Conceptually ok problem, but very difficult to solve

[Corneo]

Hi, I was given this problem on a midterm regarding a photon colliding with a free electron. I need to find the angle of the scattered photon $\theta$, the new wavelength $\lambda'$, and its energy.

It states that the electron scatters at an angle of 60 degrees (relative and below the initial photon momentum) and at a velocity of 4 x 10^7 m/s.

I already set up the three equations given but I simply have no idea how to solve this systerm, substitution seems to be suicide due to the limited time on a midterm. I do not think matrices would help either. I just would like to know how to solve for the photon's scattered angle, because that seems like the most difficult part of this problem.

From conservation of momentum I can write.

[tex]x: \frac {h}{\lambda} = \gamma m u \cos \phi + \frac {h}{\lambda'} \cos \theta \qquad \phi = 60^\circ, u = 4 \times 10^7 m/s[/itex]<br /> $$y: 0 = \frac {h}{\lambda'} \sin \theta - \gamma m u \sin \phi$$<br /> <br /> From conservation of energy I can write.<br /> <br /> $$\frac {hc}{\lambda} + mc^2 = \frac {hc}{\lambda'} + \gamma m c^2$$<br /> <br /> From here I simply do not know solve to solve this system. The unknowns are $\lambda, \lambda', \theta$[/tex]

 

[member 553083]

, and \gamma. Please help! The best way to solve this system of equations is to use the quadratic formula.First, solve the momentum equation for γm:$\gamma m = \frac{\frac{h}{\lambda} - \frac{h}{\lambda'}\cos\theta}{u\cos\phi}$Now substitute this expression into the energy equation:$\frac{hc}{\lambda} + mc^2 = \frac{hc}{\lambda'} + \left(\frac{\frac{h}{\lambda} - \frac{h}{\lambda'}\cos\theta}{u\cos\phi}\right)c^2$This can be rearranged to give a quadratic equation in λ':$\lambda'^2 + \left( \frac{hc}{u\cos\phi} - \frac{hc\cos\theta}{u\cos\phi} \right)\lambda' - \frac{hc}{u\cos\phi}\lambda = 0$The solution to this equation can be found using the quadratic formula:$\lambda' = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where$a = 1 \qquad b = \frac{hc}{u\cos\phi} - \frac{hc\cos\theta}{u\cos\phi} \qquad c = \frac{hc}{u\cos\phi}\lambda$Once you have the values of \lambda' you can then solve for θ using the momentum equation.

 

[thepatientmental]

, \gamma, and \phi.

First of all, I want to commend you for recognizing the difficulty of this problem. It shows that you are thinking critically and not just trying to blindly solve it.

In terms of solving the system of equations, substitution may indeed be a difficult and time-consuming method. One alternative approach could be to use a numerical method, such as Newton's method or the bisection method, to iteratively solve for the unknown variables. This may be more efficient and accurate than trying to solve the system algebraically.

Another option could be to use a computer program or online calculator that can solve systems of equations for you. This would save you time and allow you to focus on understanding the concepts rather than getting bogged down in the calculations.

In terms of finding the scattered angle, you could try using the conservation of momentum equation in the x-direction to solve for \theta. This would involve rearranging the equation and using trigonometric identities to simplify it.

Overall, it is important to remember that some problems may not have a straightforward analytical solution and may require more advanced techniques or the use of technology. The key is to understand the concepts and be able to apply them, rather than getting caught up in the calculations. I hope this helps and good luck on your midterm!