Black Body Radiation: Wien Displacement Law

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.

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 radiation emitted by the blackbody is known as blackbody radiation.

The characteristics of the blackbody radiation are explained with the help of the following laws:

• Wien’s displacement law
• Planck’s law
• Stefan-Boltzmann law

Wien’s Displacement Law

Wien’s displacement law states that

The blackbody radiation curve for different temperatures peaks at a wavelength is inversely proportional to the temperature.

Wien’s Law Formula

 Wien’s Law Formula $$\begin{array}{l}\lambda_{max}=\frac{b}{T}\end{array}$$ T is the temperature in kelvins b is the Wien’s displacement constant = 2.8977*103 m.K

Planck’s Law

Using Planck’s law of blackbody radiation, the spectral density of the emission is determined for each wavelength at a particular temperature.

Planck’s Law Formula

 Planck’s law $$\begin{array}{l}E_{\lambda }=\frac{8\pi hc}{\lambda ^{5}(e^{\frac{hc}{\pi KT}}-1)}\end{array}$$ Eλ is the wavelength T is the absolute temperature

Stefan-Boltzmann Law

The Stefan-Boltzmann law explains the relationship between total energy emitted and the absolute temperature.

Stefan-Boltzmann Law Formula

 Stefan-Boltzmann Law E ∝ T4 E is the total energy emitted T is the absolute temperature

Wien’s Displacement Law Example

• 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. It turns out that our eyes are highly sensitive to this particular wavelength of visible light. We really should be appreciative 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. These are the shorter wavelengths dominating the radiation.