 Suppose we have a magnetic field directed into the screen, maybe due to some giant magnets, and let's say somebody comes along, takes a conducting rod of length L, and starts spinning it this way, about one of its ends, with some speed. Our goal is to figure out what is the induced EMF, or what is the potential difference induced between the ends of this rod due to the spinning motion. Now you may be wondering, why are we spinning a rod in a magnetic field? Well, that's what physics is all about. We come up with some unrealistic situations, but to help you strengthen your concepts. All right, so the rod is spinning, let's say it's spinning with some angular speed omega. The immediate question I have is, why would there be a potential difference induced? What's going on over here? This is something we've talked about in previous videos called Motional EMF, and the whole idea is, let's take, let's quickly recap. If you have a rod, which is, let's say, not spinning, but let's say it's moving in a magnetic field, then this conducting rod has electrons inside of it, and now these electrons are moving inside a magnetic field, and we know that moving charges experience a Lorentz force given by f equals q times v cross b. And so these negatively charged electrons inside the conductors are all going to experience a Lorentz force. Can you quickly pause the video and think about what direction that Lorentz force would be in? Okay, so the force would be in the direction of v cross b. So v is upwards, and b is into the screen, so if I were to draw that to show you from some angle, upwards, into the screen, then when you do v cross b, you have to take your right hand and you use your four fingers to cross it in this direction, v cross b. And when you do that, you get your thumb points towards the left, which means the force is towards the left. But that's for a positive charge, because electrons are negative charge, this means the force would be towards the right. And so the Lorentz force acting on them would be towards the right. And as a result of that, all these electrons would, so all these electrons will start migrating to the left side, giving you a negative charge, and leaving behind positive charges on the left side. And there you go, because there's a charge separation, there's an electric field generated, induced, and that produces a potential difference. And in the previous video, we derived the expression for that induced potential difference. We called it the motional EMF, but it's the same thing. It's actually the potential difference to charge separation. And we derived it to be the product of b l, the length of the rod times v, where v is the speed of the rod. The simple expression only works when the velocity is perpendicular to the magnetic field, which is the case that we are dealing with right now. And if you need clarity on where this comes from, feel free to go back and check out our videos on motional EMF and videos on induced current inside a wire moving inside a magnetic field. Anyways, the same concept applies over here. Even here, the rod is moving in a magnetic field, charge separation will take place, and as a result, there'll be a potential difference generated. The problem over here is, instead of moving linearly, this rod is rotating. So the problem, the complication over here is when things are rotating, different parts are moving with different speeds. So again, let me show you this. Let's look at the animation. So as we spin this, notice the center hardly moves. It's at rest. Look at that. Center hardly moves. However, if you look at the tip, the tip is the one that's moving the most distance in a given time. And as a result, it has the highest velocity, which means as you go away from the center, particles are moving faster and faster. So the way to think about this is different parts of this rod are moving with different velocities. And so what do we do now? How do we figure out what's the EMF? What do I substitute for the velocity? Well, the way to think about this, at least one of the ways to think about this is instead of trying to figure out what the EMF, what the voltage is, potential differences for the entire rod, we can divide this rod into tiny, tiny pieces. Here we go. And because each piece is having its own velocity, we will see that each piece will generate its own EMF or the potential difference across each piece will be different. It'll be very lower here. And as I go towards the right, the potential difference keeps increasing, increasing more, more, more, more, more, maximum over here. But what I want to calculate is the total potential difference from here to here, which means I have to add up all these potential differences. In other words, I need to set up an integral. And it'll be a great idea now to pause again and see if you can set up that integral yourself. It's okay if you go wrong, don't worry about it, but try setting up that integral yourself. See how you would do it. Okay, let's set up the integral. So the whole idea behind setting up the integral is because different pieces are having different potential differences generated, let's just concentrate on one of these piece, find out how much potential difference that tiny infinitesimal piece generates, and then integrate it. So let's pick one piece at random. So let's say I want to concentrate on this piece. All right? This one, let's say, okay. At some distance, so this piece is at some distance from the center, let's call that distance X, the most favorite variable, okay? And now the length of this piece matters in our formula. That length would be the length of this piece. We're only concentrating. So since this is an increment in X, we'll call the length to be DX. So if you use the variable to be, say, R over here, this increment would be DR, okay? You have to use the same variable. All right, so now let's only focus on this piece and think about what's the potential difference generated over there. That potential difference, it's the same thing as the EMF, different words, but same thing, that potential difference is going to be the magnetic field B times the speed at which this thing is moving. And we don't know what that speed is, but let's just put that, times the length of that. And the length of that is just DX. And now since we want the total potential difference, we integrate this. So the total potential difference will be an integral, integral