# Summary

A blackbody is an idealized volume which emits and absorbs the maximum possible amount of radiation at a given temperature in all directions over a wide range of wavelengths. Blackbodies are perfect emitters and absorbers of radiation and therefore useful as a standard when studying radiative heat transfer systems where the amount of radiation emitted and absorbed is a also a function of material properties. This article describes the basics of a black body and presents equations to describe its emissive characteristics.

# Definitions

: | The spectral emissive power inside a blackbody cavity (W.m^{-3}) | |

: | A radiation constant (C_{1} = 2. π.h.c^{2}_{0} = 3.7421 x 10^{-16} W.m^{2}) | |

: | A radiation constant (C_{2} = h.c_{0}/k = 1.4388 x 10^{-2} m.K) | |

: | A radiation constant (C_{3} = 2.8978 x 10^{-3} m.K) | |

: | The wavelength of maximum emission (m) | |

: | Stefan-Boltzmann constant (σ = 5.6704 x 10^{-8} W.m^{-2}.K^{-4}) | |

: | The spectral intensity of a surface (J.m^{-3}) |

# Introduction

In simple terms a blackbody is a volume which operates as a perfect absorber and emitter of thermal radiation. More specifically a blackbody is a volume which:

- Absorbs all incident radiation independent of wavelength and direction.
- Will emit more energy than any other surface at a given temperature.
- Emits radiation independent of direction, i.e. a diffuse emitter.

Real surfaces never exhibit the ideal behaviour of a black body, however blackbody behaviour may be closely approximated by a cavity whose inner walls are maintained at a uniform temperature and where radiation is permitted to enter and leave the cavity via a small aperture.

This particular geometry exhibits blackbody behaviour as all radiation entering the cavity will experience many reflections and be entirely absorbed by the cavity (rule 1), all energy emitted by the walls in the cavity will be emitted from the aperture as the walls are iso-thermal (rule 2) and radiation emitted from the aperture will be diffuse due to the multiple reflections it will undergo within the cavity before emission (rule 3).

# Total Emissive Power

The total emissive power of a blackbody was determined experimentally in 1879 by Joseph Stefan and verified theoretically in 1884 by Ludwig Boltzmann and may be expressed as a function of absolute temperature as follows:

This equation is known as the Stefan-Boltzmann law and may be used to calculate the emissive power of a blackbody over all wavelengths, in all directions per unit time and area.

# Spectral Emissive Power

The Stefan-Boltzmann law permits the calculation of the total emissive power of a blackbody, however it is often useful to know the emissive power about a particular wavelength i.e. for monochromatic radiation.

In 1901 Max Planck developed the relationship which describes the spectral emissive power of a blackbody as a function of temperature and wavelength. This equation, displayed below, allows the calculation of the energy emitted from a blackbody for monochromatic radiation.

This relationship is known as the Planck distribution. Perhaps the best visualisation of the emission characteristics of a black body are observed when plotting the Plank distribution over a wide range of wavelengths and temperatures:

From the figure above the following emissive characteristics of blackbody’s become apparent:

- The emitted radiation varies continuously with wavelength
- The peak emissive power increases with temperature
- As the temperature of the body increases the wavelength at the point of peak emissive power reduces

# Wavelength of Maximum Emissive Power

The relationship between the wavelength at which maximum emission occurs and the absolute temperature of the blackbody is characterised by Wein’s Displacement Law, which is obtained by differentiating Equation with respect to and solving for

Wein’s Displacement law states that the wavelength for maximum emission varies inversely with the absolute temperature of the source. This can be seen in practice in the effect of the Earth’s atmosphere on incoming radiation from the sun. The gases in the Earth’s atmosphere will transmit the short wavelength radiation originating from the sun (acting as a blackbody at ~5800 K) but fail to transmit the longer wavelength radiation from the Earth which is at a significantly lower temperature.

# Further Reading

- Heat Transfer Handbook
- Heat Transfer A Practical Approach
- Chemical Engineering Volume 1, Sixth Edition: Fluid Flow, Heat Transfer and Mass Transfer
- Radiative Heat Transfer, Third Edition

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