NationMaster - Encyclopedia: Planck's law

For a general introduction, see black body. Image File history File links Wiens_law. ... Image File history File links Wiens_law. ... As the temperature decreases, the peak of the black body radiation curve moves to lower intensities and longer wavelengths. ...

In physics, Planck's law describes the spectral radiance of electromagnetic radiation at all wavelengths from a black body at temperature $T$ . As a function of frequency $ν$ , Planck's law is written as: Physics (Greek: (phÃºsis), nature and (phusikÃ©), knowledge of nature) is the branch of science concerned with the fundamental laws of the Universe. ... Radiance and spectral radiance are radiometric measures that describe the amount of light that passes through or is emitted from a particular area, and falls within a given solid angle in a specified direction. ... It has been suggested that this article or section be merged with light. ... 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. ... FreQuency is a music video game developed by Harmonix and published by SCEI. It was released in November 2001. ...

$I(nu,T) =frac{2 hnu^{3}}{c^2}frac{1}{ e^{frac{hnu}{kT}}-1}.$ ^[1]

As a function of wavelength λ it is written as:

$I(lambda,T) =frac{2 hc^2}{lambda^5}frac{1}{ e^{frac{hc}{lambda kT}}-1}.$ ^[2]

Note that the two functions have different units - the first is radiance per unit frequency interval while the second is radiance per unit wavelength interval. Hence, the quantities $I (ν, T)$ and $I (λ, T)$ are not equivalent to each other. To derive one from the other, they cannot simply be set equal to each other. However, the two equations are related through:

$I(nu,T)dnu=I(lambda,T)dlambda,$

The following table provides the definition and SI units of measure for each symbol: Look up si, Si, SI in Wiktionary, the free dictionary. ...

Symbol	Meaning	SI units
$I ,$	spectral radiance or energy per unit time per unit surface area per unit solid angle per unit frequency or wavelength (as specified)	J•s^-1•m^-2•sr^-1•Hz^-1, or J•s^-1•m^-2•sr^-1•m^-1
$nu ,$	frequency	hertz
$lambda ,$	wavelength	meter
$T ,$	temperature of the black body	kelvin
$h ,$	Planck's constant	joules per hertz
$c ,$	speed of light	meters per second
$e ,$	base of the natural logarithm, 2.718281...	dimensionless
$k ,$	Boltzmann's constant	joules per kelvin

1 Overview
2 Derivation
3 History
4 Appendix
5 Notes
6 References
7 Further reading
8 External links

A pocket watch, a device used to tell time Look up time in Wiktionary, the free dictionary. ... An open surface with X-, Y-, and Z-contours shown. ... A solid angle is the three dimensional analog of the ordinary angle. ... FreQuency is a music video game developed by Harmonix and published by SCEI. It was released in November 2001. ... The hertz (symbol: Hz) is the SI unit of frequency. ... The wavelength is the distance between repeating units of a wave pattern. ... The metre, or meter (symbol: m) is the SI base unit of length. ... Fig. ... The kelvin (symbol: K) is a unit increment of temperature and is one of the seven SI base units. ... A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ... The joule (IPA pronunciation: or ) (symbol: J) is the SI unit of energy. ... A line showing the speed of light on a scale model of Earth and the Moon The speed of light in a vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness. It is the speed of all electromagnetic radiation... Metre per second (U.S. spelling: meter per second) is an SI derived unit of both speed (scalar) and velocity (vector), defined by distance in metres divided by time in seconds. ... e is the unique number such that the value of the derivative (slope of a tangent line) of f (x)=ex (blue curve) at the point x=0 is exactly 1. ... 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... The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ...

Overview

The wavelength is related to the frequency by The wavelength is the distance between repeating units of a wave pattern. ...

$lambda = { c over nu }.$ ^[3]

The law is sometimes written in terms of the spectral energy density^[4] Energy density is the amount of energy stored in a given system or region of space per unit volume or per unit mass, depending on the context. ...

