Black body

Black body

Overview
A black body is an idealized physical body
Physical body
In physics, a physical body or physical object is a collection of masses, taken to be one...

 that absorbs all incident electromagnetic radiation
Electromagnetic radiation
Electromagnetic radiation is a form of energy that exhibits wave-like behavior as it travels through space...

. Because of this perfect absorptivity at all wavelengths, a black body is also the best possible emitter of thermal radiation
Thermal radiation
Thermal radiation is electromagnetic radiation generated by the thermal motion of charged particles in matter. All matter with a temperature greater than absolute zero emits thermal radiation....

, which it radiates incandescently
Incandescence
Incandescence is the emission of light from a hot body as a result of its temperature. The term derives from the Latin verb incandescere, to glow white....

 in a characteristic, continuous spectrum that depends on the body's temperature. At Earth-ambient temperatures this emission is in the infrared
Infrared
Infrared light is electromagnetic radiation with a wavelength longer than that of visible light, measured from the nominal edge of visible red light at 0.74 micrometres , and extending conventionally to 300 µm...

 region of the electromagnetic spectrum
Electromagnetic spectrum
The electromagnetic spectrum is the range of all possible frequencies of electromagnetic radiation. The "electromagnetic spectrum" of an object is the characteristic distribution of electromagnetic radiation emitted or absorbed by that particular object....

 and is not visible. The object appears black, since it does not reflect or emit any visible light.

The thermal radiation from a black body is energy converted electrodynamically from the body's pool of internal thermal energy
Thermal energy
Thermal energy is the part of the total internal energy of a thermodynamic system or sample of matter that results in the system's temperature....

 at any temperature greater than absolute zero
Absolute zero
Absolute zero is the theoretical temperature at which entropy reaches its minimum value. The laws of thermodynamics state that absolute zero cannot be reached using only thermodynamic means....

.
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Encyclopedia
A black body is an idealized physical body
Physical body
In physics, a physical body or physical object is a collection of masses, taken to be one...

 that absorbs all incident electromagnetic radiation
Electromagnetic radiation
Electromagnetic radiation is a form of energy that exhibits wave-like behavior as it travels through space...

. Because of this perfect absorptivity at all wavelengths, a black body is also the best possible emitter of thermal radiation
Thermal radiation
Thermal radiation is electromagnetic radiation generated by the thermal motion of charged particles in matter. All matter with a temperature greater than absolute zero emits thermal radiation....

, which it radiates incandescently
Incandescence
Incandescence is the emission of light from a hot body as a result of its temperature. The term derives from the Latin verb incandescere, to glow white....

 in a characteristic, continuous spectrum that depends on the body's temperature. At Earth-ambient temperatures this emission is in the infrared
Infrared
Infrared light is electromagnetic radiation with a wavelength longer than that of visible light, measured from the nominal edge of visible red light at 0.74 micrometres , and extending conventionally to 300 µm...

 region of the electromagnetic spectrum
Electromagnetic spectrum
The electromagnetic spectrum is the range of all possible frequencies of electromagnetic radiation. The "electromagnetic spectrum" of an object is the characteristic distribution of electromagnetic radiation emitted or absorbed by that particular object....

 and is not visible. The object appears black, since it does not reflect or emit any visible light.

The thermal radiation from a black body is energy converted electrodynamically from the body's pool of internal thermal energy
Thermal energy
Thermal energy is the part of the total internal energy of a thermodynamic system or sample of matter that results in the system's temperature....

 at any temperature greater than absolute zero
Absolute zero
Absolute zero is the theoretical temperature at which entropy reaches its minimum value. The laws of thermodynamics state that absolute zero cannot be reached using only thermodynamic means....

. It is called blackbody radiation and has a frequency distribution with a characteristic frequency of maximum radiative power that shifts to higher frequencies with increasing temperature. As the temperature increases past a few hundred degrees Celsius
Celsius
Celsius is a scale and unit of measurement for temperature. It is named after the Swedish astronomer Anders Celsius , who developed a similar temperature scale two years before his death...

, black bodies start to emit visible wavelengths, appearing red, orange, yellow, white, and blue with increasing temperature. When an object is visually white, it is emitting a substantial fraction as ultraviolet radiation.

Blackbody emission provides insight into the thermodynamic equilibrium
Thermodynamic equilibrium
In thermodynamics, a thermodynamic system is said to be in thermodynamic equilibrium when it is in thermal equilibrium, mechanical equilibrium, radiative equilibrium, and chemical equilibrium. The word equilibrium means a state of balance...

 state of cavity radiation. If each Fourier mode of the absolutely stable equilibrium radiation in a cavity with perfectly reflective walls were considered as a degree of freedom, and if all those degrees of freedom could freely exchange energy, then, according to the equipartition theorem
Equipartition theorem
In classical statistical mechanics, the equipartition theorem is a general formula that relates the temperature of a system with its average energies. The equipartition theorem is also known as the law of equipartition, equipartition of energy, or simply equipartition...

 in classical physics, each degree of freedom would have one and the same quantity of energy. This approach led to the paradox known as the ultraviolet catastrophe
Ultraviolet catastrophe
The ultraviolet catastrophe, also called the Rayleigh–Jeans catastrophe, was a prediction of late 19th century/early 20th century classical physics that an ideal black body at thermal equilibrium will emit radiation with infinite power....

