Wien's displacement law - encyclopedia article about Wien's displacement law.

“Wien's Law” redirects here. For the historical distribution law, see Wien's Distribution Law.

Wien's displacement law is a law of physics that states that there is an inverse relationship between the wavelength of the peak of the emission of a black body and its temperature.

$\text{[math]}$

where

$\text{[math]}$ is the peak wavelength in meters,

$\text{[math]}$ is the temperature of the blackbody in kelvins (K), and

b is a constant of proportionality, called Wien's displacement constant and equals 2.897 768 5(51) × 10^–3 m K (2002 CODATA recommended value)

The two digits between the parentheses denotes the uncertainty (the standard deviation at 68.27% confidence level) in the two least significant digits of the mantissa.

For optical wavelengths, it is often more convenient to use the nanometer in place of the meter as the unit of measure. In this case…
b = 2.897768 5(51) × 10⁶ nm K.

Explanation and familiar approximate applications

The law is named for Wilhelm Wien, who formulated the relationship in 1893 based on a thermodynamic argument. Wien considered adiabatic expansion of a cavity containing waves of light in thermal equilibrium. He showed that under adiabatic expansion or contraction, the energy of light changes in the exact same way as the frequency. This means that the peak frequency should change with temperature as the energy goes. Wien did not interpret his constant b as a new fundamental constant of nature. This was done by Planck.

Wien's displacement law states that the hotter an object is, the shorter the wavelength at which it will emit most of its radiation, and further that the frequency for maximal or peak radiation power is found by dividing Wien's constant by the temperature in kelvins.

Examples:

Light from the Sun and Moon. The surface temperature (or more correctly, the effective temperature) of the Sun is 5778 K. Using Wien's law, this temperature corresponds to a peak emission at a wavelength of 2.89777 million nm K/ 5778 K = 502 nm = about 5000 Å. This wavelength is (not by accident) fairly in the middle of the most sensitive part of land animal visual spectrum acuity. Even nocturnal and twilight-hunting animals must sense light from the waning day and from the moon, which is reflected sunlight with this same wavelength distribution. Also, the average wavelength of starlight maximal power is in this region, due to the sun being in the middle of a common temperature range of stars.

[See for example the article color, because of the spread resulting in white light. Due to the Rayleigh scattering of blue light by the atmosphere this white light is separated somewhat, resulting in a blue sky and a yellow sun].

Wien's constant may be used in different units, and many examples to calculate familiar situation types of radiation required use of only one or two significant figures:

Light from incandescent bulbs and fires. A lightbulb has a glowing wire with a somewhat lower temperature, resulting in yellow light, and something that is "red hot" is again a little less hot. It is easy to calculate that a wood fire at 1500 K puts out peak radiation at 3 million nm K /1500 K = 2000 nm = 20,000 Å. This is far more energy in the infrared than in the visible band, which ends about 7500 Å.
Radiation from mammals and the living human body. Mammals at roughly 300 K emit peak radiation at 3 thousand μm K / 300 K = 10 μm, in the far infrared. This is therefore the range of infrared wavelengths that pit viper snakes and passive IR cameras must sense.
The wavelength of radiation from the Big Bang. A typical application of Wien's law would also be to the blackbody radiation resulting from the Big Bang. Remembering that Wien's displacement constant is about 3 mm K, and the temperature of the Big Bang background radiation is about 3 K (actually 2.7 K), it is apparent that the microwave background of the sky peaks in power at 2.9 mm K / 2.7 K = just over 1 mm wavelength in the microwave spectrum. This provides a convenient rule of thumb for why microwave equipment must be sensitive on both sides of this frequency band, in order to do effective research on the cosmic microwave background.

Frequency form

In terms of frequency f (in hertz), Wien's displacement law becomes

$\text{[math]}$

where

$\text{[math]}$ is a constant resulting from the numerical solution of the maximization equation,

k is Boltzmann's constant,

h is Planck's constant, and

T is temperature (in kelvin).

Because the spectrum resulting from Planck's law of black body radiation takes a different shape in the frequency domain from that of the wavelength domain, the frequency location of the peak emission does not correspond to the peak wavelength using the simple relationship between frequency, wavelength, and the speed of light.

Derivation

Wilhelm Wien first derived this law in 1893 by applying the laws of thermodynamics to electromagnetic radiation^[1]. As is typically the case with thermodynamic arguments, Wien's derivation determines the functional form of the relationship but does not specify the values of the constants b (in the temperature form) or $\text{[math]}$ (in the frequency form.) A modern variant of Wien's derivation can be found in the textbook by Wannier ^[2]. Today, the usual practice is to derive the relationship from Planck's law of black body radiation, as this procedure also yields expressions for the constants b and $\text{[math]}$ in terms of fundamental constants.

From Planck's law, we know that the spectrum of black body radiation is

$\text{[math]}$

The value of $\text{[math]}$ for which this function is maximized is sought. To find it, we differentiate $\text{[math]}$ with respect to $\text{[math]}$ and set it equal to zero

$\text{[math]}$

If we define

$\text{[math]}$

then

$\text{[math]}$

This equation cannot be solved in terms of elementary functions. It can be solved in terms of Lambert's Product Log function but an exact solution is not important in this derivation. One can easily find the numerical value of $\text{[math]}$

$\text{[math]}$ (dimensionless)

Solving for the wavelength $\text{[math]}$ in units of nanometers, and using units of kelvins for the temperature yields:

$\text{[math]}$ .

The frequency form of Wien's displacement law is derived using similar methods, but starting with Planck's law in terms of frequency instead of wavelength.

