Black body

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In physics, a black body is an object that absorbs all electromagnetic radiation that falls onto it. No radiation passes through it and none is reflected, yet it theoretically radiates every possible wavelength of energy. Despite the name, black bodies are not actually black as they radiate energy as well. The amount and type of electromagnetic radiation they emit is directly related to their temperature. Black bodies below around 700 K produce very little radiation at visible wavelengths and appear black (hence the name). Black bodies above this temperature, however, begin to produce radiation at visible wavelengths starting at red, going through orange, yellow, and white before ending up at blue as the temperature increases.

The term "black body" was introduced by Gustav Kirchhoff in 1862. The light emitted by a black body is called black-body radiationWhen used as a compound adjective, the term is typically hyphenated, as in "black-body radiation", or combined into one word, as in "blackbody radiation". The hyphenated and one-word forms should not generally be used as nouns, however..

Explanation

In the laboratory, the closest thing to black-body radiation is the radiation from a small hole entrance to a larger cavity. Any light entering the hole would have to reflect off the walls of the cavity multiple times before it escaped and is almost certain to be absorbed by the walls in the process, regardless of what they are made of or the wavelength of the radiation (as long as it is small compared to the hole). The hole, then, is a close approximation of a theoretical black body and, if the cavity is heated, the spectrum of the hole's radiation (i.e., the amount of light emitted from the hole at each wavelength) will be continuous, and will not depend on the material in the cavity (compare with emission spectrum). By a theorem proved by Kirchhoff, this curve depends only on the temperature of the cavity walls.

Calculating this curve was a major challenge in theoretical physics during the late nineteenth century. At that time, the newly-developed theories of electromagnetism and statistical thermodynamics predicted infinite brightness at high frequencies (or, equivalently, short wavelengths), a physical impossibility. This prediction came to be called the ultraviolet catastrophe. As a result, the best-known theories at that time could not explain the observed spectrum of black-body radiation.

The problem was finally solved in 1900 by Max Planck as Planck's law of black-body radiation. By making changes to Wien's Radiation Law (not to be confused with Wien's displacement law) consistent with Thermodynamics and Electromagnetism, he found a mathematical formula fitting the experimental data in a satisfactory way. To find a physical interpretation for this formula, Planck had then to assume that the energy of the oscillators in the cavity was quantized (i.e., integral multiples of some quantity). Einstein built on this idea and proposed the quantization of electromagnetic radiation itself in 1905 to explain the photoelectric effect. These theoretical advances eventually resulted in the replacement of classical electromagnetism by quantum mechanics. Today, these quanta are called photons. In addition, it led to the development of quantum versions of statistical mechanics, called Fermi-Dirac statistics and Bose-Einstein statistics, each applicable to a different class of particles. See also fermions and bosons.

The wavelength at which the radiation is strongest is given by Wien's displacement law, and the overall power emitted per unit area is given by the Stefan-Boltzmann law. So, as temperature increases, the glow color changes from red to yellow to white to blue. Even as the peak wavelength moves into the ultra-violet enough radiation continues to be emitted in the blue wavelengths that the body will continue to appear blue. It will never become invisible—indeed, the radiation of visible light increases monotonically with temperature.

The radiance or observed intensity is not a function of direction. Therefore a black body is a perfect Lambertian radiator.

Real objects never behave as full-ideal black bodies, and instead the emitted radiation at a given frequency is a fraction of what the ideal emission would be. The emissivity of a material specifies how well a real body radiates energy as compared with a black body. This emissivity depends on factors such as temperature, emission angle, and wavelength. However, it is typical in engineering to assume that a surface's spectral emissivity and absorptivity do not depend on wavelength, so that the emissivity is a constant. This is known as the grey body assumption.

When dealing with non-black surfaces, the deviations from ideal black body behavior are determined by both the geometrical structure and the chemical composition, and follow Kirchhoff's Law: emissivity equals absorptivity, so that an object that does not absorb all incident light will also emit less radiation than an ideal black body.

In astronomy, objects such as stars are frequently regarded as black bodies, though this is often a poor approximation. An almost perfect black-body spectrum is exhibited by the cosmic microwave background radiation. Hawking radiation is black-body radiation emitted by black holes.

Equations governing black bodies

Planck's law of black-body radiation

:[I(\nu) = \frac}\frac)-1}]

where

*[I(\nu)d\nu \,] is the amount of energy per unit surface per unit time per unit solid angle emitted in the frequency range between ν and ν+dν;

*[T \,] is the temperature of the black body;

*[h \,] is Planck's constant;

*[c \,] is the speed of light; and

*[k \,] is Boltzmann's constant.