, that there would be an infinite amount of energy in any continuous field. The study of the laws of black bodies helped to establish the foundations of quantum mechanics
History of quantum mechanics
The history of quantum mechanics, as it interlaces with the history of quantum chemistry, began essentially with a number of different scientific discoveries: the 1838 discovery of cathode rays by Michael Faraday; the 1859-1860 winter statement of the black body radiation problem by Gustav...

.

The term black body was introduced by Gustav Kirchhoff
Gustav Kirchhoff
Gustav Robert Kirchhoff was a German physicist who contributed to the fundamental understanding of electrical circuits, spectroscopy, and the emission of black-body radiation by heated objects...

 in 1860. When used as a compound adjective, the term is typically written as one word in English
English language
English is a West Germanic language that arose in the Anglo-Saxon kingdoms of England and spread into what was to become south-east Scotland under the influence of the Anglian medieval kingdom of Northumbria...

, e.g. in blackbody radiation, but sometimes also hyphenated, as in black-body radiation.

Explanation


All matter emits electromagnetic radiation when it has a temperature above absolute zero
Absolute zero
Absolute zero is the theoretical temperature at which entropy reaches its minimum value. The laws of thermodynamics state that absolute zero cannot be reached using only thermodynamic means....

. The radiation represents a conversion of a body's thermal energy into electromagnetic energy, and is therefore called thermal radiation
Thermal radiation
Thermal radiation is electromagnetic radiation generated by the thermal motion of charged particles in matter. All matter with a temperature greater than absolute zero emits thermal radiation....

. It is a spontaneous process
Spontaneous process
A spontaneous process is the time-evolution of a system in which it releases free energy and moves to a lower, more thermodynamically stable energy state...

 of radiative distribution of entropy
Entropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...

.

Conversely all matter absorbs electromagnetic radiation to some degree. An object that absorbs all radiation falling on it, at all wavelength
Wavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...

s, is called a black body. When a black body is at a uniform temperature, its emission has a characteristic frequency distribution that depends on the temperature. Its emission is called blackbody radiation.

The concept of the black body is an idealization, as perfect black bodies do not exist in nature. Graphite
Graphite
The mineral graphite is one of the allotropes of carbon. It was named by Abraham Gottlob Werner in 1789 from the Ancient Greek γράφω , "to draw/write", for its use in pencils, where it is commonly called lead . Unlike diamond , graphite is an electrical conductor, a semimetal...

 and lamp black
Carbon black
Carbon black is a material produced by the incomplete combustion of heavy petroleum products such as FCC tar, coal tar, ethylene cracking tar, and a small amount from vegetable oil. Carbon black is a form of amorphous carbon that has a high surface-area-to-volume ratio, although its...

, with emissivities greater than 0.95, however, are good approximations to a black material. Experimentally, blackbody radiation may be established best as the ultimately stable steady state equilibrium radiation in a cavity in a rigid body, at a uniform temperature, that is entirely opaque and is only partly reflective. A closed box of graphite walls at a constant temperature with a small hole on one side produces a good approximation to ideal blackbody radiation emanating from the opening.

Blackbody radiation has the unique absolutely stable distribution of radiative intensity that can persist in thermodynamic equilibrium in a cavity.

Blackbody radiation becomes a visible glow of light if the temperature of the object is high enough. The Draper point
Draper point
The Draper point is the approximate temperature above which almost all solid materials visibly glow as a result of blackbody radiation. It was established at 977˚ F by John William Draper in 1847....

 is the temperature at which all solids glow a dim red, about 798 K. At 1000 K, the opening in the oven looks red; at 6000 K, it looks white. No matter how the oven is constructed, or of what material, as long as it is built such that almost all light entering is absorbed, it will be a good approximation to a blackbody, so the spectrum, and therefore color, of the light that comes out will be a function of the cavity temperature alone. A graph of the amount of energy inside the oven per unit volume and per unit frequency interval plotted versus frequency, is called the blackbody curve. Different curves are obtained by varying the temperature.

Two things that are at the same temperature stay in equilibrium, so a body at temperature T surrounded by a cloud of light at temperature T on average will emit as much light into the cloud as it absorbs, following Prevost's exchange principle, which refers to radiative equilibrium
Radiative equilibrium
Radiative equilibrium is one of the several requirements for thermodynamic equilibrium, but it can occur in the absence of thermodynamic equilibrium. There are various types of radiative equilibrium, which is itself a kind of dynamic equilibrium....

. The principle of detailed balance
Detailed balance
The principle of detailed balance is formulated for kinetic systems which are decomposed into elementary processes : At equilibrium, each elementary process should be equilibrated by its reverse process....

 says that there are no strange correlations between the process of emission and absorption: the process of emission is not affected by the absorption, but only by the thermal state of the emitting body. This means that the total light emitted by a body at temperature T, black or not, is always equal to the total light that the body would absorb were it to be surrounded by light at temperature T.