External links

Eric Weisstein's World of Physics

References and notes

1. ^ Mehra, J. and Rechenberg, H, "The Historical Development of Quantum Theory", Volume 1 Chapter 1, Springer, 1982
2. ^ Wannier, G. H. "Statistical Physics", Dover, 1987; Chapter 10-2

Wien's approximation (also sometimes called Wien's law or the Wien distribution law) is a law of physics used to describe the spectrum of thermal radiation (frequently called the blackbody function). This law was first derived by Wilhelm Wien in 1896.
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Physics is the science of matter^[1] and its motion^[2]^[3], as well as space and time^[4]^[5] —the science that deals with concepts such as force, energy, mass, and charge.
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In physics, wavelength is the distance between repeating units of a propagating wave of a given frequency. It is commonly designated by the Greek letter lambda (λ). Examples of wave-like phenonomena are light, water waves, and sound waves.
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black body is an object that absorbs all electromagnetic radiation that falls onto it. No radiation passes through it and none is reflected. It is this lack of both transmission and reflection to which the name refers.
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trillion fold).]]

Temperature is a physical property of a system that underlies the common notions of hot and cold; something that is hotter generally has the greater temperature. Temperature is one of the principal parameters of thermodynamics.
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1 metre =
SI units
1000 mm 0 cm
US customary / Imperial units
0 ft 0 in

The metre or meter^[1](symbol: m) is the fundamental unit of length in the International System of Units (SI).
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The kelvin (symbol: K) is a unit increment of temperature and is one of the seven SI base units. The Kelvin scale is a thermodynamic (absolute) temperature scale where absolute zero — the coldest possible temperature — is zero kelvins
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proportionality, see Proportionality (disambiguation).

In mathematics, two quantities are called proportional if they vary in such a way that one of the quantities is a constant multiple of the other, or equivalently if they have a constant ratio.
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Uncertainty is a term used in subtly different ways in a number of fields, including philosophy, statistics, economics, finance, insurance, psychology, engineering and science.
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In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is a measure of the spread of its values. It is usually denoted with the letter σ (lower case sigma).
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The significand (also coefficient or mantissa) is the part of a floating-point number that contains its significant digits. Depending on the interpretation of the exponent, the significand may be considered to be an integer or a fraction.
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Optics (ὀπτική appearance or look in Ancient Greek) is a branch of physics that describes the behavior and properties of light and the interaction of light with matter.
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1 nanometre =
SI units
010⁻⁹ m 010⁻³ μm
US customary / Imperial units
010⁻⁹ ft 010⁻⁹ in

A nanometre (American spelling: nanometer, symbol nm
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Wilhelm Wien

Wilhelm Carl Werner Otto Fritz Franz Wien
Born January 13 1864
Fischhausen, East Prussia
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18th century - 19th century - 20th century
1860s 1870s 1880s - 1890s - 1900s 1910s 1920s
1890 1891 1892 - 1893 - 1894 1895 1896

:
Subjects: Archaeology - Architecture -
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An adiabatic invariant is a property of a physical system which stays constant when changes are made slowly.

In thermodynamics, an adiabatic process is a change that occurs without heat flow and slowly compared to the time to reach equilibrium.
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Star

The effective temperature of a star is the temperature of a black body with the same luminosity per surface area () as the star and is defined according to the Stefan-Boltzmann law .
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The Sun

Observation data
Mean distance
from Earth 1.49610¹¹ m
(8.31 min at light speed)
Visual brightness (V) −26.74^m ^[1]
Absolute magnitude 4.
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Color or colour^[1] (see spelling differences) is the visual perceptual property corresponding in humans to the categories called red, yellow, blue, black, etc.
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White is the combination of all the colors of the visible light spectrum.^[1]. It is sometimes described as an achromatic color, like black.

White is technically achromatic, and not a color, since it has no hue.
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Light is electromagnetic radiation of a wavelength that is visible to the eye (visible light). In a scientific context, the word "light" is sometimes used to refer to the entire electromagnetic spectrum.
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Rayleigh scattering (named after Lord Rayleigh) is the scattering of light or other electromagnetic radiation by particles much smaller than the wavelength of the light. It can occur when light travels in transparent solids and liquids, but is most prominently seen in gases.
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incandescent light bulb (also spelled lightbulb) or incandescent lamp is a source of artificial light that works by incandescence. An electrical current passes through a thin filament, heating it until it produces light.
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Crotalinae
Oppel, 1811

Synonyms

Crotalini - Oppel, 1811
Crotales - Cuvier, 1817
Crotalidae - Gay, 1825
Crotaloidae - Fitzinger, 1826
Cophiadae - Boie, 1827
Crotaloidei - Eichwald, 1831
Crotalina - Bonaparte, 1831

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Big Bang is the cosmological model of the universe whose primary assertion is that the universe has expanded into its current state from a primordial condition of enormous density and temperature.
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cosmic microwave background radiation (most often abbreviated CMB but occasionally CMBR, CBR or MBR, also referred to as relic radiation) is a form of electromagnetic radiation discovered in 1965 that fills the entire universe ^[1].
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FreQuency is a music video game developed by Harmonix and published by SCEI. It was released in November 2001. A sequel, titled Amplitude was released in 2003.
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hertz (symbol: Hz) is the SI unit of frequency. Its base unit is cycle/s or s^-1 (also called inverse seconds, reciprocal seconds). In English, hertz is used as both singular and plural.
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The Boltzmann constant (k or k_B) is the physical constant relating temperature to energy.

It is named after the Austrian physicist Ludwig Boltzmann, who made important contributions to the theory of statistical mechanics, in which this
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Planck constant (denoted ) is a physical constant that is used to describe the sizes of quanta. It plays a central role in the theory of quantum mechanics, and is named after Max Planck, one of the founders of quantum theory.
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