Cyclotron working; semicircle time constant

Principle

A charged particle moving at right angles to a uniform magnetic field travels on a circle, and its period of revolution does not depend on its speed. A small oscillating electric field of the same frequency, applied across a gap, gives the particle an energy kick every half turn, so it spirals outward to very high energy without needing a high voltage.

Construction

        B (into page, uniform)
       ______   ______
      /      \ /      \
     |   D1   | |  D2   |
     |    ~ ~ ~~~ ~ ~   |   <- gap with RF
     |  ion   |gap|     |      voltage
      \______/ \______/
          ~ oscillator ~

Two hollow D-shaped metal boxes (dees) in an evacuated chamber between the poles of an electromagnet. An ion source is at the centre; a high-frequency oscillator is connected across the dees.

Working

1. An ion (charge q, mass m) at the centre is attracted to the dee of opposite polarity and accelerates across the gap.
2. Inside the dee there is no electric field. The magnetic field bends it into a semicircle.
3. When it returns to the gap, the oscillator has reversed polarity, so the ion is accelerated again, gaining energy qV.
4. The radius increases each time; after many turns the ion reaches the rim and leaves through a deflector with energy K = q²B²R²/(2m).

Time for a semicircle

1. Magnetic force provides the centripetal force: qvB = mv²/r
2. r = mv/(qB), so the time for a semicircle is

t = πr/v = π (mv/qB)/v = πm/(qB)

This has no r or v in it. Hence the time is the same for every semicircle, whatever the radius and speed, so a fixed frequency f = qB/(2πm) keeps the ion in step with the oscillator (the resonance condition).