Black body
(Redirected from Black-body radiation)
As the temperature decreases, the peak of the black body radiation curve moves to lower intensities and
longer wavelengths. The black-body radiation graph is also compared with the classical model that
preceded it.
In physics a black body is an object that absorbs all light that falls onto it: no light passes through it nor is
reflected. Despite the name, black bodies do produce thermal radiation such as light.
The term "black body" was introduced by Gustav Kirchhoff in 1862. The light emitted by a black body is
called black-body radiation.
When 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.
In the laboratory, the closest thing to a black body radiation is the radiation from a small hole in a cavity:
it 'absorbs' little energy from the outside if the hole is small, and it 'radiates' all the energy from the inside
which is black. However, the spectrum (i.e. the amount of light emitted at each wavelength) of its radiation
will not be continuous, and only rays will appear whose wavelengths depend on the material in the cavity
(see Emission spectrum). By extrapolating the spectrum curve for other frequencies, a general curve can be
drawn, and any black-body radiation will follow it. This curve depends only on the temperature of the cavity
walls.
The observed spectrum of black-body radiation could not be explained with Classical electromagnetism and
statistical mechanics: it predicted infinite brightness at low wavelength (i.e. high frequencies), a prediction
often called the ultraviolet catastrophe.
This theoretical problem was solved by Max Planck, who had to assume that electromagnetic radiation
could propagate only in discrete packets, or quanta. This idea was later used by Einstein to explain the
photoelectric effect. These theoretical advances eventually resulted in the replacement of classical
electromagnetism by quantum mechanics. Today, the quanta are called photons.
The intensity of radiation from a black body at temperature T is given by Planck's law of black body radiation:
where
is the amount of energy per unit surface per unit time per unit solid angle emitted in the frequency
range between ν and ν+δν;
h is Planck's constant
c is the speed of light
k is Boltzmann's constant.
The wavelength at which the radiation is strongest is given by Wien's 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 wavelenths 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, a typical engineering assumption is 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.
References

Planck, Max, "On the Law of Distribution of Energy in the Normal Spectrum". Annalen der Physik, vol. 4, p.
553 ff (1901).

Descriptions of radiation emitted by many different objects
Black-Body Radiation
return of classical physics by computation
A new explanation of the spectrum of black-body radiation is
presented based on finite precision computation
instead of statistics
All these fifty years of conscious brooding have brought me no nearer to the answer to the question,
'What are light quanta?' Nowadays every Tom, Dick and Harry thinks he knows it, but he is
mistaken...The quanta really are a hopeless mess. (Einstein 1954)
The whole procedure was an act of despair because a theoretical interpretation (of black-body radiation)
had to be found at any price, no matter how high that might be...I was ready to sacrifice any of my
previous convictions about physics...(Planck 1900)
Black Hole of Classical Physics
The starting point of modern physics was that classical physics failed to explain black body radiation:
A black body absorbs all incoming light of all frequencies, but only emits low frequencies with a
cut-off of high frequency depending on temperature. This is illustrated in the following figures showing
that the cut-off depends on the temperature and occurs for higher frequencies/shorter wave-lengths for higher
temperature:
which is a graphical representation of
showing the exponential cut-off factor.
This is why the Earth absorbing light from the Sun of all frequencies from low to high, does not itself glow like
the Sun, but only radiates low frequency infrared light:
The trouble with classical wave mechanics facing the physicists in the late 19th century, was that it seemed to
be fully reversible, and thus whatever was absorbed ultimately had to be emitted. Classical physics
predicted that the Earth would shine like the Sun, with no cut-off of high frequencies as indicated in the
following figure:
But the Earth does not glow like the Sun and the question to be answered was (and is):

Why is formally reversible wave mechanics, in reality irreversible with cut-off of high
frequencies?
To answer this question Planck used a form of statistical mechanics, where the high-frequency cut-off was
explained as an effect of smaller probability of high-frequency quanta. But Planck was not happy with his
resolution, which he believed was only a mathematical trick without physical meaning, and Planck became a
revolutionary against his will.
Finite Precision Computation as Explanation of Cut-Off.
