Magnetic Field Produced by Loops and Solenoids

Hey guys. So this year we're gonna talk about the magnetic field produced by loops and solid noise. Let's check it out. Alright, so remember if you have a wire that is carrying current, it's going to produce a magnetic field away from itself. So it looks kind of like this, you have a wire with a current. I it's going to produce a magnetic field with strength be some distance away from itself, distance R. And you can find that magnitude by using the equation. You not I divided by two pi R. Where R. Is a distance from the wire. Okay. And this is important. You gotta remember this but this only works for straight, this only works for straight wires. Straight wires. In this video this video we're going to deal with is not straight wires but loops. So imagine you have a long straight wire, you can actually turn that wire to look like a loop. Something like this. Right? And in this situation you're gonna get slightly different equations, but also the right hand will be a little bit different. Let's check it out. So in a straight wire you may remember that our current is obviously if you have a straight wire you have a straight current. So we use our thumb to indicate current, that's what we've always done. Um And and when you grab the wire, your fingers were the direction of the magnetic field around it. So because your magnetic field was curving was was curving, you would curl or curve your fingers. Okay, so B was curved. So you would curl you use your curled fingers to represent B. Because it makes more sense. It's better visually to curl your fingers than curling your thumb, like it's easier to see this, right? So that's why we did it that way. But now when you are in a loop in a wire loop, your current is going to be going around. So it's gonna be easier to use your fingers curved as your current. So you're gonna use your fingers as your current. And guess what what's gonna happen is that your magnetic field is going to be straight? So you're gonna use your thumb. So those two things flip and you can think that in loops the right hand rule is backwards. Or you can also think that there's sort of this this greater law this greater rule that you always want to curl your fingers. So whatever is curling in the problem is going to be represented by your finger. And then the other thing is going to be your the other thing is going to be represented by your thumb. Okay so let's look at single or multiple loops. And the first thing I want to talk about is direction. So let's say let's say that this um the current here is going in this direction. That's current which by the way is counterclockwise. So what we're going to do is remember if it's curling you're gonna use your fingers and I'm gonna curl my fingers in the direction of currents. So currents going this way. So I'm gonna go like this. Okay. And if you do this and you should do this yourself or you're gonna see that your thumb is pointing at yourself, which means it's coming out of the page and that's the direction of B. So in the middle of this thing here, in the center or everywhere inside actually you're gonna have a magnetic field that is jumping out of the page towards you and you're gonna have a magnetic field outside of the loop that is going into the page which is the reverse direction, right? And obviously as you would expect if you're going in the clockwise direction of current, if you have current in the clockwise direction it's just the opposite and you can do the same thing with your fingers and you can now curl them to the right like this clockwise. And notice that my thumb is pointing away from me, which means it's going into the page, which means on the outside of the wire you're gonna have an out of the page, magnetic fields everywhere. Now let's talk about the equation, the magnitude of the magnetic field of a loop that's produced by a loop is going to be meal. Not I true R and almost forgot there's an end here too are so let's talk about a few things. First of all, notice that there is no pie in here, that is not a typo, not a typo, I actually want you to write, not a typo because I don't want you to get confused and think that there's a pie there? What happened? There is no pie? Okay. The other thing is that you have are big R. Which is radius and not little R. Which is a distance. Okay. Little R. Is the distance. I wrote it here you have the radius. So what's what this number is going to be? Is the radius of that circle which is given to you or they will ask for it. Okay. And then third, what is this end here? N is going to be the number of loops. So you can have a single loop which looks could look like this. You got a straight wire and you made a little loop out of it. Or you can have multiple loops. Let's say you do this right? So in this case here you have N equals one. This end over here and in this case you have N equals three. And the idea is that if you have one loop, you produce a certain amount of magnetic fields, let's say 10. Well if you make three loops, you're gonna produce magnetic fields of strength. 