(adsbygoogle = window.adsbygoogle || []).push({}); Obtaining the Rayleigh-Jeans formula - what am I doing wrong??

1. The problem statement, all variables and given/known data

Spectral energy density = [tex] u(\nu, T) = {\[\displaystyle \,8\,\pi \,h{\nu}^{3}{c}^{-3} \left( {{\rm e}^{{\frac {h\nu}{kT}}}}-1 \right) ^{-1}\]} [/tex]

where h is Planck's constant and k is Bolzmann's constant.

Using the relation [tex] \lambda = \frac{c}{\nu} [/tex] express the spectral energy density as a function of the wavelength and determine the energy du of a blackbody radiation per unit volume, in a narrow range of wavelength [tex] \lambda + d\lambda [/tex]. Using an expansion of the exponential factor obtain the Rayleigh-Jeans formula,

[tex] du = \frac{8 \pi k T}{\lambda^4} d\lambda [/tex]

2. Relevant equations

Given some function [tex] f(x_1, x_2, ..., x_n) [/tex],

[tex] df = \left(\frac{\partial{f}}{\partial{x_1}}\right) dx_1 + ... + \left(\frac{\partial{f}}{\partial{x_n}}\right) dx_n [/tex].

Also, [tex] e^{x} \approx 1+x+\frac{x^2}{2!} + ... [/tex]

3. The attempt at a solution

It seems like it should be easy, but I can't get the equation to come out right. Substituting in lambda, I get

[tex]{\[\displaystyle {\it u}\, = \,8\,\pi \,h{\lambda}^{-3} \left( {{\rm e}^{{\frac {hc}{\lambda\,kT}}}}-1 \right) ^{-1}\]} [/tex]

Then from my equation in part b, we should have that

[tex] du = \frac{\partial{u}}{\partial{\lambda}} d\lambda [/tex]

But I get

[tex] \frac{\partial{u}}{\partial{\lambda}} = {\[\displaystyle \,-8\,\pi \,h \left( 3\,\lambda\,kT{{\rm e}^{{\frac {hc}{\lambda\,kT}}}}-3\,\lambda\,kT-hc{{\rm e}^{{\frac {hc}{\lambda\,kT}}}} \right) {\lambda}^{-5} \left( {{\rm e}^{{\frac {hc}{\lambda\,kT}}}}-1 \right) ^{-2}{k}^{-1}{T}^{-1}\]} [/tex]

Upon using my expansion for the exp. term (expanding around nu = 0, then substituting in nu = c/lambda) the most I can simplify this is to:

[tex] du ={\[\displaystyle \,8\,\pi \, \left( -2\,ckT+{\frac {h{c}^{2}}{\lambda}} \right) {\lambda}^{-3}{c}^{-2}\]}d\lambda [/tex]

so... what am I doing wrong?? It could have something to do with the [tex] \lambda + d\lambda [/tex] (though I just thought that meant I could neglect higher than 2nd order terms in my expansion)...

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# Obtaining the Rayleigh-Jeans formula - what am I doing wrong?

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