The largest particle accelerators have dimensions measured in miles. A cyclotron is a particle accelerator that is so compact that a small one could actually fit in your pocket. It makes use of electric and magnetic fields in a clever way to accelerate a charge in a small space.
A cyclotron consists of two D-shaped regions known as dees. In each dee there is a magnetic field perpendicular to the plane of the page. In the gap separating the dees there is a uniform electric field pointing from one dee to the other. When a charge is released from rest in the gap it is accelerated by the electric field and carried into one of the dees. The magnetic field in the dee causes the charge to follow a half-circle that carries it back to the gap.
While the charge is in the dee the electric field in the gap is reversed, so the charge is once again accelerated across the gap. The cycle continues with the magnetic field in the dees continually bringing the charge back to the gap. Every time the charge crosses the gap it picks up speed. This causes the half-circles in the dees to increase in radius, and eventually the charge emerges from the cyclotron at high speed.
How can you time it so the electric field reverses direction at the right time to accelerate the charge properly? Recall that for a charge following a circular path in a uniform magnetic field, the period is independent of the speed of the charge. Every half-circle in the dees takes the same amount of time. Unlike the cyclotron in the simulation, a real cyclotron is set up with a small gap so that the time to cross the gap is much smaller than the time spent in a dee. Hooking the dees up to an AC voltage source that reverses direction at regular intervals (corresponding to the time the charge takes to do a half-circle in a dee) is all that is required to produce an electric field that reverses direction at the appropriate time.
Consider these questions relating to the design of the cyclotron.
1. The first time the charge crosses the gap its kinetic energy increases by an amount ΔK. What is the change in kinetic energy associated with each half-circle in a dee.
In a dee the force on the charge comes from the magnetic field, so the force is perpendicular to the velocity. The speed, and hence the kinetic energy, stays constant, so the change is zero.
2. The first time the charge crosses the gap its kinetic energy increases by an amount ΔK. Assuming the electric field in the gap is the same magnitude at all times, what is the change in kinetic energy the second time the charge crosses the gap?
If the electric field has the same magnitude, the potential difference across the gap always has the same magnitude.
ΔK = q ΔV, so the kinetic energy increases by the same amount each time the charge crosses the gap.
3. Let's say you want to increase the speed of the particles when they emerge from the cyclotron. Which is more effective, increasing the potential difference across the gap or increasing the magnetic field in the dees?
The final kinetic energy is essentially independent of the potential difference in the gap, but the kinetic energy is proportional to the square of the magnetic field, so increasing the magnetic field is the way to increase the kinetic energy.
Note that whatever the magnitudes of the fields the final half-circle the charge passes through in the dee has a radius approximately equal to R, the radius of the dee itelf. The radius of the circular path of a charged particle in a magnetic field is:
r = mv/qB.
In this case the speed of the particle is v = RqB/m.
Therefore the final kinetic energy is:
K = 1/2 mv2 = R2q2B2/2m