Question

(1) Working with Planck's Law [45 pts] Planck's Law describes the intensity (energy per time per area per frequency per angular area) of radiation from a homogeneous, isothermal source: dE 2hc21 Integrating this relation over a spherical hemisphere (outward only) yields the flux density of blackbody radiation from a surface: dE (a) [10 pts] Integrate the above relation over all wavelengths (0 < λ < 00) to derive the total energy flux from a blackbody (the Stefan-Boltzmann Law) dE dtdA Jo

Answer 1

dE/dt dA = integral^infinity_0 pi B_lambda (T) d lambda = integral^infinity_0 pi 2 hc^2/lambda^5 1/e^hc/lambda KT - 1 d lambda Put hc/lambda KT = x = > lambda = hc/x KT = > d lambda = -hc/x^2 KT dx dE/dt dA = integral^infinity_0 pi 2 hc^2 x^5 k^5 T^5/h^5 c^5 1/e^x - 1 hc/x^2 KT dx = integral^infinity_0 2 pi k^4 T^4/h^3 C^2 x^3/e^x - 1 dx 2 pi k^4 T^4/h^3 c^2 pi^4/15 = 2 pi^5 k^4/15 c^2 h^3 T^4 sigma T^4 where sigma = 2 pi^5 k^4/15 c^2 h^3