Modern Physics I: Find Energy Loss & Wavelength

[baltimorebest]

Homework Statement

An electron with energy 10^3 MeV collides with a photon whose wavelength corresponds to the maximum wavelength of the 2.7-K cosmic background radiation. What is the maximum energy loss the electron can suffer as a result of the collision? What is the wavelength of the photon after this maximal collision?

Homework Equations

The Attempt at a Solution

I need help starting this problem, if anyone can help me. Thanks.

 

[hikaru1221]

To find the wavelength of the photon before collision, try Wien's displacement law
Assume that the electron is free electron. Then the Compton scattering equation can be applied. Note that due to the law of energy conservation, the electron loses energy the most when the wavelength of photon after collision is smallest.

 

[baltimorebest]

So I use Wien's displacement law to find the wavelength? Would that be f-max= 5.9E10*T? My teacher skipped over this section in class and I am struggling a bit to understand it.

 

[hikaru1221]

The peak wavelength does NOT correspond to the peak frequency. You should apply the formula of peak wavelength instead: http://en.wikipedia.org/wiki/Wien's_displacement_law

(by the way, the term "maximum" is somewhat inappropriate. The spectrum spreads to infinity, so there is no maximum. I prefer using the term "peak wavelength", which means the wavelength corresponding to the maximum value of spectral energy density)

 

[baltimorebest]

Ok so Lamda-max*T= 2.9E-3 where T=10^3?

 

[hikaru1221]

Ok so Lamda-max*T= 2.9E-3 where T=10^3?

I assume that bold part is just typo Yes, that's the right formula to use.

 

[baltimorebest]

Ok great. So that gives me a value for lamda-max equal to 2.9E-6.

How do I use the Compton Effect equation though? Aren't there too many unknowns?

 

[hikaru1221]

http://en.wikipedia.org/wiki/Compton_scattering
You have the initial wavelength, then there is also the condition that the loss in energy of the electron is maximum. This condition means the energy of the photon after collision must be maximum, which means the wavelength after collision is minimum. With this condition, you can find the wavelength from Compton scattering equation.

 

[baltimorebest]

I'm sorry but I am still not following you. I think I understand what you are saying, but I'm not able to work it out. Is the maximum energy loss of the electron going to be equal to the h*f-max of the photon?

 

[hikaru1221]

Nope. The law of energy conservation gives: $$E_{electron-before} + hc/\lambda_{before} = E_{electron-after} + hc/\lambda_{after}$$
Because the first 2 terms are fixed, for the 3rd term to be minimum (i.e. the loss is maximum), the 4th term must be maximum.

By the way, there is also another constraint: $$E_{electron-after}$$ must be equal or greater than 0. That means, you have to find the largest possible value of the 4th term such that it cannot be greater than $$E_{electron-before} + hc/\lambda_{before}$$.

 

[baltimorebest]

So lamda=2.9E-6 is the answer to the second part of the question?

I don't think I understand the statement "...corresponds to the maximum wavelength of the 2.7-K cosmic background radiation." I believe that will give me the answer to the wavelength before collision, and the rest of the answer.

 

[hikaru1221]

Oops, so sorry, I've just realized that the Compton scattering formula is only applied to cases where electron is initially at rest. I guess solving the energy conservation equation & the momentum conservation equation is inevitable. This problem is not easy

I don't think I understand the statement "...corresponds to the maximum wavelength of the 2.7-K cosmic background radiation." I believe that will give me the answer to the wavelength before collision, and the rest of the answer.

Yes, it gives you the wavelength before collision. But before coming to the rest of the answer (wavelength after collision), calculations must be made.

So here is the way to solve the problem:
1. Calculate the wavelength before collision; this will give you the energy & momentum of photon before collision.
2. In order that the energy loss of electron is max (i.e. to reduce the electron's momentum the most), before collision, the photon should travel in the opposite direction of electron. Besides, you can see that before collision, energy/momentum of photon is very much smaller than energy/momentum of electron. Therefore, we can expect that when the energy loss is max, after collision, the electron and the photon move in the same direction as the electron before collision.
So we know the direction of momentum. The rest is just applying the 2 conservation equations and calculating. In order to simplify the calculation, you should notice that energy of electron is much greater than photon's energy and electron's rest energy.

 

[baltimorebest]

I don't understand how to use that statement to find the wavelength before collision though. It's something we did not discuss, and I can't seem to get the chapter of my book to help me.

 

[hikaru1221]

I don't understand how to use that statement to find the wavelength before collision though. It's something we did not discuss, and I can't seem to get the chapter of my book to help me.

Assume that the cosmic background is like a black body (I'm not sure because I don't know much about cosmology, still have a look at this: http://en.wikipedia.org/wiki/Cosmic_microwave_background_radiation). Then from Wien displacement law, for a black body's spectrum, the wavelength corresponding to the maximum spectral energy density (which is referred as "maximum wavelength" here; but as I said, the word "maximum" in "maximum wavelength" is confusing and inappropriate) is lambda = b/T, where b = 2.898E-3 mK.