The above question is the question of irreversiblity in a formally reversible Hamiltonian system, which is
answered by the new version of the second law of thermodynamics based on finite precision computation
instead of statistics. The answer is the same because wave mechanics of absorption and emission of light is an
example of a formally reversible Hamiltonian system. The answer is developed in computational black-body
radiation [1]and in concentrate reads:







a black-body can be seen as a network of interacting atomic oscillators which are excited by
incoming waves and can emit waves by coordinated oscillation
irreversibility arises from finite precision computation and not from statistics
cut-off of high frequencies occurs because the required coordination of atomic oscillations cannot
be met by finite precision computation
physics is a form of analog computation of finite precision
the cut-off frequency increases with temperature because the amplitude of oscillation increases
which increases relative precision
a black-body transforms coordinated high-frequency input into low-frequency infrared heat
radiation
cut-off of high frequencies is a well-known phenomenon in computational wave propagation
arising from finite precision and not from statistics.
BLACK BODY RADIATION
(i) Perfect black body: A body that absorbs all the radiation incident upon it and
has an emissivity equal to 1 is called a perfectly black body. A black body is also
an ideal radiator. It implies that if a black body and an identical another body are
kept at the same temperature, then the black body will radiate maximum power as
it is obvious from equation P = eA T4 also. Because e = 1 for a perfectly black
body while for any other body e < 1.
Radiation entering the cavity has little chance of leaving before it is completely
absorbed. (e
1)
Materials like black velvet or lamp black come close to being ideal black
bodies, but the best practical realization of an ideal black body is a small hole
Cavity approximating an ideal
leading into a cavity, as this
black body.
absorbs 98% of the radiation incident on them.
(ii) Absorptive power ‘a’: “It is defined as the ratio of the radiant energy absorbed by it in a given time to the
total radiant energy incident on it in the same interval of time.”
a = energy absorbed/energy incident
As a perfectly black body absorbs all radiations incident on it, the absorptive power of a perfectly black body is
maximum and unity.
(iii) Spectral absorptive power ‘al’: The absorptive power ‘a ’ refers to radiations of all wavelengths (or the
total energy) while the spectral absorptive power is the ratio of radiant energy absorbed by a surface to the
radiant energy incident on it for a particular wavelength . It may have different values for different
wavelengths for a given surface. Let us take an example, suppose a = 0.6, a = 0.4 for 1000 Å and a = 0.7 for
2000 oA for a given surface. Then it means that this surface will absorbs only 60% of the total radiant energy
incident on it. Similarly it absorbs 40% of the energy incident on it corresponding to 1000 oA and 70%
corresponding to 2000 oA. The spectral absorptive power a is related to absorptive power a through the
relation
(iv) Emissive power ‘e’: (Don’t confuse it with the emissivity e which is different from it, although both have
the same symbol e).
“For a given surface it is defined as the radiant energy emitted per second per unit area of the surface.” It has
the units of W/m2 or J/s–m2. For a black body e= T4.
(v) Spectral emissive power ‘e ’: “It is emissive power for a particular wavelength
.” Thus,
Kirchoff’s law: “According to this law
the ratio of emissive power to
absorptive power is same for all
surfaces at the same temperature.”
Hence,
e1/a1 = e2/a2 = (e/a)
but
(a)black body = 1 and
(e)black body = E
(say)
Then,
(e/a)for any surface = constant = E,
Similarly for a particular wavelength ,
(e /a ) = E
Here
E = emissive power of black body at temperature T = T4
From the above expression, we can see that
e
a
i.e., good absorbers for a particular wavelength are also good emitters of the same wavelength.
COOLING BY RADIATION
Consider a hot body at temperature T placed in an environment at a lower temperature T0. The body emits more
radiation than it absorbs and cools down while the surrounding absorb radiation from the body and warm up. The body
is losing energy by emitting radiations at a rate.
P1 = eA T4
and is receiving energy by absorbing radiations at a rate
P2 = aA T04
Here ‘a’ is a pure number between 0 and 1 indicating the relative ability of the surface to absorb radiation from
its surroundings. Note that this ‘a’ is different from the absorptive power ‘a’. In thermal equilibrium, both the
body and the surrounding have the same temperature (say Tc) and,
P1 = P2
or
eA Tc4 = aA Tc4
or
e=a
Thus, when T > T0, the net rate of heat transferred from the body to the surroundings is,
dQ/dt = eA (T4 – T04) or
mc = (–dT/dt) = eA (T4 – T04)
=>
Rate of cooling,
(– dT/dt) = eA /mc (T4 – T04)
or
– dT/dt (T4 – T04)
Solved Example 1:
Each square metre of the sun’s surface radiates energy at the rate of 6.3 × 107 W
–2
m . Assuming that Stefan’s law applies to the radiation, calculate the temperature of the
sun’s surface. Given Stefan’s constant s = 5.7 × 10–8 W m–2 K–4.