When the body is black, the absorption is obvious: the amount of light absorbed is all the light that hits the surface. For a black body much bigger than the wavelength, the light energy absorbed at any wavelength λ per unit time is strictly proportional to the blackbody curve. This means that the blackbody curve is the amount of light energy emitted by a black body, which justifies the name. This is the condition for the applicability of Kirchhoff's law of thermal radiation
Kirchhoff's law of thermal radiation
In thermodynamics, Kirchhoff's law of thermal radiation, or Kirchhoff's law for short, is a general statement equating emission and absorption in heated objects, proposed by Gustav Kirchhoff in 1859, following from general considerations of thermodynamic equilibrium and detailed balance.An object...

: the blackbody curve is characteristic of thermal light, which depends only on the temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...

 of the walls of the cavity, provided that the walls of the cavity are completely opaque and are not very reflective, and that the cavity is in thermodynamic equilibrium
Thermodynamic equilibrium
In thermodynamics, a thermodynamic system is said to be in thermodynamic equilibrium when it is in thermal equilibrium, mechanical equilibrium, radiative equilibrium, and chemical equilibrium. The word equilibrium means a state of balance...

. When the black body is small, so that its size is comparable to the wavelength of light, the absorption is modified, because a small object is not an efficient absorber of light of long wavelength, but the principle of strict equality of emission and absorption is always upheld in a condition of thermodynamic equilibrium.

In the laboratory, blackbody radiation is approximated by the radiation from a small hole in a large cavity, a hohlraum
Hohlraum
In radiation thermodynamics, a hohlraum is a cavity whose walls are in radiative equilibrium with the radiant energy within the cavity. This idealized cavity can be approximated in practice by making a small perforation in the wall of a hollow container of any opaque material...

, in an entirely opaque body that is only partly reflective, that is maintained at a constant temperature. (This technique leads to the alternative term cavity radiation.) Any light entering the hole would have to reflect off the walls of the cavity multiple times before it escaped, in which process it is nearly certain to be absorbed. Absorption occurs regardless of the wavelength
Wavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...

 of the radiation entering (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 depend only on the opacity and partial reflectivity of the walls, but not on the particular material of which they are built nor on the material in the cavity (compare with emission spectrum
Emission spectrum
The emission spectrum of a chemical element or chemical compound is the spectrum of frequencies of electromagnetic radiation emitted by the element's atoms or the compound's molecules when they are returned to a lower energy state....

).

Calculating the blackbody curve was a major challenge in theoretical physics
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...

 during the late nineteenth century. The problem was solved in 1901 by Max Planck
Max Planck
Max Karl Ernst Ludwig Planck, ForMemRS, was a German physicist who actualized the quantum physics, initiating a revolution in natural science and philosophy. He is regarded as the founder of the quantum theory, for which he received the Nobel Prize in Physics in 1918.-Life and career:Planck came...

 in the formalism now known as Planck's law of blackbody radiation.
By making changes to Wien's radiation law
Wien's Radiation Law
Wien's approximation is a law of physics used to describe the spectrum of thermal radiation . This law was first derived by Wilhelm Wien in 1896...

 (not to be confused with Wien's displacement law
Wien's displacement law
Wien's displacement law states that the wavelength distribution of thermal radiation from a black body at any temperature has essentially the same shape as the distribution at any other temperature, except that each wavelength is displaced on the graph...

) consistent with thermodynamics
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...

 and electromagnetism
Electromagnetism
Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...

, he found a mathematical expression fitting the experimental data satisfactorily. Planck had to assume that the energy of the oscillators in the cavity was quantized, i.e., it existed in integer multiples of some quantity. Einstein
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...

 built on this idea and proposed the quantization of electromagnetic radiation itself in 1905 to explain the photoelectric effect
Photoelectric effect
In the photoelectric effect, electrons are emitted from matter as a consequence of their absorption of energy from electromagnetic radiation of very short wavelength, such as visible or ultraviolet light. Electrons emitted in this manner may be referred to as photoelectrons...

. These theoretical advances eventually resulted in the superseding of classical electromagnetism by quantum electrodynamics
Quantum electrodynamics
Quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...

. These quanta were called photon
Photon
In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...

s and the blackbody cavity was thought of as containing a gas of photons
Photon gas
In physics, a photon gas is a gas-like collection of photons, which has many of the same properties of a conventional gas like hydrogen or neon - including pressure, temperature, and entropy...

. In addition, it led to the development of quantum probability distributions, called Fermi–Dirac statistics and Bose–Einstein statistics
Bose–Einstein statistics
In statistical mechanics, Bose–Einstein statistics determines the statistical distribution of identical indistinguishable bosons over the energy states in thermal equilibrium.-Concept:...

, each applicable to a different class of particles, fermion
Fermion
In particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....

s and boson
Boson
In particle physics, bosons are subatomic particles that obey Bose–Einstein statistics. Several bosons can occupy the same quantum state. The word boson derives from the name of Satyendra Nath Bose....

s.

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
Monotonic function
In mathematics, a monotonic function is a function that preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory....

 with temperature.

The radiance
Radiance
Radiance and spectral radiance are radiometric measures that describe the amount of radiation such as light or radiant heat that passes through or is emitted from a particular area, and falls within a given solid angle in a specified direction. They are used to characterize both emission from...

 or observed intensity is not a function of direction. Therefore a black body is a perfect Lambertian
Lambert's cosine law
In optics, Lambert's cosine law says that the radiant intensity observed from a Lambertian surface or a Lambertian radiator is directly proportional to the cosine of the angle θ between the observer's line of sight and the surface normal. A Lambertian surface is also known as an ideal diffusely...

