Hey, guys. So in this video, we're going to 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 EMF. Now, what'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 φ_{B}. 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 a relationship of delta t, which is the amount of time that it takes. So there's clearly a relationship between these variables and induced EMF or 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 EMF is the rate at which the magnetic flux changes over time. Now this EMF is actually responsible for producing an induced current. And by the way, this is just v equals I r. This is just Ohm's law, which we've seen before. But this EMF, more specifically and more importantly, is related to n, 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 n | Δ φ φ t is Faraday's law and it's super important. We're going to be talking about this in the next couple of videos. Now the units for this are actually just volts because remember, at the end of the day, this is just an EMF. 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 EMF that is across these coils. So it induces some EMF, which produces an induced current.

So in other words, 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? Well, remember that the flux will change depending on 3 variables. The flux is b a cosine of theta, which means that there are 3 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 are going to go is that in all of these problems, one variable will change, whether it's b, a, or theta, while the other 2 remain constant. So what 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. Alright. 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 going to have more field lines, which means that the magnetic field has changed. So when this situation happens, is that the b field has changed, so this is a changing b field. 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, 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 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 2 are remaining 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 in 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 theta. 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 cosine of theta 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 a problem falls under. Alright. 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 EMF in the following circuit. We're told what the area of this loop of the circuit is, which by the way is in that blue line. And we're told that the magnetic field is going to change from 3 tesla to 6 tesla in 5 seconds. So let's take a look at the first part of the problem. We're supposed to figure out what is the induced e m f in the circuit. So that induced e m f is our variable e induced. So we're going to relate this to n, 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, or this 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 in the circuit, that n is just equal to 1. So n is 1 in this case, which just means that we can just replace this with a one. And we know the amount of time that it takes in order for some change to happen. That's going to be 5 seconds. So this is delta t right here. So we have that. So the key things in these problems is we have to figure out what the magnetic flux change is. So let's go ahead and figure that out. So delta φ_{B} is actually what we're trying to find. Now, we know that φ is equal to b a cosine theta, so that means delta φ is going to be delta b a cosine of theta. Right? So we have 3 variables, and we just have to figure out which is the one that's changing. Well, let's take a look. The magnetic field is changing from 3 Tesla to 6 Tesla. So that means that our b is actually the changing variable, so this changes. And then these guys, the area and the cosine of theta, these are actually going to be constants. 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 cosine of this theta 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 like flat on the page. What that actually means is that the cosine of the angle is just equal to 1. So in other words, both of these things, the area vector and the magnetic field, they point in the same direction. So we can kinda just eliminate that and that just goes to 1. 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 going to be the area times the final magnetic field minus the initial magnetic field. So now what we do is we can just plug this back into our expression for the induced EMF. So our induced EMF is going to be the absolute value of the area, and we're told that the area is 50 centimeters squared. So we actually have to do point 0 5 meters squared. And now we have to do the in the final magnetic field, which is 6 Tesla, minus the initial magnetic field 3 Tesla, and then the delta t is equal to 5 seconds. So that is going to be our induced EMF, and what we get is we get an induced EMF that is equal to 0.03 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 going to be positive. Alright. 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 of 2 ohms? So let's take a look here. So remember that we're our induced EMF can always be related to an induced current using Ohm's law. E equals IR or epsilon equals IR, because really this is just v equals I r right here. Right? So all you have to do is just move the resistance over, and so the induced current is just going to be the induced voltage or the induced EMF of 0.03 divided by 2. So we just get an induced current of 0.06, and that's going to be in amps. So these are our two answers. Alright, guys. So we're going to get some more practice in the next couple of videos. Let me know if you guys have any questions.