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Lesson 11.1 • Competency 11.1Complete Lesson
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Black-Body Radiation and Quantum Theory

Black-body radiation curves, Wien’s displacement law λ_max T = b, Stefan-Boltzmann law E = σT⁴, classical failure (ultraviolet catastrophe), and Planck’s quantum hypothesis E = nhf.

Marking a lesson complete represents finishing your study of the topic; it does not denote tested mastery.

Learning Objectives

  • Explain the spectral distribution of black-body radiation and apply Wien’s displacement law and the Stefan-Boltzmann law.
  • Describe how Planck’s quantization hypothesis resolved the ultraviolet catastrophe and initiated modern quantum physics.

Core Theory & Concepts

A black body absorbs all incident electromagnetic radiation regardless of frequency or angle of incidence. The spectral distribution of its emitted radiation at thermodynamic temperature T could not be explained by 19th-century classical electromagnetism, compelling Max Planck in 1900 to introduce energy quantization.

Wien’s Displacement Law

The wavelength of peak emissive power λ_max is inversely proportional to absolute thermodynamic temperature T: λ_max T = b, where Wien constant b ≈ 2.898 × 10⁻³ m K.

\lambda_{\max} T = b = 2.898 \times 10^{-3} \text{ m K}

Stefan-Boltzmann Law

The total energy radiated per unit surface area per second across all wavelengths is proportional to the fourth power of absolute temperature: E = σ T⁴, where σ = 5.670 × 10⁻⁸ W m⁻² K⁻⁴.

E = \sigma T^4

Planck’s Quantum Hypothesis

Atomic oscillators in cavity walls can only absorb or radiate thermal energy in discrete bundles (quanta): E = n h f, where n = 1, 2, 3... and h = 6.626 × 10⁻³⁴ J s.

E = n h f = \frac{n h c}{\lambda}

Common Misconceptions

Misconception

As a black body gets hotter, its peak radiation wavelength shifts toward longer wavelengths (red).

Physical Clarification:By Wien’s law λ_max = b/T, higher temperatures shift peak emission to SHORTER wavelengths (blue/UV). This is why a heating iron glows red, then orange, then bright white.

Syllabus Topic Outline

Competency 11.1
  • 1Concept of a perfect black body and experimental cavity radiator
  • 2Spectral energy distribution curves at different temperatures
  • 3Wien’s displacement law: λ_max T = constant (2.898 × 10⁻³ m K)
  • 4Stefan-Boltzmann law: total emissive power E = σ T⁴
  • 5Rayleigh-Jeans law failure and the Ultraviolet Catastrophe
  • 6Max Planck’s quantum hypothesis: discrete energy packets E = hf

Practice Problems