Question

Photoemission of electrons occurs from a metal (\( \phi_0 = 1.96 \, \text{eV} \)) when light of frequency \( 6.4 \times 10^{14} \, \text{Hz} \) is incident on it. Calculate: Energy of a photon in the incident light, The maximum kinetic energy of the emitted electrons, and The stopping potential.

Hint:

Shortcut:

Convert photon energy directly to eV
\( K_{\max} = h\nu - \phi \)
Stopping potential (in volts) = kinetic energy (in eV)

Answer 1

Photoelectric Effect: Calculation of Photon Energy, Electron Kinetic Energy, and Stopping Potential
We are given:
- Work function of the metal: \( \phi_0 = 1.96 \, \text{eV} \) - Frequency of incident light: \( f = 6.4 \times 10^{14} \, \text{Hz} \) 
- Electron charge: \( e = 1.602 \times 10^{-19} \, \text{C} \) 
We are asked to calculate: photon energy, maximum kinetic energy of emitted electrons, and stopping potential. 
Step 1: Energy of a Photon
Energy of a photon is given by:
\[ E_\text{photon} = h f \]
\[ E_\text{photon} = (6.626 \times 10^{-34}) (6.4 \times 10^{14}) \, \text{J} = 4.24 \times 10^{-19} \, \text{J} \]
Convert to eV: \[ E_\text{photon} = \frac{4.24 \times 10^{-19}}{1.602 \times 10^{-19}} \approx 2.65 \, \text{eV} \]
Step 2: Maximum Kinetic Energy of Emitted Electrons
Using Einstein’s photoelectric equation:
\[ K_\text{max} = E_\text{photon} - \phi_0 \]
\[ K_\text{max} = 2.65 - 1.96 = 0.69 \, \text{eV} \]
Step 3: Stopping Potential
Stopping potential \( V_0 \) is related to maximum kinetic energy:
\[ K_\text{max} = e V_0 \implies V_0 = \frac{K_\text{max}}{e} \]
Since \( K_\text{max} \) is already in eV:
\[ V_0 = 0.69 \, \text{V} \]
Step 4: Summary
- Energy of the photon: \( E_\text{photon} \approx 2.65 \, \text{eV} \) - Maximum kinetic energy of electrons: \( K_\text{max} \approx 0.69 \, \text{eV} \) - Stopping potential: \( V_0 \approx 0.69 \, \text{V} \) Conclusion:
The photoemission occurs because the photon energy exceeds the work function of the metal. The electrons are emitted with a maximum kinetic energy of 0.69 eV, requiring a stopping potential of 0.69 V to halt them.