hw5 - PHYS126 Spring 2011 Homework 5 Due March 11 beginning...

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PHYS126 Spring 2011 Homework 5, Due March 11, beginning of lecture. In the following, h = Planck’s constant; c = Speed of light; B k = Boltzmann constant. In lecture, we described the Planck distribution function for blackbody radiation using the frequency ν as variable. If, instead, we use the wavelength λ , the Planck distribution is given by: 2 / 5 2 1 1 B hc k T hc I e λ λ π λ = - The Rayleigh-Jeans distribution function (which is based on classical idea and turned out to be incorrect) is 4 2 B ck T I λ π λ = 1. (a) Show that for long wavelengths, Planck’s function can be approximated as Rayleigh-Jeans function. (Hint: Use Taylor expansion for x e ). (b) Sketch the two distribution functions on the same graph. 2. You should notice that Planck’s curve in 1(b) has a maximum peak. Let max λ be the wavelength correspond to this peak. In this question, you will have to express max λ as a function of T . (a) Consider a function y similar to Planck function. Define 5 1 x x y e = - , where B hc x k T λ = . Suppose max x x = when y is a maximum. By differentiation, show that
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