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Hellogiraffe

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## Homework Statement

Using Wien's law, show the following:

(a) If the spectral distribution function of black body radiation, ρ(λ,T), is known at one temperature, then it can be obtained at any temperature (so that a single curve can be used to represent black body radiation at all temperatures).

(b) The total emissive power is given by R=[itex]\sigma[/itex][itex]T^{4}[/itex] (the Stefan-Boltzmann law), where [itex]\sigma[/itex] is a constant.

(c) The wavelength [itex]λ_{max}[/itex] at which ρ(λ,T) -- or R(λ,T) -- has its maximum is such that [itex]λ_{max}[/itex]T=b (Wien's displacement law), where b is a constant.

## Homework Equations

1) Wien's law: ρ(λ,T) = f(λT)/[itex]λ^{5}[/itex] with f being a function of the product of λT

2) ρ(λ,T) = (4/c)*R(λ,T)

3) R(T) = [itex]\sigma[/itex][itex]T^{4}[/itex]

4) [itex]λ_{max}[/itex]T=b

## The Attempt at a Solution

So I mostly know how to derive Wien's Law from Planck's, but this question is asking to derive portions of his work without using Planck's. I got part (b) by integrating Eqn 1 from my list of relevant equations using a substitution for λT. However, I'm not even fully sure of what part (a) is even asking and where to start. For part (c), I'm assuming I'll need to take one of the forms of the functions and set its derivative (with respect to λ) equal to 0, but after substituting all different forms of the equations that I know, I cannot quite find anything that will give me what I'm looking for. I finally found another form of Wien's Law on the internet:

ρ(λ,T) = [itex]\frac{hc^{2}}{λ^{5}}[/itex][itex]e^{-hc/(λkT)}[/itex]

This particular equation I was able to differentiate to give me the answer I was looking for, but there is no mention of it in the textbook I am using (Quantum Mechanics by Bransden and Joachain). If I need to, I'll just use this equation anyway, though some kind of explanation/derivation of where this came (not using Planck) from would be very helpful.