Given
Q = 6.3 × 1078 W m–2
= 5.7 × 10–8 W m–2 K–4
T=?
Now
Q = T4
T = (Q/ )1/4 = (6.3 × 107/5.7 × 10–8)1/4 = 5765 K
The temperature of sun’s surface is 5765 K.
Solved Example 2:
The tungsten filament of an electric lamp has a length of 0.25 m and a diameter 6
× 10–5 m. The power rating of the lamp is 100 W. If the emissivity of the filament is 0.8,
estimate the steady temperature of the filament. Stefan’s constant = 5.67 × 10–8 W m–2 K–4.
Solution:
Area of the filament = 2p × (radius) × (length)
A = 2p × (3 × 10–5) × 0.25 = 4.71 × 10–5 m2
Now Q =
T4, where Q is the energy radiated per second per unit area at absolute
temperature T. Therefore, the energy radiated per second (or power radiated) from the
filament of area A is,
P=A
T4
When the temperature is steady, power radiated from filament = power received = 100 W
4
A
T = 100 W
Now A = 4.71 × 10–5 m2, = 0.8 and
= 5.67 × 10–8 W m–2 K–4
Substituting these values, we have, T = (100/4.71 × 10–5 × 0.8 × 5.67 × 10–8)1/4 = 2616 K
Assumptions
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:
The Earth only has an absorbing area equal to a two dimensional circle, rather than the surface of a sphere.
:
where
: is the Stefan-boltzmann constant,
:
is the surface temperature of the Sun, and
:
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:
:
where
:
is the radius of the Earth and
: is the distance between the Sun and the Earth.
Even though the earth only absorbs as a circular area
:
where
, it emits equally in all directions as a sphere:
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:
:
So plug in equations 1, 2, and 3 into this and we get
:
Many factors cancel from both sides and this equation can be greatly simplified.
The result
After canceling of factors, the final result is
:
where
is the surface temperature of the Sun,
is the radius of the Sun,
is the distance between the Sun and the Earth, and
is the average surface temperature of the Earth.
In other words, the temperature of the Earth depends only 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 substitute in the measured values for Earth,
:
:
:
we'll find the effective temperature of the Sun to be
:
This is within three percent of the standard measure of 5780 kelvins which makes the formula valid for
most scientific and engineering applications.
See also





Effective temperature
Color temperature
Infrared thermometer
Photon polarization
Ultraviolet catastrophe
References
1. ^ When 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.
2. ^ Huang, Kerson (1967). Statistical Mechanics. New York: John Wiley & Sons.
3. ^ Planck, Max (1901). "On the Law of Distribution of Energy in the Normal Spectrum" (HTML). Annalen
der Physik 4: 553.
4. ^ Landau, L. D.; E. M. Lifshitz (1996). Statistical Physics, 3rd Edition Part 1, Oxford: ButterworthHeinemann.
5. ^ Infrared Services. Emissivity Values for Common Materials. Retrieved on 2007-06-24.
6. ^ Omega Engineering. Emissivity of Common Materials. Retrieved on 2007-06-24.
7. ^ Elert, G. (ed.). Temperature of a Healthy Human. Retrieved on 2007-06-24.
8. ^ Lee, B.. Theoretical Prediction and Measurement of the Fabric Surface Apparent Temperature in a
Simulated Man/Fabric/Environment System. Retrieved on 2007-06-24.
9. ^ Harris J, Benedict F (1918). "A Biometric Study of Human Basal Metabolism.". Proc Natl Acad Sci
U S A 4 (12): 370-3. PMID 16576330.
10. ^ Levine, J (2004). "Nonexercise activity thermogenesis (NEAT): environment and biology". Am J
Physiol Endocrinol Metab 286: E675-E685.
11. ^ DrPhysics.com. Heat Transfer and the Human Body. Retrieved on 2007-06-24.