$u(nu,T) = { 4 pi over c } I(nu,T) = frac{8pi hnu^3 }{c^3}~frac{1}{e^{frac{hnu}{kT}}-1}$

which has units of energy per unit volume per unit frequency (joule per cubic meter per hertz). Integrated over frequency, this expression yields the total energy density. The radiation field of a black body may be thought of as a photon gas, in which case this energy density would be one of the thermodynamic parameters of that gas. The volume of a solid object is the three-dimensional concept of how much space it occupies, often quantified numerically. ... FreQuency is a music video game developed by Harmonix and published by SCEI. It was released in November 2001. ... In physics, a photon gas is a gas-like collection of photons, which come together to form something that has the properties of a conventional gas like hydrogen or neon - including pressure, and temperature. ...

The spectral energy density can also be expressed as a function of wavelength:

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

as shown in the derivation below.

Max Planck originally produced this law in 1900 (published^[5] in 1901) in an attempt to improve upon an expression proposed by Wilhelm Wien which fit the experimental data at short wavelengths but deviated from it at long wavelengths. He found that the above function, Planck's function, fit the data for all wavelengths remarkably well. In constructing a derivation of this law, he considered the possible ways of distributing electromagnetic energy over the different modes of charged oscillators in matter. Planck's law emerged when he assumed that the energy of these oscillators was limited to a set of discrete, integer multiples of a fundamental unit of energy, E, proportional to the oscillation frequency ν: Max Karl Ernst Ludwig Planck (April 23, 1858 â€“ October 4, 1947 in GÃ¶ttingen, Germany) was a German physicist. ... Year 1900 (MCM) was an exceptional common year starting on Monday (link will display the full calendar) of the Gregorian calendar, but a leap year starting on Saturday of the Julian calendar. ... 1901 (MCMI) was a common year starting on Tuesday (link will display calendar) of the Gregorian calendar (or a common year starting on Wednesday of the 13-day-slower Julian calendar). ... Wilhelm Carl Werner Otto Fritz Franz Wien (January 13, 1864 â€“ August 30, 1928) was a German physicist who, in 1893, used theories about heat and electromagnetism to compose Wiens displacement law, which relates the maximum emission of a blackbody to its temperature. ...

$E=hnu,$ .

Planck made this quantization assumption five years before Albert Einstein hypothesized the existence of photons as a means of explaining the photoelectric effect. At the time, Planck believed that the quantization applied only to the tiny oscillators that were thought to exist in the walls of the cavity (what we now know to be atoms), and made no assumption that light itself propagates in discrete bundles or packets of energy. Moreover, Planck did not attribute any physical significance to this assumption, but rather believed that it was merely a mathematical device that enabled him to derive a single expression for the black body spectrum that matched the empirical data at all wavelengths. In physics, quantization is a procedure for constructing a quantum field theory starting from a classical field theory. ... Albert Einstein ( ) (March 14, 1879 â€“ April 18, 1955) was a German-born theoretical physicist who is best known for his theory of relativity and specifically mass-energy equivalence, . He was awarded the 1921 Nobel Prize in Physics for his services to Theoretical Physics, and especially for his discovery of the... The word light is defined here as electromagnetic radiation of any wavelength; thus, X-rays, gamma rays, ultraviolet light, infrared radiation, microwaves, radio waves, and visible light are all forms of light. ... A diagram illustrating the emission of photoelectrons from a metal plate, requiring energy gained from an incoming photon to be more than the work function of the material. ... Properties In chemistry and physics, an atom (Greek á¼„Ï„Î¿Î¼Î¿Ï‚ or Ã¡tomos meaning indivisible) is the smallest particle still characterizing a chemical element. ...

Although Planck's formula predicts that a black body will radiate energy at all frequencies, the formula is only applicable when many photons are being measured. For example, a black body at room temperature (300 kelvin) with one square meter of surface area will emit a photon in the visible range once about once every thousand years or so, meaning that for most practical purposes, a black body at room temperature does not emit in the visible range.

Ultimately, Planck's assumption of energy quantization and Einstein's photon hypothesis became the fundamental basis for the development of Quantum Mechanics. Fig. ...

Derivation

The following derivation of Planck's law can be found, e.g., in ^[4]. See also the gas in a box article for a general derivation. The results of the quantum particle in a box can be used to look at the equilibrium situation for a quantum ideal gas in a box which is a box containing a large number particles which do not interact with each other except for instantaneous thermalizing collisions. ...

