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Derivations Involving Planck's Law of Blackbody Radiation

Planck's formula for spectral distribution of the flux emitted by a blackbody is:

S_v = [(2*pi*h)/(c^2)][(v^3)/((e^hv/kT)-1)]

a) from this formula deduce that the totl flux is proportional to the fourth power of the temperture, that is:

integral from 0 to infinity S_v dv (proportionality symbol) T^4

and deduce that the max in the spectral distribution occurs at a frequency proportional to the temperature, that is:

v_max (proportionality symbol) T

I cannot find the first formula S_v in any other book or the internet. I am using ohanian. I have found that the distribution formula as a function of frequency is similar, but without pi! Is this wrong or whatI am misunderstanding?

Solution Preview

integral from 0 to infinity S_v dv = [(2*pi*h)/(c^2)] integral from 0 to infinity [(v^3)/((e^hv/kT)-1)] dv.

Let x = hv/kT. Then we have v = (kT/h) x and dv = (kT/h) dx, so

integral from 0 to infinity S_v ...

Solution Summary

Using Planck's law of blackbody radiation, we show that the peak frequency is proportional to the temperature and that the total intensity of radiation is proportional to the fourth power of the temperature.

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