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Physics

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30. Induction and Inductance

Faraday's Law

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concept

Faraday's Law

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Hey, guys. So in this video, we're gonna talk about Faraday's law, which is the mathematical equation for electromagnetic induction. This is a super important, very critical topic in electromagnetism, so pay attention and let's get to it. So we saw that changing a magnetic field through conducting loops was able to create an induced E M F. Now it's actually happening. Here is when you're changing the magnetic field through a conducting loop. So we had some loops like this, and you're changing the magnetic field. What you're actually changing is you're changing the magnetic flux. And so we saw that the higher that you change the magnetic flux, the larger the induced currents. So that's that Delta Phi be. That's that magnetic flux right there. We also saw that faster changes produced higher induced currents and higher EMFs. So that means there's also relationship of Delta T, which is the amount of time that it takes. So there's clearly a relationship between these variables and induced e. M. For in the induced current in a coil of wire. And the mathematical relationship that describes those two variables is called Faraday's Law. Faraday's law tells us that the induced E M f is the rate at which the magnetic flux changes over time. Now this IMF is actually responsible for producing an induced currents, and by the way, this is just V equals IR. This is just homes law, which we've seen before. But this E m f. More specifically and more importantly, is related to end, which is the number of turns in a coil times, the absolute value of the change in the magnetic flux over the change in time. This is Faraday's law, and it's super important. We're gonna be talking about this next couple of videos. Now the units for this eyes actually just volts, because remember, at the end of the day, this is just an E. M F. It's just a voltage. So basically, what it tells us is that depending on the number of turns, which is n in a turn in a coil of wire, if you have a magnetic field that is going through this coil that produces a flux. If that flux changes with time, then it creates an E M F that is across these coils. So it induces some E. M f, which produces, um, induced currents So in other words, it have you have some current that will go around the coil like this. Okay, so this is very, very important, and you definitely need to know this. So how does this actually work? Will remember that the flux will change depending on three variables. The flux is be a cosine of fada, which means that there are three ways to change the magnetic flux. You can either change the magnetic field. You can change the area, or you can change the angle of the magnetic field and the area. So the way that these problems they're gonna go is that in all of these problems, one variable will change, whether it's B A or theta, while the other two remain constant. So we have to do is we just have to identify what is that changing variable in each of the problems and then just go ahead and use that equation for the changing flux. All right, let's get to it. Let's get let's take a look at a couple of really, really common examples that you might see. So, in one example, you'll have a sort of loop given by this square like this you'll have a magnetic field. In this case, the magnetic field is pointing outwards like this. So you have some magnetic field and then at some later time delta T, you're gonna have Mawr field lines, which means that the magnetic field has changed. So in this situation happens is that the Byfield has changed. So this is a changing Byfield. But if you take a look here, the area of the loop has remained constant. And the angle at which the magnetic field and the area, uh, the angle between those two has also remained constant. So this is the variable that changes. Let's take a look at a different example. So now what we have is we have the same amount of magnetic field lines given the by those little circles, the area is still coming out at you. There, the magnetic field is. But the thing that's different is that the loop has now changed area. So you have some initial area of the loop, and then at some later time you have a final area of the loop. So what happens is the area in which the magnetic field line has are coming out is changing while the other two are remained constant. The magnetic field is the same, and the angle between them is the same as well. So this is an example of changing area. Now let's take a look at the third example here. What happens is we have the same amount of magnetic field lines and the before and the after, and the size of the loop is the same, so you have the same amount of area. But what's different between these is that in this case we have some angle initial that the magnetic field in the area are making. And then at some later time delta t you have some other angle. So that's data. So what's happening between here is that the angle is changing. So this is an example of changing angles. And so now what happens is your co sign of data is going to change while the other two remain constant. So all of these things are constant here. So it's your job to kind of figure out which one of these three scenarios ah, problem falls under. All right, So let's go. Let's take a look at an actual example of this and see how this works. So in this example here, we've got an E m f in the following circuit. We're told with the area of this loop of the circuit is which, by the way, isn't that blue line? And we're told that the magnetic field is gonna change from three Tesla to six Tesla and five seconds. So let's take a look at the first part of the problem. We're supposed to figure out what is the induced I m f in the circuit. So that induced EMF is our variable e induced. So we're gonna relate this to end, which is the number of turns times the absolute value of the change in the magnetic flux divided by the change in time. And so the key thing here is let's take a look at our variables. We have this loop here are the circuit and we're told that it just basically goes around once What? We're not actually told that, but we can sort of infer that because it doesn't tell us the amount of turns and circuit that end is just equal toe one So And is one in this case, which just means that we can just replace this with one, and we know the amount of time that it takes. In order for some change to happen, that's going to be five seconds. So this is Delta T right here. So we have that. So the key things in these problems, we have to figure out what the magnetic flux change is. So let's go ahead and figure that out. So Delta Phi B is actually what we're trying to find. Now we know that FIEs equal to be a cosine theta. So that means Delta Phi is going to be Delta be a cosine of theta. Right? So we have three variables we just have to figure out which is the one that's changing. Let's take a look. The magnetic field is changing from three Tesla to six Tesla. So that means that our B is actually the changing variable. So this changes and then these guys, the area and the co sign of theta these air actually going to be constant. So what we do is we actually just pull those things out of the delta. But first, we can also just figure out what the co sign of this data angle is now let's see the magnetic field points into the page, right? So into the page, like this and the area or of this circuit here is kind of like flat on the page. What that actually means is that the cosine of the angle is just equal to one. So in other words, both of these things the area vector and the magnetic field, they point in the same direction. So we can kind of just eliminate that and that just goes toe one. So that means that the change in the magnetic flux is going to be the area times the Delta B. So, in other words, this is gonna be the area times the final magnetic field minus the initial magnetic field. So now we do is we can just plug this back into our expression for the induced E m f So are induced. E m f is gonna be the absolute value of the area. And we're told that the area is 50 centimeters squared, so we actually have to do 500.5 m squared. And now we have to do the the final magnetic field, which is six Tesla minus the initial magnetic field. Three Tesla and then the delta T is equal to five seconds. So that is gonna be our induced e m f. And what we get is we get an induced E M F that is equal to 0.3 volts. And by the way, that is a positive number because the absolute value takes care of that, right? Even if it was negative or positive, it's always gonna be positive. All right, so that is the answer to part A. So part B now asks us what is the induced current If the resistor has a resistance off to OEMs So let's take a look here. So remember that we're are induced. PMF can always be related to, um, induced current using alms law equals ir are excellent, equals ir because really, this is just V equals ir right here. Right? So all you have to do is just move the resistance over. And so the induced currents is just going to be the induced voltage or the induced E m f of 0.3 divided by two. So we just get an induced current of 0.6 and that's gonna be in amps So these are our two answers. Alright, guys. So we're gonna get more practice in the next couple videos. Let me know if you guys have any questions.
2
Problem