Consider a cube of side $L$ with conducting walls filled with electromagnetic radiation. At the walls of the cube, the parallel component of the electric field and the orthogonal component of the magnetic field must vanish. Analogous to the wave function of a particle in a box, one finds that the fields are superpositions of periodic functions. The wavelength $λ i$ in the three directions $i=1ldots 3$ orthogonal to the walls can be: In physics, the particle in a box (also known as the infinite potential well or the infinite square well) is a very simple problem consisting of a single particle bouncing around inside of an immovable box, from which it cannot escape, and which loses no energy when it collides with...

$lambda_{i} = frac{2L}{n_{i}}$

where the $n i$ are integers. For each set of integers $n i$ there are two linear independent solutions (modes). According to quantum theory, the energy levels of a mode are given by:

$E_{n_{1},n_{2},n_{3}}left(rright)=left(r+frac{1}{2}right)frac{hc}{2L}sqrt{n_{1}^{2}+n_{2}^{2}+n_{3}^{2}}mbox{ (1)}$

The quantum number $r$ can be interpreted as the number of photons in the mode. The two modes for each set of $n i$ correspond to the two polarization states of the photon which has a spin of 1. Note that for $r = 0$ the energy of the mode is not zero. This vacuum energy of the electromagnetic field is responsible for the Casimir effect. In the following we will calculate the internal energy of the box at temperature $T$ relative to the vacuum energy. In physics, the Casimir effect is a physical force exerted between separate objects, which is due to neither charge, gravity, nor the exchange of particles, but instead is due to resonance of all-pervasive energy fields in the intervening space between the objects. ...

According to statistical mechanics, the probability distribution over the energy levels of a particular mode is given by: Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ...

$P_{r}=frac{expleft(-beta Eleft(rright)right)}{Zleft(betaright)}$

Here

$beta stackrel{mathrm{def}}{=} 1/left(kTright)$ .

The denominator $Zleft(betaright)$ , is the partition function of a single mode and makes $P r$ properly normalized: In statistical mechanics, the partition function Z is an important quantity that encodes the statistical properties of a system in thermodynamic equilibrium. ...

$Zleft(betaright)=sum_{r=0}^{infty}expleft[-beta Eleft(rright)right]=frac{1}{1-expleft[-betavarepsilonright]}$

Here we have defined

$varepsilon stackrel{mathrm{def}}{=} frac{hc}{2L}sqrt{n_{1}^{2}+n_{2}^{2}+n_{3}^{2}}$

which is the energy of a single photon. As explained here, the average energy in a mode can be expressed in terms of the partition function: In statistical mechanics, the partition function Z is an important quantity that encodes the statistical properties of a system in thermodynamic equilibrium. ...

$leftlangle Erightrangle=-frac{dlogleft(Zright)}{dbeta}=frac{varepsilon}{expleft(betavarepsilonright)-1}$

This formula is a special case of the general formula for particles obeying Bose-Einstein statistics. Since there is no restriction on the total number of photons, the chemical potential is zero. In statistical mechanics, Bose-Einstein statistics determines the statistical distribution of identical indistinguishable bosons over the energy states in thermal equilibrium. ... In thermodynamics and chemistry, chemical potential, symbolized by Î¼, is a term introduced in 1876 by the American mathematical physicist Willard Gibbs, which he defined as follows: Gibbs noted also that for the purposes of this definition, any chemical element or combination of elements in given proportions may be considered a...

The total energy in the box now follows by summing $leftlangle Erightrangle$ over all allowed single photon states. This can be done exactly in the thermodynamic limit $Lrightarrowinfty$ . In this limit, $varepsilon$ becomes continuous and we can then integrate $leftlangle Erightrangle$ over this parameter. To calculate the energy in the box in this way, we need to evaluate how many photon states there are in a given energy range. If we write the total number of single photon states with energies between $varepsilon$ and $varepsilon + dvarepsilon$ as $gleft(varepsilonright)depsilon$ , where $gleft(varepsilonright)$ is the density of states which we'll evaluate in a moment, then we can write: Density of states (DOS) is a property in statistical and condensed matter physics that quantifies how closely packed energy levels are in some physical system. ...

