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Rayleigh-Jeans approxomation

by the riddick25
Tags: blackbody, planck, rayleigh
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the riddick25
#1
Jan27-10, 05:23 PM
P: 29
i have a quick question, which hopefully someone can straighten out for me.

i was reading this http://en.wikipedia.org/wiki/Rayleigh-Jeans_law
at the part titled 'Consistency of frequency and wavelength dependent expressions'

i follow the logic, but am confused about where the - sign goes after the differentiation.
they differentiate, and get the negative, but when this is multiplied by the frequency dependent form, it disappears.
and i now know this cannot be a typo, because i have found other references which also have the answer as a positive.

could someone please explain where this negative goes, and for what reason.

thanks :)
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cepheid
#2
Jan27-10, 06:58 PM
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The negative sign arises because wavelength and frequency are inversely related, so that an increase in one quantity corresponds to a decrease in the other quantity. However, the total power (per unit solid angle per unit area) contained in the frequency interval bounded by ν1 and ν2 is the same regardless of whether you measure the change in frequency as "going from" ν1 to ν2 or as "going from" ν2 to ν1. In other words, the "change in" frequency (or wavelength) is not really relevant, only the width of the interval is. We don't care about the direction in which you integrate. I suppose they could have written

|Bνdν| = |Bλdλ|

or something like that, in order to make things explicitly clear.
the riddick25
#3
Jan28-10, 03:25 AM
P: 29
Thank you!

I couldn't find anywhere which told me that, they either just had the final equation, or just did the step without explaining why.

thanks again :)


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