Wien's displacement law

The relationship between the temperature T of a black body, and wavelength [\lambda_] at which the intensity of the radiation it produces is at a maximum is

[T \lambda_\mathrm = 2.898... \times 10^6 \ \mathrm. \,]

The nanometer is a convenient unit of measure for optical wavelengths. Note that 1 nanometer is equivalent to 10⁻⁹ meters.

Stefan-Boltzmann law

The total energy radiated per unit area per unit time [j^] (in watts per square meter) by a black body is related to its temperature T (in kelvins) and the Stefan-Boltzmann constant [\sigma] as follows:

:[j^ = \sigma T^4.\,]

Temperature relation between a planet and its star

Here is an application of black-body laws. It is a rough derivation that gives an order of magnitude answer. See p. 380-382 of Planetary Science, for further discussion.

Assumptions

The surface temperature of a planet depends on a few factors:

• Incident radiation (from the sun, for example)
• The albedo effect (the fraction of light a planet reflects)
• The greenhouse effect (for planets with an atmosphere)
• Energy generated internally by a planet itself (This is more important for planets like Jupiter)

For the inner planets, incident radiation has the most significant impact on surface temperature. This derivation is concerned mainly with that.

If we assume the following:

# The Sun and the Earth both radiate as spherical black bodies in thermal equilibrium with themselves.

# The Earth absorbs all the solar energy that it intercepts from the Sun.

then we can derive a formula for the relationship between the Earth's surface temperature and the Sun's surface temperature.

Derivation

To begin, we use the Stefan-Boltzmann law to find the total power (energy/second) the Sun is emitting:

:[P_ = \left( \sigma T_^4 \right) \left( 4 \pi R_^2 \right) \qquad \qquad (1)]

where

:[\sigma \,] is the Stefan-boltzmann constant,

:[T_S \,] is the surface temperature of the Sun, and

:[R_S \,] is the radius of the Sun.

The Sun emits that power equally in all directions. Because of this, the Earth is hit with only a tiny fraction of it. This is the power from the Sun that the Earth absorbs:

:[P_ = P_ \left( \frac^2} \right) \qquad \qquad (2)]

where

:[R_ \,] is the radius of the Earth and

:[D \,] is the distance between the Sun and the Earth.

Even though the earth only absorbs as a circular area [\pi R^2], it emits equally in all directions as a sphere:

:[P_ = \left( \sigma T_^4 \right) \left( 4 \pi R_^2 \right) \qquad \qquad (3)]

where [T_] is the surface temperature of the earth.

Now, in the first assumption the earth is in thermal equilibrium, so the power absorbed must equal the power emitted:

:[P_ = P_\,]

So plug in equations 1, 2, and 3 into this and we get

:[\left( \sigma T_^4 \right) \left( 4 \pi R_^2 \right) \left( \frac^2} \right) = \left( \sigma T_^4 \right) \left( 4 \pi R_^2 \right).\,]

Many factors cancel from both sides and this equation can be greatly simplified.

The result

After canceling of factors, the final result is

:

[T_\sqrt}} = T_ ]

where

[T_S \,] is the surface temperature of the Sun,

[R_S \,] is the radius of the Sun,

[D \,] is the distance between the Sun and the Earth, and

[T_E \,] is the average surface temperature of the Earth.

In other words, the temperature of the Earth only depends on the surface temperature of the Sun, the radius of the Sun, and the distance between the Earth and the Sun.

Temperature of the Sun

If we plug in the measured values for Earth,

:[T_ \approx 14 \ \mathrm} = 9500 \ \mathrm \,]

This, presumably, would be the wavelength that infrared goggles would be designed to be most sensitive to.

A few historical examples of black body radiation

Blast furnaces before 1700 heated with charcoal could only produce "red hot" pig iron. The introduction of coke for heating in English ironworks in 1709 enabled "yellow hot" iron, required for the more advanced products of the industrial revolution.

See also

• Color temperature
• Infrared thermometer

Footnotes

References

• Planck, Max, "[On the Law of Distribution of Energy in the Normal Spectrum]". Annalen der Physik, vol. 4, p. 553 ff (1901).

External links

• [Cooling Mechanisms for Human Body] - From Hyperphysics
• [Descriptions of radiation emitted by many different objects]
• [Black Body Emission Calculator]
• [Quantum Chemistry Lecture]

 

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