30. Induction and Inductance
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Hey guys. So in the last two videos we saw that a changing magnetic flux through a surface produced an E. M. F. And that was called Faraday's law. Well, in this video we're gonna talk about the direction of those induced E. M. S. And currents. So let's check it out. So again, Faraday's Law gives us the magnitude of the induced E. M. F. And the current. So in other words, we could relate the E. M. F. To the current by using Homes Law. But one of the things we haven't talked about yet is the direction. So we're gonna use another law called Lenz's Law to find the direction of the induced currents. And basically what lenses Law says is that the direction of the induced current creates an induced B. Field that opposes any change in magnetic flux. So the key words here are opposes the change. So whatever the change in the magnetic flux is, lenses Law says that the direction of the induced current is going to set up something that counteracts that change. So we're always gonna be looking for what the system is doing and then we want to reverse that change. Now we're gonna be using magnetic fields and we're also gonna be using directions. So we're gonna be using a right hand rule for circular currents. So, if you use your right hand, so you take your right hand like this, our thumb is going to point in the direction of the B. Field because that's gonna be the straight line. And so that's gonna be our magnetic field here and then what happens is our fingers are going to wrap around in the direction of the induced current. So that's the right hand rule we're gonna be doing for this. We're gonna take it really slow in each one of these examples, because I really want you guys to understand this now. In fact, you might actually see Faraday's law represented as epsilon equals negative end times delta fire for delta T. Where this negative sign. Remember negative science and physics just usually indicate a direction. This this negative sign is really what lenses law is all about. It just says that whatever the change in the magnetic flux is, the epsilon is gonna be the negative that in the opposite direction. So this sort of wraps up Faraday's law and Lenz's law all into one neat equation. Okay, so let's check out a quick example here, we've got a bar magnet that is moving downwards and something we saw in the first couple of videos we saw that a moving bar magnet will generate an E. M. F. Well now we're actually gonna talk about the direction. So we have a bar magnet that's moving downwards. So remember that the north pole of a bar magnet means that the magnetic field lines go down here like this. So they actually form these little loops like that. So that's what a magnetic field looks like from a bar magnet. Now, what happens is through the loop itself, the magnetic field is pointing downwards because these magnetic field lines sort of spread out like that. Right? So that means that the magnetic field through the surface points in the downward direction. So these are gonna be the steps that we follow for every single one of these problems. So the magnetic field is gonna point downwards. Now what happens is the change in the magnetic flux comes from the fact that the bar magnet is moving with some velocity. So as the bar magnet is moving downwards, this magnetic field is getting stronger and stronger through the loop. So that means that the magnetic flux which remember depends on B. Times A. Times cosine of theta which the variables is changing. Well that's actually gonna be our magnetic field variable. That is changing because it's getting stronger. So what happens is this magnetic field gets stronger and the change in the magnetic flux is positive. So lenses Law says that there's gonna be an induced current that wants to counteract that change. So that means that for the direction of be induced, what we have to do because we have to figure out what direction would counteract a change what what direction of the magnetic field induced will want to bring the system back to the way it was before? Well, the magnetic field changed from here and then it got stronger in this direction. So that means that the induced magnetic field sort of wants to fight that change. So this is the direction of our be induced. It points upwards to counteract that change the magnetic flux. So that means that the be induced is going to be upwards. Now. What does that tell us about the direction? Well in order to get the direction from a magnetic field we're gonna have to take our right hand and use the right hand rule. So here's what I want you guys to do. I want you guys to follow along with me as I do this. So I want you to put your hand over your page like this and then we're gonna change it so that our thumb points in the direction of that magnetic field induced. And then I want you to wrap your fingers around like that. So what you should see. And I want you guys to do this on your page every single time. Is that your fingers sort of curl away from you on the right and then towards you on the left. So what that means is that if we actually take a look at this magnetic field so what this magnetic field is doing, the induced current is gonna go in this direction and then around this way sort of on the front side, right. So that means that if you were to look at this from the top, this actually is going to be a counterclockwise currents. So this is gonna be counterclockwise. Alright. And that's how lenses law works. We're gonna check out another example in which we don't have a bar magnet that's moving but we have a loop that's moving in or out of a magnetic field. Okay. So now what happens is we have a bunch of magnetic field lines that are passing through the surface but the loop is sort of going outside of that magnetic field. So we'll start off with the same two questions where is the magnetic field pointing through the surface? Well, clearly it's pointing downwards so that we're gonna do that. That's downwards. Now. How about the change in the magnetic flux? What's the change of magnetic flux? We'll remember. This depends on three variables. So we have B. Times A. And then cosine of theta all the magnetic field strength remains the