Exercises on Radiation Laws

#Exercises-on-Radiation-Laws

This tutorial explores some of the radiation laws: Planck's law, Sefan-Boltzmann-law, Wien's law.

Some constants ...

Planck's law

#Planck's-law

Planck's law for power radiated normally from a unit area (energy flux density) of a blackbody radiator at temperature T into a unit solid angle within a wavelength band of unit width centered at wavelength @@0@@ is given by

@@1@@

where k = 1.381 @@2@@ 10@@3@@ is the vacuum speed of light.

T in K, wl in @@4@@m

Same, but within a frequency band of unit width centered at frequency f. Units: W/(m@@0@@·sr·Hz)

T in K, f in Hz

Stefan-Boltzmann law

#Stefan-Boltzmann-law

The total power emitted per unit area (i.e., radiant flux density) at the surface of a blackbody at a temperature T (i.e., by integrating Planck's function over the whole spectrum and over the solid angles corresponding to a hemisphere above the surface. Units: Wm@@0@@ (T in K)

This radiant flux density is reduced when moving from the Sun's surface to the Earth. The ratio (@@0@@ then leads to the solar constant:

plotting ...

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and with logarithmic axes:

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...and everything normalized to 1:

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Wien’s Displacement Law

#Wien’s-Displacement-Law

The maximum of the Planck function, i.e., the wavelength of the maximum intensity of blackbody radiation, is obtained by differentiating with respect to wavelength and by setting the result equal to zero:

@@0@@

We then obtain the wavelength of the maximum

@@1@@

with a = 2.897 @@2@@ 10@@3@@K.

This is Wien’s displacement law stating that the wavelength of the maximum intensity of blackbody radiation is inversely proportional to the temperature. From this relationship, the temperature of a blackbody can be determined from the measurement of the maximum monochromatic intensity.