Chapter3_13

# Chapter3_13 - at a given temperature T is given by β β...

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PHY3063 R. D. Field Department of Physics Chapter3_13.doc University of Florida The “Ultraviolet Catastrophe” Rayleigh-Jeans Theory: The classical prediction for ρ (T,f) consists the following two pieces: ( ) > < × = E V df f N df f T ) ( ) , ( ρ . (energy/volume) = (number of modes/volume) × (ave energy/mode) where df c Vf df f N 3 2 8 ) ( π = and <E> = kT . Thus, ( ) 3 2 3 2 8 8 ) , ( c kT f kT c f f T π π ρ = × = . Also, 4 2 8 ) , ( ) , ( λ π ρ λ λ ρ kT f T c T = = so that 4 4 5 8 ) ( 8 ) ( ) ( x k T k T x Y T classical π λ π λ ρ = = = where x = λ T . At any fixed temperature T , the classical theory agrees at large wavelength. However, the data goes to zero at small x ( i.e. short wavelength), while the classical prediction goes to infinity! Furthermore, the total energy in the cavity (
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Unformatted text preview: ) at a given temperature T is given by β β = = β« β« β β β 4 8 ) , ( ) ( d kT d T T U . Classical theory predicts an infinite energy when integrated over all wavelengths! The is refered to as the βultraviolet catastropheβ! The problem is that every mode is assigned an average energy of kT ( independent of the wavelength ) and the number of modes becomes infinitely large as Ξ» β 0. Black Body Radiation: Y(x) versus x 0.002 0.004 0.006 0.008 0.01 x = Ξ»Ξ€ (in m K) Y(x) 0.002898 Experimental Data Classical Theory Maxwell-Boltzmann...
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