Chapter3_13 - at a given temperature T is given by ∞ →...

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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 ∞ → = = ∫ ∫ ∞ − ∞ 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 “ultraviolet catastrophe”! 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 λ → 0. Black Body Radiation: Y(x) versus x 0.002 0.004 0.006 0.008 0.01 x = λΤ (in m K) Y(x) 0.002898 Experimental Data Classical Theory Maxwell-Boltzmann...
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