 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
Emissivity
The emissivity of a material is the relative ability of its surface to emit energy by radiation. It is the ratio of energy radiated by a particular material to energy radiated by a black body at the same temperature...

 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 gray body assumption.
With non-black surfaces, the deviations from ideal blackbody behavior are determined by both the surface structure, such as roughness or granularity, and the chemical composition. On a "per wavelength" basis, real objects in states of local thermodynamic equilibrium still 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; the incomplete absorption can be due to some of the incident light being transmitted through the body or to some of it being reflected at the surface of the body.

In astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

, objects such as star
Star
A star is a massive, luminous sphere of plasma held together by gravity. At the end of its lifetime, a star can also contain a proportion of degenerate matter. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth...

s are frequently regarded as black bodies, though this is often a poor approximation. An almost perfect blackbody spectrum is exhibited by the cosmic microwave background radiation
Cosmic microwave background radiation
In cosmology, cosmic microwave background radiation is thermal radiation filling the observable universe almost uniformly....

. Hawking radiation
Hawking radiation
Hawking radiation is a thermal radiation with a black body spectrum predicted to be emitted by black holes due to quantum effects. It is named after the physicist Stephen Hawking, who provided a theoretical argument for its existence in 1974, and sometimes also after the physicist Jacob Bekenstein...

 is the hypothetical blackbody radiation emitted by black hole
Black hole
A black hole is a region of spacetime from which nothing, not even light, can escape. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon that...

s.

A black body radiates energy at all frequencies, but its intensity rapidly tends to zero at high frequencies (short wavelengths). For example, a black body at room temperature (300 K) with one square meter of surface area will emit a photon in the visible range (390–750 nm) at an average rate of one photon every 41 seconds, meaning that for most practical purposes, such a black body does not emit in the visible range.

Blackbody simulators


A black body is an idealized object, and by definition has an emissivity of . In practice, common applications define all sources of infrared radiation as a black body when the object approaches that emissivity and is greater than approximately 0.99. A source with lower emissivity is often referred to as a gray body.

An example of a nearly perfect black body is super black
Super black
Super black is a surface treatment developed at the National Physical Laboratory in the United Kingdom. It reflects much less light than the darkest conventional matte black paints available previously....

, produced from a nickel
Nickel
Nickel is a chemical element with the chemical symbol Ni and atomic number 28. It is a silvery-white lustrous metal with a slight golden tinge. Nickel belongs to the transition metals and is hard and ductile...

-phosphorus
Phosphorus
Phosphorus is the chemical element that has the symbol P and atomic number 15. A multivalent nonmetal of the nitrogen group, phosphorus as a mineral is almost always present in its maximally oxidized state, as inorganic phosphate rocks...

 alloy
Alloy
An alloy is a mixture or metallic solid solution composed of two or more elements. Complete solid solution alloys give single solid phase microstructure, while partial solutions give two or more phases that may or may not be homogeneous in distribution, depending on thermal history...

. In 2009, a team of Japanese scientists created a material even closer to an ideal black body, based on vertically aligned single-walled carbon nanotubes. This absorbs between 98% and 99% of the incoming light in the spectral range from the ultra-violet to the far-infrared regions.

Planck's law of blackbody radiation


Planck's law states that
where
I(ν,T) is the energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

 per unit time
Time
Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....

 (or the power
Power (physics)
In physics, power is the rate at which energy is transferred, used, or transformed. For example, the rate at which a light bulb transforms electrical energy into heat and light is measured in watts—the more wattage, the more power, or equivalently the more electrical energy is used per unit...

) radiated per unit area of emitting surface in the normal
Surface normal
A surface normal, or simply normal, to a flat surface is a vector that is perpendicular to that surface. A normal to a non-flat surface at a point P on the surface is a vector perpendicular to the tangent plane to that surface at P. The word "normal" is also used as an adjective: a line normal to a...

 direction per unit solid angle
Solid angle
The solid angle, Ω, is the two-dimensional angle in three-dimensional space that an object subtends at a point. It is a measure of how large that object appears to an observer looking from that point...

 per unit frequency
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...

 by a black body at temperature T;
h is the Planck constant
Planck constant
The Planck constant , also called Planck's constant, is a physical constant reflecting the sizes of energy quanta in quantum mechanics. It is named after Max Planck, one of the founders of quantum theory, who discovered it in 1899...

;
c is the speed of light
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...

 in a vacuum;
k is the Boltzmann constant;
ν is the frequency
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...

 of the electromagnetic radiation; and
T is the temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...

 of the body in kelvin
Kelvin
The kelvin is a unit of measurement for temperature. It is one of the seven base units in the International System of Units and is assigned the unit symbol K. The Kelvin scale is an absolute, thermodynamic temperature scale using as its null point absolute zero, the temperature at which all...

s.

Wien's displacement law


Wien's displacement law
Wien's displacement law
Wien's displacement law states that the wavelength distribution of thermal radiation from a black body at any temperature has essentially the same shape as the distribution at any other temperature, except that each wavelength is displaced on the graph...

 shows how the spectrum of black body radiation at any temperature is related to the spectrum at any other temperature. If we know the shape of the spectrum at one temperature, we can calculate the shape at any other temperature.

