Weins law and Rayleigh-Jeans law

The Rayleigh-Jeans Law:

The Rayleigh-Jeans Law was a useful but not completely successful attempt at establishing the functional form of the spectra of thermal radiation. The energy density u_ν per unit frequency interval at a frequency ν is, according to the The Rayleigh-Jeans Radiation,

u_ν = 8πν²kT/c²

where k is Boltzmann’s constant, T is the absolute temperature of the radiating body and c is the speed of light in a vacuum.

This formula fits the empirical measurements for low frequencies but fails increasingly for higher frequencies. The failure of the formula to match the new data was called the ultraviolet catastrophe. The significance of this inadequate so-called law is that it provides an asymptotic condition which other proposed formulas, such as Planck’s, need to satisfy. It gives a value to an otherwise arbitrary constant in Planck’s thermal radiation formula.

The Derivation of the Rayleigh-Jeans Radiation Law

Consider a cube of edge length L in which radiation is being reflected and re-reflected off its walls. Standing waves occur for radiation of a wavelength λ only if an integral number of half-wave cycles fit into an interval in the cube. For radiation parallel to an edge of the cube, this requires

L/(λ/2) = m, an integer

or, equivalently

λ = 2L/m

Between two endpoints there can be two standing waves, one for each polarization. In the following, the matter of polarization will be ignored until the end of the analysis and there the number of waves will be doubled to take into account the matter of polarization.

Since the frequency ν is equal to c/λ, where c is the speed of light

ν = cm/(2L)

It is convenient to work with the quantity q, known as the wave number, which is defined as

q = 2π/λ

and hence

q = 2πν/c

In terms of the relationship for the cube,

q = 2πm/(2L) = π(m/L)

and hence

q² = π²(m/L)²

Another convenient term is the radian frequency ω=2πν. From this it follows that q=ω/c.

If mX, mY and mZ denote the integers for the three different directions in the cube then the condition for a standing wave in the cube is that

q² = π²[(mX/L)² + (mY/L)² +(mZ/L)²]

which reduces to

mX² + mY² + mZ² = 4L²ν²/c²

Now the problem is to find the number of nonnegative combinations of (mX, mY, mZ) that fit between a sphere of radius R and and one of radius R+dR. First the number of combinations ignoring the nonnegativity requirement can be determined.

The volume of a spherical shell of inner radius R and outer radius R+dR is given by

dV = 4πR²dR

If

R = (mX²+mY²+mZ²)½

then

R = (4L²ν²/c²)½ = 2Lν/c

and hence

dR=2Ldν/c.

This means that

dV = 4π(2Lν/c)²(2L/c)dν = 32π(L³ν²/c³)dν

Now the nonnegativity require for the combinations (mX, mY, mZ) must be taken into account. For the two dimensional case the nonnegative combinations are approximately those in one quadrant of circle. The approximation arises from the matter of the combinations on the boundaries of the nonnegative quadrant. For the three dimensional case the nonnegative combinations consistute approximately one octant of the total. Thus the number dN for the nonnegative combinations of (mX, mY, mZ) in this volume is equal to (1/8)dV and hence

dN = 4πν²dν

The average kinetic energy per degree of freedom is ½kT, where k is Boltzmann’s constant. For harmonic oscillators there is an equality between kinetic and potential energy so the average energy per degree of freedom is kT. This means that the average radiation energy E per unit frequency is given by

dE/dν = kT(dN/dν) = 4πkT(L³/c³)ν²

and the average energy density, uν, is given by

duν/dν = (1/L³)(dE/dν) = 4πkTν²/c³

The previous only considered one direction of polarization for the radiation. If the two directions of polarization are taken into account a factor of 2 must be included in the above formula; i.e.,

duν/dν = 8πkTν²/c³

This is the Raleigh-Jeans Law of Radiation. It holds empirically as the frequency goes to zero.

 

What is Wien’s approximation?

Wien’s approximation (also sometimes called Wien’s law or the Wien distribution law) is a law of physics used to describe the spectrum of thermal radiation (frequently called the blackbody function). This law was first derived by Wilhelm Wien in 1896.[1][2][3] The equation does accurately describe the short wavelength (high frequency) spectrum of thermal emission from objects, but it fails to accurately fit the experimental data for long wavelengths (low frequency) emission.[3]

Wien took the wavelength of black body radiation and combined it with the Maxwell–Boltzmann distribution for atoms. The exponential curve was created by the use of Euler’s number e raised to the power of the temperature multiplied by a constant. Fundamental constants were later introduced by Max Planck.

T⁴ Radiation Law may be written as:

I(v,T)=2v³e^-x

^where:

 ^{I(v,T) is the amount of energy per unit surface area per unit time per unit solid angle per unit frequency emitted at a frequency ν.}

^{T is the temperature of the black body.}

^{x is the ratio temperature over frequency.}

^{h is the Planck constant.}

^{c is the speed of light.}

^{k is the Boltzmann constant.}

 

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