# Cyclotron - show that the speed of the particle after it crosses the gap a 2nd time

1. Jul 25, 2011

### blueyellow

1. The problem statement, all variables and given/known data

A cyclotron is a particle accelerator made of two electrodes shaped as two empty half-circles separated by a small gap. We consider the dees as lying on the x-y plane, having radius R, both with centre in the origin, and each occupying the semiplane of positive and negative x coordinates, respectively. The cyclotron is immersed in a uniform magnetic field B parallel to the z axis. A particle with charge q and mass m is placed at rest halfway between the two electrodes at the origin (0, 0, 0). a potential difference V applied between the two electrodes then accelerated the particle while it is in the gap. The gap is so small that the influence of the magnetic field on the motion of the particle in the gap is negligible, so the magnetic field can be neglected when considering the motion of the particle in the gap.

The speed of the particle after it enters the electrode on the left for the first time is sqrt(qV/m)

once the particle enters the first half-circle, the electric field there is negligible, and the particle proceeds in uniform circular motion due to the presence of the magnetic field. After a half-circle, the particle crosses the gap again, towards the second electrode. When this happens, the potential difference between the two electrodes is reversed, and the particle is accelerated again in the gap by the potential difference V. Show that the speed of the particle after it crosses the gap a second time is sqrt(3) times larger than the speed after the first acceleration along the positive x direction.

3. The attempt at a solution

i have looked at my course notes and the internet but i could not find anything to help me do the question. please help.

PLEASE DO NOT DELETE THIS POST, MODERATORS. IT IS NOT THE SAME QUESTION AS I MAY HAVE POSTED BEFORE, IT IS A DIFFERENT PART OF THE QUESTION. THANKS

2. Jul 25, 2011

### Staff: Mentor

Re: cyclotron - show that the speed of the particle after it crosses the gap a 2nd ti

How much kinetic energy does a particle with mass m and charge q gain falling through an electric potential difference E? How would you determine the velocity of that particle?