$U = int_{0}^{infty}frac{varepsilon}{expleft(betavarepsilonright)-1}gleft(varepsilonright)dvarepsilon mbox{ (2)}$

To calculate the density of states we rewrite equation (1) as follows:

$varepsilon stackrel{mathrm{def}}{=} frac{hc}{2L}n$

where $n$ is the norm of the vector $vec{n}=left(n_{1},n_{2},n_{3}right)$ :

$n=sqrt{n_{1}^{2}+n_{2}^{2}+n_{3}^{2}}$

For every vector $n$ with integer components larger or equal than zero there are two photon states. This means that the number of photon states in a certain region of n-space is twice the volume of that region. An energy range of $dvarepsilon$ corresponds to shell of thickness $dn= frac{2L}{hc}dvarepsilon$ in n-space. Because the components of $vec{n}$ have to be positive, this shell spans an octant of a sphere. The number of photon states $gleft(varepsilonright)depsilon$ in an energy range $dvarepsilon$ is thus given by:

$gleft(varepsilonright)depsilon=2frac{1}{8}4pi n^{2}dn=frac{8pi L^{3}}{h^{3}c^{3}}varepsilon^{2}dvarepsilon$

Inserting this in Eq. (2) gives:

$U =L^{3}frac{8pi}{h^{3}c^{3}}int_{0}^{infty}frac{varepsilon^{3}}{expleft(betavarepsilonright)-1}dvarepsilonmbox{ (3)}$

From this equation one easily derives the spectral energy density as a function of frequency $u (ν, T)$ and as a function of wavelength $u (λ, T)$ :

$frac{U}{L^3} = int_0^{infty}u(nu,T) dnu$

where:

$u(nu,T) = {8pi hnu^3over c^3}{1over e^{hnu/kT}-1}$

$u (ν, T)$ is known as the black body spectrum. It is a spectral energy density function with units of energy per unit frequency per unit volume.

And:

$frac{U}{L^3} = int_0^infty u(lambda,T) dlambda$

where

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

This is also a spectral energy density function with units of energy per unit wavelength per unit volume. Integrals of this type for Bose and Fermi gases can be expressed in terms of polylogarithms. In this case, however, it is possible to calculate the integral in closed form. Let's first make the integration variable in Eq. (3) dimensionless by substituting $varepsilon = k T x$ : The polylogarithm (also known as de JonquiÃ¨res function) is a special function Lis(z) that is defined by the sum The above definition is valid for all complex numbers s and z where |z|< 1. ...

$u(T) =frac{8pi (kT)^{4}}{(hc)^{3}} J$

where $J$ is given by:

$J=int_{0}^{infty}frac{x^{3}}{expleft(xright)-1}dx = frac{pi^{4}}{15}$

We prove this result in the Appendix below. The total electromagnetic energy inside the box is thus given by:

${Uover V} = frac{8pi^5(kT)^4}{15 (hc)^3}$

where $V = L 3$ is the volume of the box. (Note - This is not the Stefan-Boltzmann law, which is the total energy radiated by a black body. See that article for an explanation.) Since the radiation is the same in all directions, and propagates at the speed of light (c), the spectral intensity (energy/time/area/solid angle/frequency) is The Stefan-Boltzmann law, also known as Stefans law, states that the total energy radiated per unit surface area of a black body in unit time (known variously as the black-body irradiance, energy flux density, radiant flux, or the emissive power), j*, is directly proportional to the fourth...

$I(nu,T) = frac{u(nu,T),c}{4pi}$

which yields

$I(nu,T) = frac{2 hnu^3 }{c^2}~frac{1}{e^{hnu/kT}-1}$

History

Many popular science accounts of quantum theory, as well as some physics textbooks, contain some serious errors in their discussions of the history of Planck's Law. Although these errors were pointed out over forty years ago by historians of physics, they have proved to be difficult to eradicate. An article by Helge Kragh^[6] gives a lucid account of what actually happened.

