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## 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 R

_{0}) 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 p

_{0}in momentum space. If some focusing system reduces the beam radius from R

_{0}to R

_{1}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}=πr

_{1}r

_{2}

## 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 R

_{0}to R

_{1}.

Does this sound right?