2. The Photoelectric Effect

Learning Outcomes
  • I can describe the photoelectric effect.
  • I can explain why light below a threshold frequency cannot eject electrons.
  • I can relate photon energy to electron emission.
  • I can explain how the photoelectric effect supports the photon model of light.
  • I can solve simple photoelectric-effect problems.

Key Topics:
  • de Broglie wavelength formula: λ=hp
  • Applications to everyday particles

Let's calculate the de Broglie wavelength (

λ\lambda

) of particles using the equation:

λ=hp=hmv\lambda = \frac{h}{p} = \frac{h}{mv}

where:

  • λ\lambda = de Broglie wavelength (m)
  • hh = Planck’s constant =6.63×10−346.63 \times 10^{-34} J·s
  • p=mvp = mv = momentum (kg·m/s)
  • mm = mass of the particle (kg)
  • vv = velocity of the particle (m/s)

1. Example Calculations

Let's compute the de Broglie wavelength for:
1️⃣ An electron moving at

5.0×1065.0 \times 10^6

 m/s
2️⃣ A proton moving at

3.0×1053.0 \times 10^5

 m/s
3️⃣ A baseball (

m=0.145m = 0.145

 kg) moving at 40 m/s

I'll perform these calculations now.

Results: de Broglie Wavelengths

1️⃣ Electron moving at

5.0×1065.0 \times 10^6

 m/s:

λ=1.46×10−10 m(Comparable to atomic spacing)\lambda = 1.46 \times 10^{-10} \text{ m} \quad (\text{Comparable to atomic spacing})

2️⃣ Proton moving at

3.0×1053.0 \times 10^5

 m/s:

λ=1.32×10−12 m(Smaller than X-ray wavelengths)\lambda = 1.32 \times 10^{-12} \text{ m} \quad (\text{Smaller than X-ray wavelengths})

3️⃣ Baseball (

0.1450.145

 kg) moving at 40 m/s:

λ=1.14×10−34 m(Negligible, classical motion dominates)\lambda = 1.14 \times 10^{-34} \text{ m} \quad (\text{Negligible, classical motion dominates})

📌 Conclusion:
✔ Quantum effects are significant for electrons and protons (small masses).
✔ Macroscopic objects like baseballs have extremely tiny wavelengths, making quantum effects undetectable.

Activities:

  • Solving numerical problems on de Broglie wavelengths
  • Diagramming de Broglie waves for different particles

Assessment:

  • Worksheet on de Broglie wavelength calculations

Einstein's profound explanation of the photoelectric effect revolves around the concepts of the work function and the maximum kinetic energy of photoelectrons, offering a compelling framework to calculate the work function of metals and unravel the mysteries of photon-electron interactions in the radiant symphony of physics.

Einstein's explanation of the photoelectric effect can be summarized as follows:

  1. Work Function (

    ΦΦ

    ): The work function of a metal represents the minimum energy required to liberate an electron from the surface of the material. It is denoted by the symbol

    ΦΦ

    and is measured in electronvolts (eV). The work function determines the threshold energy that photons must possess to release photoelectrons from the metal surface.

  2. Maximum Kinetic Energy of Photoelectrons (

    KmaxK_{\text{max}}

    ): According to Einstein's theory, the maximum kinetic energy of photoelectrons emitted in the photoelectric effect is determined by the difference between the energy of incident photons and the work function of the metal. The equation for calculating the maximum kinetic energy is:

Kmax=Ephoton−ΦK_{\text{max}} = E_{\text{photon}} - Φ

Where:

KmaxK_{\text{max}}

 = Maximum kinetic energy of photoelectrons

EphotonE_{\text{photon}}

= Energy of incident photon

ΦΦ

 = Work function of the metal

By measuring the maximum kinetic energy of photoelectrons and knowing the frequency or wavelength of incident light, one can calculate the work function of the metal using Einstein's photoelectric equation.

To calculate the work function of a metal, one can rearrange the equation for maximum kinetic energy as:

Φ=Ephoton−KmaxΦ = E_{\text{photon}} - K_{\text{max}}

By substituting the energy of the incident photon and the measured maximum kinetic energy of photoelectrons into the equation, the work function of the metal can be determined. This approach allows physicists to experimentally determine the work function of various metals and validate Einstein's theory of the photoelectric effect through quantitative measurements and calculations.