6. This problem will explore Blackbody radiation and its relationship

Question

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.