NationMaster - Encyclopedia: Wien's displacement 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. A black hole concept drawing by NASA. Physics (from the Greek, Ï†Ï…ÏƒÎ¹ÎºÏŒÏ‚ (physikos), natural, and Ï†ÏÏƒÎ¹Ï‚ (physis), nature) is the science of the natural world dealing with the fundamental constituents of the universe, the forces they exert on one another, and the results produced by these forces. ... The wavelength is the distance between repeating units of a wave pattern. ... As the temperature decreases, the peak of the black body radiation curve moves to lower intensities and longer wavelengths. ... Temperature is the physical property of a system which underlies the common notions of hot and cold; the material with the higher temperature is said to be hotter. ...

$lambda_{max} = frac{2.898 ldots times 10^6 mathrm{nm cdot K} }{T}$

where $T ,$ is the temperature of the blackbody in kelvins (K) and $lambda_{max} ,$ is the peak wavelength in nanometers. The number 2.898... × 10⁶ is a proportionality constant with units nanometer-kelvins (nm·K). It is sometimes expressed in SI units as 0.002898... m·K, or in cgs units as 0.2898... cm·K. The kelvin (symbol: K) is the SI unit of temperature, and is one of the seven SI base units. ... A nanometre (American spelling: nanometer) is 1. ... 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. ... The International System of Units (abbreviated SI from the French language name SystÃ¨me International dUnitÃ©s) is the modern form of the metric system. ... The centimetre-gram-second system (CGS) is a system of physical units. ...

The nanometer is a convenient unit of measure for optical wavelengths. To convert nanometers to meters, which is the SI unit of distance, note that 1 nanometer is equivalent to 10⁻⁹ meters. A nanometre (American spelling: nanometer) is 1. ... See also: List of optical topics Optics (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. ... metre or meter, see meter (disambiguation) The metre is the basic unit of length in the International System of Units. ... The International System of Units (abbreviated SI from the French language name SystÃ¨me International dUnitÃ©s) is the modern form of the metric system. ...

Basically, the hotter an object is, the shorter the wavelength at which it will emit radiation. For example, the surface temperature of the sun is 5780 K. Using Wien's law, this temperature corresponds to a peak emission at a wavelength of 500 nm. As can be seen in the article Color, this is fairly in the middle of the visual spectrum, due to 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. The Sun is the star at the center of our Solar system. ... Color is an important part of the visual arts. ... White is a color (more accurately it contains all the colors of the visible spectrum and is sometimes described as an achromatic colorâ€”black is the absence of color) that has high brightness but zero hue. ... Prism splitting light Light is electromagnetic radiation with a wavelength that is visible to the eye (visible light) or, in a technical or scientific context, electromagnetic radiation of any wavelength. ... Rayleigh scattering causing a reddened sky at sunset Rayleigh scattering (named after Lord Rayleigh) is the scattering of light by particles much smaller than the wavelength of the light. ...

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. The incandescent light bulb uses a glowing wire filament heated to white-hot by electrical resistance, to generate light (a process known as thermal radiation). ...

Wilhelm Wien formulated this law based entirely on empirical observations, prior to the development of Planck's law of black body radiation. With the benefit of hindsight, however, it is now possible to derive Wien's law as a direct consequence of Planck's more general law. Wilhelm Wien Wilhelm Wien (January 13, 1864 â€“ August 30, 1928) was a German physicist who, in 1893, used theories about heat and electromagnetism to compose Wiens Law, which relates the maximum emission of a blackbody to its temperature. ... Black body spectrum as a function of wavelength In physics, the spectral intensity of electromagnetic radiation from a black body at temperature T is given by the Plancks law of black body radiation: where: I(Î½) is the amount of energy per unit time per unit surface area per unit...

From the relationship between wavelength and frequency,

$f = { c over lambda }$

where c = 2.998 × 10⁸ meters per second is the speed of light in free space, we can convert Wien's law to a relationship between the frequency (in hertz) of maximum radiation to the temperature (in kelvin): Cherenkov effect in a swimming pool nuclear reactor. ... In physics, free space is a concept of electromagnetic theory, corresponding roughly to the vacuum, the baseline state of the electromagnetic field, or the replacement for the electromagnetic aether. ... Sine waves of various frequencies; the lower waves have higher frequencies than those above. ... The hertz (symbol: Hz) is the SI unit of frequency. ...

$f_{max} = (1.0346... times 10^{11} mathrm{Hz/K}) cdot T$

Derivation

From Planck's law of black body radiation we know that Black body spectrum as a function of wavelength In physics, the spectral intensity of electromagnetic radiation from a black body at temperature T is given by the Plancks law of black body radiation: where: I(Î½) is the amount of energy per unit time per unit surface area per unit...

$u(lambda) = {8pi h cover lambda^5}{1over e^{h c/lambda kT}-1}$

The value of $λ$ for which this function is maximized is sought. To find it, we differentiate $u (λ)$ with respect to $λ$ and set it equal to zero

${ partial u over partial lambda } = 8pi h cleft( {hcover kT lambda^7}{e^{h c/lambda kT}over left(e^{h c/lambda kT}-1right)^2} - {1overlambda^6}{5over e^{h c/lambda kT}-1}right)=0$

${hcoverlambda kT }{1over 1-e^{-h c/lambda kT}}-5=0$

If we define

$xequiv{hcoverlambda kT }$

then

${xover 1-e^{-x}}-5=0$

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 $x$ In mathematics, Lamberts W function, named after Johann Heinrich Lambert, also called the Omega function or product log, is the inverse function of where ew is the exponential function and w is any complex number. ...

$x = 4.965114231744276ldots$ (dimensionless)

Solving for the wavelength $λ$ in units of nanometers, and using units of kelvins for the temperature yields: In the physical sciences, a dimensionless number (or more precisely, a number with the dimensions of 1) is a quantity which describes a certain physical system and which is a pure number without any physical units; it does not change if one alters ones system of units of measurement...

$lambda_{max} = {hcover kx }{1over T} = {2.89776829ldots times 10^6 mathrm{nm cdot K} over T}$ .

Using a similar derivation it can be shown that

${ partial u over partial nu}= (3-{{hnu_{max}}over kT})e^{{{hnu_{max}}over kT}}-3=0$

This gives

${ nu_{max}= {2.8kT over h}}$

It turns out that

${Clambda_{max} over nu_{max}} approx 1.76$

External links

PlanetPhysics

Categories: Statistical mechanics | Foundational quantum physics | Optics | Astrophysics | Eponymous laws

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	Wiens thermodynamical derivation performed the useful role of allowing all fl body spectra measured at different temperatures to be discussed theoretically in terms of the single function f (lT).
	Rayleigh-Jeans law : A law giving the intensity of radiation emitted by a flbody within a narrow band of wavelengths; it states that this intensity is proportional to the temperature divided by the fourth power of the wavelength; it is a good approximation to the experimentally verified Planck radiation formula only at long wavelengths.
	Wien's displacement law : A law for flbody radiation which states that the wavelength at which the maximum amount of radiation occurs times the temperature is a constant equal to approximately 2898 micron-degrees.

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