Derive Stefan-Boltzmann Law from Planck Distribution for blackbody radiation

[omegas]

Homework Statement

Starting with the Planck distribution R(\lambda,T) for blackbody radiation.

(a) Derive the blackbody Stefan-Boltzmann law (ie total flux is proportional to T⁴) by integrating the above expression over all wavelengths. Thus show that
R(T) = (2\pi⁵k⁴)T⁴ / (15h³c²

(b) Show that the maximum value of R(\lambda,T) occurs for \lambda_mT = 2.898 * 10^-3 mK (this is called Wien's displacement law).

(c) Use Wien's displacement law to determine for the cosmic background radiation with T = 2.7 K
(i) the value of \lambda_m for peak intensity
(ii) the energy in eV of photons at this peak intensity, and
(iii) the region of the electromagnetic spectrum corresponding to the peak intesnsity.

Homework Equations

Planck distribution for blackbody radiation:

R(\lambda,T) = (c/4)(8\pi/\lambda⁴[(hc/\lambda)1/e^hc/\lambdakT-1]

The Attempt at a Solution

That was a mouthful. Help please.

 

[lanedance]

here's how to write it in tex (click on it to see how)
R(\lamda,T) <br /> = \frac{c}{4} \frac{8 \pi}{\lambda^4} <br /> (\frac{hc}{\lambda})<br /> (\frac{1}{e^{hc/(\lambda kT)}-1})<br />
ps - check i got it correct

now - how about simplifying and trying the intergal?

 

[zachzach]

Homework Statement

Starting with the Planck distribution R(\lambda,T) for blackbody radiation.

(a) Derive the blackbody Stefan-Boltzmann law (ie total flux is proportional to T⁴) by integrating the above expression over all wavelengths. Thus show that
R(T) = (2\pi⁵k⁴)T⁴ / (15h³c²

(b) Show that the maximum value of R(\lambda,T) occurs for \lambda_mT = 2.898 * 10^-3 mK (this is called Wien's displacement law).

(c) Use Wien's displacement law to determine for the cosmic background radiation with T = 2.7 K
(i) the value of \lambda_m for peak intensity
(ii) the energy in eV of photons at this peak intensity, and
(iii) the region of the electromagnetic spectrum corresponding to the peak intesnsity.

Homework Equations

Planck distribution for blackbody radiation:

R(\lambda,T) = (c/4)(8\pi/\lambda⁴[(hc/\lambda)1/e^hc/\lambdakT-1]

The Attempt at a Solution

That was a mouthful. Help please.

(a) Simplify the expression and integrate it with respect to \lambda over all wavelengths (so the minimum wavelength is 0, what is the maximum wavelength?). You are going to have to make a substitution to get it in a certain form of an integral you can look up.

(b) How do you find the maximum of a function?