A cyclotron is a circular accelerator. It has two hollow, D-shaped electrodes called dees, with a gap between them, in an evacuated chamber. A uniform magnetic field acts perpendicular to the dees, and an alternating p.d. is connected across them.
• Charged particles are released near the centre.
• Inside a dee there is no electric field. The magnetic force BQv provides the centripetal force, so the particle moves in a semicircle at constant speed.
• Each time the particle crosses the gap, the p.d. has reversed, so it is accelerated.
• As its momentum increases, the radius increases (r = p/BQ), so the particle spirals outwards until it is deflected out at the edge.Since r = mv/BQ, the time for each half circle is t = πr/v = πm/BQ, which does not depend on the speed. So the alternating p.d. can have a fixed frequency:f = BQ / 2πm At speeds approaching the speed of light, relativistic effects make the particles take longer to go round, so they fall out of step with the p.d. This limits the energy a simple cyclotron can reach.Worked example: protons (m = 1.67 × 10-27 kg) in B = 1.5 T need
f = (1.5 × 1.6 × 10-19) / (2π × 1.67 × 10-27) = 2.3 × 107 Hz.
At the edge, radius 0.50 m: p = BQr = 1.5 × 1.6 × 10-19 × 0.50 = 1.2 × 10-19 kg m s-1.
• Charged particles are released near the centre.
• Inside a dee there is no electric field. The magnetic force BQv provides the centripetal force, so the particle moves in a semicircle at constant speed.
• Each time the particle crosses the gap, the p.d. has reversed, so it is accelerated.
• As its momentum increases, the radius increases (r = p/BQ), so the particle spirals outwards until it is deflected out at the edge.Since r = mv/BQ, the time for each half circle is t = πr/v = πm/BQ, which does not depend on the speed. So the alternating p.d. can have a fixed frequency:
f = (1.5 × 1.6 × 10-19) / (2π × 1.67 × 10-27) = 2.3 × 107 Hz.
At the edge, radius 0.50 m: p = BQr = 1.5 × 1.6 × 10-19 × 0.50 = 1.2 × 10-19 kg m s-1.