Question

1. Blackbody radiation ranges over the entire spectrum 0 < λ < ∞. According to Plank’s law, a blackbody (the Sun, our bodies, incandescent light bulb filaments) with a surface temperature T, emit electromagnetic radiation with intensity

I (T) = h c² ∫ 1/[λ⁵ (e^W -1) dl

where W = h c/λ k T. Here k is Boltzmann’s constant, 1.38(10)^-23 J/K and h is Plank’s constant. The integrand as a function of λ is well approximated by the graph f(λ) = λ e^-λ/5for λ > 0. a) Verify that the units of I(T) are watts/m² hint: dλ has units. b) The change of variables u = λ T gives λ = u/T and dλ = du/T. Carry out this substitution to obtain an I(T) as a product of h c² times T⁴ times an integral involving u, h, c and k. Evaluating this integral numerically (don’t even think about it!) yields the Stephan Boltzmann law for blackbody radiation. c) Up to a multiplicative constant, the above integrand is equal to the function

φ(W) = W⁵/(e^W-1)

Differentiate φ(W) and set the derivative equal to 0. Using a graphing calculator solve (approximately) the resulting equation for the maximum value W_max. Since W = h c/λ k T we have λ_max T = h c/k W_max. Compute h c/k W_max to obtain Wein’s law for the maximum frequency of a blackbody that has a temperature T.

d) As an application of c), find your own λ_max given a surface body temperature 300 K.

Answer 1