Transcription of Chapter 10
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Chapter 10
Chapter 10 The Derivation ofthe planck FormulaTopicsThe planck formula for black-body radiation. Revision of waves in a box. Radiationin thermal equilibrium. The equipartition theorem and the ultraviolet photoelectric effect. The wave-particle duality. Quantisation of radiation andthe derivation of the planck spectrum. The Stefan-Boltzmann IntroductionIn the first lecture, we stated that the energy den-sity of radiation per unit frequency intervalu( )for black-body radiation is described by thePlanckformula(Figure ),u( ) d =8 h 3c31(eh /kT 1)d ( )wherePlanck s constant,h= 10 34J this lecture, we demonstrate why quantum con-cepts are necessary to account for this formula. Theprogramme to derive this formula is as follows. First, we consider the properties of waves ina box and work out an expression for the ra-diation spectrum in thermal equilibrium Spectrum of black body radiationf(x) dx x3ex 1dxin terms of the dimensionless 1 2 3 4 5 6 7 8 9 (x).
The Derivation of the Planck Formula 3 is sin!t, where! = 2…” is the angular frequency of the wave. Therefore, the expression for the wave is A(r;t) = A0 sin(kr ¡!t); and the speed of the wave is c =!=k. 10.3 Electromagnetic Modes in a Box
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