A cyclotron is a particle accelerator that uses a magnetic field and an alternating electric voltage to speed up charged particles. It is important because it can give ions high kinetic energy in a relatively compact circular machine. Cyclotrons are used in nuclear physics research, medical isotope production, and some cancer treatments.
The basic idea is to make a particle cross the same accelerating gap many times instead of using one very long accelerator.
Inside the cyclotron, two hollow D-shaped metal electrodes called dees sit in a uniform magnetic field. The magnetic field bends the moving charged particle into a circular path, while the electric field in the gap between the dees accelerates it each time it crosses. As the particle gains speed, its circular path grows into an outward spiral.
For nonrelativistic particles, the cyclotron frequency stays nearly constant, so the alternating voltage can be timed to push the particle forward on every crossing.
Key Facts
- Magnetic force on a moving charge: F = qvB when v is perpendicular to B.
- Centripetal force in the cyclotron: qvB = mv^2/r.
- Orbit radius: r = mv/(qB).
- Cyclotron angular frequency: omega = qB/m.
- Cyclotron frequency: f = qB/(2 pi m).
- Kinetic energy after acceleration: K = 1/2 mv^2, so larger radius means larger speed and energy.
Vocabulary
- Cyclotron
- A device that accelerates charged particles in a spiral path using a magnetic field and an alternating electric voltage.
- Dee electrode
- A hollow D-shaped metal electrode in a cyclotron where particles move in curved paths between acceleration gaps.
- Accelerating gap
- The narrow space between the dees where an electric field speeds up the charged particle.
- Ion source
- The central part of a cyclotron that produces the charged particles to be accelerated.
- Cyclotron frequency
- The frequency at which a charged particle circles in a uniform magnetic field when relativistic effects are small.
Common Mistakes to Avoid
- Using the electric field to explain the circular motion, which is wrong because the magnetic force provides the centripetal force inside the dees.
- Forgetting that acceleration happens mainly in the gap, which is wrong because the electric field is shielded inside the metal dees.
- Thinking the particle moves in a perfect circle forever, which is wrong because its speed and radius increase after each gap crossing.
- Using f = qB/m instead of f = qB/(2 pi m), which is wrong because qB/m is the angular frequency in radians per second.
Practice Questions
- 1 A proton moves in a cyclotron with magnetic field B = 0.80 T. Using q = 1.60 x 10^-19 C and m = 1.67 x 10^-27 kg, calculate its cyclotron frequency f = qB/(2 pi m).
- 2 An ion of mass 6.64 x 10^-27 kg and charge 3.20 x 10^-19 C moves at 2.0 x 10^6 m/s in a 1.5 T magnetic field. Calculate the radius of its path using r = mv/(qB).
- 3 Explain why the alternating voltage must reverse direction every half orbit for the particle to keep gaining energy.