Max wavelength of CMB radiation

[Magma828]

I'm doing OCR A-Level Physics, and in my textbook it states "They (Penzias and Wilson) made a calculation to find the temperature of the source of the radio waves, which had a maximum intensity at wavelength 1.1 cm, and found it to be 2.7K".

This was all good and well, until I answered a question in the book which asked me to find the max wavelength of a body with temperature 2.7K using given data for other temperatures, and it came out as roughly 1mm not 1cm. The book has no answer to the question, so now I'm confused as to what the actual value is for the wavelength of the CMB radiation at it's maximum intensity.

I double checked my workings over and over, and I'm sure I've done it correctly. Either the data they gave in the question is wrong, or their original statement of the wavelength is wrong...

 

[CaptainMarvel]

If we use the rough energy/temperature relationship:

$$E \approx k_B T$$

And we know the energy of a photon is related to wavelength as so:

$$E=\frac{h c} {\lambda}$$

Then combining these two formula we get:

$$\lambda \approx \frac{h c} {k_B T}$$

Plugging in for $$T=2.7K$$

That gives us $$\lambda \approx 5.27 \times 10^{-3}$$ meters.

Which is is millimeters and not centimeters.

So it looks like your notes were incorrect and your working out was right :-)

Hope this helps.

 

[nicksauce]

Disagree with the above.

Wien's displacement law is (http://en.wikipedia.org/wiki/Wien's_displacement_law)

$$\lambda_{\textnormal{max}} \approx \frac{hc}{5kT}$$

If you include this factor of 5, then you get 10^-3 m for [itex]\lambda[/itex, or 1mm.

 

[CaptainMarvel]

I purposely indicated that mine was a rough calculation only.

I agree that for the exact wavelength then Wien's law is the correct one to use.

 

[Magma828]

Ahh okay thanks a lot guys! :D