(a) If the cyclotron of Example 43-2 accelerated αparticles,...

Key Concepts

Cyclotron Operation

A cyclotron is a type of particle accelerator in which charged particles are injected into a region with a constant magnetic field and accelerated by an oscillating electric field. The particles spiral outward as they gain kinetic energy each time they cross the gap between the electrodes (usually called 'dees').

Charged Particle Acceleration

The acceleration mechanism relies on the resonant timing of the electric field so that each time the particle crosses the gap it receives a boost in energy. This process continues until the particles reach the maximum radius of the magnet, which limits their energy based on the size of the device and the strength of the magnetic field.

Cyclotron Frequency

The cyclotron frequency is the rate at which a charged particle orbits in the magnetic field and is determined by the particle's charge-to-mass ratio and the magnetic field strength. Proper synchronization of the applied voltage with this frequency is critical for efficient acceleration.

Charge-to-Mass Ratio

This ratio (q/m) is fundamental in cyclotron operation because it directly influences the cyclotron frequency. Different particles, such as alpha particles and deuterons, have different charge-to-mass ratios, which result in different orbits, energies, and required frequencies of the alternating voltage.

Energy and Speed Relationship

The maximum kinetic energy that a particle can attain in a cyclotron is related to the work done by the accelerating electric field. Once the maximum energy is known, the particle's speed can be determined using kinetic energy formulas. In many cyclotron applications, non-relativistic approximations are used, though relativistic corrections may be necessary at very high speeds.

Transcript

00:01 For all of these problems, we're going to be using the relationship that the maximum kinetic energy for a particle being accelerated in a cyclotron is q squared, b squared, r squared over 2m, which of course is also equal to 1 half m v squared, so that the velocity, of a particle is the square root of 2km.

00:40 The example that we are comparing to had a kinetic energy of, call it k -0, a kinetic energy of 8 .7 m -ev.

01:05 In our first part, we we look at accelerating an alpha particle instead of a proton.

01:16 Well, the charge has increased by a factor of two, going from a proton to an alpha particle, and the mass has increased by a factor of four.

01:30 Looking at our equation up here, where the kinetic energy is proportional to q squared over, m, doubling the charge and increasing the mass by a factor of four means that the new kinetic energy is equal to the old kinetic energy.

01:53 There's no change.

01:56 So k is equal to 8 .7 m e v.

02:04 Then the velocity, which is the square root of 2k over m or the square root of, now remember we want a velocity in meters per second.

02:16 We're going to have a mass in kilograms, so we need kinetic energy in joules.

02:21 So this is going to be 8 .7 times 10 to the, let's leave it in megavolt, 8 .7 mev, and then we can use the conversion 1 .6 times 10 to the minus 13.

02:44 Joules per mega electron volt.

02:50 Divided by the mass of an alpha particle is four times the mass of a proton, 1 .67 times 10 to the minus 27 kilograms, gives us a speed of about 2 .0 times 10 to the 7 meters per second.

03:16 So we see that even though we have the same connect.

03:19 Energy as we did in our previous example, we have a different velocity because the mass has increased.

03:31 Let's now consider a deuteron where we're going to say that the charge is unchanged from our previous example.

03:48 Deuteron's have the same charge as a proton, but the mass has increased because we now, have a neutron attached.

03:56 So the mass is two m not.

04:00 Well, k is proportional to q squared over m.

04:05 We haven't changed q.

04:07 We've doubled m.

04:09 So the new kinetic energy is going to be half of our old kinetic energy, which is equal to about 4 .3 m .e .v...

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