How does Planck's radiation law relate to the Stefan-Boltzmann law?

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In summary, the conversation discusses using Planck's radiation law to show that the total energy flow is given by Stefan-Boltzmann law, J=@T^4, and provides hints on how to approach the integration required for this proof. The conversation also discusses the need to evaluate the integral for the proportionality constant.
  • #1
Eivind
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Hi,
I have been thinking at this for quite a while now and I just can`t figure it out.
1)Use Plancks radiation law to show that the total energy flow is given by Stefan-Boltzmann law, J=@T^4, where @ is a constant.
Plancks radiation law:I(v,T)=(2piv^2/c^2)(hv/e^(hv/kT)-1)
I don't know of anything to do, can anybody give me a hint?
 
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  • #2
Eivind said:
Hi,
I have been thinking at this for quite a while now and I just can`t figure it out.
1)Use Plancks radiation law to show that the total energy flow is given by Stefan-Boltzmann law, J=@T^4, where @ is a constant.
Plancks radiation law:I(v,T)=(2piv^2/c^2)(hv/e^(hv/kT)-1)
I don't know of anything to do, can anybody give me a hint?
Try integrating to find the area under the Planck Law radiation distribution curve. (It is not a trivial integration). This should give you the energy emitted by the black body (per solid angle unit). Since it is radiating equally in all directions, multiply by [itex]4\pi[/itex] to get the total energy radiated.

AM
 
  • #3
You'll need

[tex] \int_{0}^{\infty} \frac{x^{3}}{e^{x}-1} \ dx =\Gamma (4)\zeta (4) [/tex]

Daniel.
 
  • #4
As has been said, you have write down the integral of Planck's law over all frequencies. As also has been said, the integral is tricky. However, you don't actually have to evaluate this integral to get the answer for which your questions asks, i.e., to show proportionality to [itex]T^4[/itex]. If you need the proportionality constant, then you do need to evaluate the integral.

Hint: in the integral, get rid of the mess in argument of the exponential, i.e., make the change of variable

[tex]x = \frac{h \nu}{kT}.[/tex]

Regards,
George
 

What is the Planck's law?

Planck's law, also known as the Planck radiation law, is a fundamental principle in physics that describes the spectral energy density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature. It is named after Max Planck, who first proposed the law in 1900.

How is the Planck's law used in science?

The Planck's law is used to calculate the amount of radiation emitted by a black body at different wavelengths and temperatures. It is also used to explain the observed behavior of objects at high temperatures, such as the radiation of stars and the cosmic microwave background radiation. It is an important tool in astrophysics, cosmology, and other fields of science.

What is the Stefan-Boltzmann law?

The Stefan-Boltzmann law, also known as the Stefan's law, is a physical law that describes the total energy emitted per unit surface area of a black body as a function of its temperature. It is named after Josef Stefan and Ludwig Boltzmann, who derived the law in the late 19th century.

How is the Stefan-Boltzmann law related to the Planck's law?

The Stefan-Boltzmann law is a consequence of the Planck's law, as it is derived from integrating the Planck's law over all possible wavelengths. It states that the total energy radiated by a black body is proportional to the fourth power of its temperature and the surface area of the body. This relationship is important in studying the thermal radiation of objects in the universe.

What are some applications of the Planck's law and Stefan-Boltzmann law?

Both laws have a wide range of applications in various fields of science, such as astronomy, astrophysics, thermodynamics, and materials science. They are used to study the radiation emitted by stars and other celestial objects, to calculate the thermal radiation of objects at different temperatures, and to design and improve technologies such as solar cells and thermal imaging devices. They are also important in understanding the behavior of matter and energy in extreme environments, such as black holes and the early universe.

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