Black Body Radiation: Wien Displacement Law

What is a Blackbody?

A black body is an idealization in physics that pictures a body that absorbs all electromagnetic radiation incident on it irrespective of its frequency or angle. Through the second law of thermodynamics that a body always tries to stay in thermal equilibrium.

To stay in thermal equilibrium, a black body must emit radiation at the same rate as it absorbs and so it must also be a good emitter of radiation, emitting electromagnetic waves of as many frequencies as it can absorb i.e. all the frequencies.

The black body radiation is governed by the Planck radiation law. The Planck radiation law explains the various electromagnetic radiations emitted by a black body at thermal equilibrium at a fixed temperature. Through the Planck radiation law, we were able to understand that radiation emitted by a body is completely independent of the body’s shape, size or composition and is dependent only on the Temperature of the body.

Wien’s Displacement Law

The Wien Displacement law is the predecessor to the all-encompassing Planck’s Law. In the graph below, you will see that each curve seems to peaks at a considerably different wavelength at different temperatures. The Wien displacement law explained the shift of the peaks of the curve in terms of temperature. The Wien displacement law states that the blackbody radiation curve (seen in the graph below) for different temperatures peaks at a corresponding wavelength which is inversely proportional to that temperature.

Wien’s law formula

Quantity Formula Notations
Wien’s Law Formula \(\lambda_{max}=\frac{b}{T}\)
  • T: temperature in kelvins
  • b: Wien’s displacement constant = 2.8977*103 m.K
Frequency-dependent formula: \(U_{max}=\frac{\alpha }{h}kT\approx(5.879*10^{10}\frac{Hz}{k})T\)
  • k: Boltzmann constant
  • h: Planck’s constant
  • T: temperature in Kelvin
  • 𝛼: equivalent value = 2.821
From Planck’s law: \(\lambda_{max}=\frac{h_{c}}{x}\frac{1}{kT}=\frac{2.8977*10^{6}nm.K}{T}\)

Black Body Radiation

We see the examples of the Wien Displacement Law in real life all the time.

  • We can easily deduce that a wood fire which is approximately 1500K hot, gives out peak radiation at 2000 nm. This means that the majority of the radiation from the wood fire is beyond the human eye’s visibility. This is why a campfire is an excellent source of warmth but a very poor source of light.
  • The temperature of the sun’s surface is 5700 K. Using the Wien displacement law; we can calculate the peak radiation output at a wavelength of 500 nm. This lies in the green portion of the visible light spectrum. Turns out, our eyes are highly sensitive to this particular wavelength of visible light. We really should be appreciative of the fact that a rather unusually large portion of the sun’s radiation falls in a fairly small visible spectrum.
  • When a piece of metal is heated, it first becomes ‘red hot’. This is the longest visible wavelength. One further heating, it moves from red to orange and then yellow. At its hottest, the metal will be seen to be glowing white. This is the shorter wavelengths dominating the radiation.

Stay tuned with BYJU’S to learn more about black body radiation, light sources and much more.

Practise This Question

A spherical black body of radius 12 cm radiates 450 W power at 500 K. If the radius were halved and the temperature doubled, the power radiated would be