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I have just picked up the book Elementary Quantum Mechanics by Peter Fong, and on the first page he writes Planck's law as $$u(\nu)d\nu=\frac{8\pi h \nu^3}{c^3}\frac{1}{e^{h\nu/kT}-1}d\nu$$ where $u(\nu)$ is the "energy per unit volume of the blackbody radiation in the frequency range from $\nu$ to $\nu + d\nu.$ This statement of the law differs from it statement on Wikipedia by a factor of $4\pi/c.$

My two questions are: First, what is the reason for this difference? I have checked the units for $h$ and $k$ and they seem to be the same as those used on Wikipedia, so I can't think of another reason for the difference in the constants. Second, what does $d\nu$ mean here, and how does $u(\nu)d\nu$ equal $B(\lambda,T)?$ (is the latter just the limit as $d\nu\to 0?$)

For background, I am a first-year Math Ph.D. student, however, I have mostly focused on algebra/discrete math, and my calculus is embarrassingly rusty. I did take AP Physics A and B in high school (my classes were algebra-based), but none in college, to my regret. I am mostly looking at this book because of personal curiosity, plus trying to refresh my calc skills while learning something new.

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    $\begingroup$ You're conflating energy density $u=4\pi B/c$ with spectral radiance $B$. Also, a $d\nu$ factor should appear on both sides of the equation or neither, not just the left. (If Fong did it that way, it was a misprint.) $\endgroup$
    – J.G.
    Commented Mar 31, 2022 at 17:46
  • $\begingroup$ @J.G. Ahhh, that explains it! Thx! Mind if I ask, what does the dv mean here then? $\endgroup$
    – Conan G.
    Commented Mar 31, 2022 at 17:49
  • $\begingroup$ @ConanG See edit. $\endgroup$
    – J.G.
    Commented Mar 31, 2022 at 17:50
  • $\begingroup$ @J.G. It does, that was just my mistype $\endgroup$
    – Conan G.
    Commented Mar 31, 2022 at 17:52

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