Equating Units in Plank's Blackbody Equation

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SUMMARY

The discussion focuses on the necessity of including the differential element dλ when substituting frequency (ν) with wavelength (λ) in Plank's Blackbody Equation. It is established that ε_B is a density function, which requires careful handling of units. The units of ε_B(λ,T)dλ and ε_B(ν,T)dν are equivalent, allowing for their equating, but a direct substitution without the differential element is incorrect. This highlights the importance of maintaining unit consistency in physical equations.

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My question is, why, when changing the variable from "v" to "lambda", does the d\lambda need to be included? Why can't a simple substitution c = v * lambda do the trick?
 
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Because \varepsilon_B is a density. If it's in terms of frequency, its units are something per unit frequency. Similarly, if it's in terms of wavelength, its units are something per unit wavelength. So you can't simply equate the functions. But the units of \varepsilon_B(\lambda,T)\,d\lambda and \varepsilon_B(\nu,T)\,d\nu are the same, so you can equate those.
 

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