Lecture Notes

Black Body Radiation

Definition

• Black Body: An object that absorbs all kinds of electromagnetic radiation (all wavelengths and frequencies).
• Black Body Radiation: Discussed energy distribution at different temperatures in previous video.

Key Points

• Uses Planck's Quantum Theory to explain energy distribution curve.
• Planck's Theory: Radiation made of tiny packets of energy called photons.
  • Photon Energy: Given by E = hν (where h is Planck's constant and ν is frequency).
  • Properties of Photons: Indistinguishable, identical, integral spin angular momentum.
• Distribution Curve: Plotted between wavelength (λ) and energy.

Planck's Quantum Theory

• Energy Distribution Curve: Explains using Planck's theory.
• Photons: Follow Bose-Einstein statistics; radiation considered both emitted and absorbed.
• Energy Distribution Law: Important for deriving related formulas.

Bose-Einstein Energy Distribution Law

Formula Derivation

• Isolated Gaseous System: Compartmentalized into elementary cells for calculations.
• Energy Range: Calculation of number of particles with mean energy U.
• General Derivation: Nᵢ = Gᵢ / (e^(α + βUᵢ) - 1)

Applying to Photons

• Replace general variables with photon-specific terms.
• Photons in Black Body: Treated similarly to gas molecules (random movement).
• Photon Gas: Conceptual framework within the black body

Number of Photons

• Energy Range: Calculation within the energy interval U to U + dU.
• Energy and Absorption: Black body does not maintain a constant number of photons.
• Mathematical Adjustment: Introduce alpha = 0 and beta = 1/kT (Boltzmann constant).
• Formula: N(U) dU = G(U) dU / (e^(hν / kT) - 1)

Planck's Radiation Law

In Terms of Frequency

• Energy Density Calculation: Multiplying the photon count by hν.
• Energy Density Expression: For frequency range ν to ν + dν

Derivation

• Convert phase-space cell count expression in momentum terms to frequency terms.
• Photon Momentum: P = hν / c
• Derived Expression: n(ν) dν = 8πVν² dν / (c³ (e^(hν/kT) - 1))

Final Steps

• Unit Volume Condition: Divide by volume to get energy density per unit volume.
• Key Derivation: u(ν) dν = 8πhν³ / (c³ (e^(hν / kT) - 1))

Conversion to Wavelength Terms

Frequency to Wavelength

• Use ν = c/λ and its differential form.
• Energy Density Expression: u(λ) dλ = 8πhc / (λ⁵ (e^(hc / λkT) - 1))

Conclusion

• Planck's Law: Derived in terms of wavelength for range λ to λ + dλ
• Exam Preparation: Important to know how to derive and use these formulas correctly for full marks.