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Rayleigh–Jeans law

Comparison of Rayleigh–Jeans law with Wienapproximation and Planck's law , for a body of 8 mKtemperature .

From Wikipedia, t he free ency clopedia

In physics , the Raylei gh–Jeans la w attemptsto describe the spectral radian ce ofelectromagnetic radiation at all wavelengthsfrom a black body at a given temperature

through classical arguments. For wavelength λ , it is:

where c is the spe ed of light , k is th eBoltzmann c onstant and T is the temperaturein kelvins. For frequency ν, the expression isinstead

The Ra yleigh–Jeans law agrees withexperim ental results at large wavelength s (low frequ encies) but st rongly disagrees at short

wavelen ths hi h fre uencies . This inconsistenc between observations and the redictions o f

(T ) = ,B λ2ckT

λ 4

(T ) = .B ν 2 kT ν 2

c 2

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classical physics is commonly known as the ultraviolet catastrophe ,[1][2] and its resolution was afoundational aspect of the development of quantum mechanics in the early 20th century.

Contents [hide ]

1 Historical development2 Comparison to Planck's law3 Consistency of frequency and wavelength dependent expressions4 Other forms of Rayleigh–Jeans law5 See also6 References7 External links

Historical development [edit ]

In 1900, the British physicist Lord Rayleigh derived the λ−4 dependence of the Rayleigh–Jeans lawbased on classical physical arguments. [3] A more complete derivation, which included theproportionality 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 energyoutput 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 [edit ]

In 1900 Max Planck empirically obtained an expression for black-body radiation expressed in termsof wavelength λ = c/ν (Planck's law ):

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where h is the Planck constant and k the Boltzmann constant . The Planck law does not suffer froman 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,

then in the limit of very high temperatures or long wavelengths, the term in the exponentialbecomes small, and the exponential is well approximated with the Taylor polynomial's first-orderterm,

So,

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 expressed in terms of frequencyν = c/ λ . In the limit of small frequencies, that is ,

(T ) = ,B λ2hc 2

λ 5

1

− 1ehc

λkT

= 1 + x + + +⋯ .e x x 2

2!x 3

3!

≈ 1 + .ehc

λkT

hc

λkT

≈ = .1

− 1ehc

λkT

1hc

λkT

λkT

hc

(T ) = ,B λ2ckT

λ 4

hν ≪ kT

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This last expression is the Rayleigh–Jeans law in the limit of small frequencies.

Consistency of frequency and wavelength dependent expressions[edit ]

When comparing the frequency and wavelength dependent expressions of the Rayleigh–Jeans lawit is important to remember that

, and

Therefore,

even after substituting the value , because has units of energy emitted per unittime per unit area of emitting surface, per unit solid angle, per unit wavelength , whereashas units of energy emitted per unit time per unit area of emitting surface, per unit solid angle, pe r

unit frequency . To be consistent, we must use the equality

where both sides now have units of power (energy emitted per unit time) per unit area of emittingsurface, per unit solid angle.

Starting with the Rayleigh–Jeans law in terms of wavelength we get

(T ) = ≈ ⋅ = .B ν 2h /ν 3 c 2

− 1ehν

kT

2hν 3

c 2

kT

hν

2 kT ν 2

c 2

dP / dλ = ( T )B λ

dP / dν = ( T )B ν

(T ) ≠ ( T )B λ B ν

λ = c / ν (T )B λ

(T )B ν

dλ =

dP =

dν Bλ

Bν

(T ) = ( T ) ×B λ B ν

dν

dλ

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where

.

This leads us to find:

.

Other forms of Rayleigh–Jeans law [edit ]

Depending on the application, the Planck Function can be expressed in 3 different forms. The firstinvolves energy emitted per unit time per unit area of emitting surface, per unit solid angle, per unitfrequency. 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 powerintegrated over all solid angles. In this form, the Planck Function and associated Rayleigh–Jeanslimits are given by

or

= ( )= −dν

dλ

d

dλ

c

λ

c

λ 2

(T ) = × =B λ

2kT

( )c

λ

2

c 2

c

λ 2 2ckT

λ 4

(T ) = ≈B λ2c 2

λ 5

h

− 1ehc

λkT

2ckT

λ 4

(T ) = ≈B ν 2h /ν 3 c 2

− 1ehν

kT

2kTν 2

c 2

I (ν , T ) = π (T )B ν

I (λ , T ) = ≈2πc 2

λ 5

h

− 1ehc

λkT

2πckT

λ 4

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In other cases, Planck's Law is written as for energy per unit volume (energydensity). In this form, the Planck Function and associated Rayleigh–Jeans limits are given by

or

See also [edit ]

Stefan–Boltzmann lawWien's displacement lawSakuma–Hattori equation

References [edit ]

1. ^ Ast ronomy: A Physical Perspective , Mark L. Kutner pp. 152. ^ Radiative Processes in Astrophysics , Rybicki and Lightman pp. 20–283. ^ Ast ronomy: A Physical Perspective , Mark L. Kutner pp. 15

External links [edit ]

Derivation at HyperPhysics

I (ν , T ) = ≈2πh /ν 2 c 2

− 1ehν

kT

2πkTν 2

c 2

u (ν , T ) = ( T )4π

c B ν

u(

λ,

T ) = ≈

8πc

λ 5

h

− 1ehc

λkT

8πkT

λ 4

u (ν , T ) = ≈8πh /ν 3 c 3

− 1ehν

kT

8πkTν 2

c 3

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