Black-Body Radiation Laws

Every object with a temperature above absolute zero radiates energy in the form of electromagnetic waves. A perfect absorber and emitter of radiation is called a black body — it absorbs all incoming radiation regardless of wavelength and re-emits energy purely as a function of its temperature. Real objects approximate this ideal to varying degrees, but the black-body model gives us powerful, exact laws that describe how much energy is radiated and at which wavelengths the emission peaks. Two landmark laws govern black-body radiation. The Stefan-Boltzmann law tells us the total power radiated per unit surface area across all wavelengths, and it turns out this depends only on the fourth power of the absolute temperature. Wien's displacement law tells us the wavelength at which emission is most intense, and it shows that hotter objects radiate at shorter (bluer) wavelengths. Together these laws explain why a heating element glows from dull red to bright orange as it gets hotter, why the Sun's surface temperature can be inferred from its yellow-white colour, and why night-vision cameras detect the infrared glow of warm bodies. These results were established empirically in the late 19th century and later explained from first principles by Max Planck's quantum theory in 1900. Understanding black-body radiation is therefore not just a thermal-physics topic — it is historically the gateway to quantum mechanics. For the high school physicist, the focus is on applying the two laws quantitatively and developing physical intuition for how temperature drives radiation.

Key takeaways
  • What Makes a Black Body?
  • The Stefan-Boltzmann Law: Total Power Radiated
  • Wien's Displacement Law: Peak Wavelength
  • The Planck Curve and Why ν-space vs. λ-space Matters

What Makes a Black Body?

A black body is an idealised object that absorbs 100% of any electromagnetic radiation that hits it — no reflection, no transmission. Because it is in thermal equilibrium, it must also be the best possible emitter. The emissivity ε of a real surface runs from 0 (perfect mirror) to 1 (perfect black body). Most physics problems either treat surfaces as perfect black bodies (ε = 1) or multiply the result by ε for real materials.

The Stefan-Boltzmann Law: Total Power Radiated

Josef Stefan (1879, experimentally) and Ludwig Boltzmann (1884, theoretically) showed that the total power radiated per unit area by a black body is proportional to the fourth power of its absolute temperature T. This T⁴ dependence is dramatic: doubling the temperature increases radiated power by a factor of 2⁴ = 16. The Stefan-Boltzmann constant σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴ connects the formula to measurable quantities. For a real body with emissivity ε and surface area A, the total radiated power is P = εσAT⁴. Net power lost by a body at temperature T surrounded by an environment at T₀ is P_net = εσA(T⁴ − T₀⁴).

Wien's Displacement Law: Peak Wavelength

A black body does not radiate equally at all wavelengths. The Planck curve (intensity vs. wavelength) has a peak at a wavelength λ_max that depends inversely on temperature. Wilhelm Wien discovered this in 1893: λ_max × T = b, where b = 2.898 × 10⁻³ m·K is Wien's displacement constant. 'Displacement' refers to the peak being displaced to shorter wavelengths as temperature rises. The Sun's surface (~5 778 K) peaks near 500 nm (visible green-yellow), matching peak human eye sensitivity. A human body (~310 K) peaks near 9.4 μm — deep infrared, invisible to the eye but detectable by thermal cameras. A light-bulb filament at ~2 700 K peaks around 1 070 nm (near-infrared), which is why incandescent bulbs are inefficient: most energy is emitted as infrared heat, not visible light.

The Planck Curve and Why ν-space vs. λ-space Matters

The full description of black-body emission is given by Planck's radiation law, which gives spectral irradiance as a function of wavelength. The Stefan-Boltzmann and Wien laws are both derived from it: integrating the Planck curve over all wavelengths gives the σT⁴ result, and differentiating to find the peak gives Wien's displacement law. At the high school level you apply the two simpler laws directly, but knowing they come from a single underlying formula shows the unity of the theory.

The Key Equation

P=εσAT4,λmax⁡T=bP = \varepsilon \sigma A T^{4}, \qquad \lambda_{\max} T = b
  • P — Radiated Power (W): Total electromagnetic power emitted by the surface.
  • \varepsilon — Emissivity (dimensionless (0 to 1)): Fraction of black-body emission the real surface achieves; ε = 1 for an ideal black body.
  • \sigma — Stefan-Boltzmann Constant (W m⁻² K⁻⁴): Universal constant equal to 5.67 × 10⁻⁸ W m⁻² K⁻⁴.
  • A — Surface Area (m²): Total radiating surface area of the object.
  • T — Absolute Temperature (K): Temperature measured from absolute zero on the Kelvin scale.
  • \lambda_{\max} — Peak Wavelength (m): Wavelength at which spectral intensity is maximum.
  • b — Wien's Displacement Constant (m·K): Universal constant equal to 2.898 × 10⁻³ m·K.

Worked Example

Problem. The surface temperature of the star Sirius A is approximately 9 940 K and its radius is 1.19 × 10⁹ m. Treating it as a black body (ε = 1): (a) Find the wavelength of peak emission. (b) Calculate the total power radiated by the star.

λmax⁡≈292 nm,P≈9.9×1027 W\lambda_{\max} \approx 292 \text{ nm}, \quad P \approx 9.9 \times 10^{27} \text{ W}

Answer. Peak wavelength ≈ 292 nm (ultraviolet); total radiated power ≈ 9.9 × 10²⁷ W.

Figure
Diagram
Planck Radiation Curves for Three TemperaturesWavelength (nm)Spectral Intensity05001000150020006000 K (Sun)4000 K2700 K (bulb)~483nm~725nmpeak>2000nmArea under each curve ∝ T⁴ (Stefan-Boltzmann)Peak shifts left as T rises (Wien's law)
Origins · 1 minJosef Stefan and how this was discovered
JS
Josef Stefan
1835–1893 · Austrian

Josef Stefan was an Austrian physicist who in 1879 empirically deduced from experimental data that the total radiant energy emitted by a black body is proportional to the fourth power of its absolute temperature. His student Ludwig Boltzmann later derived this result theoretically from thermodynamics in 1884, leading to the law bearing both their names. Stefan also used his law to make the first reasonably accurate estimate of the Sun's surface temperature. Stefan was born to Slovenian-speaking parents and grew up in modest circumstances, yet rose to become director of the Institute for Experimental Physics in Vienna. His student Boltzmann credited him as the greatest influence on his scientific career.

  • Modes of Heat Transfer — Black-body radiation is one of the three mechanisms of heat transfer; understanding it requires first distinguishing it from conduction and convection.
  • Planck's Quantum Hypothesis — The Stefan-Boltzmann and Wien laws are both derived from Planck's radiation law, which introduced the concept of quantised energy and launched quantum mechanics.
  • The Photoelectric Effect — Einstein's explanation of the photoelectric effect used Planck's quantum ideas about light, directly building on the black-body radiation problem.
  • The Electromagnetic Spectrum — Wien's law predicts where on the EM spectrum a body's peak emission falls — spanning infrared, visible, and ultraviolet depending on temperature.