1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Integrate Plancks Function to find Stefans Law

  1. Sep 3, 2012 #1
    1. The problem statement, all variables and given/known data
    See uploaded file

    2. Relevant equations
    See Uploaded file

    3. The attempt at a solution
    Ok problem 1 I have with the solution why is (KT/HC)^4 on the outside of the integral and why is it to the 4th power. I assume it is to find (lamda)^5 but I would think if that was the case it would be to the 5th power. My second issue is when I apply the series formula 1/(1-epsilon) and make it 1/(1-e^-x) I end up x^3e^-nx but it appears to be missing another e^(-x). Then they do this derivative deal and get 1/a but I think it should be -1/a(e^-ax). Anyways they continue on this path until they get 6/a^4 which is the answer to the series but why do they stop there I am confused . Then how did the get (pi)^4/90 in there and how do I get the actual answer? Sorry I know its a ton of questions but I am hopelessly lost.

    Attached Files:

  2. jcsd
  3. Sep 4, 2012 #2
    They essentially do [itex]u = a\lambda[/itex], where [itex]a[/itex] is that mix of constants. Express [itex]\lambda = u/a[/itex] and [itex]d\lambda = du/a[/itex], and see what you get, then convert a to that constant mix.

    No, it's not. Observe that [itex]\frac 1 {1 - e^{-x}} = 1 + e^{-x} + e^{-2x} + ... [/itex]. That is, the first term of that is just 1. But when they integrate, the first term is [itex]e^{-x}[/itex], because they multiply the series by [itex]e^{-x}[/itex]
    This is the indefinite integral. Now plug in the integration limits - what do you get?
    They have found that all the integrals they need to compute are [itex]\frac 6 {n^4}[/itex].

    [itex]\sum_{n = 0}^{\infty}\frac 1 {n^4} = \frac {\pi^4} {90}[/itex]. This is known because on the left you haven the value of [itex]\zeta[/itex]-function at 4, which can be computed in a number of ways.
  4. Sep 4, 2012 #3
    Sweet thanks man you cleared up all my issues I get it now.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook