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## Homework Statement

A beam of particles of charge q and velocity v is emitted from a point source, roughly parallel with a magnetic field B, but with a small angular dispersion. Show that the effect of the field is to focus the beam to a point at a distance

[tex]z=\frac{2\pi mv}{qB}[/tex]

from the source

**2. The attempt at a solution**

All that I know for sure is that

[tex]\vec{F}=q \vec{v} \wedge \vec{B}[/tex]

I also think that, due to the angular dispersion, I'll have to somehow find an equation that will relate to all particles in the beam. We know that if v is initially perpendicular to B then the particle describes a circle with radius r = mv/qB. But what happens when the velocity is NOT perpendicular to B (nor parallel, in which case the particle will experience no force)?

My understanding is that with "focus" they mean that particles will move in a spiral around the z-axis (set in the direction of B) from where they were emitted to where they are focused, i.e. if distance r between the particle and the z-axis is initially r, then the final distance at point z will be 0. Is this correct?

So far, however, all attempts at a solution have been unsuccessful.

Could someone please give me a hint?

Thanks!

phyz

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