 We've seen how cyclotrons can accelerate heavy nuclear using just a few thousand volts. The whole idea is that it uses oscillating electric fields to accelerate the charge particle a little bit. But then the magnetic fields makes it turn and re-enter the electric field so that it can keep accelerating over and over and over again. And so that eventually when it reaches the maximum radius it can be shot out with very high speed. Now in this video we want to talk about how do we ensure that the electric field flips at the right moment. I mean think about it as an engineer if you were to design this oscillator a couple of questions that come to my mind is one should this flipping be at a constant frequency or should that frequency keep changing because we have a very complicated spiral motion happening over here. The second thing is how do I calculate at what frequency this electric field should flip so that my cyclotron works properly. Alright let's think about this. Now because I want to calculate the oscillator frequency let's start thinking about how long it would take for my electric field to flip and then flip back. How long it would take to finish that one cycle. Okay and for that let's follow along this proton. So right now if the proton is over here my electric field would be to the right so that I can speed it up to the right and then as my proton speeds up it'll it'll turn upwards because of the magnetic field and then as it's about to re-enter that's when I need to flip my electric field. So this is the first time I flip my electric field so that I can accelerate it again because our goal is to keep accelerating it. So my proton is going to the left I need to accelerate it to again so I flip it my flip my electric field to the left. As a result the proton speeds up speeds up and then it now keeps turning due to the magnetic field and now when it's about to re-enter this time I'm going to flip my electric field back and I've finished one oscillation. So what I see is that the time it takes for my proton to finish one spiral this is one spiral then this is the second spiral so the time it takes to finish one spiral is exactly the time it takes for my electric field to finish one oscillation that seems important let me write that down. So time for one oscillation which is basically the time period by definition that's a time period of the oscillator should equal the time period time for one spiral let's say that time for one spiral. So if I can figure out how long it takes for my proton to finish one spiral I'm done I have my time period but the second question that I'm really interested in is does that time to finish one spiral does that stay the same or does it change so would it take now for example if I compare this time with the time taken to finish this spiral would that be the same or would it be different can you pause the video a little bit and think about this we have to invoke stuff that we've seen earlier so think about the time taken so okay let's let's take an example let's say the time it takes for it to complete this one spiral or you know what let's keep things easier let's say the time it takes to complete this half a circle okay let's say that is one millisecond the question I have for you is what is the time it takes to finish this circle half a circle would it be one millisecond would be more than that would be less than that and then let's think about this half circle and then let's think about that half circle so can you pause and think about what will be the times taken over here would it keep increasing would it keep decreasing or what just pause and think all right now my first instinct is to think that hey because it's taking longer paths the time should keep increasing so if it takes one millisecond for this curve it should take I don't know maybe two maybe this one takes three I don't know it should increase because it's taking more and more turn but remember the real reason why not three of course the reason why it's taking a longer turn is because the reason is taking a bigger radius is because the speed is increasing why does that matter because yes definitely traveling more distance but it's also traveling faster than it was over here right in fact you can see the distance which is basically the circumference or half the circumference is that is proportional to r is proportional to the velocity so what I'm trying to say is over here if the distance has traveled is twice as much as over here it automatically means the speed of the proton is twice over here as much as over here and that means the time taken which is the ratio of the two should be exactly the same the time taken here should be exactly the same here the time taken over here should be one millisecond and that means the same thing applies here when it goes from here to here sure it takes a longer path but it also goes faster and they are proportionate the increase in the path and the the speed is proportionate so the time taken here should also be one millisecond and so on and so forth in fact we can derive the expression for this we can look at it mathematically we've done it before but let's do it one more time if I want to calculate how long it takes to go from here to here this half a circle how do I do that well the time so I'm calculating this now the time for that half a circle is going to be um that speed equals distance over time time equals distance by speed okay so time will be distance which is half a circle so that is pi r to pi r is full circle so half circle is pi r divided by speed and that will be I know the expression for r r is mv by qb divide by v vv cancels and notice this time is pi m by qb the time only depends upon the mass the charge and the magnetic field which are all constants as it spirals out that does not change and so in our example this is one millisecond and so the time taken to go from here to here is also exactly the same and therefore the total time it takes to finish one spiral over here is going to be twice of this this is for half and half total time would be twice so the time period of one full spiral is two times this value two pi m divide by qb in our example that is two milliseconds and the same thing will be true for this spiral as well it's going to take two pi m divided by qb time and the same is for this spiral and what that means is that the time period of the oscillator should also be the exact same value so this will be also the time period of the oscillator because the two are exactly the same which means we have answered both of our questions so is the time period of the oscillator should that be a constant or changing the answer is it should remain a constant because even though it's spiraling outwards the time it takes to finish one spiral stays the same the reason for that is because even though the distance is increasing the speed is increasing proportionately and so the time it takes to finish it stays the same and how much is that time it takes to finish that spiral well that time it takes to finish that spiral is two pi m by qb and in that time you should finish one full oscillation and so of course if i now want to calculate the frequency of the oscillation or the frequency of this oscillator that's going to be the reciprocal of this so that's going to be qb divided by two pi m and this is often called the cyclotron frequency this is our cyclotron frequency and again notice i didn't have to remember any of these equations i didn't in fact if i just and remember my basics of the centripetal force is given by the magnetic force i can derive everything else that's what i that's what i love about physics i can derive everything else all right now before you wind up i think we're also in a position to understand the major limitation of the cyclotron you see the whole idea was that this frequency stays a constant because as the proton spirals out these values don't change but technically that's not true you see einstein's theory of relativity has shown us that mass is actually dependent on velocity that's right when you're running your mass actually increases that's true okay it's been shown now at normal speeds that usually we're dealing with are when the speeds are much smaller than the speed of light that increase is so insignificantly small that we don't care about them and that's why that's why we usually think that masses don't change when things move faster right we just think that mass is an inherent property however it turns out that in it really does change and this effect becomes really significant when particles approach the speed of light so what does that mean that means in our cyclotron as long as our particles are having speeds much smaller than the speed of light everything we just said makes sense everything we said works and the oscillator frequency stays a constant needs to be a constant however when the proton starts approaching the speed of light comes close to the speed of light its mass starts increasing and it's significantly which means the oscillator frequency can no longer stay a constant so this means that our entire cyclotrons will only work as long as the speeds we are dealing with is much smaller than the speed of light one out of percent speed of light is still fine but if you want particles to approach 50 60 or 90 99 percent the speed of light then we need to find ways to adjust the frequency of the oscillator as the particle spirals out and modern accelerators can do that they're often called the synchrotrons or synchro cyclotrons which means that the oscillators have to keep syncing to ensure that they sync with the spirals of the protons you can look it up it's pretty interesting