Contrary to popular opinion, Planck did not quantize light. This is evident from his original 1901 paper ^[5] and the references therein to his earlier work. It is also plainly explained in his book "Theory of Heat Radiation," where he explains that his constant refers to Hertzian oscillators. The idea of quantization was developed by others into what we now know as quantum mechanics. The next step along this direction was made by Albert Einstein, who, by studying the photoelectric effect, proposed a model and equation whereby light was not only emitted but also absorbed in packets or photons. Then, in 1924, Satyendra Nath Bose developed the theory of the statistical mechanics of photons, which allowed a theoretical derivation of Planck's law. Fig. ... Albert Einstein ( ) (March 14, 1879 â€“ April 18, 1955) was a German-born theoretical physicist who is best known for his theory of relativity and specifically mass-energy equivalence, . He was awarded the 1921 Nobel Prize in Physics for his services to Theoretical Physics, and especially for his discovery of the... A diagram illustrating the emission of photoelectrons from a metal plate, requiring energy gained from an incoming photon to be more than the work function of the material. ... The word light is defined here as electromagnetic radiation of any wavelength; thus, X-rays, gamma rays, ultraviolet light, infrared radiation, microwaves, radio waves, and visible light are all forms of light. ... Satyendra Nath Bose Bengali: ) (January 1, 1894 â€“ February 4, 1974) was a Indian physicist, specializing in mathematical physics. ... The results of the quantum particle in a box can be used to look at the equilibrium situation for a quantum ideal gas in a box which is a box containing a large number particles which do not interact with each other except for instantaneous thermalizing collisions. ...

Contrary to another myth, Planck did not derive his law in an attempt to resolve the "ultraviolet catastrophe", the name given to the paradoxical result that the total energy in the cavity tends to infinity when the equipartition theorem of classical statistical mechanics is applied to black body radiation. Planck did not consider the equipartion theorem to be universally valid, so he never noticed any sort of "catastrophe" — it was only discovered some five years later by Einstein, Lord Rayleigh, and Sir James Jeans. The ultraviolet catastrophe, also called the Rayleigh-Jeans catastrophe, was a prediction of early 20th century classical physics that an ideal black body at thermal equilibrium will emit radiation with infinite power. ... Figure 1. ... Albert Einstein ( ) (March 14, 1879 â€“ April 18, 1955) was a German-born theoretical physicist who is best known for his theory of relativity and specifically mass-energy equivalence, . He was awarded the 1921 Nobel Prize in Physics for his services to Theoretical Physics, and especially for his discovery of the... See also Rayleigh fading Rayleigh scattering Rayleigh number Rayleigh waves Rayleigh-Jeans law External links Nobel website bio of Rayleigh About John William Strutt MacTutor biography of Lord Rayleigh Categories: People stubs | 1842 births | 1919 deaths | Nobel Prize in Physics winners | Peers | British physicists | Discoverer of a chemical element ... Sir James Hopwood Jeans (born Ormskirk, September 11, 1877, died Dorking, September 16, 1946) was a British physicist, astronomer and mathematician who was the first to propose the theory of continuous creation of matter in the universe. ...

Appendix

A simple way to calculate the integral

$J=int_{0}^{infty}frac{x^{3}}{expleft(xright)-1}dx$

is to calculate the general case first and then compute the answer at the end. Consider the integral

$int_{0}^{infty}frac{x^{n}}{expleft(xright)-1}dx$ .

Multiplying the numerator and denominator by $expleft(-xright)$ gives us

$int_{0}^{infty}frac{x^{n} expleft(-xright)}{1-expleft(-xright)}dx$ .

We can expand the denominator in powers of $exp( - x)$ to get a convergent series $frac{1}{1-expleft(-xright)} = sum_{n=0}^{infty} exp{left(-n xright)}$ since the denominator is always less than one. Then we have

$int_{0}^{infty}x^{n} sum_{k=1}^{infty} exp{left(-kxright)}dx$ .

Since each term in the sum represents a convergent integral, remove the summation out from under the integral sign. In addition, by a change of variable such that $u = k x$ , we have

$sum_{k=1}^{infty} frac{1}{k^{n+1}} int_{0}^{infty}u^{n} exp{left(-uright)}du$ .

