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Wavelength distribution

Physics Asked by A R on February 26, 2021

I really need some help on this problem, as I seem to be extremely close to the answer, but am only differing in one spot. Here is the problem:

The function $u(f, T)$, where $$u(f, T)=frac{8pi hf^3}{c^3}frac{1}{e^{hf/kT}-1}$$ is the distribution of blackbody radiation in terms of frequency $f$ and temperature $T$; $u(f, t), df$ is the energy contained in the frequency interval from $f$ to $f+df$. Use the relation between frequency and wavelength to find the function $Upsilon(lambda, T)$ that describes the distribution in wavelength; $Upsilon(lambda, T), dlambda$ is the energy contained in a wavelength interval from $lambda$ to $lambda+dlambda$.

I seem to almost get the right answer, as I get $$Upsilon(lambda, T)=frac{8pi hc}{lambda^5}frac{1}{e^{h c/lambda kT}-1}$$However, the book says the correct answer is
$$Upsilon(lambda, T)=frac{8pi hc}{lambda^5}frac{1}{e^{hbar c/lambda kT}-1}$$ where $hbar$ is in the exponent instead of $h$. I simply used the absolute value of the differential $$df=frac{c}{lambda^2}, dlambda$$ to substitute for $df$ in $u(f, t), df$. Can someone please help me on where the $hbar$ may come from, and if my reasoning may be flawed?

One Answer

Looks like a typo in the book.

Answered by akhmeteli on February 26, 2021

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