# (Statistics) Blackbody spectrum in terms of wavelength?

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1. Mar 26, 2016

### Ryaners

This is a question about transforming a probability distribution, using the blackbody spectrum as an example.

The problem statement, all variables and given/known data

An opaque, non-reflective body in thermal equilibrium emits blackbody radiation. The spectrum of this radiation is governed by B(f) = af3 / (ebf−1) , where a and b are constants and f is the frequency of the emitted light. Work out the corresponding distribution of wavelengths B(λ) using f = c/λ .

The attempt at a solution
I tried substituting f = c/λ into the given equation - which seemed like a good place to start - and came out with the following:
ac3 / λ3(ebc/λ - 1).
Then when I looked up the actual spectrum in terms of wavelength (on HyperPhysics, here) it gives something else.
This is the HyperPhysics formula:
8πhc / λ5 ⋅ 1 / (ehc/λkT - 1)
Comparing the constants in the frequency spectrum on HPhys with a & b in the formula in the question:
a = 8πh / c3
b = h / kT
Which then turns my answer of ac3 / λ3(ebc/λ - 1) into:
8πh / c3 ⋅ c3 / λ3(ehc/λkT - 1)
= 8πh / λ3 ⋅ 1 / (ehc/λkT - 1)

It looks like I'm out by a factor of c / λ2 compared to the 'real' wavelength spectrum - have I approached this the wrong way, or has the question just simplified the given formula in some way?

2. Mar 26, 2016

### SammyS

Staff Emeritus
As a hint:

If f = c/λ, then what is the derivative, df/dλ ?

3. Mar 26, 2016

### Ryaners

Ok, that's definitely got me thinking now :) I did some digging and found this formula for performing transformations of variables:
f(x) = g(y)⋅dy/dx
where I take it f(x) is the original form of the function, and g(y) is the function in the desired 'new' terms.

So in this case that would be B(f) = B(λ)⋅df/dλ... right? ***
⇒ B(λ) = B(f) / (df/dλ)

And as f = c/λ = c⋅λ-1 ⇒ df/dλ = -c/λ2 (is that correct?)

⇒ B(λ) = ac3 / λ3(ebc/λ - 1) ⋅ -(λ2/c)
= - ac2 / λ(ebc/λ - 1)

I've definitely gone wrong there, haven't I? I can see that multiplying B(f) [= B(c/λ)] by df/dλ would get me to the formula I found on HPhys, but with a minus sign...?!

*** Edit *** Hang on, I can see I got dλ/df the wrong way around in B(f) = B(λ)⋅dλ/df above, which gives me:
B(λ) = ac3 / λ3(ebc/λ - 1) ⋅ -(c / λ2)
= - ac4 / λ5(ebc/λ - 1)
Now the only thing bugging me is that minus sign - what's going on there?

Thanks so much for your comment, it's really pointed me in the right direction!

4. Mar 26, 2016

### SammyS

Staff Emeritus
Suppose λ1 = c/f1, and λ2 = c/f2 .

So if f2 > f1 , then λ2 < λ1 , and you would need to switch the order of the limits of integration, which changes the sign back to positive.

5. Mar 26, 2016

### Ryaners

Ahhhh of course, I forgot about that! Thank you so much, it's such a relief to 'get' that now. :)