**Concept:** Toroidal Solenoids aka Toroids

Hey guys. In this video we're going to talk about the magnetic field produced by special kind of solenoids called a toroidal solenoid also referred to sometimes as a toroid so let's go. Alright, 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 mu naught, I divided by 2R. 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 mu naught, I, L, I'm sorry, N divided by L, okay? So, those are the two equations that we've seen so far for loops but some of you also need to know about the toroidal solenoid, which is a special kind, and all it is, is you get a regular solenoid like this, which you may remember as the magnetic field through the center, right? It could to left or to the right depending the direction of the current and what you're going to do, think about this as a slinky that you could just like turn it like this, right? You can make it look, you can turn it so that it looks kind of like this, so imagine that you do that and then when the current through it? what's the magnetic field will look like? Well, the magnetic field is going to follow the center of this thing, okay? And, that's exactly what a toroidal solenoid is, is you're going to get a solenoid and arrange it in the doughnut shape. So, you're going to get something like this, like a doughnut literally a doughnut 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 going to plug this into here. Now, I have to go around the doughnut like this, by the way, this is what it looks like right here, right? You might have seem one of these guys around probably not but some of you may. So, it's going to going out, it's going to the back. So, you can't see it but it's coming back around this way and then it does this and it's behind it so you can't see it and it keeps going, okay? Behind and you can't see it, keeps going, let's make them farther apart so that we want to draw so many of them, okay? Keeps going like this, keeps going like this, 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 going to get connected right here, okay? The current is going to leave this side of the battery and it's going to go into the solenoids, by the way, a lot of times you don't get shown this, you're just going to 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 actually gets arranged, okay? Now, the first thing I want to talk about is the direction. So, there's going to be a new equation here for the solenoid but I first want to talk about for the toroids doughnut but first I want to talk about the direction. So, remember I told you that V is going to be this way, it's going to follow, it's going to keep going through the inside so the magnetic field is actually going to be this way here, okay? This way. Now, it has two possibilities, it could be clockwise or it could be counterclockwise, clockwise or counter clockwise depending on which way the direction of the current flows depending on the flow of the current and it's going to be similar to this, what you're going to do is you're going to look at where the current is entering, which is this piece right here, and you're going to 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 doughnut and towards the back because it 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 thumbs sticks out to the left, okay? Now, this is actually kind of an angle like this. So, my thumb is going this way, long story short, the current immediately in the beginning here is going to go in this direction, okay? The current is going in this direction, I'm sorry, the magnetic field, I'm sorry, the magnetic field, this is the magnetic is going to go in this direction, meaning it's going to be looping in a clockwise motion, so the direction of the magnetic field is this way, okay? The magnetic field is this way. Alright, but you have to be very careful it has to do with the drawing, this is quite, these questions are annoying because you have to like inspect the drawing and figure out what to do. So, what I'm going to do now is I'm going to draw differently, I'm going to draw the same battery with the current coming out the same way but I'm going to show you, right? Current comes out this way, I'm going to show you how this drawing 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 going to 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, if you are here, look at what your first motion looks like, get your hand ready, get excited, right? So, you're going to go behind the doughnut and back, you're grabbing, instead of grabbing the doughnut like this the first grab is going to be like this. So, and 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 of V that way except that there's a bunch of loops. So, it's going to shoot B. Well, that was terrible, it's going to shoot B over here, Oh Lord pretend that that's a straight line and this is the direction of my magnetic field. So, hopefully you get a sense here, of how even though these look very similar the direction magnetic field is different depending on this, right? And there's probably not a really easy rule to memorize, like if the first wire is under or whatever, or the first wire is over, you just gotta grab it, okay? Go for the grab, grab that doughnut, that's the hard part, the equation tends to be pretty easy, so the equation is mu naught, I, N divided by 2pi, little r, all kinds of weird stuff going on here. Notice that pi is back, right? When we did regular loops there was no pi, pi is back, and remember that you have 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 we have here. So, the radius of the thing doesn't really matter so much, okay? Now, super important is that the magnetic field only exists between the inner and outer radii. So, we haven't talked about those guys. So, here's a center this distance here is my inner radius or R1 and this here is my outer radius R2 and remember a magnetic field lives inside of the loop, if the loop tightly grips this bagel or this doughnut then there can only be magnetic fields in here, okay? Magnetic field out here is 0, magnetic fields in here is 0, B in equals 0, it only exists in this blue circle, super important, so it only exists between R1 and R2, 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 R1 plus R2 divided by 2, sometimes you might see that it's just the average. So, for example, if R1 is 4 and R2 is 8 then R or R mean would be in this case 6 obviously, okay? Now, the thing, that's the only thing that R1 and R2 are good for, the only reason they're important is that you know that you can only have a B field at those little distances away, okay? We are going to do this example right away here and you're going to see what I'm talking about.

We have a 300 turn toroidal solenoid. So, N equals 300 and he has inner and outer radii 12 and 16 centimeters. So, R1 is 0.12 and R2 is 0.16 meters and the current is 5 and I want to know what is the magnitude of the field at these three different places. So, a, what is the magnitude of the field at the center of the solenoid. So, here's my toroidal of solenoid with the linings around it, the center is right around here. Well, the center. Remember, inside you have no B field, you have no B field, okay? So, in that case is always going to be 0, the magnitude of the magnetic field at the center of a toroidal solenoid is always 0, okay? The center of the toroidal or a toroid is always 0, okay? If you were, 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 R is not a radius, this R is a distance 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 0, right? R is distance for the center, if you walk at the center your distance between the center to Center is 0 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 your have to is you have to see if it falls between this range, it does fall between this range. So, you're going to 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, 20 is outside of this range, okay? This one was inside, outside of the range but inside the toroid, this is outside of the range and outside of the toroid, in other words, B will also be 0, okay? This is a trick question, the only one you have to solve is this guy here and we're just going to use the equation, mu naught, I, N divided by 2pi, r where the 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 mean radius, okay? If I was asked 13 centimeters away, that's not the mean radius, I would have plugged a 13 for r, okay? So, mu naught is 4pi times 10 to the negative 7, the current is given as a 5, the number of loops is 300 divided by 2pi and then r is 14 centimeters. So, 0.14, some stuff cancels here, but if you plug this whole thing in your calculator you get 2.14 times 10 to the negative 3 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 this one, let's keep going.

An ideal toroidal solenoid containing 1000 equally spaced coils is shown in the Figure below. How large must the current *I* be so that the magnetic field within the coils at a distance of 20.0 cm from the center is 0.0250 T?

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An ideal toroidal solenoid containing 825 equally spaced coils is shown below. What is the magnetic field strength in the region outside the coils?

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An ideal toroidal solenoid containing 825 equally spaced coils is shown below. How large must the current I be so that the magnetic field within the coils at a distance of 17 cm from the center is 0.0250 T?

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