A consequence of Wien's displacement law is that the wavelength at which the intensity of the radiation produced by a black body is at a maximum, , it is a function only of the temperature
where the constant, b, known as Wien's displacement constant, is equal to .

Note that the peak intensity can be expressed in terms of intensity per unit wavelength or in terms of intensity per unit frequency. The expression for the peak wavelength given above refers to the intensity per unit wavelength; meanwhile the Planck's Law section above was in terms of intensity per unit frequency. The frequency at which the power per unit frequency is maximised is given by.

Stefan–Boltzmann law


The Stefan–Boltzmann law states that the power emitted per unit area of the surface of a black body is directly proportional to the fourth power of its absolute temperature:
where j*is the total power radiated per unit area, T is the absolute temperature and is the Stefan–Boltzmann constant.

Human body emission

Much of a person's energy is radiated away in the form of infrared
Infrared
Infrared light is electromagnetic radiation with a wavelength longer than that of visible light, measured from the nominal edge of visible red light at 0.74 micrometres , and extending conventionally to 300 µm...

 light. Some materials are transparent in the infrared, while opaque to visible light, as is the plastic bag in this infrared image (bottom). Other materials are transparent to visible light, while opaque or reflective in the infrared, noticeable by darkness of the man's glasses.

As all matter, the human body radiates some of a person's energy away as infrared
Infrared
Infrared light is electromagnetic radiation with a wavelength longer than that of visible light, measured from the nominal edge of visible red light at 0.74 micrometres , and extending conventionally to 300 µm...

 light.

The net power radiated is the difference between the power emitted and the power absorbed:
Applying the Stefan–Boltzmann law,


The total surface area of an adult is about 2 m2, and the mid- and far-infrared emissivity
Emissivity
The emissivity of a material is the relative ability of its surface to emit energy by radiation. It is the ratio of energy radiated by a particular material to energy radiated by a black body at the same temperature...

 of skin and most clothing is near unity, as it is for most nonmetallic surfaces. Skin temperature is about 33 °C, but clothing reduces the surface temperature to about 28 °C when the ambient temperature is 20 °C. Hence, the net radiative heat loss is about
The total energy radiated in one day is about 9 MJ (megajoules), or 2000 kcal (food calorie
Calorie
The calorie is a pre-SI metric unit of energy. It was first defined by Nicolas Clément in 1824 as a unit of heat, entering French and English dictionaries between 1841 and 1867. In most fields its use is archaic, having been replaced by the SI unit of energy, the joule...

s). Basal metabolic rate
Basal metabolic rate
Basal Metabolic Rate , and the closely related resting metabolic rate , is the amount of daily energy expended by humans and other animals at rest. Rest is defined as existing in a neutrally temperate environment while in the post-absorptive state...

 for a 40-year-old male is about 35 kcal/(m2·h), which is equivalent to 1700 kcal per day assuming the same 2 m2 area. However, the mean metabolic rate of sedentary adults is about 50% to 70% greater than their basal rate.

There are other important thermal loss mechanisms, including convection
Convection
Convection is the movement of molecules within fluids and rheids. It cannot take place in solids, since neither bulk current flows nor significant diffusion can take place in solids....

 and evaporation
Evaporation
Evaporation is a type of vaporization of a liquid that occurs only on the surface of a liquid. The other type of vaporization is boiling, which, instead, occurs on the entire mass of the liquid....

. Conduction is negligible – the Nusselt number
Nusselt number
In heat transfer at a boundary within a fluid, the Nusselt number is the ratio of convective to conductive heat transfer across the boundary. Named after Wilhelm Nusselt, it is a dimensionless number...

 is much greater than unity. Evaporation via perspiration is only required if radiation and convection are insufficient to maintain a steady state temperature (but evaporation from the lungs occurs regardless). Free convection rates are comparable, albeit somewhat lower, than radiative rates. Thus, radiation accounts for about two-thirds of thermal energy loss in cool, still air. Given the approximate nature of many of the assumptions, this can only be taken as a crude estimate. Ambient air motion, causing forced convection, or evaporation reduces the relative importance of radiation as a thermal loss mechanism.

Application of Wien's Law
Wien's displacement law
Wien's displacement law states that the wavelength distribution of thermal radiation from a black body at any temperature has essentially the same shape as the distribution at any other temperature, except that each wavelength is displaced on the graph...

 to human body emission results in a peak wavelength of
For this reason, thermal imaging devices for human subjects are most sensitive in the 7000–14000 nanometer range.

Temperature relation between a planet and its star


The blackbody law may be used to estimate the temperature of a planet orbiting the Sun.


The temperature of a planet depends on several factors:
  • Incident radiation from its sun
  • Emitted radiation of the planet, e.g., Earth's infrared glow
  • The albedo
    Albedo
    Albedo , or reflection coefficient, is the diffuse reflectivity or reflecting power of a surface. It is defined as the ratio of reflected radiation from the surface to incident radiation upon it...

     effect causing a fraction of light to be reflected by the planet
  • The greenhouse effect
    Greenhouse effect
    The greenhouse effect is a process by which thermal radiation from a planetary surface is absorbed by atmospheric greenhouse gases, and is re-radiated in all directions. Since part of this re-radiation is back towards the surface, energy is transferred to the surface and the lower atmosphere...

     for planets with an atmosphere
  • Energy generated internally by a planet itself due to radioactive decay
    Radioactive decay
    Radioactive decay is the process by which an atomic nucleus of an unstable atom loses energy by emitting ionizing particles . The emission is spontaneous, in that the atom decays without any physical interaction with another particle from outside the atom...

