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Using Planck's fromula, derive the approximate expressions for the space spectral density u_(omega) of radiation (a) in the range where h omegalt lt kT (Rayleigh-Jeans fromula), (b) in the range where h omegagt gt kT (Wien's formula). |
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Answer» Solution :From Planek's formula `u_(omega) = (cancelh omega^(2))/(pi^(2)c^(3))(1)/(c^(homega//k)-1)` (a) In a range `h omega lt lt KT` (long wavelength or high temperature). `u_(omega) rarr (cancel homega^(3))/(pi^(2)c^(3))(1)/((cancelh omega)/(kT))` `= (omega^(2))/(pi^(2)c^(3))kT`, using `e^(x) = 1+x` for small `x`. (B) In the range `cancelh omega gt gtT` (high FREQUENCY ot low temperature): `(cancelh omega)/(kT)gt gt 1` so `e^((cancelh omega)/(kT)) gt gt 1` and `u_(omega) = (cancelhomega^(3))/(pi^(2)c^(3))e^(-cancelh omega//kT)` |
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