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

[Eivind]

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?

 

[Andrew Mason]

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 $4\pi$ to get the total energy radiated.

AM

 

[dextercioby]

You'll need

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

Daniel.

 

[George Jones]

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 $T^4$. 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

$$x = \frac{h \nu}{kT}.$$

Regards,
George