In a particle accelerator called the cyclotron, a charged...

In a particle accelerator called the cyclotron, a charged particle is accelerated by an electric field and bent into a circular path by a magnetic field. The magnetic field is assumed uniform and the north and south poles are separated by a small gap. The particles to be accelerated originate from a source at the centre of the bottom magnet pole (the south pole in Figure 6.42 ) and begin to move outward in a circular path. The bottom magnet pole is split into two pieces called dees. An alternating potential difference is set up between the two dees so that every time the particle crosses from one dee to the other it increases its kinetic energy and thus moves on a circle of larger radius: (a) Explain why the particle follows a spiral path. (b) Show that the operation of the cyclotron depends on the frequency of the alternating voltage source, being equal to the frequency of revolution of the particle to be accelerated. (c) If the mass and charge of the particle are $m$ and $q,$ respectively, show that the period of revolution is $$T=\frac{2 \pi m}{q B}$$ where $B$ is the magnetic field, and is thus independent of the speed of the particle. (d) Find the frequency (i.e. the number of revolutions per second) of an electron assuming that the magnetic field has a value of $0.50 \mathrm{T}$ (e) If the potential difference between the dees at the instant the electrons cross is $120 \mathrm{kV}$, what would the kinetic energy of the electrons be after 100 revolutions?

In a particle accelerator called the cyclotron, a charged particle is accelerated by an electric field and bent into a circular path by a magnetic field. The magnetic field is assumed uniform and the north and south poles are separated by a small gap. The particles to be accelerated originate from a source at the centre of the bottom magnet pole (the south pole in Figure 6.42 ) and begin to move outward in a circular path. The bottom magnet pole is split into two pieces called dees. An alternating potential difference is set up between the two dees so that every time the particle crosses from one dee to the other it increases its kinetic energy and thus moves on a circle of larger radius:
(a) Explain why the particle follows a spiral path.
(b) Show that the operation of the cyclotron depends on the frequency of the alternating voltage source, being equal to the frequency of revolution of the particle to be accelerated.
(c) If the mass and charge of the particle are $m$ and $q,$ respectively, show that the period of revolution is $$T=\frac{2 \pi m}{q B}$$ where $B$ is the magnetic field, and is thus independent of the speed of the particle.
(d) Find the frequency (i.e. the number of revolutions per second) of an electron assuming that the magnetic field has a value of $0.50 \mathrm{T}$
(e) If the potential difference between the dees at the instant the electrons cross is $120 \mathrm{kV}$, what would the kinetic energy of the electrons be after 100 revolutions?

Key Concepts

Lorentz Force in Uniform Magnetic Fields

This concept explains how a moving charged particle experiences a force when placed in a magnetic field. The force, which is perpendicular to both the particle's velocity and the magnetic field, causes the particle to move in a circular path when the magnetic field is uniform. It is the fundamental reason for the circular motion observed in devices like cyclotrons.

Resonance Condition and Synchronization

In accelerator physics, for efficient energy transfer, the frequency of the applied alternating electric field must match the natural frequency (revolution frequency) of the charged particle in its circular motion. This resonance ensures that the accelerating voltage is applied at the correct phase of the particle’s motion to continuously add energy without causing phase slippage.

Orbit Expansion and Spiral Trajectory

As energy is incrementally added to a charged particle by the alternating electric field, the particle’s speed increases. Because the magnetic force determining the circular path depends on the particle's velocity, a higher speed enlarges the radius of the orbit. As a result, the particle follows a spiral path, gradually moving outward as its energy increases.

Period of Revolution Independence from Speed

In a uniform magnetic field, the period of revolution for a charged particle is given by T = (2?m)/(qB), indicating that the period depends only on the particle’s mass, its charge, and the magnetic field strength. This independence from the particle's speed is crucial, as it allows the cyclotron to operate effectively even as the particle’s kinetic energy increases.

Incremental Kinetic Energy Gain

The mechanism of energy gain in a cyclotron involves the particle receiving a discrete boost in kinetic energy each time it crosses the gap between acceleration regions. The energy imparted per crossing is directly related to the potential difference applied, and over many revolutions, these incremental energy gains accumulate, enhancing the particle's speed and orbital radius.

Transcript

00:01 The cyclotron, like many particle accelerators, uses magnetic and electric fields to control and accelerate charged particles.

00:11 So here we're going to take a look at the parts of the cyclotron and then get into the physics of how this all works.

00:17 So there are two magnets, circular, usually semi -circular, well, completely circular in shape, divided each pole into two halves.

00:30 Each half is called a d, and the magnetic field is sandwiched in between the two poles, like usual.

00:41 Then there is an ac voltage between the two ds.

00:45 And what happens is a charged particle gets injected with very low velocity in the center, gets accelerated by the voltage, and then starts to travel in a circular path, which gets bigger and bigger and bigger in radius, the more kinetic energy the particle gains.

01:11 So it's a very compact arrangement because the particle moves around in a circle.

01:18 So in order to understand how the cyclotron works, there's just a little bit of physics.

01:24 The first is the lorentz force on a charged particle, f is equal to qv cross b, and here, the velocity of the particle is maintaining itself perpendicular to the magnetic field.

01:40 So we can just write down the magnitude of that force f equals qvb.

01:48 The second thing we need to understand is that the circular motion that arises is really happening because that force is what's called a centripetal force.

02:01 That is, it is always perpendicular to the velocity, and the force is thus holding the particle and circular motion.

02:15 And so we need newton's second law in terms of centripetal acceleration.

02:26 Some of the centripetal forces is equal to m v squared over r.

02:35 Now, supposedly, this occurs in a vacuum, so we'll just assume that there's a single force on the particle, neglect gravity, since it's so weak.

02:48 And if we put all that together and cancel some of the like sides, like pieces on each side, what we see is that the velocity and the radius are directly proportional to each other.

03:07 As the velocity goes up, the radius of the orbit goes up and vice versa.

03:22 So now we can understand why there is an accelerating potential in between the two dees.

03:29 The idea there is that you're going to be using conservation of energy to boost up the kinetic energy of the particle, and hence its kinetic energy and velocity go up.

03:45 So the third piece of physics is conservation of energy.

03:56 So we're going to make the kinetic energy go up, and that change in kinetic energy is minus the change in electric potential, which is proportional to the charge on the object in the cyclotron and the voltage, oscillating voltage amplitude.

04:20 Now that oscillating voltage amplitude, usually there are two attachments.

04:37 So there is a kick in the kinetic energy gains in two spots, one on the right -hand side and a similar thing going on on the left -hand side.

04:54 So you kind of have, if you want to think about it, quadrants.

04:57 Here i'm trying to draw some plates in there.

05:11 Actually, you can just leave the same thing connected all the way across, but you have to be careful about the phase between the two sides.

05:21 So here i'm just trying to show that you get a kinetic energy gain on the other side as well.

05:27 And that can happen if the ac potential is varying with just the right frequency so that there is a potential drop across the two plates on each side, and in between the potential isn't doing much at all.

05:52 Okay, so let's analyze that.

05:55 The oscillating voltage has to be in phase with the motion to give a boost to the kinetic energy of the particle.

06:08 So let's kind of see how that works...

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