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A black body absorbs and then re-emits all incident EM radiation. By definition it has an absorptivity and emissivity of 1, and a transmissivity and reflectivity of 0. The Planck black body equation describes the spectral exitance of an ideal black body. The study of black-body radiation was an integral step in the formulation of quantum mechanics.

## Contents

### Planck's Law: Wavelength

Formulated in terms of wavelength: where:

Symbol Units Description  Input wavelength  Input temperature  Planck's constant  Speed of light in vacuum  Boltzmann constant

Note that the input is in meters and that the output is a spectral irradiance in . Omitting the term from the numerator gives the blackbody emission in terms of radiance, with units where "sr" is steradians.

### Planck's Law: Frequency

Formulated in terms of frequency: where:

Symbol Units Description  Input frequency

All other units are the same as for the Wavelength formulation. Again, dropping the from the numerator gives the result in radiance rather than irradiance.

### Properties of the Planck Equation

Taking the first derivative leads to the wavelength with maximum exitance. This is known as the Wien Displacement Law.

A closed form solution exists for the integral of the Planck blackbody equation over the entire spectrum. This is the Stefan-Boltzmann equation. In general, there is no closed-form solution for the definite integral of the Planck blackbody equation; numerical integration techniques must be used.

The relationship between the ideal blackbody exitance and the actual exitance of a surface is given by emissivity.

An ideal blackbody at 300K (~30 Celsius) has a peak emission 9.66 microns. It has virtually no self-emission before 2.5 microns, hence self-emission is typically associated with the "thermal" regions of the EM spectrum. However, the Sun can be characterized as a 5900K blackbody and has a peak emission around 0.49 microns which is in the visible region of spectrum.

The Planck equation has a single maximum. The wavelength with peak exitance becomes shorter as temperature increases. The total exitance increases with temperature.