30. Induction and Inductance
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Hey, guys, how's it going? So in this video, we're gonna talk about a concept called magnetic flux. Now, this is gonna be extremely similar to how we talked about electric flux. Is we talked about KAOS is law and electric fields. Now, this video is gonna be extremely similar to that. So if you need a refresher, go ahead. Watch that video again. Let's check it out. So when we talked about electric flux, it was basically just the amount of electric field lines that passed through a surface. So if you have these electric field lines that air passing through the surface right here, then we have three variables. We had the amount of electric field lines right here. We also have the area or the size of the surface that it was passing through. So that's the area. And then we had also the angle between them and the electric flux was basically just the relationship between all of those three variables. E eight times co sign of theta. We also have the units associate ID. Well, the idea is that magnetic flux is the exact same thing, but instead of the electric field, it's the amount of magnetic field that passes through a surface so literally this diagram is the exact same. The only thing that's different is that we have a B field of magnetic field instead of an electric fields. And so what we do is we just replace these letters or the E with a B. So that means that the magnetic flux, which is given by five B, is just going to be be times a times the cosine of the angle where they have the strength of the magnetic field. We have the actual size of the object in this data. This angle right here represents the angle between B and the normal of the surface. So remember that the normal of the surface is that if if I have the back of my hand right here, this pen sort of points in the perpendicular direction. So in any surface, the normal is always going to be perpendicular to that surface. And if you have that, then this angle basically represents the angle between the B field, which points out this way and the area vector. So that's gonna be right here. That angle right here. Okay, so there's the last thing that we need to know is that the magnetic flux is are always going to be positive, or as when we talk about electric flux is it could be positive or negative, depending where we're going outside or inside. So magnetic fields or magnetic flux? Is there always gonna be positive? That's basically it. So let's go ahead and check it out because it's gonna be very similar to how we dealt with electric flux is cool. So what is the magnetic flux through this square surface that's depicted below? So we have the strength of the magnetic field. We have a square surface that's over here. We're sort of looking at it from the side, and we're told with side length is so we're gonna start off with our equation. The magnetic flux, which is given by five B, is just going to be be times a times the cosine of the angle in which this angle represents thes sort of angle between B and A. Now, if we're looking at this object here on the surface, then the normal vector is always going to be perpendicular to that surface. In other words, is going to point towards the right. So if you have a surface like this, then that means that the perpendicular vector it's gonna point outwards like this. You're always also going to assume that it points sort of along the same direction as the Byfield because then you'll get a positive number there. So you have sort of a choice. You could have written it like this, but we're actually always gonna stick to the rights or alongside the magnetic field lines because that would be perpendicular so that that would be positive. Okay, so we have with the electric, we have the magnetic field strength what it is. And then how about this area? Well, this area right here, the area of a square is just given as side squared. So in other words, if the side length is 5 m, then the area is just five squared, which is 25 m squared. OK, so we have the area. Now we just have to figure out the cosine of the angle. So we're given this angle of 30 degrees. The problem with this angle is that this is actually the angle between the plane of the surface and the magnetic fields. It's not actually the angle between the area and the magnetic field. This is the area. So that means that this right here is the angle theta that we need. So it's not going to be this 30 degrees, but we actually need is we need this angle right here, which is actually going to be 60 degrees, because 60 plus 30 would be this right angle right here. All right, so you have to be very careful with how you choose that angle. So let's go ahead and plug it in. So we've got this five b is equal to We've got 50. Tesla, and then we're gonna multiply this by 25 now we have the co sign of 60. If you work this all out, you should get 600.6 to 5. And the units for that are actually Weber's. So I kind of talked. I think I forgot to discuss it up there. But that's the unit for these. It's called Weber's. Or that's equal to a Tesla Times A meter squared. Alright, so that's it for this one. This is the answer. And, uh oops. Let me know if you guys have any questions and I'll see you, man. And I'll see you guys in the next one
A ring of radius 0.5m lies in the xy-plane. If a magnetic field of magnitude 2 T points at an angle of 22° above the x-axis, what is the magnetic flux through the ring?
Magnetic Flux of a Rotating Ring
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Alright, guys, let's work this one out together. So we're told that you have a radius of some ring of some radius that is in the presence of a magnetic field. And the bring begins with its plane parallel to the magnetic field and ends perpendicular. We're gonna talk about the just in just a second we're supposed to be figuring out is what is the change in the magnetic flux. Now, the key word in this problem is the change. So in other words, we're looking for some magnetic flux. So that's fi B, but we're looking for the change and were represented by a Delta symbol, remember? So that means that we just need to find out what thief I be final is minus whatever fi be initial is. So we're basically gonna have to separate sort of initial and final states. We're gonna have to figure out those two. So the initial and the final I'm gonna go ahead and draw diagrams for those two 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 So we have the magnetic field that points in this direction, and I'm just gonna assume that 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 gonna rotate like this. And we're told that this ring begins with the 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. All right, so we have our five b final are five the initial, so we can go ahead and write out the equations. For those that's Delta Phi B is equal to this is gonna be be a and then we have the co sign of feta. 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 air constant right here. So let me write that. So these are constants, and what's actually happening is that we have an angle so theta final and then minus b a times cosine of theta 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 with the final case. Right. So we know that if the ring sort of sits vertically, that means the plane of it is gonna be or sort of the normal is gonna be perpendicular to that surface. So that means it's gonna point out the right there's always wanted on appointing alongside the magnetic field. So what's the angle between B and A. Well, these things points in the same direction. That means that this theta here, which is actually fate a final, is gonna be equal to zero degrees, right, because they point in the same direction. And so the co sign of zero degrees is just equal to one Now, what that does for our equation is we say Okay, well, if co sign of zeros, one That means this term right here just equals one. And we can sort of just get it out of there because one doesn't do anything to the equation. So that means this is actually just equal to B times A. Now we have to look at what the initial angle is. 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 gonna be pointing up in this case because now we have the ring sort of lying flat like that, and it's always gonna be perpendicular. So this is our area Vector and theta angle represents the area between B and A. So that's actually gonna be this guy right here. Now, hopefully, you guys realize that this angle right here, feta initial is actually equal to 90 degrees. And if you go ahead and work out what the co sign of 90 degrees in your calculator is, you're gonna get zero. So what happens is this whole entire term gets wiped out because this is just equal to zero. Okay, so that means that the change in the magnetic field is literally just be times a and then minus zero because we have nothing there. Let's just go ahead. Multiply those two out. So we have a 20.6 Tesla magnetic field, and then the radius is two centimeters so that we have We have pie times 20.2 squared. And so you work this out, you're gonna get 7.54 times 10 to the minus four, and that's gonna be 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.