12. ^ Cole, George H. A.; Woolfson, Michael M. (2002). Planetary Science: The Science of Planets
Around Stars (1st ed.). Institute of Physics Publishing. ISBN 0-7503-0815-X.
Other textbooks


Kroemer, Herbert; Kittel, Charles (1980). Thermal Physics (2nd ed.). W. H. Freeman Company.
ISBN 0-7167-1088-9.
Tipler, Paul; Llewellyn, Ralph (2002). Modern Physics (4th ed.). W. H. Freeman. ISBN 0-71674345-0.
Black body
From Wikipedia, the free encyclopedia
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As the temperature decreases, the peak of the blackbody radiation curve moves to lower intensities and longer
wavelengths. The blackbody radiation graph is also compared with the classical model of Rayleigh and Jeans.
The color (chromaticity) of blackbody radiation depends on the temperature of the black body; the locus of such
colors, shown here in CIE 1931 x,y space, is known as the Planckian locus.
In physics, a black body is an idealized object that absorbs all electromagnetic radiation falling on it. Blackbodies
absorb and incandescently re-emit radiation in a characteristic, continuous spectrum. Because no light (visible
electromagnetic radiation) is reflected or transmitted, the object appears black when it is cold. However, a black
body emits a temperature-dependent spectrum of light. This thermal radiation from a black body is termed
blackbody radiation. In the blackbody spectrum, the shorter the wavelength, the higher the frequency, and the
higher frequency is related to the higher temperature. Thus, the color of a hotter object is closer to the blue end
of the spectrum and the color of a cooler object is closer to the red.
At room temperature, black bodies emit mostly infrared wavelengths, but as the temperature increases past a few
hundred degrees Celsius, black bodies start to emit visible wavelengths, appearing red, orange, yellow, white,
and blue with increasing temperature. By the time an object is white, it is emitting substantial ultraviolet
radiation.
The term black body was introduced by Gustav Kirchhoff in 1860. When used as a compound adjective, the term is
typically written as one word in blackbody radiation, but sometimes also hyphenated, as in black-body
radiation.
Blackbody emission gives insight into the thermal equilibrium state of a continuous field. In classical physics,
each different Fourier mode in thermal equilibrium should have the same energy. This approach led to the
paradox known as the ultraviolet catastrophe, that there would be an infinite amount of energy in any continuous
field. Black bodies could test the properties of thermal equilibrium because they emit radiation which is
distributed thermally. Studying the laws of the black body historically led to quantum mechanics.
Explanation
Blackbody radiation is electromagnetic radiation in thermal equilibrium with a black body at a given
temperature. Experimentally, it is established as the steady state equilibrium radiation in a rigid-walled cavity.
There are no ideal (perfect) black bodies in nature, but graphite is a good approximation, and a closed box with
graphite walls at a constant temperature gives a good approximation to an ideal black body.[1][2][3].
An object at temperature T emits radiation, which is a visible glow if T is high enough. The Draper point is the
name given to the point at which all solids glow a dim red (about 798 K).[4][5]
A black body is an object that absorbs all light that falls on it, and emits light in a wavelength spectrum
determined solely by its temperature. A black body can be approximated by, for example, an oven: a cavity
surround by walls at temperature T and with a small opening through which light can enter and leave. 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 that enters 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 almost
entirely a function of its temperature alone. A plot of the amount of energy inside the oven per unit volume per
unit frequency interval versus frequency (or per unit wavelength interval, versus wavelength), at a temperature
T, is called the blackbody curve.
The temperature of a Pāhoehoe lava flow can be estimated by observing its color. The result agrees well with
measured temperatures of lava flows at about 1000 to 1200 °C.
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. The principle of detailed balance 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 Kirchhoff's law of thermal radiation: the
blackbody emission curve is a thermal characteristic of light, which depends only on the temperature of the
walls of the cavity, provided that the cavity is in radiative equilibrium.[6] 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 the laboratory, blackbody radiation is approximated by the radiation from a small hole entrance to a large
cavity, a hohlraum, that has reached and 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. This occurs regardless of
the wavelength 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 not depend on the
material in the cavity (compare with emission spectrum).
Calculating the blackbody curve was a major challenge in theoretical physics during the late nineteenth century.
The problem was solved in 1901 by Max Planck in the formalism now known as Planck's law of blackbody
radiation.[7] 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 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 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 superseding of classical electromagnetism by quantum electrodynamics.
