Photoelectric Effect: Cutoff Wavelength, Frequency, and Kinetic Energy

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Photoelectric Effect

Introduction to the Photoelectric Effect

The photoelectric effect is the phenomenon where electrons are emitted from a material when it is exposed to electromagnetic radiation of sufficient frequency. This effect provided crucial evidence for the quantum nature of light and led to the development of quantum mechanics.

Work Function ($ \phi $): The minimum energy required to remove an electron from the surface of a material.
Threshold Frequency ($ f_c $): The minimum frequency of incident light that can cause electron emission.
Cutoff Wavelength ($ \lambda_c $): The maximum wavelength of light that can cause electron emission.

Photoelectric Effect Equation

The maximum kinetic energy of emitted electrons is given by:

$KE_{\text{max}} = E_{\text{photon}} - \phi = hf - \phi$

$ h $: Planck's constant ($ 6.63 \times 10^{-34} \ \text{J} \cdot \text{s} $)
$ f $: Frequency of incident light
$ \phi $: Work function of the material

Cutoff Wavelength Calculation

At the cutoff wavelength, the maximum kinetic energy is zero ($ KE_{\text{max}} = 0 $), so:

$KE_{\text{max}} = hf_c - \phi = 0 \implies hf_c = \phi$

Since $ c = f \lambda $, the cutoff wavelength is:

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

Example calculation:

$\lambda_c = \frac{(6.63 \times 10^{-34} \ \text{J} \cdot \text{s})(3.00 \times 10^8 \ \text{m/s})}{2.24 \ \text{eV}} \left( \frac{1 \ \text{eV}}{1.60 \times 10^{-19} \ \text{J}} \right) = 5.55 \times 10^{-7} \ \text{m} = 555 \ \text{nm}$

Cutoff wavelength calculation example

Cutoff Frequency Calculation

The lowest frequency of light that will free electrons is:

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

Example calculation:

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

Cutoff frequency calculation example

Work Function from Wavelength and Kinetic Energy

The work function can be determined if the cutoff wavelength and the maximum kinetic energy are known:

$\phi = \frac{hc}{\lambda} - KE_{\text{max}}$

Example calculation:

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

Work function calculation example

Maximum Kinetic Energy of Ejected Electrons

The maximum kinetic energy of photoelectrons is given by:

$KE_{\text{max}} = E_{\text{photon}} - \phi$

Example calculation:

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

Maximum kinetic energy calculation example

Stopping Potential

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

$eV_s = KE_{\text{max}} \implies V_s = \frac{KE_{\text{max}}}{e}$

Example calculation:

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

Stopping potential calculation example

Photon Energy from Wavelength

The energy of a photon can be calculated from its wavelength:

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

Example calculation:

$E_{\text{photon}} = \frac{(6.63 \times 10^{-34} \ \text{J} \cdot \text{s})(3.00 \times 10^8 \ \text{m/s})}{400 \times 10^{-9} \ \text{m}} \left( \frac{1 \ \text{eV}}{1.60 \times 10^{-19} \ \text{J}} \right) = 3.11 \ \text{eV}$

Photon energy calculation example

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 threshold frequency. The slope of the line is Planck's constant, and the x-intercept gives the threshold frequency.

Graph of maximum kinetic energy vs frequency

Summary Table: Key Photoelectric Effect Equations

Quantity	Equation	Description
Maximum Kinetic Energy	$KE_{\text{max}} = hf - \phi$	Energy of ejected electrons
Cutoff Frequency	$f_c = \frac{\phi}{h}$	Minimum frequency for emission
Cutoff Wavelength	$\lambda_c = \frac{hc}{\phi}$	Maximum wavelength for emission
Stopping Potential	$V_s = \frac{KE_{\text{max}}}{e}$	Voltage to stop photoelectrons
Photon Energy	$E_{\text{photon}} = \frac{hc}{\lambda}$	Energy of a photon

Additional info:

The photoelectric effect demonstrates the particle-like properties of light, supporting the concept of photons.
The work function is material-dependent and typically measured in electron volts (eV).
Only photons with energy greater than the work function can eject electrons from the material.