 We've already seen how cyclotrons make use of oscillating electric fields to speed up charged particles and then the magnetic fields to make them turn and re-enter those electric fields so that keep on accelerating, keep on speeding up over and over and over again to obtain very high speeds. Now the question we want to answer in this video is which of the two fields decides the speed at which the particle comes out? In other words, if I want to increase, say the increase the speed at which the particle is shot out, which of the two fields should I be concentrating or should I be increasing? Should I increase the electric field or should I be increasing the magnetic field or should I increase both of them? That's the question that I want to talk about. Alright so let's start with my intuition, my instincts. Because I know that it's the electric field that is speeding up my charged particle and the magnetic field is not speeding it up, is not doing any work, it's only making it turn, my initial instincts tell me that hey, it must be the electric field that decides the speed at which it gets shot out. If I increase the electric field then it will speed up more and probably because of that my particle will have more speed when it gets shot out. And probably the magnetic field has no effect because it does not change the speed of the particle, it only makes it turn, so changing the magnetic field probably has no effect is what my initial instincts say. But my instincts can be wrong and most of the times it is wrong. So one of the best ways to check this is to actually derive an equation for that. So let's go ahead and do that. So let's say that the speed at which the particle is shot out, let's call that as the maximum speed. Okay, can I build an equation for the maximum speed is the question I have? Well I do have an equation that connects the speed of the particle and the radius of that particle, right? So if I could just know what is the radius of this proton when it has the maximum speed that is this radius. If I knew the radius of this curve then I can substitute in this equation and I can rearrange and I can get the speed. So do I know what the radius of this curve is? Yes, this radius has to be the radius of this metallic D itself. Why? Because it's when the radius equals the radius of the metallic D that's when we say hey we can't increase the radius even more, we can't increase the speed even more because if it did, if it spiraled even more then it would go and hit the walls of that metal and that's not what we want and that's why we shoot it out. So by design we only shoot it out when the radius of that particle has equaled the radius of the metallic D. So by design when the particle has the maximum speed it should be having the radius of the D which means if I substitute R to be R of the D, the metallic D then in this equation V has to be the maximum speed and from that I can figure out what the maximum speed is so let's quickly go ahead and do that. So for R I substitute radius of the D and then that means that M times V should be the maximum speed divided by QB and from this if I rearrange I get the maximum speed to be QBRD divided by M. Now let's look at this and see if this makes sense to us. So what does the maximum speed depend on? Well, I'll not worry about the charge and the mass because they are constants for a given particle so forget about that. I see that it depends upon the radius of the D. It says that that means if I increase the radius of the D making my cyclotron bigger the maximum speed increases and that makes sense to me because if I have a larger cyclotron I can spiral it to higher speeds. I can accelerate it to higher speeds before shooting it out so this makes sense. The second thing I see is that it also depends upon the magnetic field. That means if I increase the magnetic field even then my maximum speed with which it gets shot out that increases and that's surprising to me because my instincts told me that it shouldn't depend upon the magnetic field. That's actually fine. I find that a little weird because magnetic field doesn't actually speed up the charge particle. The only job is to make it turn back yet the speed maximum speed depends on the magnetic field. And more surprisingly I find that even though it's the electric field that is actually doing the work and speeding it up that is not in my equation. That means even if I were to increase the electric field if I were to increase the oscillator voltage or decrease the oscillator voltage that would have no effect on the speed at which my particle gets thrown out how does that make any sense? Can you pause the video and just think about this and just ponder upon this and see what is going on. Alright, so let's carefully think about this. So first let's see what happens if I were to increase the magnetic field. Why would the speed increase? So what's the effect of increasing the magnetic field based on whatever we have learned? Well in this equation this equation makes sense to me, right? So if I increase the magnetic field I can see that if everything else stays the same the radius starts decreasing. For a given speed I would have a smaller radius. What does that mean? That means that my proton as it accelerates for that same speed as it had before it'll have smaller radius so it'll have a small it'll take smaller spirals. Let me try and draw this nicely. You'll take smaller spirals. This is what would happen if you were to increase the strength of the magnetic field. You'll take smaller spirals. Does that make sense? That kind of makes sense. Now if it takes smaller spirals this means it enters into this electric field more number of times before exiting out compared to before. Because it is entering my field more number of times compared to earlier that's why it ends up getting accelerated to higher speeds. Does that make sense? So increasing the magnetic field is increasing the number of revolutions it's having having in the cyclotron and as a result of that it's increasing the number of times it enters and accelerates in the electric field and as a result of that when it gets shot out it now ends up having more speed because it has been accelerated more number of times. That makes sense to me. Okay, now let's move on to the more interesting one. Why doesn't the electric field matter? What's really going on? So again, let's think about this. Let's now get rid of this. Let's think about what would have happened if I were to say increase the strength of my electric field over here. How would things be different? Well, if I were to increase the strength of my electric field then every time a proton gets accelerated from here to here it would now have more speed compared to earlier. But if it has more speed compared to earlier it'll end up having more radius. Now the magnetic field is not changing. It would have more radius compared to earlier. What that means is what would my new spiral look like? Can you now think about what would my new spiral look like? Since it'll have more speed, more radius compared to earlier, my new spiral would look like this. Like this. Okay, I could have drawn that better. And probably it's hard to see. Let me make it easier for you. This is what it would look like. This is an ugly spiral. But the point to note is that this means that there will be less number of rounds that it would take before it reaches the maximum speed. Very quickly reach that maximum speed. Ah, so can you see there are two effects happening over here? Let me write that down because it's pretty subtle. So what are the effects of increasing the strength of the electric field? One of course, as we expected, now the protons would gain more speed per round. Sure, every time it goes to the electric field it'll gain more speed, more speed per round. But there's a second effect compared to before, okay? This is compared to before. But the second effect is it will now have less number of rounds compared to before. So less number of accelerations compared to before. And you can see these two are counteracting each other and it turns out that the two effects nullify each other and that's why increasing the electric field has no effect on the maximum speed. The same thing would happen if you were to decrease the electric field. Now it would gain less speed per round but because it is gaining less speed per round it will now have more spirals so it'll have more number of rounds, so more accelerations compared to before and again the two of them two will counteract and the maximum speed will have no effect. It'll be independent of the electric field. Such a beautiful and subtle concept that the electric field has no effect even though it's the one that is doing all the work and speeding up our charged particles. It's not the electric field that commands and tells us what's the final speed, it's the magnetic field that does it. What a subtle and beautiful effect. So the summary is the speed at which the particle gets shot out depends on the magnetic field and not on the electric field. Why? Because when you increase the magnetic field you ensure that it has more number of revolutions before it gets shot out which means more number of accelerations so more speed gained. And when you increase the electric field sure per acceleration per round you gain more speed but the number of rounds itself decreases the two counteract each other and the speed remains the same. It took me some time to wrap my head around this so if you don't get it the first time please don't be discouraged. This is really a subtle effect. A final challenge for you would be to actually see if you can derive an expression for the number of revolutions that a particle would take in a cyclotron. How would you go about it? Here's a clue for you think in terms of energies. See if n is the number of revolutions then I know after taking n revolutions I know the energy that it has gained. If I know the speed I know the kinetic energy so I know the energy it gains after n revolution. But I also know the energy it gains after one revolution. If the field is electric field is E we can figure out how much energy it gains using basic electrostatics in one revolution and then I can use that to figure out what the expression for n should be. So that's a good challenge for you to work out and from that you can actually convince yourself how n depends upon the magnetic field and the electric field.