Derive Stefan-Boltzmann Law from Planck Distribution for blackbody radiation

Use the result from part (a) to find the maximum of R(\lambda,T) and solve for \lambda_mT = 2.898 * 10^-3 mK.(c) Use Wien's displacement law, \lambda_mT = 2.898 * 10^-3 mK, to determine the values requested in parts (i)-(iii). In summary, we start with the Planck distribution R(\lambda,T) for blackbody radiation and use it to derive the blackbody Stefan-Boltzmann law, which shows that the total flux is proportional to T^4. We then use this law to show that the maximum value of R(\lambda,T) occurs for \lambdamT = 2.
  • #1
omegas
9
0

Homework Statement


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.

Homework 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]


The Attempt at a Solution


That was a mouthful. Help please.
 
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  • #2
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?
 
  • #3
omegas said:

Homework Statement


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.

Homework 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]

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?
 

Related to Derive Stefan-Boltzmann Law from Planck Distribution for blackbody radiation

1. What is the Stefan-Boltzmann Law?

The Stefan-Boltzmann Law is a fundamental law in physics that describes the relationship between the temperature and total radiation of a blackbody object. It states that the total energy radiated per unit surface area of a blackbody is proportional to the fourth power of its absolute temperature.

2. What is the Planck Distribution for blackbody radiation?

The Planck Distribution is a mathematical function that describes the spectrum of electromagnetic radiation emitted by a blackbody object at a given temperature. It is based on the quantum theory and provides a more accurate model for blackbody radiation compared to classical physics.

3. How do you derive the Stefan-Boltzmann Law from the Planck Distribution?

To derive the Stefan-Boltzmann Law from the Planck Distribution, one must integrate the Planck function over all wavelengths and then equate it to the Stefan-Boltzmann Law. This involves using mathematical techniques such as substitution and integration by parts.

4. What are the assumptions made in deriving the Stefan-Boltzmann Law from the Planck Distribution?

The derivation assumes that the blackbody is in thermal equilibrium, that it is a perfect absorber and emitter of radiation, and that the radiation is in thermal equilibrium with the walls of the container. It also assumes that the blackbody is a homogeneous and isotropic emitter, and that the radiation is in the form of electromagnetic waves.

5. Why is the Stefan-Boltzmann Law important in physics?

The Stefan-Boltzmann Law is important because it helps us understand the behavior of blackbody objects and their radiation. It has numerous applications in various fields, such as astrophysics, thermodynamics, and climate science. It also serves as a basis for other important laws, such as Wien's displacement law and the Rayleigh-Jeans Law.

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