Understanding the Rayleigh-Jeans Approximation: Where Does the Negative Sign Go?

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The discussion clarifies the confusion surrounding the negative sign in the Rayleigh-Jeans approximation related to frequency and wavelength differentiation. The negative sign appears due to the inverse relationship between wavelength and frequency, where an increase in one results in a decrease in the other. However, the total power in a frequency interval remains constant regardless of the direction of measurement, meaning the change in frequency or wavelength is not significant. The focus should be on the width of the interval rather than the direction of integration. This explanation emphasizes the importance of understanding the underlying relationships in the equations rather than just the final results.
the riddick25
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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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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.
 
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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