How Does Integrating Planck's Formula Relate to the Stefan-Boltzmann Law?

[mathlete]

1. Show that the integral over all frequencies of the Planck formula given by U(t) = blah blah (Planck's Formula) gives a result that is of the form (a constant) [Hint: Change variables from f to ] The energy emitted per unit area per unit time, P(T), is proportional to U(T), and thus P(T) is also proportional to as in the Stefan–Boltzmann formula

Don't even know where to start on that one. The hint doesn't help me at all.

 

[dextercioby]

Do you want to prove that
$$U(V,T)\sim VT^{4}$$

Daniel.

 

[mathlete]

Basically that's what we're asked to do. I'm not sure where to start integrating though, and since everything seems to hinge on understanding the hint (which I don't get) I'm stuck.

 

[dextercioby]

Can u prove that the volumic density of electromagnetic energy within a blackbody is proportional to the 4-th power of the temperature??My guess is,no.What formula would you have to use...??(HINT:It bears the name of the German physicist who won the Nobel Prize in 1918)...

Daniel.

 

[vincentchan]

you crazy? almost all textbook for upper division therma/statistics class has this proof... the integral is easy... do you what me to type it here or what? if you textbook doesn't have this integral... go to your school library or google... let me see...
http://farside.ph.utexas.edu/teaching/sm1/lectures/node84.html
this is the second hit in google... (i seached for stefan Boltzmann integral)

 

[dextercioby]

Well,Vincentchan,have you forgotten the policy we have for the homework section?

Besides,the link is useless.It doesn't show a proof for the integral evaluation...

Daniel.

 

[vincentchan]

Well,Vincentchan,have you forgotten the policy we have for the homework section?

Besides,the link is useless.It doesn't show a proof for the integral evaluation...

Daniel.

I remember... that's why I chose this link for him... he has to do the maths himself... but at least he can check if his answer is correct or not...

 

[dextercioby]

I just remembered.He needn't do that integral.He needs to show that the emissivity of a BB is proportional to the 4-th power of the absolute temperature...
BTW:
The integral is:
$$D_{3}=\Gamma(4)\zeta(4)$$

Daniel.

 

[mathlete]

First off, thanks for the responses everyone

I just remembered.He needn't do that integral.He needs to show that the emissivity of a BB is proportional to the 4-th power of the absolute temperature...
BTW:
The integral is:
$$D_{3}=\Gamma(4)\zeta(4)$$

Daniel.

Right, that's what I have to show (and the problem says by integrating Planck's Formula). I'm afraid I don't exactly understand what the formula you gave me is and how I'm supposed to use it.

 

[dextercioby]

That's the value of the integral.First if all u must write the integral in its initial variables (involving physical quantities) and then do an appropriate substitution.

Daniel.


P.S.The link contains the substitution...