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Derive Stefan-Boltzmann Law from Planck Distribution for blackbody radiation

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omegas
#1
Apr27-10, 06:31 AM
P: 9
1. The problem statement, all variables and given/known data
Starting with the Planck distribution R([tex]\lambda[/tex],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[tex]\pi[/tex]5k4)T4 / (15h3c2

(b) Show that the maximum value of R([tex]\lambda[/tex],T) occurs for [tex]\lambda[/tex]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 [tex]\lambda[/tex]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.

2. Relevant equations

Planck distribution for blackbody radiation:

R([tex]\lambda[/tex],T) = (c/4)(8[tex]\pi[/tex]/[tex]\lambda[/tex]4[(hc/[tex]\lambda[/tex])1/ehc/[tex]\lambda[/tex]kT-1]


3. The attempt at a solution
That was a mouthful. Help please.
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lanedance
#2
Apr28-10, 01:11 AM
HW Helper
P: 3,307
here's how to write it in tex (click on it to see how)
[tex] R(\lamda,T)
= \frac{c}{4} \frac{8 \pi}{\lambda^4}
(\frac{hc}{\lambda})
(\frac{1}{e^{hc/(\lambda kT)}-1})
[/tex]
ps - check i got it correct

now - how about simplifying and trying the intergal?
zachzach
#3
Apr28-10, 02:37 AM
P: 258
Quote Quote by omegas View Post
1. The problem statement, all variables and given/known data
Starting with the Planck distribution R([tex]\lambda[/tex],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[tex]\pi[/tex]5k4)T4 / (15h3c2

(b) Show that the maximum value of R([tex]\lambda[/tex],T) occurs for [tex]\lambda[/tex]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 [tex]\lambda[/tex]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.

2. Relevant equations

Planck distribution for blackbody radiation:

R([tex]\lambda[/tex],T) = (c/4)(8[tex]\pi[/tex]/[tex]\lambda[/tex]4[(hc/[tex]\lambda[/tex])1/ehc/[tex]\lambda[/tex]kT-1]


3. The attempt at a solution
That was a mouthful. Help please.
(a) Simplify the expression and integrate it with respect to [tex] \lambda [/tex] 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?


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