Chapter 5

Quantum mechanics

Black-body Radiation

Fondamental

What is a black-body?

The law of the black-body describes its electromagnetic radiation behavior when it is in thermal equilibrium with matter.

It specifies the spectral repartition of total electromagnetic energy density.

Let be the spectral density, so that the density of electromagnetic energy between the frequencies and is written :

.

Physicist Planck established at the end of the XIXth century that the spectral density could be written as :

The figure below shows the graphs of the curves associated to at different temperatures.

We establish that these curves all go through a maximum, which depends on temperature, for a frequency .

It follows Wien's law :

Black-body spectrum

It is remarkable to notice that the spectrum of a black-body radiation is continuous and does not depend on the physical-chemical nature of the latter, but only depends on its temperature.

These properties are a direct consequence of the thermal equilibrium in which the body is.

The emission and absorption spectra of atoms are discontinuous.

When a body is heated, it emits a radiation in its spectrum of emission.

However, the emission and absorption spectra of an atom are identical.

Hence, at thermal equilibrium, atoms located inside the black-body absorb all of the radiation they emit.

There is only one radiation left in the body. It results from its thermodynamic state, and its spectrum is continuous.

ExempleThe Sun is a black body

The maximum solar light visible at the surface of the Earth is yellow-green, for a wave-length around , hence a frequency :

This radiation is the one of a black-body of temperature :

This matches the temperature of the surface of the Sun.

ComplémentA 7 minutes lesson about black-body radiation

Black-body radiation (7 minutes lesson)
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