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Current in a Circuit with a Changing Magnetic Field

Patrick Ford
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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