Effect of a magnetic field on a bean of particles.

[felixthecat]

Homework Statement

a beam of particles of charge q with velocity v is emitted from a point source, roughly parallel with a magnetic field B, but with a small angle of angular dispersion. Show that the effect of the field is to focus the beam to a point at a distance z = 2[/pi]mv/|q|B.

Homework Equations

We know the force F= q vxB
dv/dt = -q/m Bxv

The Attempt at a Solution

I try to solve the problem with the force. However, since the beam is roughly parallel to B, the cross product tends to 0. I am now like clueless...

 

[gabbagabbahey]

The initial velocity is approximately parallel to B, not exactly parallel. Try calling the angle between v and B, $\theta$ and calculating r(t) for a particle on the beams outer edge. Then use a Taylor approximation for your expression for small $\theta$ to find the location where r(t) lies on the z-axis (assume that B points in the z-direction).