Focusing a beam of electrons with a charged cylinder

A uniformly charged cylinder of small radius extends from z=z=-\infty to z=z=\infty through the origin. Electrons are emitted towards the left from a source located at point A=(0,r0,0)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 α01\alpha_{0}\ll 1 (α01\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 r0r_{0} . Show that the beam focuses at certain point B and find the angle Θ=AOB\Theta=\angle AOB in degrees. Hint: You may use the approximation (1+x)n1+nx(1+x)^{n}\approx 1 +n x for x1 x\ll 1 .

×

Problem Loading...

Note Loading...

Set Loading...