Fundamentals of Plasma Physics

22 Chapter 1. Basic concepts

1.13 Assignments

b

vf vi





r

x

y

trajectory

Figure 1.4: Geometry of scattering in center of mass frame. Scattering center is at the origin
andθis the scattering angle. Note symmetries of velocities before and after scattering.

1. Rutherford Scattering: This assignment involves developing a derivation for Ruther-
   ford scattering which uses geometrical arguments to take advantage of the symmetry
   of the scattering trajectory.
   (a) Show that the equation of motion in the center of mass frame is

μ

dv

dt

=

q 1 q 2

4 πε 0 r^2

ˆr.

The calculations will be done using the center of mass frame geometry shown in

Fig.1.4 which consists of a cylindrical coordinate systemr,φ,zwith origin at the

scattering center. Letθbe the scattering angle, and letbbe the impact parameter

as indicated in Fig.1.4. Also, define a Cartesian coordinate systemx,yso that

y=rsinφetc.;these Cartesian coordinates are also shown in Fig.1.4.

(b) By taking the time derivative ofr× ̇rshow that the angular momentumL=

μr× ̇ris a constant of the motion. Show thatL=μbv∞=μr^2 φ ̇so that

φ ̇=bv∞/r^2.

(c) Letviandvfbe the initial and final velocities as shown in Fig.1.4. Since energy

is conserved during scattering the magnitudes of these two velocities must be the

same, i.e.,|vi|=|vf|=v∞.From the symmetry of the figure it is seen that

thexcomponent of velocity at infinity is the same before and after the collision,

even though it is altered during the collision. However, it is seen thatthey

component of the velocity reverses direction as a result of the collision. Let∆vy