Photoelectric Effect: Cutoff Wavelength, Frequency, Work Function, and Kinetic Energy

Photoelectric Effect

Introduction to the Photoelectric Effect

The photoelectric effect describes the emission of electrons from a material when it is exposed to electromagnetic radiation of sufficient frequency. This phenomenon provided crucial evidence for the quantum nature of light and led to the development of quantum mechanics. The key equations relate the energy of incident photons, the work function of the material, and the kinetic energy of emitted electrons.

Key Equations and Concepts

Photoelectric Effect Equation

The maximum kinetic energy (KEmax) of photoelectrons is given by:

• Equation:

$KE_{\text{max}} = E_{\text{photon}} - \phi = h f - \phi = \frac{hc}{\lambda} - \phi$

• Where:

• h = Planck's constant ($6.63 \times 10^{-34}$ J·s)

• c = speed of light ($3.00 \times 10^8$ m/s)

• \phi = work function (minimum energy to remove an electron, in eV or J)

• \lambda = wavelength of incident light (in m)

• f = frequency of incident light (in Hz)

Cutoff Wavelength and Frequency

The cutoff wavelength ($\lambda_c$) is the longest wavelength (lowest energy) that can eject electrons from the material. The cutoff frequency ($f_c$) is the lowest frequency that can cause photoemission.

• Cutoff Wavelength:

$\lambda_c = \frac{hc}{\phi}$

• Cutoff Frequency:

$f_c = \frac{c}{\lambda_c} = \frac{\phi}{h}$

Work Function Calculation

The work function ($\phi$) can be determined if the cutoff wavelength or frequency is known:

$\phi = \frac{hc}{\lambda_c}$

Stopping Potential

The stopping potential ($V_s$) is the minimum voltage needed to stop the most energetic photoelectrons:

$eV_s = KE_{\text{max}}$

Thus, $V_s = \frac{KE_{\text{max}}}{e}$

Photon Energy

The energy of a photon with wavelength $\lambda$ is:

$E_{\text{photon}} = \frac{hc}{\lambda}$

Worked Examples

Example 1: Calculating Cutoff Wavelength

Given a work function $\phi = 2.24$ eV, the cutoff wavelength is:

$\lambda_c = \frac{(6.63 \times 10^{-34} \ \text{J} \cdot \text{s})(3.00 \times 10^8 \ \text{m/s})}{2.24 \ \text{eV}} \left(\frac{1 \ \text{eV}}{1.60 \times 10^{-19} \ \text{J}}\right) = 555 \ \text{nm}$

Example 2: Calculating Cutoff Frequency

Given $\lambda_c = 2.88 \times 10^{-7}$ m, the cutoff frequency is:

$f_c = \frac{3.00 \times 10^8 \ \text{m/s}}{2.88 \times 10^{-7} \ \text{m}} = 1.04 \times 10^{15} \ \text{Hz}$

Example 3: Calculating Work Function from Wavelength

Given $\lambda = 350 \times 10^{-9}$ m and photon energy $E_{\text{photon}} = 1.31$ eV, the work function is:

$\phi = \frac{(6.63 \times 10^{-34} \ \text{J} \cdot \text{s})(3.00 \times 10^8 \ \text{m/s})}{350 \times 10^{-9} \ \text{m}} \left(\frac{1 \ \text{eV}}{1.60 \times 10^{-19} \ \text{J}}\right) - 1.31 \ \text{eV} = 2.24 \ \text{eV}$

Example 4: Maximum Kinetic Energy of Ejected Electrons

If $E_{\text{photon}} = 5.50$ eV and $\phi = 4.31$ eV, then:

$KE_{\text{max}} = 5.50 \ \text{eV} - 4.31 \ \text{eV} = 1.19 \ \text{eV}$

Example 5: Stopping Potential

If $KE_{\text{max}} = 2.15$ eV, then the stopping potential is:

$V_s = \frac{2.15 \ \text{eV}}{e} = 2.15 \ \text{V}$

Example 6: Maximum Electron Speed

Given $KE = 2.50$ eV, the maximum speed of the electron is:

$KE = \frac{1}{2} m_e v_{\text{max}}^2$

$v_{\text{max}} = \sqrt{\frac{2 \times 4.00 \times 10^{-19} \ \text{J}}{9.11 \times 10^{-31} \ \text{kg}}} = 9.37 \times 10^5 \ \text{m/s}$

Example 7: Graphical Representation of the Photoelectric Effect

The relationship between the maximum kinetic energy of photoelectrons and the frequency of incident light is linear above the cutoff frequency. The slope of the line is Planck's constant, and the x-intercept gives the cutoff frequency.

Summary Table: Key Photoelectric Effect Quantities

Quantity	Symbol	Formula	Units
Work Function	$\phi$	$\phi = \frac{hc}{\lambda_c}$	eV or J
Cutoff Wavelength	$\lambda_c$	$\lambda_c = \frac{hc}{\phi}$	m or nm
Cutoff Frequency	$f_c$	$f_c = \frac{\phi}{h}$	Hz
Maximum Kinetic Energy	$KE_{\text{max}}$	$KE_{\text{max}} = hf - \phi$	eV or J
Stopping Potential	$V_s$	$V_s = \frac{KE_{\text{max}}}{e}$	V

Additional info:

• The photoelectric effect demonstrates the particle-like properties of light, as only photons with energy greater than the work function can eject electrons.

• The cutoff wavelength and frequency are material-dependent and are determined by the work function of the surface.

• Stopping potential measurements are a classic experimental method for determining the work function of a material.