These quanta were called photons and the blackbody cavity was thought of as containing a gas of photons. In
addition, it led to the development of quantum probability distributions, called Fermi-Dirac statistics and BoseEinstein statistics, each applicable to a different class of particles, 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.[8]
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.
WMAP image of the cosmic microwave background radiation anisotropy. It has the most precise thermal
emission spectrum known and corresponds to a temperature of 2.725 K with an emission peak at 160.2 GHz.
When dealing with non-black surfaces, the deviations from ideal blackbody behavior are determined by both the
geometrical structure and the chemical composition. On a "per wavelength" basis, real objects 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.
In astronomy, objects such as stars 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. Hawking radiation is the hypothetical blackbody radiation emitted by black holes.
A typical industrial "extended source plate" type black body.
A black body radiates energy at all frequencies, but its intensity rapidly tends to zero at high frequencies (low
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.[9]
[edit] Blackbody simulators
Although a black body is a theoretical object (i.e. emissivity e = 1.0), common applications define a source of
infrared radiation as a black body when the object approaches an emissivity of 1.0, (typically e = 0.99 or better).
A source of infrared radiation less than 0.99 is referred to as a "grey body".[10] Applications for black body
simulators typically include the testing and calibration of infrared systems and infrared sensor equipment.
Super black
is an example of such a material, made from a nickel-phosphorus alloy. More recently, a team of
Japanese scientists created a material even closer to a black body, based on vertically aligned single-walled
carbon nanotubes, which absorbs between 98% and 99% of the incoming light, in the spectral range from UV to
far infrared.[11]
[edit] Equations governing black bodies
[edit] Planck's law of blackbody radiation
Main article: Planck's law
Planck's law states that
where
I(ν,T) dν is the amount of energy per unit surface area per unit time per unit solid angle emitted in the
frequency range between ν and ν + dν by a black body at temperature T;
h is the Planck constant;
c is the speed of light in a vacuum;
k is the Boltzmann constant;
ν is frequency of electromagnetic radiation; and
T is the temperature in kelvins.
[edit] Wien's displacement law
Main article: Wien's displacement law
Wien's displacement law 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, λmax, it is a function only of the temperature
where the constant, b, known as Wien's displacement constant, is equal to 2.8977685(51)×10−3 m K.
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
.[12]
[edit] Stefan–Boltzmann law
Main article: Stefan–Boltzmann law
This 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. That is
where j*is the total power radiated per unit area, T is the temperature (specified in a temperature system where 0
is at absolute zero, such as the kelvin scale) and σ = 5.67×10−8 W m−2 K−4 is the Stefan–Boltzmann constant.
[edit] Radiation emitted by a human body
Much of a person's energy is radiated away in the form of
infrared energy. Some materials are transparent to infrared
light, while opaque to visible light (note the plastic bag). Other
materials are transparent to visible light, while opaque or
reflective to the infrared (note the man's glasses).
Blackbody laws can be applied to human beings. For example, some of a person's energy is radiated away in the
form of electromagnetic radiation, most of which is infrared.
The net power radiated is the difference between the power emitted and the power absorbed:
Pnet = Pemit − Pabsorb.
Applying the Stefan–Boltzmann law,
The total surface area of an adult is about 2 m², and the mid- and far-infrared emissivity of skin and most
clothing is near unity, as it is for most nonmetallic surfaces.[13][14] Skin temperature is about 33°C,[15] but
clothing reduces the surface temperature to about 28 °C when the ambient temperature is 20 °C.[16] Hence, the
net radiative heat loss is about
The total energy radiated in one day is about 9 MJ (megajoules), or 2000 kcal (food calories). Basal metabolic rate
for a 40-year-old male is about 35 kcal/(m2·h),[17] 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.[18]
There are other important thermal loss mechanisms, including convection and evaporation. Conduction is
negligible since the Nusselt 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.[19] 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 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.
[edit] Temperature relation between a planet and its star
The blackbody law may be used to estimate the temperature of a planet orbiting the Sun.
