Wien’s Distribution Law, Rayleigh Jeans Law and Planck’s Radiation Law - Civil Engineering (CE) PDF Download

Wein’s distribution law , Rayleigh-Jeans law and Planck’s Radiation Law
According to Wien’s distribution law the energy emitted by the blackbody per unit volume in the range of wavelength from λ to λ + dλ is given by

where C₁ and C₂ are constants and T is absolute temperature. It was found that Wein’s distribution law explains energy distribution only in the shorter wavelength region. Also, the energy emitted by the blackbody tends to zero at very high temperature which is in contradiction with the experimental results.

Rayleigh-Jeans law 
According to Rayleigh-Jeans law the energy emitted by the blackbody per unit volume in the range of wavelength from λ to λ + dλ is given by

Where k is Boltzmann constant whose value is 1.38 × 10⁻²³JK⁻¹ . Rayleigh-Jeans formula was found to explain energy distribution only in the longer wavelength region of the blackbody radiation spectrum. Further, Rayleigh-Jeans law predicts that blackbody radiates enormous amount of energy in the shorter wavelength region so that no energy is available for emission in the longer wavelength region. But blackbody radiates mostly in the visible and infrared region. This discrepancy between experimental results and theoretical predictions was known as ‘ultraviolet catastrophe’.

 Planck’s Radiation Law
According to Planck’s radiation law whenever radiation is emitted or absorbed by a blackbody energy exchange between matter and radiation takes place only in discrete steps, i.e emission or absorption of radiation of frequency ν takes place in steps of integrals multiples of packets of energy called quanta. Thus Energy emitted or absorbed by the body is given by

E = nhν
Where h = Planck’s constant =6.254 × 10⁻³⁴Js and n = 1, 2, 3 . . .

According to Planck’s radiation law the energy emitted by the blackbody per unit volume in the range of wavelength from λ to λ + dλ is given by

Where c = velocity of light = 3 × 10⁸ms⁻¹ .

Planck’s radiation law explains energy distribution in the spectrum of blackbody radiation over the wide range of wavelength, i.e., it is in agreement with the experimental results. Further, it can be shown that Planck’s law reduces to Wein’s distribution law in the shorter wavelength region while reduces to Rayleigh-Jeans law in the longer wavelength

Reduction of Planck’s law to Wein’s law and Rayleigh-Jeans law 
Planck’s radiation law is given by equation (1.3). (i) For shorter wavelengths, e ^hc/λkT is very large ⇒ e ^hc/λkT >> 1 therefore

The equation (1.6) is the expression for Wein’s law of radiation.
(ii) For longer wavelengths, Expanding e x as power series we have e ^x = 1 + x + x ²/2! + x³/3! + ......
If x is small then its higher power terms could be neglected thus e^x = 1 + x here x = hc/λkT and for longer wavelengths λ is large making x small

Using equation (1.7) in (1.3) we get

This equation is Rayleigh-Jeans law of radiation. Thus, Wein’s and Rayleigh-Jeans law turn out to be special case of Planck’s raiation law.