Question 10.6: Elastic collision between two electrons In an elastic collis...
Elastic collision between two electrons
In an elastic collision between an electron with kinetic energy E and an electron at rest, the incoming electron is observed to be deflected through an angle of 30°. What are the energies of the two electrons after the collision?
Since the collision is elastic and the electrons have equal mass, the opening angle between the emerging paths must be 90°. The target electron must therefore recoil at an angle of 60° to the initial direction of the incoming electron. Let the speed of the incoming electron be u and speeds of the electrons after the collision be u_{1} \text { and } u_{2} respectively. Then conservation of linear momentum implies that
\begin{aligned} m u &=m u_{1} \cos 30^{\circ}+m u_{2} \cos 60^{\circ}, \\ 0 &=m u_{1} \sin 30^{\circ}-m u_{2} \sin 60^{\circ},\end{aligned}
hich gives u_{1}=\frac{1}{2} \sqrt{3} u \text { and } u_{2}=\frac{1}{2} u . Hence, after the collision, the electrons have energies \frac{3}{4} E \text { and } \frac{1}{4} E respectively.