Earth's longwave thermal radiation intensity, from clouds, atmosphere and ground
The temperature of a planet depends on a several factors:
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Incident radiation from its sun
Emitted radiation of the planet, e.g., Earth's infrared glow
The albedo effect causing a fraction of light to be reflected by the planet
The greenhouse effect for planets with an atmosphere
Energy generated internally by a planet itself due to radioactive decay, tidal heating, 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 (energy/second) the Sun is emitting:
The Earth only has an absorbing area equal to a two dimensional circle, rather than the surface of a sphere.
where
is the Stefan–Boltzmann constant,
is the surface temperature of the Sun, and
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.
The power from the Sun that strikes the Earth (at the top of the atmosphere) is:
where
is the radius of the Earth and
is the astronomical unit, the distance between the Sun and the Earth.
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 Earth reflects a fraction α of this energy where α is the albedo or reflectance
of the Earth in the UV-Vis range. In other words, the Earth absorbs a fraction 1 − α of the sun's light, and
reflects the rest. The power absorbed by the Earth and its atmosphere is then:
Even though the Earth only absorbs as a circular area πR2, it emits equally in all directions as a sphere. If the
Earth were a perfect black body, it would emit according to the Stefan-Boltzmann law
where TE is the temperature of the Earth. The Earth, 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 Earth and its
atmosphere is then:
Assuming that the Earth is in thermal equilibrium, the power absorbed must equal the power emitted:
Substituting the expressions for solar and Earth power in equations 1-6 and simplifying yields:
In other words, given the assumptions made, the temperature of Earth depends only on the surface temperature
of the Sun, the radius of the Sun, the distance between Earth and the Sun, the albedo and the IR emissivity of
the Earth.
[edit] Temperature of Earth
Substituting the measured values for the Sun and Earth yields:
[20]
[20]
[20]
[21]
With the average emissivity set to unity, the effective temperature of the Earth is:
TE = 254.356 K 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, and assuming an unchanging albedo. The Earth in fact radiates almost 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[22] and
0.95[23] 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 (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.[24] Similarly using albedo 0.3 and solar constant of 1372 W
m−2), one obtains an effective temperature of 255 K.[25][26]
[edit] Doppler effect for a moving black body
The 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.[27] 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, which exhibits a dipole
anisotropy from the Earth's motion relative to this black body radiation field.
[edit] See also
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Bolometer
Color temperature
Effective temperature
Emissivity
Infrared thermometer
Photon polarization
Pyrometry
Rayleigh-Jeans law
Super black
Thermal radiation
Thermography
Ultraviolet catastrophe
Sakuma–Hattori equation
[edit] References
1.
^
G. Kirchhoff (1860). On the relation between the Radiating and Absorbing Powers of different Bodies
for Light and Heat, translated by F. Guthrie in Phil. Mag. Series 4, volume 20, number 130, pages 1-21,
original in Poggendorff's Annalen, vol. 109, pages 275 et seq.
2. ^ M. Planck (1914). The theory of heat radiation, second edition, translated by M. Masius, Blackiston's
Son & Co, Philadelphia.
3. ^ Robitaille, P. (2003). "On the validity of Kirchhoff's law of thermal emission". IEEE Transactions on
Plasma Science 31: 1263. doi:10.1109/TPS.2003.820958.
4. ^ "Science: Draper's Memoirs". The Academy (London: Robert Scott Walker) XIV (338): 408. Oct. 26,
1878. http://books.google.com/?id=hZINAAAAQAAJ&pg=PA408&dq=draper+point+red+1847.
5. ^ J. R. Mahan (2002). Radiation heat transfer: a statistical approach (3rd ed.). Wiley-IEEE. p. 58.
ISBN 9780471212706. http://books.google.com/?id=y9zUEzA7iN0C&pg=PA58&dq=draper-point+red.
6. ^ Huang, Kerson (1967). Statistical Mechanics. New York: John Wiley & Sons.
7. ^ Planck, Max (1901). "On the Law of Distribution of Energy in the Normal Spectrum" ([dead link]). Annalen
der Physik 4: 553. http://dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Planck-1901/Planck-1901.html.
8. ^ Landau, L. D.; E. M. Lifshitz (1996). Statistical Physics (3rd Edition Part 1 ed.). Oxford: ButterworthHeinemann.
9. ^ Mathematica:Planck intensity (energy/sec/area/solid angle/wavelength) is:
i[w_, t_] = 2*h*c^2/(w^5*(Exp[h*c/(w*k*t)] - 1))
The number of photons/sec/area is:
NIntegrate[2*Pi*i[w, 300]/(h*c/w), {w, 390*10^(-9), 750*10^(-9)}] = 0.0244173...
