Beam of particles in momentum space

[Habeebe]

I'm mostly concerned with whether or not I understand this problem intuitively in order to answer the final part of this problem.

Homework Statement

Discuss the implications of Liouville's theorem on the focusing of beams of charged particles by considering the following case. An electron beam of circular cross section (radius R₀) is directed along the z-axis. The density of electrons across the beam is constant, but the momentum components transverse to the beam are disctributed uniformly over a circle of radius p₀ in momentum space. If some focusing system reduces the beam radius from R₀ to R₁ find the resulting distribution of the transverse momentum components. What is the physical meaning of this result? (Consider the angular divergence of the beam.)

Homework Equations

A_ellipse=πr₁r₂

The Attempt at a Solution

I answered that the circle in momentum space would become an ellipse of equal area, thereby satisfying the equation [itex]R_0^2=R_1R_p[/itex] where [itex]R_p[/itex] is the radius of the ellipse along the momentum axis. The next part is what I'm feeling sketchy on: the physical meaning. It seems like the focusing causes an increased tendency of the beam to want to converge/diverge, that is, the divergence of the beam is increased proportional to the change of radius from R₀ to R₁.

Does this sound right?

 

[Nate810]

Yes, that is correct. The focusing of the beam increases its angular divergence, which means that the particles will have a greater tendency to spread out over a wider area in space. This is because the transverse momentum components are distributed uniformly over an ellipse rather than a circle.