The cyclotron, like many particle accelerators uses magnetic and electric fields to control and accelerate charged particles. 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. So there are two magnets uh circular usually uh semicircular well completely circular in shape divided each pole into two halves. Each half is called A. D. And the magnetic field is sandwiched in between the two poles. Like usual. Then there is an A. C. Voltage between the two Ds. And what happens is charged particle gets injected with very low velocity in the centre um 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. So it's a very compact arrangement because the particle moves around in a circle. Um So in order to understand how the cyclotron works, there's just a little bit of physics. Um The first is the Lorentz force on a charged particle F. Is equal to Q. V. Crosby. And here the velocity of the particle is maintaining itself perpendicular to the magnetic field. So we can just write down the magnitude of that force F equals Q. B. B. Uh 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 that is it is always perpendicular to the velocity. Um And the force is thus holding the particle in circular motion. And so we knew needs Newton's second law. In terms of uh huh centripetal acceleration, some of the centripetal forces is equal to mm B squared over R. 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. 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. As the velocity goes up, the radius of the orbit goes up and vice versa. So now we can understand why there is an exhilarating potential in between the two Ds. 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. So the third piece of physics is conservation of energy. 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. Now that oscillating voltage amplitude, usually there are two attachments. So there is a kick in the kinetic energy gain. The kinetic energy gains in two spots, one on the right hand side, and a similar thing going on on the left hand side. So you kind of have, if you want to think about it quadrants here, I'm trying to draw some plates in there. Um Yeah, actually you can just sleep the same thing connected all the way across. But you have to be careful about the phase between the two sides. So here I'm just trying to show that you get it a kinetic energy gain on the other side as well. And that can happen if the A. C. 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. Okay, so let's analyze that. The oscillating voltage has to be in phase with the motion to give a boost to the the kinetic energy of the particle. So let's kind of see how that works. And so what we're going to go back and do is just slightly rewrite equation number two because we know that the angular velocity of the particle times its radius is equal to be and we can substitute that in and going back to that equation. Then we have Q. B. R over em is equal to omega times are and notice that the angular velocity does not depend on the radius at all. And this is really at the heart of how the cyclotron is able to do. What it does is if you oscillate the A. C voltage at this so called cyclotron frequency, you will be as long as you set it up correctly, you will be accelerating the particle at each time it crosses from one day to the other. Um So this is very important relationship. There are different ways to write it. For example, omega is two pi times the frequency. And so that cyclotron frequency is a product of QB over two pi am does not depend on the radius. The time it takes for the particle to go all the way around is equal to one over the frequency. And we can just invert that. So it depends what you want to do. But this is basically the design parameter that you need in order to make your cyclotron operate correctly. So has a quick finish. Let's do the case of an electron in a half a Tesla magnetic field. We know the mass of the electron and we know it's charge of course. And we can calculate that psych latin frequency using those parameters. Yeah, I don't I think I need to put those numbers in but we get about 1.4 zero times 10 to the 10th hurts. Which is in the gigahertz range. It sounds like microwaves to me. Certainly doable. Mhm. And we can ask the question, how much energy does the particle have after 100 revolutions? Well, each revolution the kinetic energy gain is the absolute value of the change in the electric potential energy. So we need to know the operating voltage. And here the operating voltage amplitude is 120 kilovolts. That's quite a substantial voltage. Um So each revolution um you're passing through the DS twice and then you have, what's that, kilovolts? So you have two times 120 times 10 to the third electron volts are 240 times tender. The third electron volts. And so after 100 revolutions rounds will call them. You just multiply that by 100 and that is a sex essentially 24 million electron volts, 24 mega electron volts of energy. Okay, so to get a feel for that, I just want to remind you that the rest energy of the electron is a half a million electron volts. So definitely usually what you do in relativity is you compare the amount of energy to the rest energy of the particle and when that energy starts to be about equal, you know, you have relativistic effects coming in. So I'll just note that you have definitely achieved relativistic energy at this point.