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physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, the Rayleigh–Jeans law is an approximation to the
spectral radiance In radiometry, spectral radiance or specific intensity is the radiance of a surface per unit frequency or wavelength, depending on whether the Spectral radiometric quantity, spectrum is taken as a function of frequency or of wavelength. The Internat ...
of
electromagnetic radiation In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic field, electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, inf ...
as a function of
wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tro ...
from a
black body A black body or blackbody is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. The name "black body" is given because it absorbs all colors of light. A black body ...
at a given temperature through classical arguments. For wavelength λ, it is: B_ (T) = \frac, where B_ is the
spectral radiance In radiometry, spectral radiance or specific intensity is the radiance of a surface per unit frequency or wavelength, depending on whether the Spectral radiometric quantity, spectrum is taken as a function of frequency or of wavelength. The Internat ...
, the power emitted per unit emitting area, per
steradian The steradian (symbol: sr) or square radian is the unit of solid angle in the International System of Units (SI). It is used in three-dimensional geometry, and is analogous to the radian, which quantifies planar angles. Whereas an angle in radian ...
, per unit wavelength; c is the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
; k_ is the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
; and T is the
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...
in
kelvin The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based engineer and phys ...
. For
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
\nu, the expression is instead B_(T) = \frac. The Rayleigh–Jeans law agrees with experimental results at large wavelengths (low frequencies) but strongly disagrees at short wavelengths (high frequencies). This inconsistency between observations and the predictions of
classical physics Classical physics is a group of physics theories that predate modern, more complete, or more widely applicable theories. If a currently accepted theory is considered to be modern, and its introduction represented a major paradigm shift, then the ...
is commonly known as the
ultraviolet catastrophe The ultraviolet catastrophe, also called the Rayleigh–Jeans catastrophe, was the prediction of late 19th century/early 20th century classical physics that an ideal black body at thermal equilibrium would emit an unbounded quantity of energy ...
. Its resolution in 1900 with the derivation by
Max Planck Max Karl Ernst Ludwig Planck (, ; 23 April 1858 – 4 October 1947) was a German theoretical physicist whose discovery of energy quanta won him the Nobel Prize in Physics in 1918. Planck made many substantial contributions to theoretical p ...
of Planck's law, which gives the correct radiation at all frequencies, was a foundational aspect of the development of
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
in the early 20th century.


Historical development

In 1900, the British physicist
Lord Rayleigh John William Strutt, 3rd Baron Rayleigh, (; 12 November 1842 – 30 June 1919) was an English mathematician and physicist who made extensive contributions to science. He spent all of his academic career at the University of Cambridge. Am ...
derived the ''λ''−4 dependence of the Rayleigh–Jeans law based on classical physical arguments and empirical facts. A more complete derivation, which included the proportionality constant, was presented by Rayleigh and Sir
James Jeans Sir James Hopwood Jeans (11 September 187716 September 1946) was an English physicist, astronomer and mathematician. Early life Born in Ormskirk, Lancashire, the son of William Tulloch Jeans, a parliamentary correspondent and author. Jeans was ...
in 1905. The Rayleigh–Jeans law revealed an important error in physics theory of the time. The law predicted an energy output that diverges towards
infinity Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions amo ...
as wavelength approaches zero (as frequency tends to infinity). Measurements of the spectral emission of actual black bodies revealed that the emission agreed with the Rayleigh–Jeans law at low frequencies but diverged at high frequencies; reaching a maximum and then falling with frequency, so the total energy emitted is finite.


Comparison to Planck's law

In 1900
Max Planck Max Karl Ernst Ludwig Planck (, ; 23 April 1858 – 4 October 1947) was a German theoretical physicist whose discovery of energy quanta won him the Nobel Prize in Physics in 1918. Planck made many substantial contributions to theoretical p ...
empirically obtained an expression for
black-body radiation Black-body radiation is the thermal electromagnetic radiation within, or surrounding, a body in thermodynamic equilibrium with its environment, emitted by a black body (an idealized opaque, non-reflective body). It has a specific, continuous spect ...
expressed in terms of wavelength ( Planck's law): B_\lambda(T) = \frac~\frac, where ''h'' is the
Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
and the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
. The Planck's law does not suffer from an ultraviolet catastrophe, and agrees well with the experimental data, but its full significance (which ultimately led to quantum theory) was only appreciated several years later. Since, e^x = 1 + x + + + \cdots. then in the limit of high temperatures or long wavelengths, the term in the exponential becomes small, and the exponential is well approximated with the Taylor polynomial's first-order term, e^ \approx 1 + \frac. So, \frac \approx \frac = \frac. This results in Planck's blackbody formula reducing to B_(T) = \frac, which is identical to the classically derived Rayleigh–Jeans expression. The same argument can be applied to the blackbody radiation expressed in terms of frequency . In the limit of small frequencies, that is h \nu \ll k_\mathrm T , B_\nu(T) = \frac\frac \approx \frac \cdot \frac = \frac. This last expression is the Rayleigh–Jeans law in the limit of small frequencies.


Consistency of frequency and wavelength dependent expressions

When comparing the frequency and wavelength dependent expressions of the Rayleigh–Jeans law it is important to remember that \frac = B_(T), and \frac = B_(T) Therefore, B_(T) \neq B_(T) even after substituting the value \lambda = c /\nu, because B_(T) has units of energy emitted per unit time per unit area of emitting surface, per unit solid angle, per unit wavelength, whereas B_(T) has units of energy emitted per unit time per unit area of emitting surface, per unit solid angle, per unit frequency. To be consistent, we must use the equality B_ \, d\lambda = dP = B_ \, d\nu where both sides now have units of power (energy emitted per unit time) per unit area of emitting surface, per unit solid angle. Starting with the Rayleigh–Jeans law in terms of wavelength we get B_(T) = B_(T) \, \frac where \frac = \frac\left(\frac\right) = -\frac. This leads us to find: B_(T) = \frac \times \frac = \frac.


Other forms of Rayleigh–Jeans law

Depending on the application, the Planck function can be expressed in 3 different forms. The first involves energy emitted per unit time per unit area of emitting surface, per unit solid angle, per spectral unit. In this form, the Planck function and associated Rayleigh–Jeans limits are given by B_\lambda(T) = \frac~\frac \approx \frac or B_\nu(T) = \frac\frac \approx \frac Alternatively, Planck's law can be written as an expression I(\nu,T) = \pi B_\nu(T) for emitted power integrated over all solid angles. In this form, the Planck function and associated Rayleigh–Jeans limits are given by I(\lambda,T) = \frac~\frac \approx \frac or I(\nu,T) = \frac\frac \approx \frac In other cases, Planck's law is written as u(\nu,T) = \frac B_\nu(T) for energy per unit volume (energy density). In this form, the Planck function and associated Rayleigh–Jeans limits are given by u(\lambda,T) = \frac~\frac \approx \frac or u(\nu,T) = \frac\frac \approx \frac


See also

*
Stefan–Boltzmann law The Stefan–Boltzmann law describes the power radiated from a black body in terms of its temperature. Specifically, the Stefan–Boltzmann law states that the total energy radiated per unit surface area of a black body across all wavelengths ...
*
Wien's displacement law Wien's displacement law states that the black-body radiation curve for different temperatures will peak at different wavelengths that are inversely proportional to the temperature. The shift of that peak is a direct consequence of the Planck r ...
* Sakuma–Hattori equation


References


External links


Derivation at HyperPhysics
{{DEFAULTSORT:Rayleigh-Jeans law Foundational quantum physics Obsolete theories in physics