    , tidal heating
    Tidal heating
    Tidal heating occurs through the tidal friction processes: orbital and rotational energy are dissipated as heat in the crust of the moons and planets involved. Io, a moon of Jupiter, is the most volcanically active body in the solar system, with no impact craters surviving on its surface...

    , and adiabatic contraction due by cooling.


This example is concerned with the balance of incident and emitted radiation, which is the most important impact for the inner planets in the Solar System.

The Stefan–Boltzmann law gives the total power
Power (physics)
In physics, power is the rate at which energy is transferred, used, or transformed. For example, the rate at which a light bulb transforms electrical energy into heat and light is measured in watts—the more wattage, the more power, or equivalently the more electrical energy is used per unit...

 (energy/second) the Sun is emitting:


where is the Stefan–Boltzmann constant, is the effective temperature of the Sun, and is the radius of the Sun.

The Sun emits that power equally in all directions. Because of this, the planet is hit with only a tiny fraction of it. The power from the Sun that strikes the planet (at the top of the atmosphere) is:

where is the radius of the planet and is the astronomical unit
Astronomical unit
An astronomical unit is a unit of length equal to about or approximately the mean Earth–Sun distance....

, the distance between the Sun
Sun
The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...

 and the planet.

Because of its high temperature, the sun emits to a large extent in the ultraviolet and visible (UV-Vis) frequency range. In this frequency range, the planet reflects a fraction of this energy where is the albedo
Albedo
Albedo , or reflection coefficient, is the diffuse reflectivity or reflecting power of a surface. It is defined as the ratio of reflected radiation from the surface to incident radiation upon it...

 or reflectance of the planet in the UV-Vis range. In other words, the planet absorbs a fraction of the sun's light, and reflects the rest. The power absorbed by the planet and its atmosphere is then:


Even though the planet only absorbs as a circular area , it emits equally in all directions as a sphere. If the planet were a perfect black body, it would emit according to the Stefan–Boltzmann law


where is the temperature of the planet. This temperature, calculated for the case of the planet acting as a black body by setting , is known as the effective temperature
Effective temperature
The effective temperature of a body such as a star or planet is the temperature of a black body that would emit the same total amount of electromagnetic radiation...

. The actual temperature of the planet will likely be different, depending on its surface and atmospheric properties. Ignoring the atmosphere and greenhouse effect, the planet, since it is at a much lower temperature than the sun, emits mostly in the infrared (IR) portion of the spectrum. In this frequency range, it emits of the radiation that a black body would emit where is the average emissivity in the IR range. The power emitted by the planet is then:


For a body in radiative exchange equilibrium with its surroundings, the rate at which it emits radiant energy is equal to the rate at which it absorbs it:


Substituting the expressions for solar and planet power in equations 1–6 and simplifying yields the estimated temperature of the planet, ignoring greenhouse effect, TP:


In other words, given the assumptions made, the temperature of a planet depends only on the surface temperature of the Sun, the radius of the Sun, the distance between the planet and the Sun, the albedo and the IR emissivity of the planet.

Temperature of Earth


Substituting the measured values for the Sun and Earth yields:

With the average emissivity set to unity, the effective temperature
Effective temperature
The effective temperature of a body such as a star or planet is the temperature of a black body that would emit the same total amount of electromagnetic radiation...

 of the Earth is:

or −18.8 °C.

This is the temperature of the Earth if it radiated as a perfect black body in the infrared, ignoring greenhouse effects (which can raise the surface temperature of a body above what it would be if it were a perfect black body in all spectrums, with an albedo of zero), and assuming an unchanging albedo. The Earth in fact radiates not quite as a perfect black body in the infrared which will raise the estimated temperature a few degrees above the effective temperature. If we wish to estimate what the temperature of the Earth would be if it had no atmosphere, then we could take the albedo and emissivity of the moon as a good estimate. The albedo and emissivity of the moon are about 0.1054 and 0.95 respectively, yielding an estimated temperature of about 1.36 °C.

Estimates of the Earth's average albedo vary in the range 0.3–0.4, resulting in different estimated effective temperatures. Estimates are often based on the solar constant
Solar constant
The solar constant, a measure of flux density, is the amount of incoming solar electromagnetic radiation per unit area that would be incident on a plane perpendicular to the rays, at a distance of one astronomical unit...

 (total insolation power density) rather than the temperature, size, and distance of the sun. For example, using 0.4 for albedo, and an insolation of 1400 W m−2, one obtains an effective temperature of about 245 K.
Similarly using albedo 0.3 and solar constant of 1372 W m−2, one obtains an effective temperature of 255 K.

Doppler effect for a moving black body


The relativistic Doppler effect
Relativistic Doppler effect
The relativistic Doppler effect is the change in frequency of light, caused by the relative motion of the source and the observer , when taking into account effects described by the special theory of relativity.The relativistic Doppler effect is different from the non-relativistic Doppler effect...

 causes a shift in the frequency f of light originating from a source that is moving in relation to the observer, so that the wave is observed to have frequency f:
where v is the velocity of the source in the observer's rest frame, θ is the angle between the velocity vector and the observer-source direction measured in the reference frame of the source, and c is the speed of light
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...