same. We have the same amount of magnetic field lines. But the area is actually changing in this one because the amount of area that the magnetic field passes through is this, you know, big surface here in the initial case and then it's the smaller surface here in the final. So what happens is this is our changing variable which is the A. And this area is getting smaller. So that means that the change in the magnetic flux is actually negative. So this is gonna be negative right here. So the induced field. So the B. Field that's induced wants to counteract that change Now the magnetic field points downwards but the flux is getting smaller. So the magnetic field wants to do the opposite of that. So it actually wants to induce a field that is going to sort of strengthen that weakening magnetic field in this direction. So it sort of wants to bring it back to where it was before. So if it's downwards and getting smaller the magnetic field wants to be downwards to bring it back to the way it was before. Okay so that means that the induced field is going to point downwards. So what does that tell us about the induced currents? Well now we have to use our right hand rule again, so get out your right hand. Now put it over the page like this and what you should see is that when you point your thumb in the downwards direction exactly like what I have here and wrap your fingers around, you should see that they point away from you on the left and then curl towards you on the right. So they're gonna be doing this right. So go ahead and work that on your papers, you're gonna be doing this on your test and it's gonna look a little silly. So point your thumb downwards, your fingers curl around away from you on the left and towards you on the right. So what that means is that this magnetic field is actually pointing in this direction on the left and this direction sort of on the right side. So if you were to view that from the top, this actually would look like a clockwise current, right? So if you view it from the top it's clockwise. Alright guys, that's basically it. So we're gonna do a couple more examples because really this is all just about practice. It's very straightforward when you actually get the hang of it. Cool. So in the following scenarios we're gonna find the direction of the current, of the current induced on the conducting wires. Okay, So we got a bar magnet again. So we always want to identify the north pole and that's where the magnetic field is gonna be coming out of right. So now we have a magnetic field that points in this direction. So that means that through the surface the magnetic field points up. So as this as this bar magnet is moving upwards, how is the magnetic flux changing? We have B. We have A. And we have cosine theta. And what happens is as the bar magnet gets closer, the magnetic field is going to get stronger and stronger. So it's gonna keep getting stronger like this. So that means that the B field is are changing variable and it's going to be positive because it's getting getting stronger. So the induced field wants to do the opposite of that, it's upwards and getting stronger. So the magnetic field actually wants to point downwards to bring it back to the way it was before. So to bring it back to the original state. So that means our downward magnetic field is going to be, the downward field's gonna be induced. So we have this downward and we just get out of right hands again and we say okay well we have a downward pointing magnetic field. So that means that the current is gonna go away from you on the left and then towards you on the right, so it's gonna be doing this okay, so that means that the magnetic field lines are gonna be going this way and this way, so that's I induced and that is actually going to be a clockwise current when viewed from the top. And for this final example right here we've got a bar magnet that's moving now away. The magnetic field is going to come out of the north pole like this. So that means that through the surface is gonna be pointing downwards like that. So we have this. And as the bar magnet is now moving away from this loop the magnetic flux is actually going to be decreasing. So in this case when the bar magnet was moving toward the magnetic flux was increasing. So if it's moving away then that means the magnetic field gets weaker and the flux is smaller. So the magnetic field induced is going to want to counteract that change. So it's it's downwards and the flux is getting smaller. So the induced currents or the induced b field actually wants to bring it back to the way it was. By strengthening or sort of by reinforcing the existing magnetic field. So that means that the magnetic field induced points downward and we already know what direction that that current actually goes in. A downwards induced magnetic field produces a current that goes in this direction. Using the same exact right hand rule that we just used for the other example. Okay, so that means that this is I induced and this is going to be clockwise. Alright, so hopefully you guys got the hang of this. There's actually a pattern here if you haven't noticed. So when we had a strengthening magnetic field so in this case when the B field was downwards and the flux was increasing the magnetic field did the opposite of that here did the exact same thing when the magnetic field was upwards and the flux was positive. So in other words getting stronger, the magnetic field did the opposite. So that means that the induced magnetic field is always going to do the opposite and be pointed opposite the increasing magnetic field. And here in these situations when the magnetic field was downwards and decreasing the induced magnetic field was going to sort of reinforce that. So in other words, the induced magnetic field is always going to be directed a long a decreasing magnetic field to restore it back to the way it once was. Alright guys, that wraps up our discussion on lenses law, we're gonna do a couple more practice problems. Let me know if you guys have any questions.
An outer ring is connected to a variable voltage source. If the battery’s voltage is continuously INCREASING, what is the direction of the induced current in the inner ring, centered inside of the outer ring?
