Alright, guys. Let's work this one out together. So we're told that you have a radius, some ring of some radius that is in the presence of a magnetic field, and the ring begins with its plane parallel to the magnetic field and then ends perpendicular. We're gonna talk about it just in just a second. We're supposed to be figuring out what is the change in the magnetic flux. Now the keyword in this problem is the change. So in other words, we're looking for some magnetic flux, so that's ϕb, but we're looking for the change and we represent that by a delta symbol, remember? So that means that we just need to find out what the ϕb final is minus whatever ϕb initial is. So we're basically going to have 2 separate sort of initial and final states, and we're going to have to figure out those 2. The initial and the final, I'm going to go ahead and draw diagrams for those 2 right here. So let's say the magnetic field just happens to point to the right. It doesn't really matter where we choose to draw it as long as we're just consistent. Right? We have the magnetic field that points in this direction, and I'm just going to assume that it points to the right like this. We have the exact same magnetic field over here. Now what happens in the between the initial and the final is that the ring is going to rotate like this. And we're told that this ring begins with a plane parallel to the magnetic field. Now what that means is that the actual ring itself, the lines of the ring, the plane of it is parallel to the magnetic field, not the normal. And then what happens is, finally, it ends up with the plane of the ring perpendicular to the magnetic field. So that means it's actually going to be vertical like this. We're supposed to figure out what the change in magnetic flux is. Alright. So we have our ϕb final, our ϕb initial. So we can go ahead and write out the equations for those. That's Δϕb is equal to B·A·cos(θ) but what happens is we know that the angle is changing, whereas B and A are going to remain constant. So these guys don't actually change. These are constants right here. So, let me write that. So, these are constants. And what's actually happening is that we have an angle, so θ final and then minus B·A·cos(θ initial). And these angles right here represent the angle between B and A. So let's go ahead and find out what those are. Let's start out with the final case. Right? So we know that if the ring sort of sits vertically, then that means the plane of it is going to be, or sorry. The normal is going to be perpendicular to that surface. So that means it's going to point out to the right. There's always one we want it pointing alongside the magnetic field. So, what's the angle between B and A? Well, if these things point in the same direction, then that means that this θ here, which is actually θ final, is going to be equal to 0 degrees. Right? Because they point in the same direction. And so, the cosine of 0 degrees is just equal to 1. Now, what that does for our equation is we say, okay. Well, if the cosine of 0 is 1, then that means this term right here just equals 1 and we can sort of just, you know, get it out of there because one doesn't do anything to the equation. So that means this is actually just equal to B·A. Now we have to look at what the initial angle is. So, we have a minus sign right here. Right? So let's look at what the initial angle is. Well, the area is, again, perpendicular to the surface, so it's going to be pointing up in this case because now we have the ring sort of lying flat like that and it's always going to be perpendicular. So this is our area vector and the θ angle represents the area between B and A. So that's actually going to be this guy right here. Now, hopefully, you guys realize that this angle right here, θ initial, is actually equal to 90 degrees. And if you go ahead and work out what the cosine of 90 degrees in your calculator is, you're going to get 0. So what happens is this whole entire term gets wiped out because this is just equal to 0. Okay? So that means that the change in the magnetic field is literally just B·A and then minus 0, because we have nothing there. So, let's just go ahead and multiply those 2 out. So we have a 0.6 Tesla magnetic field, and then the radius is 2 centimeters, so that we have ∏(0.2 ²). And so if you work this out, you're going to get 7.54×10-4 Weber's. Okay? So that's the change in the magnetic field. Let me know if you guys have any questions, and I'll see you in the next one.
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30. Induction and Inductance
Magnetic Flux
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