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Emf induced in rod traveling through magnetic field

Video transcript

- [Voiceover] I have an interesting set up over here. I have a magnetic field that is constant, and it's going straight out of the surface of this loop. The magnitude of the magnetic field at any point of the surface is going to be 'B'. What's interesting here is this loop that we have, This right part of the loop is movable. You can imagine it's a cylinder that can roll to the right, and the magnitude of it's velocity, we're going to say is lowercase 'v'. And this cylinder, lets say it has length 'L' Given that, you can see that we're going to have a change in magnetic flux. Why are we gonna have a change in magnetic flux, or a change in magnetic flux through the surface? Well if this thing is moving to the right, when it's speed is 'v' it can be any units, meters per second, or whatever it is. Even though the magnetic field itself is constant, you're actually going to be changing the area, the area that is, I guess you could, contained by this loop, and so that is going to give us a change in flux And if you have a change in flux that is going to induce an electromotive force, or it's going to induce a voltage in this loop which will cause a current to flow. Let's think about what that electromotive force that's going to be induced is going to be. I'm just gonna rewrite Faraday's law right over here. Faradays law: negative 'N' times our change in flux, and we're talking about our change in flux over change in, let me write that a little bit neater, our change in flux over change in time. The 'N' is the number of loops we're talking about. So in this case, 'N' is just going to be one and the negative, I've already complained a little bit about in previous videos. This is really just reminding us the math, when you're not using proper vector mathematics, this just reminds us that this 'Emf' is going to cause a current to go through this loop, and the magnetic field that is induced by that current will go against the direction of our change in flux. That's just something to remind us. What we really care about is our change in flux over change in time. What is that going to be? Let me just write it here, our change in flux over change in time. Our change in flux in this case is going to be our change in our, the magnitude of our magnetic field that is going perpendicular to the surface times the area of our surface over our change in time. What is this going to be equal to? 'B' is constant, it's not going to change over the time that we care about. So the change in 'B' times 'A' is really going to be 'B' times our change in area. 'B' times our change in area over how much time goes by. So what is our change in area going to be? Let's say we let delta 't' , let's say delta 't' happens, what is going to be our change in area? Let's let delta 't' go on. If delta 't' goes on, this cylinder, or if we wait for 't' units of time, or if 't' units of time go by, let me do this in another color, I'm gonna use this color right over here. How far will we go after delta 't'? We know the magnitude of our velocity if you multiply the magnitude of your velocity times your change in time, that will give you your distance. Your change in area is going to be the amount of distance it's travelled times the length of the rod, which is just that. Our change in area, which is that area right over there, our change in area over that time is going to be the distance this rod goes. Notice this rod is going in a direction that is perpendicular to the direction of the magnetic field that's an important thing to realize. We have our change in area is going to be this dimension which is how far the rod travels times the length of the rod, that's how much area we gain, that's how much our area increases, times the length of the rod, that's our change in area. We can substitute that back over here. This is going to be equal to, we got our 'B', we have our change in time, and our change in area we just said is going to be, Let me just write it this way, I could write is as the length of our rod times the magnitude of our velocity, or our speed times our change in time. Well, change in time divided by change in time, those cancel out. Our change in flux over that time or our average rate of change in flux we see simplifies to the length of our rod times the magnitude of our velocity, or our speed, times the magnitude of the magnetic field that is going perpendicular to the surface. This is something that you will see many times in your physics class, this whole notion of if your rod going in a perpendicular direction to a magnetic field, it induces an electromotive force of 'LvB' This is where it's coming from, it's coming directly from Faraday's law. You can say, okay, if that's happening, what direction is the current going to actually flow in? If the magnetic field isn't changing, but since the area is increasing, the flux is increasing in the upward direction. You can say the flux is increasing in that direction. The current that gets induced, and the magnitude of the current is going to be, based on how much resistance we have, but the current is going to induce a magnetic field that goes against our change in flux. Let's see, if the current went that way, what will happen? I'm going to take my right hand out, put my thumb in that direction, and if I were to loop my fingers around, this would, if we went that way, this would describe, this would induce a magnetic field that goes like this. That actually would enhance the flux going through the, or change in flux, going the same direction of the change in flux, We've already talked about this, would violate the law of conservation of energy. The current's gonna go in the other direction. It's gonna go, in this case is going to go clockwise, because the current is going to induce a magnetic field. Once again, I could take my right hand out, and try to draw my right hand, and we can see now, the direction my fingers go in are that way. Notice we're looking at the inside of the surface, the surface area was increasing the flux in the upward direction, getting more of the magnetic field in the upward direction, being contained in the area. The magnetic field induced by the current that is induced, or that current that's being caused by the electromotive force and the magnitude of the force is going to be dependent on our resistance, that's going to go in the other direction, it's going to go down that way. Or it's going to induce a magnetic field that is gonna go downwards. So the current needs to be going clockwise.