A cyclotron in a magnetic field of 9 T is used to accelerate protons to 50 % of the speed of li

A cyclotron in a magnetic field of 9 T is used to...

A cyclotron in a magnetic field of $9 \mathrm{~T}$ is used to accelerate protons to $50 \%$ of the speed of light. What is the cyclotron frequency of these protons? What is the radius of their trajectory in the cyclotron? What is the cyclotron frequency and trajectory radius of the same protons in the Earth's magnetic field? Assume that the Earth's magnetic field is about $0.5 \mathrm{G}$.

Key Concepts

Cyclotron Frequency

The cyclotron frequency is the rate at which a charged particle orbits in a perpendicular magnetic field. It is determined by balancing the Lorentz force with the centripetal force needed to keep the particle in circular motion, leading to the formula ? = qB/m for non?relativistic cases. This concept is central to accelerator physics and is used to design cyclotrons, which exploit the fact that the frequency is independent of the particle’s speed in the non-relativistic limit.

Cyclotron Radius

The radius of the trajectory of a charged particle moving in a magnetic field is found by equating the magnetic Lorentz force to the centripetal force, resulting in the expression r = (m v)/(qB) for non-relativistic speeds. This parameter plays a key role in the design and operation of cyclotrons, determining the size of the device, and it becomes larger when the particle’s momentum increases or when the magnetic field strength decreases.

Relativistic Corrections in Cyclotron Motion

When the speeds of the particles approach a significant fraction of the speed of light, relativistic effects become important. The effective mass of the particle increases by a Lorentz factor ?, which modifies both the cyclotron frequency and the radius of the orbit. For instance, at 50% of the speed of light, the Lorentz factor is about 1.15, meaning that the particle appears heavier, thereby slightly reducing the cyclotron frequency and increasing the orbital radius compared to the non-relativistic prediction.

Magnetic Field Strength and Unit Conversions

Magnetic field strength is a critical parameter in determining the cyclotron frequency and radius. In many physics problems, magnetic fields may be given in different units, such as Tesla or Gauss; 1 Tesla is equal to 10,000 Gauss. Understanding and converting between these units is essential when comparing environments with very different magnetic field strengths, such as a laboratory cyclotron and the Earth’s naturally weak magnetic field.

Transcript

00:01 Hello.

00:04 So the radius of the protons part is given by the mass times the velocity, by the charge times the magnetic field.

00:18 So that is the radius.

00:22 And from the radius we can get the frequency from the formula.

00:26 I'm going to rewrite this so r equals m over qb.

00:35 Now the velocity, since this is a circular, motion is also can also be 2 pi r over t right that's going to be the v right there and you know that the period which is t is one over frequency right so this is going to be m over qb two pi r over one over f so this is just going to go and become two pi m r f over q b alright, so let's rewrite this.

01:28 So this is the same.

01:29 So this is going to be 2r equals 2 pi m r f or qb right.

01:38 And this r is going to cancel out this r.

01:41 So you can find f to be qb pi m.

01:51 So these are the two expressions we're going to be using.

01:54 So if you want to find a frequency, that's going to be the electron charge.

02:00 1 .6 .02 times down to the negative 19 times the magnetic 5x0 is 9 tesla divided by 2 pi with the mass 1 .67 10 to the negative 27.

02:23 The frequency is going to be approximately 1 .37 megahertz.

02:36 They're going to find the frequency, i mean the radius that's going to be the mass time the velocity over the charge times magnetic field.

02:44 So the mass is 1 .6, 7, 10 to the negative 27 times the velocity, which is almost the speed of light.

02:58 Let's say 2 .998 times 10 to the 8.

03:08 So that's half of that...

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