A tightly-wound 200-turn rectangular loop has dimensions of 40cm by 70cm. A constant magnetic field of 3.5T points in the same direction as the normal of the loop. If the dimensions of the loop change to 20cm by 35cm over 0.5s, with the number of turns remaining the same, what is the induced EMF on the rectangular loop?

3
example

Current in a Circuit with a Changing Magnetic Field

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Alright, guys, we're gonna get some more practice with Faraday's law and we're gonna work this one out together. So we have a small circular loop of some radius with some resistance. It's centered inside of a larger loop with some bigger radius articles five, we're told the initial current and then what happens is this large loop gets disconnected from its vaulted source and the current steadily decreases. And we're supposed to figure out what is the change in the magnetic flux throughout this time. So for party, what we're really trying to find is not the induced e m f. We're actually just looking for What is the change in the magnetic flux? So, in other words, what is the change in B a times cosine of theta? That's gonna be the first part. But before we dive into the math, well, let's just go ahead and draw a diagram of what's going on. So we have this larger circular loop that we're told is gonna look something like that, and then we have an inner circular loop that's gonna be inside of it. So obviously it's not gonna be to scale or anything like that, and we're told that the that the radius of this small loop is five centimeters. So in other words, I'm gonna say I'm gonna say this is our s for our small. This is equal to 0.5 and then the larger ring here, which is gonna have our big and that's gonna equal 5 m. Okay, so we have to figure out what the changing variable is in order to figure out the magnetic flux. So let's take it from the top or the left. I guess. What is the magnetic field? Where does the magnetic field even appear in this equation? We never told anything about a magnetic field. So we're told that the larger loop carries an initial currents of six amps. So in other words, it's gonna be either in this direction or this direction. And this is the current, and that's equal to six amps. Okay, so what do we get our magnetic field From that, we'll remember that current carrying wires always produce magnetic fields. So in order to find out what the direction of that magnetic field is, we're gonna use our right hand rule. So there is a B field that is generated somewhere inside of this loop, and we're gonna be using the right hand rule for this. So if you take your right hand and you curl your fingers in the direction of the currents and by the way, the direction doesn't matter, So we're actually just gonna go ahead and choose clockwise to be the direction. So we're not told the direction of that current it could be clockwise or counterclockwise, but the truth is, it doesn't matter because we're only just looking for the change in the magnetic flux. So I'm just gonna curl my fingers in the direction off that current, and then my thumb should actually be pointing away from me. It's gonna look exactly how my hand looks on the screen right now. So what happens is that this magnetic field actually points into the page like this. So we have a magnetic field that points inwards, and that magnetic field is gonna be constant on the inside of the wires that points inwards. Now, what about the formula for the magnetic field? Well, that magnetic field is gonna be mu knots. Times I divided by two times are. But which radius are we gonna use what we're talking about The larger loop. So we're actually this is gonna use this are big equation right here. And by the way, this equation we've used before for the center of a loop of current, um, so you should have this into your own somewhere. Cool. So now that we have the magnitude of this Byfield and the direction now we just need to figure out how the magnetic flux changes. So we need to identify which one is our changing variable. Okay, well, what's happening is that those larger loop is disconnected from its vaulted source, and the current is going to decrease to zero over sometime. So what's happening during this 20 microseconds? Is the area changing? Is the angle changing? Or is the Byfield changing? Well, what happens is the area is gonna be the area of the smaller loop, but we're not told anything about that changing area, right? That area is just, um, equal to the radius small, which is at five centimeters, and the angle doesn't change either were not told that this ring rotates or anything like that. So what happens is our magnetic field is the variable that's going to be changing because if you take a look at this equation, the magnetic field strength is proportional to the current that is going through the larger loop. So what happens is as this current