. This can be simplified for the special cases of objects moving directly towards (θ = π) or away (θ = 0) from the observer, and for speeds much less than c.

Through Planck's law the temperature spectrum of a black body is proportionally related to the frequency of light and one may substitute the temperature (T) for the frequency in this equation.

For the case of a source moving directly towards or away from the observer, this reduces to
Here v > 0 indicates a receding source, and v < 0 indicates an approaching source.

This is an important effect in astronomy, where the velocities of stars and galaxies can reach significant fractions of c. An example is found in the cosmic microwave background radiation
Cosmic microwave background radiation
In cosmology, cosmic microwave background radiation is thermal radiation filling the observable universe almost uniformly....

, which exhibits a dipole anisotropy from the Earth's motion relative to this black body radiation field.

Balfour Stewart


In 1858, Balfour Stewart described his experiments on the thermal radiative emissive and absorptive powers of polished plates of various substances, compared with the powers of lamp-black surfaces, at the same temperature. Stewart chose lamp-black surfaces as his reference because of various previous experimental findings, especially those of Pierre Prevost
Pierre Prévost
Pierre Prévost was a Swiss philosopher and physicist. In he showed that all bodies radiate heat, no matter how hot or cold they are.-Life:...

 and of John Leslie
John Leslie (physicist)
Sir John Leslie was a Scottish mathematician and physicist best remembered for his research into heat.Leslie gave the first modern account of capillary action in 1802 and froze water using an air-pump in 1810, the first artificial production of ice.In 1804, he experimented with radiant heat using...

. He wrote "Lamp-black, which absorbs all the rays that fall upon it, and therefore possesses the greatest possible absorbing power, will possess also the greatest possible radiating power." More an experimenter than a logician, Stewart failed to point out that his statement presupposed an abstract general principle, that there exist either ideally in theory or really in nature bodies or surfaces that respectively have one and the same unique universal greatest possible absorbing power, likewise for radiating power, for every wavelength and equilibrium temperature.

Stewart measured radiated power with a thermo-pile and sensitive galvanometer read with a microscope. He was concerned with selective thermal radiation, which he investigated with plates of substances that radiated and absorbed selectively for different qualities of radiation rather than maximally for all qualities of radiation. He discussed the experiments in terms of rays which could be reflected and refracted, and which obeyed the Stokes-Helmholtz reciprocity
Helmholtz reciprocity
The Helmholtz reciprocity principle describes how a ray of light and its reverse ray encounter matched optical adventures, such as reflections, refractions, and absorptions in a passive medium, or at an interface....

 principle (though he did not use an eponym for it). He did not in this paper mention that the qualities of the rays might be described by their wavelengths, nor did he use spectrally resolving apparatus such as prisms or diffraction gratings. His work was quantitative within these constraints. He made his measurements in a room temperature environment, and quickly so as to catch his bodies in a condition near the thermal equilibrium in which they had been prepared by heating to equilibrium with boiling water. His measurements confirmed that substances that emit and absorb selectively respect the principle of selective equality of emission and absorption at thermal equilibrium.

Stewart offered a theoretical proof that this should be the case separately for every selected quality of thermal radiation, but his mathematics was not rigorously valid. He made no mention of thermodynamics in this paper, though he did refer to conservation of vis viva. He proposed that his measurements implied that radiation was both absorbed and emitted by particles of matter throughout depths of the media in which it propagated. He applied the Helmholtz reciprocity principle to account for the material interface processes as distinct from the processes in the interior material. He did not postulate unrealizable perfectly black surfaces. He concluded that his experiments showed that in a cavity in thermal equilibrium, the heat radiated from any part of the interior bounding surface, no matter of what material it might be composed, was the same as would have been emitted from a surface of the same shape and position that would have been composed of lamp-black. He did not state explicitly that the lamp-black-coated bodies that he used as reference must have had a unique common spectral emittance function that depended on temperature in a unique way.

Gustav Kirchhoff


In 1859, not knowing of Stewart's work, Gustav Robert Kirchhoff reported the coincidence of the wavelengths of spectrally resolved lines of absorption and of emission of visible light.

Kirchhoff then went on to consider bodies that emit and absorb heat radiation, in an opaque enclosure or cavity, in equilibrium at temperature .

Here is used a notation different from Kirchhoff's. Here, the emitting power denotes a dimensioned quantity, the total radiation emitted by a body labeled by index at temperature . The total absorption ratio of that body is dimensionless, the ratio of absorbed to incident radiation in the cavity at temperature . (In contrast with Balfour Stewart's, Kirchhoff's definition of his absorption ratio did not refer in particular to a lamp-black surface as the source of the incident radiation.) Thus the ratio of emitting power to absorption ratio is a dimensioned quantity, with the dimensions of emitting power, because is dimensionless. Also here the wavelength-specific emitting power of the body at temperature is denoted by and the wavelength-specific absorption ratio by . Again, the ratio of emitting power to absorption ratio is a dimensioned quantity, with the dimensions of emitting power.

