# Focusing a beam of electrons with a charged cylinder

**Electricity and Magnetism**Level 5

A uniformly charged cylinder of small radius extends from \(z=-\infty\) to \(z=\infty\) through the origin. Electrons are emitted towards the left from a source located at point \(A=(0,r_0,0)\) in (x,y,z) coordinates and move in the x-y plane. The beam is not perfectly focused however, so that all the electrons do not come out in the exact same direction. Instead, the beam has a angular spread above and below the horizontal of \(\alpha_{0}\ll 1\) (\(\alpha_{0}\ll 1\) is the half-angle of divergence). All the electrons do have equal initial speed however. The initial speed is such that the electrons emitted with zero radial velocity (exactly to the left) move in a circular orbit of radius \(r_{0} \). Show that the beam focuses at certain point B and find the angle \(\Theta=\angle AOB \) **in degrees**. Hint: You may use the approximation \((1+x)^{n}\approx 1 +n x \) for \( x\ll 1 \).