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Plancks law: Relationship between peak wavelength and peak frequency

  1. Oct 9, 2012 #1
    1. The problem statement, all variables and given/known data

    What is the relationship between vmax and λmax and why isn't it just vmaxλmax = c

    2. Relevant equations

    vλ = c

    vmax = 5.88 x 1010T s-1 K-1

    λmax = 0.290T-1 cm K


    3. The attempt at a solution

    I worked out that Bλ = -dv/dλ Bv

    where -dv/dλ = c/λ2

    So I thought that λmax = -dv/dλ vmax but that doesn't seem to work.

    Any idea what I have missed? I have been playing around with various rearrangements for a while . . . do I need to do some more calculus?
     
  2. jcsd
  3. Oct 9, 2012 #2
    How do you determine the peak wavelength and the peak frequency? Are the two conditions satisfied simultaneously?
     
  4. Oct 9, 2012 #3

    rude man

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    Where did you dig this up?
    This is correct. And vmax = c/λmax unless n > 1 (refraction index).
     
    Last edited: Oct 9, 2012
  5. Oct 12, 2012 #4
    Hi and thanks for your reply - I've been working on another problem but I am back on this one now!

    Peak wavelength occurs when dBv / dv = 0 and I already showed that vmax = 5.88 x 1010T s-1 K-1 in part one of the question (taken straight from the lecture notes).

    Peak frequency occurs when dBλ / dλ = 0 and in the question I am given the expression λmax = 0.290T-1 cm K as the solution to this.

    So aren't these two conditions satisfied when

    dBv / dv = dBλ / dλ = 0

    Is that the same as saying

    5.88 x 1010T s-1 K-1 = 0.290T-1 cm K

    I am more confused than ever now as I am pretty sure that makes no sense!
     
  6. Oct 12, 2012 #5
    Hi rude man and thanks for your reply. The thing I dug up is from my lecture notes and in the question paper, we work in cgs units in this class, which might be why you don't recognise it?

    Hmmm, how can vmax = c/λmax if my question is "Show that vmaxλmax ≠ c ? I don't get it . . .
     
  7. Oct 12, 2012 #6

    rude man

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    That's a good question, and my only answer would be that the index of refraction of the medium is not 1. Example: if you're looking at the radiation from inside a swimming pool, the frequency would be the same as in air but the wavelength would be reduced to c/nf, f = freq in Hz. Water has n = 1.3 or something like that.
     
  8. Oct 12, 2012 #7

    rude man

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    No, of course not. You don't want to equate a frequency with a wavelength!

    What should be true is that λ = c/f or 0.29T-1 cm-K = c/5.88e10T s-1K-1 where c =3e10 cm s-1. But it isn't ...

    I believe some kind of semantic confusion is at the heart of this, as dickfore suggests. If the issue is n > 1 then I calculate n = 1.76.
     
  9. Oct 12, 2012 #8

    TSny

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    Sub second relation into first to get λ2Bλ = c Bv

    So, does Bv have its max when Bλ has its max? Or, does Bv have its max when λ2Bλ has its max?
     
  10. Oct 12, 2012 #9

    rude man

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    OK, time for me to eat crow again.

    Here's the story: the Planck radiation law may be written either as a function of wavelength or as a function of frequency. If you take the wavelength formulation you get λmax = 0.29T-1 cm-K whereas if you start with the frequency expression you get hfmax /kT = 2.821439 which corresponds to fmax = 5.88e10T s-1 K-1.

    As to why these don't agree, the answer is in the formulation of the two versions of the law.
    One is not directly derived from the other by λf = c. In fact, as I see it, they contradict each other and I assume both are models of the real thing and therefore inaccurate. Good question for your prof!

    If you want to know whichever version of the Planck radiation law you don't have, they can both be found on the Web.

    Ya live and learn!
     
  11. Oct 13, 2012 #10
    I showed in my first post that I worked out how to go from one form to the other. They are not "models" and they are both accurate.
     
  12. Oct 13, 2012 #11
    Hi TSny and thanks for your reply.

    I think I want to show that

    dBv / dv = 0 when dλ2 c-1 Bλ /dλ = 0

    However, using a similar substitution method I used to derive dBv /dv = 0, I just end up getting exactly the same result for both expressions! So that equation is true, which again makes me think that the conversion factor should just be λ2/c but as soon as I put a value in for T and actually compare numbers that doesn't work. What am I missing?
     
  13. Oct 13, 2012 #12

    TSny

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    We have cBv = λ2Bλ where v on the left is related to λ on the right by c = vλ. Bv will take on its maximum value when λ2Bλ has its maximum value. So, the value of λ that corresponds to the value of vmax (where Bv takes on its max) would be obtained from Bλ by the condition d(λ2Bλ)/dλ = 0. Show that this leads to dBλ/dλ = -(2/λ)Bλ.

    So, the value of λ that corresponds to vmax will not be the value of λ that makes Bλ a maximum, but to the value of λ that makes dBλ/dλ = -(2/λ)Bλ. So, dBλ/dλ will be negative at this point, which means that λ will be on the "downhill" side of the peak of Bλ. In other words, the value of λ that corresponds to vmax will be greater than λmax (where dBλ/dλ = 0).
     
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