The summation on the left is the Riemann zeta function $zeta{left(n+1right)}$ , while the integral on the right is the Gamma function $Gamma{left(n+1right)}$ , and we are finally left with the general result In mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of significant importance in number theory, because of its relation to the distribution of prime numbers. ... The Gamma function along part of the real axis In mathematics, the Gamma function is an extension of the factorial function to complex numbers. ...

$int_{0}^{infty}frac{x^{n-1}}{expleft(xright)-1}dx = zeta{left(nright)} Gamma{left(nright)}$ .

For our problem, the numerator contains $x 3$ , leaving us with our specific result

$J=zeta{left(4right)} Gamma{left(4right)} = frac{pi^{4}}{90} times 6 = frac{pi^4}{15}.$

Here we have used that $sum_{n=1}^{infty}frac{1}{n^{4}}$ is the Riemann zeta function evaluated for the argument 4, which is given by $frac{pi^{4}}{90}$ . This fact can be proven by considering the contour integral In mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of significant importance in number theory, because of its relation to the distribution of prime numbers. ... In complex analysis, the evaluation of integrals of real-valued functions along intervals on the real line, is not readily found with certain integrands and methods involving only real variables. ...

$oint_{C_{R}}frac{picot(pi z)}{z^{4}}.$

Where $C R$ is a contour of radius $R$ around the origin. In the limit $Rrightarrowinfty$ the integral approaches zero. Using the residue theorem the integral can also be written as a sum of residues at the poles of the integrand. The poles are at zero, the positive and negative integers. The sum of the residues yields precisely twice the desired summation plus the residue at zero. Because the integral approaches zero, the sum of all the residues must be zero. The summation must therefore equal minus one half times the residue at zero. From the series expansion of the cotangent function The residue theorem in complex analysis is a powerful tool to evaluate path integrals of meromorphic functions over closed curves and can often be used to compute real integrals as well. ... All of the trigonometric functions of an angle Î¸ can be constructed geometrically in terms of a unit circle centered at O. Trigonometric functions: , , , , , In mathematics, the trigonometric functions (also called circular functions) are functions of an angle; they are important when studying triangles and modeling periodic phenomena, among many other...

$cot(x)=frac{1}{x} - frac {x}{3} - frac {x^3} {45} +ldots,$

we see that the residue at zero is $-frac{pi^{4}}{45}$ which yields the desired result. The evaluation of the Gamma function can be done by recognizing that for integral values of $n$ , $Gamma{left(n+1right)} = n!$ . In the appendix of the article Stefan-Boltzmann law we give a different derivation of this integral. (See also the polylogarithm article.) The Gamma function along part of the real axis In mathematics, the Gamma function is an extension of the factorial function to complex numbers. ... The Stefan-Boltzmann law, also known as Stefans law, states that the total energy radiated per unit surface area of a black body in unit time (known variously as the black-body irradiance, energy flux density, radiant flux, or the emissive power), j*, is directly proportional to the fourth... The polylogarithm (also known as de JonquiÃ¨res function) is a special function Lis(z) that is defined by the sum The above definition is valid for all complex numbers s and z where |z|< 1. ...

Notes

^ Rybicki, p. 22.
^ Rybicki, p. 22.
^ Rybicki, p. 1.
^ ^a ^b Brehm, J.J. and Mullin, W.J., "Introduction to the Structure of Matter: A Course in Modern Physics," (Wiley, New York, 1989) ISBN 047160531X.
^ ^a ^b Planck, Max, "On the Law of Distribution of Energy in the Normal Spectrum". Annalen der Physik, vol. 4, p. 553 ff (1901) Planck's original 1901 paper.
^ Kragh, Helge Max Planck: The reluctant revolutionary Physics World, December 2000

Annalen der Physik is one of the best-known and oldest (it was founded in 1799) physics journals worldwide. ...

References

Rybicki, G. B., A. P. Lightman (1979). Radiative Processes in Astrophysics. New York: John Wiley & Sons. ISBN 0-471-82759-2.

External links

Interactive Blackbody Radiance Calculator - Includes all common units and an in-depth description of formulas
Radiation of a Blackbody - interactive simulation to play with Planck's law
Scienceworld entry on the Planck Law

Categories: Statistical mechanics | Foundational quantum physics

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