Rayleigh-Jeans law
Encyclopedia
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, the Rayleigh–Jeans law attempts to describe the spectral radiance of electromagnetic radiation
Electromagnetic radiation
Electromagnetic radiation is a form of energy that exhibits wave-like behavior as it travels through space...
at all wavelengths from a black body
Black body
A black body is an idealized physical body that absorbs all incident electromagnetic radiation. Because of this perfect absorptivity at all wavelengths, a black body is also the best possible emitter of thermal radiation, which it radiates incandescently in a characteristic, continuous spectrum...
at a given temperature through classical arguments. For wavelength λ, it is;
where c is the speed of light
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
, k is the Boltzmann constant and T is the temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
in kelvin
Kelvin
The kelvin is a unit of measurement for temperature. It is one of the seven base units in the International System of Units and is assigned the unit symbol K. The Kelvin scale is an absolute, thermodynamic temperature scale using as its null point absolute zero, the temperature at which all...
s. For frequency
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...
ν, the expression is instead
The Rayleigh–Jeans law agrees with experimental results at large wavelengths (or, equivalently, low frequencies) but strongly disagrees at short wavelengths (or high frequencies). This inconsistency between observations and the predictions of classical physics
Classical physics
What "classical physics" refers to depends on the context. When discussing special relativity, it refers to the Newtonian physics which preceded relativity, i.e. the branches of physics based on principles developed before the rise of relativity and quantum mechanics...
is commonly known as the ultraviolet catastrophe
Ultraviolet catastrophe
The ultraviolet catastrophe, also called the Rayleigh–Jeans catastrophe, was a prediction of late 19th century/early 20th century classical physics that an ideal black body at thermal equilibrium will emit radiation with infinite power....
, and its resolution was a foundational aspect of the development of quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
in the early 20th century.
Historical development
In 1900, the British physicist Lord RayleighJohn Strutt, 3rd Baron Rayleigh
John William Strutt, 3rd Baron Rayleigh, OM was an English physicist who, with William Ramsay, discovered the element argon, an achievement for which he earned the Nobel Prize for Physics in 1904...
derived the λ−4 dependence of the Rayleigh–Jeans law based on classical physical arguments. A more complete derivation, which included the proportionality constant, was presented by Rayleigh and Sir James Jeans 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 as wavelength approaches zero (as frequency tends to infinity) and measurements of energy output at short wavelengths disagreed with this prediction.
Comparison to Planck's law
In 1900 Max PlanckMax Planck
Max Karl Ernst Ludwig Planck, ForMemRS, was a German physicist who actualized the quantum physics, initiating a revolution in natural science and philosophy. He is regarded as the founder of the quantum theory, for which he received the Nobel Prize in Physics in 1918.-Life and career:Planck came...
empirically obtained an expression for blackbody radiation
Planck's law of black body radiation
In physics, Planck's law describes the amount of energy emitted by a black body in radiation of a certain wavelength . The law is named after Max Planck, who originally proposed it in 1900. The law was the first to accurately describe black body radiation, and resolved the ultraviolet catastrophe...
expressed in terms of wavelength :
where h is the Planck constant
Planck constant
The Planck constant , also called Planck's constant, is a physical constant reflecting the sizes of energy quanta in quantum mechanics. It is named after Max Planck, one of the founders of quantum theory, who discovered it in 1899...
. The Planck 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. In the limit of very high temperatures or long wavelengths, the term in the exponential becomes small, and so the exponential is well approximated by its first-order Taylor polynomial
Taylor series
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
:
Then
This results in Planck's blackbody formula reducing to
which is identical to the classically derived Rayleigh–Jeans expression.
The same argument can be applied to the Blackbody Radiation
Planck's law of black body radiation
In physics, Planck's law describes the amount of energy emitted by a black body in radiation of a certain wavelength . The law is named after Max Planck, who originally proposed it in 1900. The law was the first to accurately describe black body radiation, and resolved the ultraviolet catastrophe...
expressed in terms of frequency in the limit of small frequency:
This last expression is the Rayleigh–Jeans law in the limit of small frequency.
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
because
has units of energy emitted per unit time per unit area of emitting surface, per unit solid angle, per unit wavelength, whereas 
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
where both sides now have units of 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
where
.This leads us to find:
.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 unit frequency. In this form, the Planck Function and associated Rayleigh–Jeans limits are given by
or
Alternatively, Planck's law can be written as an expression
for emitted power integrated over all solid angles. In this form, the Planck Function and associated Rayleigh–Jeans limits are given by
or
In other cases, Planck's Law is written as
for energy per unit volume (energy density). In this form, the Planck Function and associated Rayleigh–Jeans limits are given by
or
See also
- Stefan–Boltzmann law
- Wien's displacement lawWien's displacement lawWien's displacement law states that the wavelength distribution of thermal radiation from a black body at any temperature has essentially the same shape as the distribution at any other temperature, except that each wavelength is displaced on the graph...
- Sakuma–Hattori equation