Blackbody Radiation

A blackbody is an ideal surface that absorbs all incident electromagnetic radiation and, in thermal equilibrium, emits the maximum possible spectral radiance at each wavelength for its temperature. A cavity with a small hole provides a close experimental approximation.

1. Definitions and quantities

• Absolute temperature, T (K), sets the shape and scale of the spectrum.

• Spectral intensity describes how emission is distributed over wavelength or frequency. Always identify which variable is on the horizontal axis.

• Emissivity, e, is the ratio of a real surface’s total emissive power to that of a blackbody at the same temperature; 0 ≤ e ≤ 1.

• Stefan–Boltzmann law:

  P = eσ AT⁴

  where σ = 5.67 × 10⁻⁸ W m⁻²K⁻⁴.

• Wien displacement law:

  λₘₐₓT = b

  where b = 2.90 × 10⁻³ m K for a wavelength spectrum.

Blackbody spectra and the Rayleigh–Jeans failure

Hotter and cooler blackbody spectra each rise to a peak and fall at short wavelength. The hotter peak is at shorter wavelength and has greater area. A dashed Rayleigh–Jeans curve diverges toward zero wavelength.

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Values and uncertainty for Blackbody spectra and the Rayleigh–Jeans failure

Series	Wavelength (relative units)	Wavelength uncertainty	Spectral intensity (relative units)	Spectral intensity uncertainty
Hotter body	0.5		0.01
Hotter body	1		0.12
Hotter body	1.5		0.45
Hotter body	2		0.78
Hotter body	2.5		0.96
Hotter body	3		1
Hotter body	3.5		0.94
Hotter body	4		0.83
Hotter body	5		0.62
Hotter body	6		0.46
Hotter body	7		0.34
Hotter body	8		0.25
Hotter body	9		0.18
Hotter body	10		0.13
Cooler body	0.5		0
Cooler body	1		0.01
Cooler body	1.5		0.04
Cooler body	2		0.1
Cooler body	2.5		0.2
Cooler body	3		0.32
Cooler body	3.5		0.44
Cooler body	4		0.53
Cooler body	5		0.6
Cooler body	6		0.57
Cooler body	7		0.49
Cooler body	8		0.4
Cooler body	9		0.32
Cooler body	10		0.25
Rayleigh–Jeans prediction	0.5		1.45
Rayleigh–Jeans prediction	1		0.9
Rayleigh–Jeans prediction	1.5		0.58
Rayleigh–Jeans prediction	2		0.4
Rayleigh–Jeans prediction	2.5		0.29
Rayleigh–Jeans prediction	3		0.22
Rayleigh–Jeans prediction	4		0.14
Rayleigh–Jeans prediction	5		0.1
Rayleigh–Jeans prediction	6		0.07
Rayleigh–Jeans prediction	8		0.04
Rayleigh–Jeans prediction	10		0.025

Values and uncertainty for Blackbody spectra and the Rayleigh–Jeans failure

Series	Wavelength (relative units)	Wavelength uncertainty	Spectral intensity (relative units)	Spectral intensity uncertainty
Hotter body	0.5		0.01
Hotter body	1		0.12
Hotter body	1.5		0.45
Hotter body	2		0.78
Hotter body	2.5		0.96
Hotter body	3		1
Hotter body	3.5		0.94
Hotter body	4		0.83
Hotter body	5		0.62
Hotter body	6		0.46
Hotter body	7		0.34
Hotter body	8		0.25
Hotter body	9		0.18
Hotter body	10		0.13
Cooler body	0.5		0
Cooler body	1		0.01
Cooler body	1.5		0.04
Cooler body	2		0.1
Cooler body	2.5		0.2
Cooler body	3		0.32
Cooler body	3.5		0.44
Cooler body	4		0.53
Cooler body	5		0.6
Cooler body	6		0.57
Cooler body	7		0.49
Cooler body	8		0.4
Cooler body	9		0.32
Cooler body	10		0.25
Rayleigh–Jeans prediction	0.5		1.45
Rayleigh–Jeans prediction	1		0.9
Rayleigh–Jeans prediction	1.5		0.58
Rayleigh–Jeans prediction	2		0.4
Rayleigh–Jeans prediction	2.5		0.29
Rayleigh–Jeans prediction	3		0.22
Rayleigh–Jeans prediction	4		0.14
Rayleigh–Jeans prediction	5		0.1
Rayleigh–Jeans prediction	6		0.07
Rayleigh–Jeans prediction	8		0.04
Rayleigh–Jeans prediction	10		0.025

2. Reading a spectrum

For a wavelength spectrum:

• the area under the curve is proportional to total emitted power per unit area;
• the hotter curve lies higher overall and has a larger area because total power varies as T⁴;
• the hotter curve peaks at smaller λ because λₘₐₓ ∝ 1/T;
• the intensity tends to zero at both very long and very short wavelength.

Do not convert a wavelength peak to a frequency peak using f = c/λₘₐₓ. Spectral density is defined per unit interval, and the change of variable introduces a Jacobian; the two peak locations are not simple reciprocals.

3. Physical model and assumptions

A cavity’s walls absorb and emit radiation until the radiation field reaches thermal equilibrium. The small hole samples that field while disturbing it minimally. The ideal laws assume uniform temperature and negligible external irradiation when discussing emitted power.

For a real body exchanging radiation with surroundings at temperature Tₛ, the net radiative power is

Pₙₑₜ = eσ A(T⁴-Tₛ⁴)

not simply eσ AT⁴.

4. Common mistakes

• Saying a blackbody emits no radiation because it appears black. Absorption and emission are different processes; a hot blackbody is a strong emitter.
• Using degrees Celsius in fourth-power or displacement laws. Use kelvin.
• Treating the peak height as total power. Total power corresponds to area under the spectrum.
• Applying Wien’s wavelength constant to a frequency-spectrum peak.
• Forgetting emissivity or surrounding radiation for a real surface when the question supplies them.

6. Worked Examples

7. Model check

Next: Failure of Classical Theory.