Into the page
Out of the page
Lenz's Law for a Long Straight Wire
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Alright, guys, let's work this one out together. We have a long straight wire on a horizontal surface in the X Y plane, and it carries a constantly increasing currents in the plus y direction. So now if the square loop when we need to figure out what the direction is off the induced currents. So that should automatically tell you this is gonna be a lenses law problem. So the first thing I like to do in these kinds of problems were given sort of like a three d perspective. It's just draw out sort of, what's going on here. So we have this access right here, which we're told is the plus y direction and we're told directly on the side of it on the right, that is the plus X direction. So in other words, sort if I have a straight line like this and this is gonna be my ex direction, all right, so it's kind of weird, So that means that this direction it was sort of like the Z axis. So I've got some three dimensional stuff going on here. So what I always like to do in these kinds of situations to see if I can change my perspective a little bit in sort of draw, like a parallel or a different diagram. So if I were to view this current from the side, then basically what that happened, what happens is if I were sort of view along the axis of the current, then that means that the current would be going away from me like this. And then that means that square loop would be sort of on the right like that. So I'm looking sort of, like, perfectly on the side on the access like that. Okay, so we need to find out what the direction of the induced current is. We need to use lenses law. But there's a couple things we need to figure out first. So we need to figure out what the direction is off the magnetic field to the square loop. And then we need to figure out how the the magnetic flux is changing. So in other words, well, it's Delta five B that will tell us what the direction is of the induced magnetic field. Okay, so we need to do is first we need to figure out the magnetic field is so for a straight current carrying wire. We know that there is some relationship between the current in that wire and the magnetic field that it generates. Now, you don't have to necessarily remember this, but remember that the the magnetic field for a straight carrying are straight current. Carrying wire is mu, not times I divided by two pi times are okay, so the direction of our magnetic field is gonna be given by our right hand rule. So now it happens is our thumb points in the direction of the straight thing. In this case, the currents and our fingers will curl in the direction of the magnetic field. All right, so get out your right hands on. What we're gonna dio is we're gonna point our thumbs in the direction of that current. So in other words, our thumbs are gonna point in this direction. But remember that were sort of viewing this from a weird perspective. So what I like to do in this situation is we're gonna use the side view. So in other words, we're gonna be looking at what's going on in this diagram right here. So what happens is we need to point our thumbs in the direction off the current in this case. And what you should see is that you should be pointing your thumb into the page away from you, and your fingers will be curling in the direction of the magnetic field from that wire. So what that means is that our magnetic field lines actually curl clockwise. So what that means in this sort of diagram is that in the first diagram on the rights, I have magnetic field lines that are going like this, but on the side view now, what happens is they have magnetic field lines that are sort of cooling around like this. And so we know that they're given by our right hand rule. They're gonna go clockwise or sorry, they're gonna go clockwise like Wait, no, that's right. Yeah. So clockwise like that. So what that means is that the magnetic field is gonna be pointing in this direction through the square loop. So, in other words, the magnetic field to the square loop is gonna be pointing downwards. So that is the direction of B. So that's our first question. The magnetic field through the square loop points downwards. So now What we need to do is figure out what the change of the magnetic flux is. So remember that the change of the magnetic flux is gonna be given by three variables B A and CO. Sign of theta. So what is changing as this current is constantly increasing in the wire? Well, remember that we said that the relationship between I and B is that as the current increases, the magnetic field also increases. So what happens is the changing variable in this case on our magnetic flux equation is going to be be right because the area of the square loop is not changing its always just constant right there and the direction of it. So there was the co sign of data is always going to be again straight and not changing. So what happens is RB Field is changing. So as the current increases, the magnetic field is gonna get stronger. So that means that the change in the magnetic flux is positive. So if we have a positive magnetic flux change and downward pointing, be, uh, magnetic field, then we can figure out the direction of the induced magnetic field using lenses, law lenses law says that it's going to do whatever the opposite is off that change in magnetic flux. So if it's downwards and it's increasing, then that means that lenses law wants to counter act that change by producing a magnetic field. That sort of fights that. So in other words, our magnetic field is going to point upwards like that. So now we find out what the direction is of the induced currents by using our right hand rule again. But in this case, we have to use a different right hand rule. So before we had our thumb pointing in our direction of the current because that was the straight thing. But now the straight thing here is our magnetic field. So now what happens is our right thumb is going to point in the direction of the magnetic field, and our fingers will give us the direction of the induced currents. So be careful when you're using these right hand rules, because sometimes in the same problem, you might be using two different right hand rules to solve it. Okay, so now what we're gonna do is on our page, right? So just follow exactly what we're doing. We're gonna take our right hands and we're gonna point in the direction off the magnetic field. So what happens is if we have a magnetic flux are sorry if we have an induced magnetic field that points upwards like this, then that means that our fingers will curl in the direction of that induced current. So if we're looking at it from the side view so we're still looking at the side view right here, then what happens is our induced currents is going to point sort of in the clockwise direction. So we're sorry that's gonna be counterclockwise. So in other words, if we're looking at it from this perspective in the first diagram like this, uh, it might be helpful to sort of visualize this as your thumb pointing so, like, quite a like upwards and are induced. Magnetic are sorry are induced. Current will point like this. So that means that in our first diagram are induced, current is gonna be pointing in this direction. So that's the direction of our induced current. And that actually happens to be the counter clockwise direction. So that's gonna be counterclockwise. Okay, so let me know if you guys have any questions with this and I'll see you guys the next one