Photoelectric Effect: Cutoff Wavelength, Frequency, 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 effect is governed by the energy of incident photons and the material's work function (φ), which is the minimum energy required to remove an electron from the surface.

Key Equations and Concepts

Photoelectric Effect Equation

The maximum kinetic energy (KEmax) of ejected electrons 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)

• f = frequency of incident light

• λ = wavelength of incident light

• φ = work function of the material (in eV or J)

Cutoff Wavelength (λc)

The cutoff wavelength is the longest wavelength (lowest energy) of light that can eject electrons from the material. At this wavelength, $KE_{\text{max}} = 0$:

$KE_{\text{max}} = \frac{hc}{\lambda_c} - \phi = 0 \implies \lambda_c = \frac{hc}{\phi}$

• 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 Frequency (fc)

The cutoff frequency is the minimum frequency of light required to eject electrons:

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

• Example Calculation:

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

Work Function (φ)

The work function is the minimum energy needed to remove an electron from the material's surface. It can be calculated if the cutoff wavelength is known:

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

• Example Calculation:

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

Maximum Kinetic Energy of Ejected Electrons

The maximum kinetic energy is found using the energy of the incident photon and the work function:

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

• Example Calculation:

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

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

Stopping Potential (ΔVs)

The stopping potential is the voltage required to stop the most energetic photoelectrons:

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

• Example Calculation:

If $KE_{\text{max}} = 2.15$ eV:

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

Photon Energy Calculation

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

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

• Example Calculation:

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

Velocity of Ejected Electrons

The maximum velocity of ejected electrons can be found from their kinetic energy:

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

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

• Example Calculation:

If $KE = 4.00 \times 10^{-19}$ J and $m_e = 9.11 \times 10^{-31}$ kg:

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

Graphical Representation

Maximum Kinetic Energy vs. Frequency

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	φ	$\phi = \frac{hc}{\lambda_c}$	eV or J
Cutoff Wavelength	λc	$\lambda_c = \frac{hc}{\phi}$	m or nm
Cutoff Frequency	fc	$f_c = \frac{c}{\lambda_c}$	Hz
Photon Energy	Ephoton	$E_{photon} = hf = \frac{hc}{\lambda}$	eV or J
Maximum Kinetic Energy	KEmax	$KE_{max} = E_{photon} - \phi$	eV or J
Stopping Potential	Vs	$V_s = \frac{KE_{max}}{e}$	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.

• Increasing the intensity of light increases the number of ejected electrons but does not affect their maximum kinetic energy unless the frequency is also increased.