Radius & time period of charges moving in magnetic field

if a charged particle like say a proton is moving with some speed in a magnetic field we've seen that it will experience a force given by this expression which we call the lorentz force and the direction of this force is in the direction of v cross b and we get that using our right hand thumb rule where we cross our hands from v to b and the thumb represents the direction of the force and as a result the force on this would be in this direction and we saw that this force acts like a centripetal force making that charged particle go in a circular path like this and we've talked a lot about this in great detail in previous videos so if you need a clarity feel free to go back and check that out but we're going to do in this video is figure out the properties of this motion for example we want to figure out what's what will be the expression for the radius of this circle so what would be the radius of that circle and what how much time would it take to complete that circle and trust me what this has an application is a pretty cool application that we'll talk about in a few minutes so let's start with the radius how do i figure this out well whenever i want to calculate or when i want to connect force and any properties of motion like the radius or the time or whatever that is i go back to mechanics newton's second law newton's second law says force f equals mass times acceleration we know the force we can calculate the magnitude of the force from here mass we can take it as m what about the acceleration well this is going in a circular motion circular motion uniform circular motion centripetal acceleration i think i've said too much so can you pause the video i'm sure you're pumped right now can you pause the video and see if you can put all that together and figure out what the radius is going to be remember it's not the formula is not important what's important is how you get that that's the fun part of it so pause the video and see if you can figure this out all right let's make some space let's start with the force what's the magnitude of that force well the magnitude of this is going to be q q times what's the magnitude of v cross b well cross product is v b sine theta so v times b times sine of the angle theta and what is theta here theta is the angle between these two vectors v and b and b is into the board v is this way so that's 90 degrees and sine 90 is one so this number is going to be this number is going to be 1 that equals mass times acceleration mass is m what is acceleration this is the centripetal acceleration right acceleration is towards the center and therefore centripetal acceleration hope you remember the formula that's v squared over r and that's where the r comes v squared over r and now we can just do the algebra so one v cancels out and if i just rearrange i'll get this to be m v divided by qb so it's divided by q times b and there we go we have the expression for our radius and let's see if this let's quickly see if this makes sense to us so it's saying that if the mass increases it's saying that the radius is going to increase does that make sense well yeah if the proton had of this particle had more mass then it would have more inertia and so it will tend to have a more straight line velocity increasing the radius that makes sense it says if you have more charge or more magnetic field then the radius is going to be smaller why would that be well that's because if you have more charge or you have more magnetic field your force is going to be stronger if you have more force it tends to curve more strongly giving you a smaller radius that makes sense what about the velocity this is the tricky one velocity has effects in both direction in both cases velocity with a higher velocity you tend to have more force but you also tend to because it's also in your centripetal acceleration formula you also tend to have a larger radius if you think about this now because there is a v squared over here and you can literally see the derivation because you have v squared over here what we are seeing is with more velocity we are seeing that the radius tends to become larger you can also think of this as momentum with more momentum we tend to get a larger radius because it becomes harder to curve it all right before we go forward a quick question for you along with the proton imagine we also were to throw a deuteron a neutron is basically a proton and a neutron stuck together uh we have to throw a deuteron inside the magnetic field that's exactly the same velocity same magnetic field same velocity my question is which one would have a bigger radius the proton or the deutron or both would have the same radius can you pause the video and think about this all right well they have the same charge because neutron has neutral so there's still only same charge one protons charge magnetic field is same velocity is same which means the only difference between them is their mass the deuteron because it has an extra neutron over here will have twice or you know more mass think of it that way more mass which means that will have more radius oh you know what that means that means imagine we had thrown both a proton and the deuteron together into this magnetic field you know what would happen because the neutron has more mass it will end up having a bigger let me try and draw this properly it will end up having a bigger radius and that is really interesting i'll tell you why because protons will have smaller radius and deuterons will have a bigger radius this means we can now separate these two particles i mean think about what an amazing application we can have this means as long as we have charged particles with us and if they have different masses we can just throw them in the magnetic field and we can separate them out we can we can have some kind of a collection chamber somewhere over here somewhere over here um and the lower masses will get collected over here the ones which have higher masses will end up getting collected over here we can separate particles out wow just by using magnetic fields and you might say well okay but where would we actually use that well let me give you an actual application of this in the production of atomic bomb that's right nuclear bomb we required uh you know a particular kind of uranium isotope it turns out that when you actually mine uranium the uranium ore contains lots and lots of different kinds of isotopes and so to separate them one of the methods you can use is this you can heat up that uranium you can shoot ions uranium ions in a magnetic field and the lighter ones will get separated in one you know one bucket if you're using a bucket the heavier ones will get separated in another bucket and that's how you can separate the isotopes and then you can use the ones that are required to build a bomb and in fact that's exactly one of the methods which was used in the production of the first atomic bomb by the us and this method where we are able to now you know separate masses out and you know sort of like create a spectrum a spectrum means an arrangement an ordered arrangement according to the masses this method is called mass spectroscopy pretty cool name but it's all it's saying is that you can separate out particles with different masses okay one last question imagine this time i throw two protons but one with you know one with twice the speed as the other okay then this one would have twice the radius because radius is proportional to velocity everything else remains the same for the which means in this case let me get rid of this the second one would have twice twice the radius okay i can't see that but it's going to come over here the question i have for you is which one would take more time to complete the turn okay the time period if you're allowing them to go in circles which one would take more time again can you pause the video and think about this all right now my common sense is telling me that since the second particle is taking twice the radius it's going to travel twice the distance because circumference is 2 pi r twice the radius is twice the distance which means that's going to take longer right but wait this is not only going twice the distance it's also going twice the speed twice the distance twice the speed compared to this what's gonna happen to the time period well let's go back to our speed formula speed equals distance over time so time equals distance over speed and so if distance is doubled and the speed is also doubled but then that means the time taken remains the same of course twice the distance in twice the speed same time period so ooh an amazing another amazing thing that we are seeing is that the time period taken by both these charged particles remain exactly the same what if this was three times the speed that would be at three times the big circle but three times the speed again the time period would remain the same which means which means the time taken the time period is independent of the speed it does not depend upon the speed it also has an application in a device called cyclotron which will not talk about right now but that's an important result and let's see if we can now uh you know derive the expression for time period just by using this again i encourage you to pause for one last time and see if you can you know quickly derive the expression for time period yourself all right so the time period let's call it as t is going to be the distance that the particle travels the distance is going to be 2 pi r so that's going to be 2 pi times r and we know what r is that's mv by qb so let me quickly substitute mv by qb divided by speed which is v and then v cancels and sure enough time period is independent of the velocity so we also got the expression for time period again you don't have to remember these expressions you can just derive them so what we see is important is that the time period is independent of the velocity or the speed