6. This problem will explore Blackbody radiation and its relationship with thermodynamics. As you may remember form class, Planck's law for blackbody radiation is: 8AV² hv 8 TV² (E)= c³ hv/kgI -1 u(v, T) = - The first term (²) represents the density of electromagnetic waves (harmonic oscillators) in the cavity and the second term, (E), represents the average energy of per oscillator. 1 1-e* a. Show how quantization of energy gives rise to Planck's Law. Hint: Start with quantized harmonic oscillators of the form Ennhv and calculate the average energy using the Boltzmann equation. You will need the following equations: Σex 00 Σ ne = d dx 71-0 nx 1 = -√√1-e²) d et (1-e² b. Show that at a given temperature there will be a most probable wavelength given by: Amax T = 2.898 x 10³ m/K (This is known as Wien's law and determines the color of a black body at a given temperature). c. Show that in the high and low temperature limits the Plank distribution becomes the Rayleigh- Jeans u(v, I) = 8²k₂I and Wien Lawsu(v, I) = 8 exp(r) respectively.

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