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Faraday's Law

Patrick Ford
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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.