from where to where. So we are integrating with respect to X. So I asked myself, what should be the minimum and the max value of X? Well, I want to start from here. So X equals zero. And I want to end up over here where X equals L. I need to integrate this, B times V times DX. And we can remove all the constants out. So B is a constant. The magnetic field everywhere is the same. So I can pull that out. What about the velocity? Is that a constant? Is it the same for each and every piece? Not at all. The whole reason we are integrating is because the speed is not the same. So the speed stays inside the integral. And now I need to figure out, I need to figure out an expression for speed in terms of X because you're integrating with respect to X. Clearly, V is a function of X. V depends on X, right? If X is very small, V is very small. If X is very large, V is very large. So V depends on X. I need to figure out what that connection is. Can you pause the video and think about what that connection would be? And while you're at it, once you get that, try integrating and see what you end up with yourself. Okay, so the connection comes from mechanics of circular motion. We've already seen before, a long time back probably, that if an object moves some distance S in a circular path and the angle covered is theta in radians, then the connection between them S equals R theta. So if we divide now both sides by time, this becomes the linear velocity of the speed. And this now is the angles covered per second becomes omega. And so the connection becomes V equals R omega. In our case, R becomes X. And so clearly you see as R increases or as X increases, the speed increases exactly what we predict. So we can plug that in over here now. So for our piece, that velocity V is going to be R times omega. So it's gonna be X times omega times DX. So omega is a constant that comes out. And now we can integrate. We're integrating from zero to L. So if I pull that omega out, we have to integrate X DX and integral of X DX is X squared by two. So what we end up with now, and move this a little bit up. So V equals B that omega gets pulled out times X squared by two. So X squared by two. And we put the limits from zero to L. So when we substitute the upper and the lower limits and we subtract, we'll end up with B omega L squared by two minus zero. We'll end up with L squared over two. And so that is the expression for the voltage that gets induced or the EMF that gets induced or the potential difference. Basically, this is the potential difference that gets induced. Okay, shall we quickly try one more problem? And I want you to give it a shot. Very similar problem. Magnetic field, constant uniform field. And instead of a wire, this time we have a disc of radius R. And that disc is spinning. And the question is, what's going to be the potential difference between the center of that disc and the edge of that disc? Can you pause and think about what that would be? All right, we can redo all the integration if we need, but there's another way to think about this. If I just consider a tiny radial piece, like imagine I just take, concentrate on a radial piece of this disc, then that particular piece is doing exactly the same thing that Rod was doing in a previous case. It's going in a circular path about this point, which means the potential difference generated or the EMF induced in that piece is going to be exactly the same. It's going to be B omega, and the L is going to be R now. So it's going to be B omega R squared over two. And the beauty is, if I choose another radial piece like this, over here, that piece is also doing the exact same thing. It's also spinning. That should also have the same potential difference, which means if I choose various pieces, if I divide this disc into tiny, tiny radial pieces, each piece should have the exact same potential difference. So what's the answer to our original question? What's the potential difference within the center and the edge of the disc is going to be the same? B omega R squared over two. If you want to choose between this point, the center and this point on the edge, you think of the potential difference between this radial piece. If you want to consider the potential difference between center and this point on the edge, you imagine that's the potential difference across this radial piece. And so if you think of it that way, anywhere you consider between the center and the edge, the potential difference should stay the same. But a couple of questions that comes to my mind sometimes is, wait a second. All these wires are like, in this case, all these pieces are touching each other. So wouldn't that cause some charges to move and wouldn't that change things? Well, remember charges only move if there is a potential difference, right? And over here, if I go at some radius over here, let's say if I go to some radius, let me choose the same color. Okay, if I go to some radius over here, the potential here, if it's say, I don't know, maybe five volt, then at this point, the potential would be the same, five volt. At this point, the potential would be the same. So at a particular radius, the potential everywhere is gonna stay the same. And so charges will not move from here to here or here to here. There is no reason for them to move from here to here at all. And so the fact that they're touching each other is not gonna make any difference. Another question that comes to my mind is, wait a second. Shouldn't we like, I don't know, maybe add all the potential differences? And I'll ask, why should we do that? And just to convince myself, there's another way to think about this. You see, since each piece is having a potential difference, you can sort of kind of think of it as a cell, as a battery. All right, so if you imagine this to be, it's not really a battery, okay? The potential difference is coming because it's spinning, but you can imagine it's like acting like a battery. And so if you imagine these things where tiny, tiny batteries, then see how all these batteries are connected. They're connected in parallel with each other. Can you see that? Ns are all connected. And because they're all connected in parallel, in parallel, voltages don't add up. They stay the same, okay? And it's for that reason, you don't add up the voltages. And so the potential difference between the center and the edge anywhere you take is gonna be B omega R squared over two.