What is the Fraction of Energy Lost by an Electron in Elastic Scattering?

[Poirot]

Homework Statement

An electron collides with a particle with mass M at rest and scatters elastically through an angle θ (assume electron mass negligible).
Show that the fraction of energy lost by the e^- is:

(E_e - E_e')/E_e = 1/[1+ Mc²/E_e(1-cosθ)]

Homework Equations

Conservation of Energy: E_e + Mc² = E_e' + E_M
Conservation of momentum: P_e = P_e' + P_M

E² = P²c² + M²c⁴

or for electron since mass negligible, E=Pc

Previous parts of the questions required the rearrangements of these to get:
P_M² = 1/c²[E_e² +E_e'² - 2E_eE_e'cosθ]

The Attempt at a Solution

I've tried to solve this so many times but the closest I can get is:

(E_e - E_e')/E_e = E_e'(1-cosθ)/Mc²

I don't know if I'm missing some kind of relation I can use to sub in E_e' for E_e because in the final expression I'm trying to get there seems to be way more E_e's than I would expect.

I've also tried working backwards to find out what I'm missing from the answer but I just don't see it.

I think I have tunnel vision from trying this so often and can't see another way, so thank you in advance for any help, it's greatly appreciated!

 

[mfb]

Your situation is equivalent to Compton scattering (just with different particles), you should find derivations for this in textbooks and websites.

 

[vela]

I've tried to solve this so many times but the closest I can get is:

(E_e - E_e')/E_e = E_e'(1-cosθ)/Mc²

I don't know if I'm missing some kind of relation I can use to sub in E_e' for E_e because in the final expression I'm trying to get there seems to be way more E_e's than I would expect.

I've also tried working backwards to find out what I'm missing from the answer but I just don't see it.

I think I have tunnel vision from trying this so often and can't see another way, so thank you in advance for any help, it's greatly appreciated!

You're not missing any physics here. It's just algebra. Starting from the expression you got, solve for $E_e'$ in terms of $E_e$, $\theta$, and $M$. Then plug the result into $(E_e-E'_e)/E_e$ and simplify.

 

[bjnartowt]

I hope this is helpful.