In a second report made in 1859, Kirchhoff announced a new general principle or law for which he offered a theoretical and mathematical proof, though he did not offer quantitative measurements of radiation powers. His theoretical proof was and still is considered by some writers to be invalid. His principle, however, has endured: it was that for heat rays of the same wavelength, in equilibrium at at a given temperature, the wavelength-specific ratio of emitting power to absorption ratio has one and the same common value for all bodies that emit and absorb at that wavelength. In symbols, the law stated that the wavelength-specific ratio has one and the same value for all bodies, that is for all values of index . In this report there was no mention of black bodies.

In 1860, still not knowing of Stewart's measurements for selected qualities of radiation, Kirchhoff pointed out that it was long established experimentally that for total heat radiation, of unselected quality, emitted and absorbed by a body in equilibrium, the dimensioned total radiation ratio , has one and the same value common to all bodies, that is, for every value of the material index . Again without measurements of radiative powers or other new experimental data, Kirchhoff then offered a fresh theoretical proof of his new principle of the universality of the value of the wavelength-specific ratio at thermal equilibrium. His fresh theoretical proof was and still is considered by some writers to be invalid.

But more importantly, it relied on a new theoretical postulate of "perfectly black bodies". Such black bodies showed complete absorption in their infinitely thin most superficial surface. They correspond to Balfour Stewart's reference bodies, with internal radiation, coated with lamp-black. They were not the more realistic perfectly black bodies later considered by Planck. Planck's black bodies radiated and absorbed only by the material in their interiors; their interfaces with contiguous media were only mathematical surfaces, capable neither of absorption nor emission, but only of reflecting and transmitting with refraction.

Kirchhoff's proof considered an arbitrary non-ideal body labeled as well as various perfect black bodies labeled . It required that the bodies be kept in a cavity in thermal equilibrium at temperature . His proof intended to show that the ratio was independent of the nature of the non-ideal body, however partly transparent or partly reflective it was.

His proof first argued that for wavelength and at temperature , at thermal equilibrium, all perfectly black bodies of the same size and shape have the one and the same common value of emissive power , with the dimensions of power. His proof noted that the dimensionless wavelength-specific absorption ratio of a perfectly black body is by definition exactly 1. Then for a perfectly black body, the wavelength-specific ratio of emissive power to absorption ratio is again just , with the dimensions of power. Kirchhoff considered, successively, thermal equilibrium with the arbitrary non-ideal body, and with a perfectly black body of the same size and shape, in place in his cavity in equilibrium at temperature . He argued that the flows of heat radiation must be the same in each case. Thus he argued that at thermal equilibrium the ratio was equal to , which may now be denoted , a continuous function, dependent only on at fixed temperature , and an increasing function of at fixed wavelength , at low temperatures vanishing for visible but not for longer wavelengths, with positive values for visible wavelengths at higher temperatures, which does not depend on the nature of the arbitrary non-ideal body. (Geometrical factors, taken into detailed account by Kirchhoff, have been ignored in the foregoing.)

Thus Kirchhoff's law of thermal radiation can be stated: "For any material at all , radiating and absorbing in thermodynamic equilibrium at any given temperature , for every wavelength , the ratio of emissive power to absorptive ratio has one universal value, which is characteristic of a perfect black body, and is an emissive power which we here represent by ." (For our notation , Kirchhoff's original notation was simply .)

Kirchhoff announced that the determination of the function was a problem of the highest importance, though he recognized that there would be experimental difficulties to be overcome. He supposed that like other functions that do not depend on the properties of individual bodies, it would be a simple function. That function has occasionally been called 'Kirchhoff's (emission, universal) function', though its precise mathematical form would not be known for another forty years, till it was discovered by Planck in 1900. The theoretical proof for Kirchhoff's universality principle was worked on and debated by various physicists over the same time, and later. Kirchhoff stated later in 1860 that his theoretical proof was better than Balfour Stewart's, and in some respects it was so. Kirchhoff's 1860 paper did not mention the second law of thermodynamics, and of course did not mention the concept of entropy which had not at that time been established.

See also


  • Bolometer
    Bolometer
    A bolometer is a device for measuring the power of incident electromagnetic radiation via the heating of a material with a temperature-dependent electrical resistance. It was invented in 1878 by the American astronomer Samuel Pierpont Langley...

  • Color temperature
    Color temperature
    Color temperature is a characteristic of visible light that has important applications in lighting, photography, videography, publishing, manufacturing, astrophysics, and other fields. The color temperature of a light source is the temperature of an ideal black-body radiator that radiates light of...

  • Infrared thermometer
    Infrared thermometer
    Infrared thermometers infer temperature using a portion of the thermal radiation sometimes called blackbody radiation emitted by the object of measurement. They are sometimes called laser thermometers if a laser is used to help aim the thermometer, or non-contact thermometers to describe the...

  • Photon polarization
    Photon polarization
    Photon polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. Individual photons are completely polarized...

  • Planck's law
  • Pyrometry
  • Rayleigh–Jeans law
  • Thermography
    Thermography
    Infrared thermography, thermal imaging, and thermal video are examples of infrared imaging science. Thermal imaging cameras detect radiation in the infrared range of the electromagnetic spectrum and produce images of that radiation, called thermograms...

  • Sakuma–Hattori equation


External links