UY1: Planck radiation law and Wien displacement law


The thermal radiation is emitted over a broad spectrum of wavelengths. For a blackbody, this radiation follows the Planck radiation law:

$$P(\lambda) = \frac{2 \pi h c^{2}}{ \lambda^{5} \left( e^{\frac{hc}{\lambda k T}} – 1 \right) }$$

Pλ is power emitted per unit area per wavelength interval at wavelength λ. 
– It is also called spectral emission
– SI units: W m-2 m-1 (Corresponding equal to power emitted per unit area multiplied by per wavelength interval)

Importance of this formula: 
– Wien displacement law came from taking the derivative of this equation and setting to zero to find the maxima.
– Stefan-Boltzmann law came from integrating this equation over all wavelengths.

 

Thermal Radiation

Wiens_law
Black body radiation as a function of wavelength for various absolute temperatures

 

As temperature T increases:
– the thermal peak wavelength shifts proportional to T-1: Wien displacement law
– the total radiated power increases proportional to T4: Stefan-Boltzmann law

The relation between the peak wavelength and the radiant body temperature is the Wien displacement law:

$$\lambda_{peak} T = 2.898 \times 10^{-3} \, m K$$

 

Net Radiation Gain or Loss

The amount of electromagnetic energy absorbed by an object depends on the temperature of its surroundings. If an object at a temperature T and is fully embedded in a surrounding which is at temperature Tsurr, the rate of net energy gained (or lost) by the object due to emission and absorption of electromagnetic radiation is:

$$\frac{dQ}{dt} = \sigma A e (T^{4} – T_{surr}^{4} )$$
Hence, if T > Tsurr, the object radiates more energy than it absorbs, and its temperature decreases until equilibrium with its surroundings is reached.

 

Next: Equation-of-state – Ideal gas law

Previous: Radiation

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