What is a black body

A black body (Kirchhoff, 1862) is called an ideal object that absorbs all the radiation it receives, at any frequency — hence the name: since it reflects nothing, at room temperature it appears black. But a black body is not just a perfect absorber: it is also the best possible emitter. The physical reason is thermal equilibrium. If an object sits at constant temperature $T$ inside a cavity, it must re-emit exactly as much energy as it absorbs, otherwise it would heat up or cool down; a perfect absorber is therefore also a perfect emitter.

The surprising fact is that, for a black body in thermal equilibrium at temperature $T$, the emission is a universal function of $T$ and frequency $f$ (or wavelength $\lambda$): it does not depend on the material. Iron, coal or hot gas, brought to the same temperature, emit the same spectrum. It is this universality that makes the black body a crucial testing ground for theory: a curve that depends only on $T$ and on fundamental constants of nature, and on nothing else.

An ideal black body does not exist in nature, but an excellent approximation can be built with a closed cavity fitted with a small hole. Any ray of light that enters through the hole bounces off the inner walls so many times that it is almost certainly absorbed before it can get out: the hole, seen from outside, absorbs practically everything and is therefore black. For the same reason, the radiation that leaves the hole when the cavity is hot is exactly that of an ideal black body.

A small hole in a closed cavity is the best practical realisation of a black body: the light that enters bounces off the walls until it is absorbed and almost never manages to escape.

What is measured is the spectral emission density $u(\lambda,T)$: how much energy the black body radiates in each interval of wavelengths. Experimentally this function shows three universal features, valid for any black body:

• continuous spectrum: $u(\lambda,T)$ has no lines, but a continuous trend with a single peak at a certain wavelength $\lambda_\text{max}$;
• Wien’s law: as temperature increases the peak shifts towards shorter wavelengths, according to $\lambda_\text{max}\cdot T = b$;
• Stefan-Boltzmann law: the total power emitted per unit area, summed over all frequencies, grows as $T^4$.

The first two tell us where the body emits most; the third tells us how much it emits in total.

Key formula

$\text{Wien:}\quad \lambda_\text{max}\,T = b \qquad\qquad \text{Stefan-Boltzmann:}\quad \frac{P}{A} = \sigma\,T^4$ Both are experimental laws, independent of any theoretical model.

Wien's law

The wavelength at which the black body emits the maximum of radiation is inversely proportional to temperature: $\ev{\lambda_\text{max}\,T = b \approx 2{,}898\cdot 10^{-3}\;\text{m K}}$ So the hotter a body is, the “bluer” its light becomes.

Stefan-Boltzmann law

The total power (over all frequencies) emitted per unit area by a black body grows with the fourth power of the absolute temperature: $\ev{\frac{P}{A} = \sigma\,T^4}, \qquad \sigma \approx 5{,}67\cdot 10^{-8}\;\frac{\text{W}}{\text{m}^2\,\text{K}^4}$ The $T^4$ dependence is violent: doubling the absolute temperature means emitting sixteen times more energy.

Example — The surface of the Sun and of our body

The Sun has a surface temperature $T_\odot \approx 5800\;\text{K}$. From Wien’s law: $\lambda_\text{max} = \frac{b}{T_\odot} \approx \frac{2{,}898\cdot 10^{-3}}{5800}\;\text{m} \approx 5\cdot 10^{-7}\;\text{m} = 500\;\text{nm}$ that is, in the visible range (green-yellow). This is no coincidence: our eyes evolved to be sensitive exactly where the Sun emits most.

The human body is at $T \approx 310\;\text{K}$, so $\lambda_\text{max} = \frac{b}{T} \approx \frac{2{,}898\cdot 10^{-3}}{310}\;\text{m} \approx 9{,}4\;\mu\text{m}$ we therefore emit in the far infrared. This is exactly what a thermal camera sees, and what makes it possible to spot a living being in the dark.

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Topics: Quantum physics Concepts: Black body and Planck’s hypothesis · Temperature · Heat transfer

Related exercises: Metal and wood to the touch · The Sun’s temperature from Wien’s law · Problem — Black body at different temperatures