Integrate Plancks Function to find Stefans Law

[DODGEVIPER13]

Homework Statement

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Homework Equations

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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.

 

[voko]

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.

They essentially do $u = a\lambda$, where $a$ is that mix of constants. Express $\lambda = u/a$ and $d\lambda = du/a$, and see what you get, then convert a to that constant mix.

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).

No, it's not. Observe that $\frac 1 {1 - e^{-x}} = 1 + e^{-x} + e^{-2x} + ...$. That is, the first term of that is just 1. But when they integrate, the first term is $e^{-x}$, because they multiply the series by $e^{-x}$

Then they do this derivative deal and get 1/a but I think it should be -1/a(e^-ax).

This is the indefinite integral. Now plug in the integration limits - what do you get?

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 .

They have found that all the integrals they need to compute are $\frac 6 {n^4}$.

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.

$\sum_{n = 0}^{\infty}\frac 1 {n^4} = \frac {\pi^4} {90}$. This is known because on the left you haven the value of $\zeta$-function at 4, which can be computed in a number of ways.

 

[DODGEVIPER13]

Sweet thanks man you cleared up all my issues I get it now.