Are these two writings of Planck's Law the same thing?

Question

I checked one of the other questions on this - and I still seem to have a different equation than they offer (as far as I can tell). I'll use the notation the books used, btw.

In one of my reference books, they first derived the Rayleigh-Jeans formula (I can include how they derived it, if that will help):

$$u(v,T) = \frac{8\pi v^2}{c^3}k_BT,$$

($k_B$ - Boltzmann constant, $v$ - frequency, $T$ - temperature)

Then they "re-derived" Planck's formula by replacing the integral with the summation and ended up with:

$$u(v,T) = \frac{8\pi v^2}{c^3} \frac{hv}{e^{\frac{hv}{kT}}-1}$$

($h$ - Planck constant)

However, the next book I looked at puts Planck's Law in terms of $\lambda$:

$$u(\lambda) = \frac{8 \pi hc\lambda^{-5}}{e^{\frac{hc}{\lambda kT}}-1} = \frac{8\pi}{\lambda^5}\frac{hc}{e^{\frac{hc}{\lambda kT}}-1}$$

By the way, the Rayleigh-Jeans law the second book derived is:

$$u(\lambda) = \frac{8\pi kT}{\lambda^{-4}}$$

My questions are:

If both functions are valid, why are only one of them mentioned in each book? Why can't I find the function with $v^2$ on Wiki, or PSE... is the first book wrong?

..the one I linked... did you read the post ? I even ask why there's a difference between the equations in mine and that post. — galois, Aug 30 '15 at 18:29
I don't see the difference. Your first formula is the Rayleigh-Jeans law. There is no formula with $\nu^2$ for the Planck law in your post, there's one with $\nu^2$ and then $h\nu$ after it, which is the $\nu^3$ depenndence you see for frequency everywhere else. — ACuriousMind, Aug 30 '15 at 18:35
By the way, that is Greek $\nu$ (nu) for frequency, not $v$. — , Aug 30 '15 at 19:20

score 1 · Accepted Answer · answered Aug 30 '15 at 18:34

Both expressions are equivalent; they are simply calculating slightly different quantities. The first one gives the energy density per unit frequency, and the second one gives the energy density per unit wavelength. These two differ by a square of the frequency / wavelength because one unit of frequency represents a larger range of wavelengths at low frequency than at high frequency. If you do the change of variables correctly you can go back and forth between the two.

Are these two writings of Planck's Law the same thing?

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