Tricky Derivation of Blackbody Equations.

[omegas]

Homework Statement

Starting with the Planck distribution:

$$R(\lambda,T) = \frac{c}{4} \frac{8 \pi}{\lambda^4} (\frac{hc}{\lambda})(\frac{1}{e^{hc/(\lambda kT)}-1})$$

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) = $$\frac{2 \pi^5 k^4}{15h^3 c^2} T^4$$

and $$\int$$ $$\frac{x^3}{e^x -1} dx = \frac{\pi^4}{15}$$

The Attempt at a Solution

I know I need to substitute $$x = \frac{hc}{kT} \frac{1}{\lambda}$$. And somehow I think I can use the form KR($$\lambda$$,T) = A($$\lambda$$)B($$\lambda$$)

 

[zachzach]

Homework Statement

Starting with the Planck distribution:

$$R(\lambda,T) = \frac{c}{4} \frac{8 \pi}{\lambda^4} (\frac{hc}{\lambda})(\frac{1}{e^{hc/(\lambda kT)}-1})$$

Derive the blackbody Stefan-Boltzmann law (ie total flux is proportional to T⁴) by integrating the above expression over all wavelengths. Thus show that

$$R(T) = \frac{2 \pi^5 k^4}{15h^3 c^2} T^4$$

and $$\int$$ $$\frac{x^3}{e^x -1} dx = \frac{\pi^4}{15}$$

The Attempt at a Solution

I know I need to substitute $$x = \frac{hc}{kT} \frac{1}{\lambda}$$. And somehow I think I can use the form KR($$\lambda$$,T) = A($$\lambda$$)B($$\lambda$$)

Just use that substitution and try to get it into the form of the integral provided.