10. ^ Electro Optical Industries, Inc. (2008)What is a Blackbody and Infrared Radiation? In
Education/Reference
11. ^ K. Mizuno et al. (2009). "A black body absorber from vertically aligned single-walled carbon
nanotubes" (free download). Proceedings of the National Academy of Sciences 106 (15): 6044–6077.
doi:10.1073/pnas.0900155106. PMID 19339498.
12. ^ Nave, Dr. Rod. "Wien's Displacement Law and Other Ways to Characterize the Peak of Blackbody
Radiation". HyperPhysics. http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/wien3.html#c1. Provides
5 variations of Wien's Displacement Law
13. ^ Infrared Services. "Emissivity Values for Common Materials". http://infraredthermography.com/material-1.htm. Retrieved 2007-06-24.
14. ^ Omega Engineering. "Emissivity of Common Materials".
http://www.omega.com/literature/transactions/volume1/emissivityb.html .
Retrieved 2007-06-24.
15. ^ Farzana, Abanty (2001). "Temperature of a Healthy Human (Skin Temperature)". The Physics
Factbook. http://hypertextbook.com/facts/2001/AbantyFarzana.shtml. Retrieved 2007-06-24.
16. ^ Lee, B.. "Theoretical Prediction and Measurement of the Fabric Surface Apparent Temperature in a
Simulated Man/Fabric/Environment System". http://www.dsto.defence.gov.au/publications/2135/DSTOTR-0849.pdf. Retrieved 2007-06-24.
17. ^ Harris J, Benedict F (1918). "A Biometric Study of Human Basal Metabolism.". Proc Natl Acad Sci
USA 4 (12): 370–3. doi:10.1073/pnas.4.12.370. PMID 16576330.
18. ^ Levine, J (2004). "Nonexercise activity thermogenesis (NEAT): environment and biology". Am J
Physiol Endocrinol Metab 286 (5): E675–E685. doi:10.1152/ajpendo.00562.2003. PMID 15102614.
http://ajpendo.physiology.org/cgi/content/full/286/5/E675 .
19. ^ DrPhysics.com. "Heat Transfer and the Human Body".
http://www.drphysics.com/convection/convection.html. Retrieved 2007-06-24.
20. ^ a b c NASA Sun Fact Sheet
21. ^ Cole, George H. A.; Woolfson, Michael M. (2002). Planetary Science: The Science of Planets Around
Stars (1st ed.). Institute of Physics Publishing. pp. 36–37, 380–382. ISBN 0-7503-0815-X.
http://books.google.com/?id=Bgsy66mJ5mYC&pg=RA3-PA382&dq=blackbody+emissivity+greenhouse+intitle:Planetary-Science+inauthor:cole.
22. ^ Saari, J. M.; Shorthill, R. W. (1972). "The Sunlit Lunar Surface. I. Albedo Studies and Full Moon".
The Moon 5 (1-2): 161–178. doi:10.1007/BF00562111.
http://articles.adsabs.harvard.edu//full/1972Moon....5..161S/0000167.000.html .
23. ^ Lunar and Planetary Science XXXVII (2006) 2406
24. ^ Michael D. Papagiannis (1972). Space physics and space astronomy. Taylor & Francis. pp. 10–11.
ISBN 9780677040004. http://books.google.com/?id=SpgOAAAAQAAJ&pg=PA10.
25. ^ Willem Jozef Meine Martens and Jan Rotmans (1999). Climate Change an Integrated Perspective.
Springer. pp. 52–55. ISBN 9780792359968. http://books.google.com/?id=o1SELkgK6PcC&pg=RA1PA53&dq=Earth+effective-temperature+albedo+black-body+0.3.
26. ^ F. Selsis (2004). "The Prebiotic Atmosphere of the Earth". In Pascale Ehrenfreund et al..
Astrobiology: Future Perspectives. Springer. pp. 279–280. ISBN 9781402025877.
http://books.google.com/?id=bA_uR3iwzQUC&pg=PA279&dq=Earth+effective-temperature+albedo+blackbody+0.3.
27. ^ The Doppler Effect, T. P. Gill, Logos Press, 1965