I is decreasing, the magnetic field is also going to be decreasing. As the current around this loop starts to go to zero, the magnetic field strength will also start to decrease. So be is actually are changing variable. So that means that are dealt if I is going to be, you can pull these out of the delta so we can pull these to the outside like that and we're gonna have a times the co sign of fada times Delta B, which is equal to be final minus B initial. Okay, so the area as for the area so we have dealt if I is equal to the area and the area we're gonna take is the area of the small loop, because that is the flux that we're trying to evaluate. So we're actually gonna use pi. Times are small squared. Now, how about the cosine of the angle? Well, the normal of the small loop actually points into the page as well, so if the normal points into the page and the magnetic field points into the page on the co sign of this angle is equal to just one, because theta is equal to zero. And now we have is the magnetic field final. Remember that that magnetic field has an equation. Because we have a final current and an initial current. We're gonna just substitute these two equations, so I'm gonna have mu knots, and this is gonna be I initial divided by two are big minus you. Oh, sorry. Not not I initial. This is gonna be our final. So this is gonna be I final minus mu not I initial divided by two are big. So this is actually the expression for the changing magnetic flux. We're gonna have to evaluate the final magnetic field and the initial magnetic field. Well, we're told that the current will steadily decrease to zero over some time. So that means that this whole entire thing will go to zero, because I final is equal to zero. So that whole entire term drops away. All right, so that means that the change in the magnetic flux is equal to now. We've got two pi times. Uh, the radius, the small radius, which is 20.5 squared and then we have times negative New knots is just equal to four pi times 10 to the minus seven. That's the magnetic permeability. Now the initial currents is equal to three Sorry, six amps and the two times the are big are big is just 5 m. So if you work this out, you're gonna get the change in the magnetic flux is equal to negative 5. negative. 5.92 times 10 to the minus nine. And that's Weber's. So that's the change of the magnetic flux and notice how there's no absolute value that we have to take into account because it's actually just asking for the change of the magnetic flux. All right, so this is the answer to part A. What's the change of the magnetic flux? So let's move on to part B. Part B is now asking us what is the magnitude of the induced EMF. So now we actually are going to take this e m f and use Faraday's law, and that's gonna be the end, which is the number of turns in this loop times the absolute value of the change in the magnetic flux Divide by two. The change in time. Now, this circular loop here were not told that has any turns. So we're just gonna assume that the amount of turns this end is just equal toe one. So this is just a one and the induced e m f is just going to be the absolute value of the change in the magnetic flux, which is 5.92 times 10 to the minus nine. And we're gonna divide that by the change in time, which we're told is 20 microseconds. So that's 20 times 10 to the minus six, and you have to take the absolute value. So when you do that, when you work this out, you're gonna get to 60.96 times 10 to the minus four, and that's in volts. You're gonna get a negative number, but that has to be positive because of the absolute value. So that's the answer to Part B. Now we're almost done here for the last part. We just have to figure out what is the induced currents on the smaller loop. So remember when you're when you're trying to find out what an induced current is, you have to relate that back to thean deus e m f divided by the resistance. Right? And this just comes from alms law. So the induced E M F is 2.96 times 10 to the minus four, and the resistance is equal to, Let's see, we've got 10 million homes, so that's actually 100.10 And that's in homes. By the way, this is a four that's volts. So that should give us a current off 0.296 amps. So that's just kind of proportional, or that's roughly equal to 0.3 amps. Okay, guys, there's kind of a long problem. There was a lot of steps, but if you work the steps out and you kind of just work backwards with magnetic field in the currents, use Faraday's law. You should be able to figure it out. Let me know if you guys have any questions
4
Problem

A square conducting wire of side length 4 cm is in a 2 T magnetic field. It rotates such that the angle of the magnetic field to the normal of the square increases from 30° to 60° in 2 s. What is the induced current on the wire if its resistance is 5 mΩ?

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