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Toroidal Solenoids aka Toroids

Patrick Ford
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So remember that a magnetic field at the center of a loop is given by this equation. This is if you have a single or like a few loops, it's given by new. Not I divided by two are now if you have a solenoid which is just a really really really really really long loop kind of like this. It's actually exactly that. Then the equation changes a little bit where you have new knots. I l. I'm sorry. N N. Divided by L. Okay so those are the two equations that we've seen so far for loops. Um But some of you also need to know about the steroidal solenoid which is a special kind. And all it is is you get a regular solenoid like this which you may remember has a magnetic field through the center, right? It could be to the left or to the right direction the current. And what you're gonna do think about this as a slinky that you could just like turn it like this right? You can make it look um you can turn it so that it looks kind of like this. So imagine you do that and then you run a current through it. What's the magnetic field going to look like? All the magnetic field is going to follow the center of this thing. Okay. And that's exactly what it's what a toronto solenoid is, is you're going to get a solenoid and arrange it in the donut shape. Um And so you're gonna get something like this, like a donut. Um It could literally be a donut and then you just wrap wire around it. Okay so let's wrap some wire around it, let's say I have a battery here v. That's going to produce a current and then we're gonna plug this into here. Now I have to go around the donut like this by the way, this is what it looks like right here, right? You might have seen one of these guys around. Probably not but some of you may, so it's gonna go and now it's going the back so you can't see it but it's coming back around this way and then it does this and then it's behind it. So you can't see it then it keeps going okay behind you can't see it keeps going, let's make them farther apart so that we don't have to draw so many of them, okay, keeps going like this keeps going like this, let's do a few more. Just one more cool. So it's something like this, they they're usually evenly spaced. I just kind of got in a hurry there, right? And then this cable is gonna get connected right here, okay the current is going to leave this side of the battery um and it's going to go into the solenoid by the way, a lot of times you don't get shown this you're just gonna get shown the direction of the current but I just wanted to draw a battery there so you get an idea of what this might of how this actually gets arranged. Okay, now the first thing I want to talk about is the direction, so there's gonna be a new equation here um for the solenoid, but I first want to talk about for the tortoise annoyed, but I first want to talk about the direction. So remember I told you that B is going to be this way it's gonna follow, it's gonna keep going through the inside. So the magnetic field is actually gonna be this way here. Okay, this way, now it has two possibilities. It could be clockwise or it could be counter clockwise clockwise or counter clockwise depending on which way the direction of the current floats, depending on the flow of the current. And it's gonna be similar to this, that what you're going to do is you're gonna look at where the current is entering which is this piece right here, and you're gonna follow it with your hand but just now, sort of at a weird angle. Okay, so what I'm doing here is I'm following this wire. Okay, and what it first does, it goes around the donut and towards the back because he wants to go behind, right? So the first thing that my hand does is this meaning my fingers are going into the page and when I do that, my thumb sticks out to the left. Okay, now this is actually kind of at an angle like this. So my thumb is going this way long story short, the current immediately in the beginning here is gonna go in this direction. Okay, the current's gonna go in this direction and I'm sorry the magnetic fields. I'm sorry the magnetic field, this is the magnetic field is going to go in this direction, meaning it's gonna be looping in a clockwise motion. So the direction of the magnetic field is this way. Okay, the magnetic field is this way? All right. But you have to be very careful. It has to do with the drawing. These questions are annoying because you have to like inspect the joint and figure out what to do. So what I want to do now is I want to draw it differently. I'm gonna draw the same battery with the current coming out the same way. But I'm gonna show you right current comes out this way. I'm gonna show you how destroying could have looked different. You could have gone under first and then over and then under over here and over and then under and over. And I'm just gonna keep drawing a bunch of these. Right? These don't matter as much and then this would have connected over here. So still going to the left, just like here. But look what the first motion looks like. If you're here, whoops! If you are here look at what your first motion looks like. Get your hand ready. Get excited, right? So you're gonna go behind the donut and back you're