases. The kinetic energy of emitted electrons increases accordingly, provided the frequency exceeds the threshold frequency.

Answer 1

Einstein's photoelectric equation relates the energy of the incident photons to the kinetic energy of the emitted photoelectrons. The equation is: \[ K_{\text{max}} = h\nu - \Phi \] Where: - \( K_{\text{max}} \) is the maximum kinetic energy of the emitted electrons,
- \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \, \text{J} \cdot \text{s} \)),
- \( \nu \) is the frequency of the incident light (photons),
- \( \Phi \) is the work function of the metal, which is the minimum energy required to remove an electron from the metal surface.
The equation shows that the maximum kinetic energy of the emitted electrons is the difference between the energy of the incident photons and the work function of the material.
Effect of Increasing Frequency by \( n \) Times:
If the frequency of the incident light is increased by a factor of \( n \), the energy of the photons increases by the same factor, i.e., the energy of the photons becomes \( n \cdot h\nu \). The new maximum kinetic energy \( K'_{\text{max}} \) is given by: \[ K'_{\text{max}} = n \cdot h\nu - \Phi \] Thus, if the frequency is increased by \( n \) times, the maximum kinetic energy of the emitted electrons also increases by a factor of \( n \).
Threshold Wavelength and Work Function:
The threshold wavelength \( \lambda_{\text{th}} \) is the wavelength of the incident light below which no photoelectron is emitted, even if the intensity of the light is increased. The threshold wavelength is related to the work function \( \Phi \) by the equation: \[ \Phi = \frac{h c}{\lambda_{\text{th}}} \] Where: - \( c \) is the speed of light,
- \( \lambda_{\text{th}} \) is the threshold wavelength.
This equation shows that a larger work function corresponds to a shorter threshold wavelength, i.e., the more energy required to release electrons from the metal, the shorter the wavelength of light needed to release those electrons.