Derive Wien's Law from Plancks Law

[Mosaness]

1. Derive Wien's displacement law from the Planck Spectrum.
2. Planck's Law: [itex]\frac{2hv³}{c²(e^{\frac{hv}{kt}}-1)}[/itex]

Where v = frequency;
c = speed of light;
h = Plank's constant
k = Boltzmann's constant

The Attempt at a Solution

Well, the first thing I wanted to confirm was that this was in fact the correct equation which I was going to derive.

My first attempt would be to try and simplify this equation as much as I can, by plugging in c, and h. And because this gives out an incredibly small number, it can be disregarded as having not too big of an effect and can be represented by a 1, giving rise to:

[itex]\frac{v³}{(e^{\frac{hv}{kt}}-1)}[/itex]

 

[camjohn]

You might want to try re-typing this question. Some of the input didn't quite translate into mathematical notation.

 

[ehild]

1. Derive Wien's displacement law from the Planck Spectrum.



2. Planck's Law: $\frac{2hv^3}{c^2(e^{\frac{hv}{kt}} -1)}$

Where v = frequency;
c = speed of light;
h = Plank's constant
k = Boltzmann's constant

The Attempt at a Solution

Well, the first thing I wanted to confirm was that this was in fact the correct equation which I was going to derive.

My first attempt would be to try and simplify this equation as much as I can, by plugging in c, and h. And because this gives out an incredibly small number, it can be disregarded as having not too big of an effect and can be represented by a 1, giving rise to:

$\frac{v^3}{(e^{]\frac{hv}{kt}}-1)}$

A law states something so you should write Plank's Law as $$B(v,T)=\frac{2hv^3}{c^2(e^{\frac{hv}{kt}} -1)}$$ where B_v(T) is the spectral radiance at frequency v and temperature T.
You want to find the position of maximum in terms of frequency. The constant 2h/c² can be omitted, it does not influence the position of maximum.

But Wien's Displacement Law is a relation between wavelength and temperature http://hyperphysics.phy-astr.gsu.edu/hbase/wien.html so you should work with Planck's Law in terms of wavelength. http://en.wikipedia.org/wiki/Planck's_law

ehild

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