grabbing Instead of grabbing a donut like this, the first grab is gonna be like this. So when you do that your thumb sticks out to the right. So that means by the way you can think of this as a single loop that just shoots out to be that way. Except that there's a bunch of loops. So it's gonna shoot a. B. Oh that was terrible, it's gonna shoot a B. Over here. Oh Lord pretended that's a straight line. Um And this is the direction of my magnetic fields. So hopefully you get a sense here of how even though these look very similar, the direction of the magnetic field is different depending on this, right? And there's probably not a really easy rule to memorize like if the first wires under or whatever or the first wires over, you just got to grab it. Okay go for the grab, grab that donut. So that's the hard part. The equation tends to be pretty easy. Um So the equation is mu not I. N. Divided by two pi little R. All kinds of weird stuff going on here. Notice that pie is back right? When we did regular loops there was no pi pi is back. And remember that you have a little R. And not big R. That's also different from other loops. Little R. Is a distance and in this case it's a distance from the center. Okay And little R. Is radius but it doesn't matter because that's not what we have here. So the radius of this thing doesn't really matter so much. Okay now super important is that the magnetic field only exists between the inner and outer radius. So we haven't talked about those guys. So here's the center. This distance here Is my inner radius or r. one and this here is my outer radius R. Two. And remember a magnetic field lives inside of the loop. If the loop tightly grips this bagel um or this donut then there can only be magnetic fields in here. Okay, Magnetic field out here is zero magnetic fields in here is zero B. In equals zero. It only exists in this blue circle. Super important. So it only exists between R. One and R. Two. Okay sometimes you hear this thing called a mean radius mean it doesn't mean that it's a bad person. Mean just means average radius, right? Which is R. One plus R. Two divided by two. Sometimes you might see that it's just the average. So for example if our one is four and R. Two is um eight then our or our mean would be in this case six obviously. Okay now the thing that's the only thing that R. One and R. Two are good for the only reason they're important is so that you know that you can only have a B. Field at those distances away. Okay we're gonna do this example right away here and you're gonna see what I'm talking about. Um We have a 300 turned to royal solenoid. So n equals 300. And it has inner and outer radius radius 12 and 16 centimeters. So R one is 10.12 and R. Two is 20.16 m and the current is five and I want to know what is the magnitude of the fields At these three different places. So a what is the magnitude of the field at the center of the solenoid? So here's my toronto solenoid with the winding around it. The center is right around here. Well, the center remember inside you have no beef fields. You have no beef fields. Okay, so the answer in that case is always going to be zero. The magnitude of the of the magnetic field at the center of a two royal solenoid is always zero. Okay, the center of a two royal aura to roid is always zero. Okay. Um If you weren't, if you weren't paying attention, you might have tried to use the radius of this thing. The inner radius or the outer radius or the mean radius into this equation here, but this are is not a radius. This are is a distance. Um And we specifically know that this only exists between this range, meaning you have to be by the way at the center means that the distance is zero, right? R is distance from the center. If you are at the center, your distance between the center the center zero and you can only have it between 12 you can only have a b. Between these two distances. Cool, awesome. What about at 14? So again what you gotta do is you have to see and if it falls between this range it does fall between this range. So you're gonna have a magnetic field B equals. We're going to calculate that before we calculate that. I want to talk about c real quick just to get it out of the way. um 20 cm is outside of this range. Okay this one was inside um outside of the range but inside the tor Oid this is outside of the range and outside of the tor Oid in other words be will also be zero. Okay this is a trick question. Um The only one you have to solve is this guy here and we're just gonna use the equation mu knots I n divided by two pi R. Where little R is this distance here? It's not any of the other numbers. Okay. This distance happens to be the main radius but I'm not using it because it's the main radius. Okay. If I was asked 13 centimeters away that's not the main radius I would have plugged in a 13 for our okay. Um so mu not is four pi times 10 to the negative seven. The current is given as a five. The number of loops is 300 divided by two pi and then r is 14 centimeters, so 0.14. Um Some stuff cancels here, but if you plug this whole thing on your calculator, you get 2.14 times 10 to the negative three. Tesla notice that this problem didn't ask you about direction, but that's cool because we talked about direction exhaustively earlier. That's it for this one. Let's keep going.