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(Statistics) Blackbody spectrum in terms of wavelength?

  1. Mar 26, 2016 #1
    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. jcsd
  3. Mar 26, 2016 #2

    SammyS

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    As a hint:

    If f = c/λ, then what is the derivative, df/dλ ?
     
  4. Mar 26, 2016 #3
    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? :oldfrown: 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!
     
  5. Mar 26, 2016 #4

    SammyS

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    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.
     
  6. Mar 26, 2016 #5
    Ahhhh of course, I forgot about that! Thank you so much, it's such a relief to 'get' that now. :)
     
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