All right, guys. We're going to get some more practice with Faraday's law, and we're going to 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 r equals 5. We're told the initial current. And then what happens is this large loop gets disconnected from its voltage 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 part a, what we're really trying to find is not the induced EMF. We're actually just looking for what is the change in the magnetic flux. So in other words, what is the change in B times A times cosine of theta? That's going to be the first part. But before we dive into the math, let's just go ahead and draw a diagram of what's going on. We have this larger circular loop that we're told it's going to look something like that, and then we have an inner circular loop that's going to be inside of it. So, obviously, it's not going to be to scale or anything like that. And we're told that the radius of this small loop is 5 centimeters. So in other words, I'm going to say this is r_s for r small, and this is equal to 0.05 m. And then the larger ring here, which is going to have r_big, and that's going to equal 5 meters.

We have to figure out what the changing variable is in order to figure out the magnetic flux. So, let's start at the top or the left, I guess. What is the magnetic field? Where does the magnetic field even appear in this equation? We're never told anything about a magnetic field. So, we're told that the larger loop carries an initial current of 6 amps. In other words, it's going to be in either this direction or that direction. This is the current, and that's equal to 6 amps. So, where do we get our magnetic field from that? Well, remember that current-carrying wires always produce magnetic fields. So, in order to find out what direction that magnetic field is, we're going to use our right-hand rule. So, there is a B-field that is generated somewhere inside of this loop, and we're going to use the right-hand rule for this. If you take your right hand and you curl your fingers in the direction of the current, and by the way, the direction doesn't matter. So we're actually just going to go ahead and choose clockwise to be the direction. The truth is it doesn't matter because we're only looking for the change in the magnetic flux. So, I'm just going to curl my fingers in the direction of that current, and then my thumb should point away from me. It's going to 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 going to be constant inside of the wires that point inwards.

Now what about the formula for the magnetic field? That magnetic field is going to be μ₀ times I divided by 2 times r. But which radius are we going to use? We're talking about the larger loop, so we're actually just going to use this r_big equation right here. By the way, this equation we've used before for the center of a loop of current. So, you should have this in your notes somewhere.

Now that we have the magnitude of this B-field 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. What's happening is that this larger loop is disconnected from its voltage source, and the current is going to decrease to 0 over some time. So, what's happening during these 20 microseconds? Is the area changing, is the angle changing, or is the B-field changing? The area is going to be the area of the smaller loop, but we're not told anything about that changing area. That area is just equal to radius small, which is 5 centimeters, and the angle doesn't change either. We're 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 going through the larger loop. So, as this current I decreases, the magnetic field is also going to be decreasing. As the current around this loop starts to go to 0, the magnetic field strength will also start to decrease. So B is our changing variable. So, that means that our δφ is going to be. We can pull these out of the δ, so we can pull these to the outside like that. And we're going to have A times the cosine of theta times δB, which is equal to B_final minus B_initial.

As for the area, so we have δφ is equal to the area, and the area we're going to take is the area of the small loop because that is the flux that we're trying to evaluate. So, we're actually going to use π times rs squared. Now, how about the cosine of the angle? Well, the normal of this 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, then the cosine of this angle is equal to just 1 because θ is equal to 0.

And now what we have is the magnetic field final. Well, remember that that magnetic field has an equation. And because we have a final current and an initial current, we're going to substitute these two equations. So, I'm going to have μ₀ and this is going to be I_initial divided by 2 r_big minus μ₀ I_final divided by 2 r_big. So, this is actually the expression for the changing magnetic flux. We're going to have to evaluate the final magnetic field and the initial magnetic field. Well, we're told that the current will steadily decrease to 0 over some time. So that means that this whole entire thing will go to 0 because I_final is equal to 0. So that whole entire term drops away.

Alright. So that means that the change in the magnetic flux is equal to now we've got π times the radius, the small radius which is 0.05 m squared, and then we have times negative, μ₀ is just equal to 4π times 10 to the minus 7. That's the magnetic permeability. Now, the initial currents is equal to 6 amps. And the 2 times the r_big, the r_big is just 5 meters. So if you work this out, you're going to get that the change in the magnetic flux is equal to negative 5.92 times 10 to the minus 9 and that's Webers. So that's the change in 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 in the magnetic flux.

Alright, so this is the answer to part a, what's the change in 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 EMF and use Faraday's law, and that's going to be the n, which is the number of turns in this loop, times the absolute value of the change in the magnetic flux divided by the change in time. Now, this circular loop here, we're not told that it has any turns, so we're just going to assume that the amount of turns, this n, is just equal to 1. So, this is just a 1. And the induced EMF is just going to be the absolute value of the change in the magnetic flux, which is 5.92 times 10 to the minus 9. And we're going to divide that by the change in time, which we're told is 20 microseconds. So, that's 20 times 10 to the minus 6, and you have to take the absolute value. So when you do that, when you work this out, you're going to get 2.96 times 10 to the minus 4 and that's in volts. You're going to 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 current on the smaller loop. So, remember, when you're trying to find out what an induced current is, you have to relate that back to the induced EMF divided by by the resistance, which comes from Ohm's law.

So, the induced EMF is 2.96 times 10 to the minus 4, and the resistance is equal to, let's see. We've got 10 milliohms, so that's actually 0.010 and that's in ohms. By the way, this is a 4 that's volts. So, this should give us a current of 0.0296 amps. So, that's just kind of proportional or just roughly equal to 0.03 amps. Okay, guys. There's kind of a long problem. There were a lot of steps, but if you work the steps out and you kind of just work backward in the magnetic field and the currents and use Faraday's law, you should be able to figure it out. Let me know if you guys have any questions.