How Does the Peak Radiative Flux from Planck's Law Vary with Temperature?

[Callisto]

Planck's law of radiation??

Hi, can anybody help me with this problem?

Planck's law of radiation for a blackbody radiator quantifies the relation between it's radiative flux and wavelength at a particular temperature.
given by:
F(w)=C1/[w^5(exp(C2/wT)-1]

where, w=wavelength, C1 and C2 are constants and T is the absolute temperature.

My problem is , I have to show that the peak radiative flux of the Planck spectrum varies as the fourth power of temperature.
Do i need to do some integration of Plancks law to show this?
If so, where do i start?

 

[chroot]

I have to show that the peak radiative flux of the Planck spectrum varies as the fourth power of temperature.

I think you mean the "total radiative flux," not the "peak radiative flux."

If so, simply integrate your formula over all values of wavelength.

Here's a page that might help:

http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/stefan2.html#c1

- Warren

 

[M98Ranger]

Hi there,

To show that the peak radiative flux varies as the fourth power of temperature, you do not need to perform any integration of Planck's law. You can simply use the fact that the peak of the Planck spectrum occurs at the wavelength where the derivative of Planck's law with respect to wavelength is equal to zero.

To find this wavelength, we can take the derivative of Planck's law with respect to wavelength and set it equal to zero:

dF/dw = (-5C1C2/w^6) * (exp(C2/wT)-1) + (C1C2T/w^7) * (exp(C2/wT)) = 0

Simplifying this expression, we get:

5exp(C2/wT) - 5 = C2T/w

We can then rearrange this equation to solve for the wavelength at the peak of the spectrum:

w = C2T / [5ln(5) - 5ln(exp(C2/wT))]

Now, we can substitute this value of wavelength into Planck's law to find the peak radiative flux:

Fpeak = C1/[w^5(exp(C2/wT)-1)] = C1/[C2^5T^5 / (5ln(5) - 5ln(exp(C2/wT)))^5 * (exp(C2/wT)-1)]

Simplifying this expression, we get:

Fpeak = (C1C2^5T^4) / [(5ln(5)-5ln(exp(C2/wT)))^5 * (exp(C2/wT)-1)]

We can see that the peak radiative flux is directly proportional to the fourth power of temperature, as required. Therefore, we have shown that the peak radiative flux of the Planck spectrum varies as the fourth power of temperature without needing to perform any integration. I hope this helps!