30 triple that. Okay. Um in fact that's why sometimes you see um you might see some electric devices that have a ton of wires, tightly loops like this and it's because you're trying to produce a stronger magnetic field. It's like magic. You loop more, You get a stronger magnetic field. Cool. So that's it for these points. That end just goes in there. The situation is super important. There's one last point I wanna make here is that this equation is for the magnetic field be at the center at the center of the loop. So it's not at any point, it's always at the dead center. So this has to be a perfect circle. Now, remember how we could make loops that were like this? You can have multiple loops. Well, if you make a loop that's really, really, really long, you're actually going to get a different equation. That's what we have here. And I briefly mentioned this over here. Okay, So, if you have a very long loop, what's gonna happen is you're gonna have a different equation. The equation now is going to be mu not I. N. Over L. And over L. Everything is the same. This L. Means this length over here. Okay, it's this length over here. So, it's similar equation but different. All right. Now, what you see sometimes another version of this equation gets N over L. And changes into little N. Little N is big N over L. So, you can also see this equation like this. Let's move this out of the way. You can also little N. Is big and over L. So you can see this as mu knots. I and little N will replace those two guys like this. Okay, this is another version of the equation. So you can see either one of these two versions. Um and is the number of the number of loops per meter? This is loops per meter and it's kind of almost like a density. It has to do with how tight these things are. So for example, this has four loops not very tight. This has four loops super tight. Okay, so the end here is greater than the end here, let's say N. One and to this end here is greater. Okay, cool, so that's the equation. What about the direction of the magnetic fields? Well, remember what we talked about just now, which is if something curves, we're gonna use our fingers. What's carving here? Well, let's say the current is entering here, let's say the current is entry on this side. Notice what the current does. The current's gonna go in and then it's gonna go in this picture, it's going into the page and back into the page and back into the patient back. So the current is what's curving. So we're gonna use our our fingers for current as well. Okay, So it's just like this one same thing. So but what you have to be careful, you have to be careful and look at this first loop right here as soon as the current starts carving. Is it first going backwards or is it going forwards this way? Okay, so if the current starts here, notice that the cable goes into the page. So you have to get your thumb your fingers and curl into the page away from you. And when you do this, when you do this it looks like this right? When you do this your thumb is to the left. So if the currents here, let's call this I. One. It's going to create a magnetic field that is going this way B. One. Okay but you could get a current in a different direction, you could get a current in different directions. What if the current was going here? Let's call this I too notice. And this is kind of tricky to look to notice. But if you look at the wire it all really comes down to this little piece of the image guys, this is like really messed up. But you gotta be careful. The wire is going initially into the page and then it keeps doing this right? So you're gonna get your your this is your current which is curling right? You're gonna get your fingers and you're gonna go into the page and back. And when you do that, look what happens with your thumb, your thumb is pointing right? Which means you would get a magnetic field that goes like this. Be two. Very very tricky, depends on the direction and depends on what this little image looks like. Is it coming at you and then in or is it coming down and then back? Right so be very very careful with that. Um There's one other thing. You should know some questions we'll ask about the total length of wire, total length of wire. What the heck is that? Well length L. Is just the side by side length. The total length of wire means, what is the length of this entire thing here? Right of this entire thing here. And what we're going to do is we're gonna say well the length of length of a single loop, one loop has a length of circumference. The amount of why you need to make One loop is a circumference which is two pi R. Where R. Is the radius of this thing. Two pi R. Where R. Is the radius. But if you have end loops, if you have end loops the total length of wire will be two pi R. And okay so I want you to remember this that the total amount of wire is two pi R. And this doesn't have a variable. So I just write just write total wire equals two pi R. And this is one of the other things you might get asked sometimes. Let's do an example. I got one last point to make here. And this is actually really interesting point. Sullen Oid will produce magnetic fields that are very similar. Two magnets magnets. Let me show you real quick if you have a let's say we have I one over here. So we have something like this, right? And then this is I won. And there is a magnetic field to be this way. Well, guess what? The magnetic field is going to keep going and it's going to do this, right? And if you remember the magnet, the magnetic field is always going from north to south on the outside. So this means this is the north and this is the south. So it behaves very similar to a magnet. Anyway, let's do this example here. And in doing this example will kind of address this situation where you have to fuse at the same location. We already talked about this in the previous video. So a wire is twisted into five type loops four m in radius. So you got a wire and you made 12345. Obviously don't really have to draw it But M. E. equals five. And the radius of these loops is four m. A three amp current is ran through the wire in the counterclockwise direction. Let's draw the a frontal view of the wire. So the current is three In a counterclockwise. So imagine there's five wires here or five loops of wire here and it's in the counterclockwise direction like that. I want to know the magnitude and direction of the field produced by the loop in its center. So right here the magnetic field is mu not I divided by two big r New knots is four pi times 10 to the negative seven. The current is three and then two times the radius, the radius is four. So this four cancels with this four. Um There's no pies to cancel unfortunately. So we got to multiply this whole mess and we forgot the end, forgot the end over here. There's a five over here. Almost forgot that. And this is gonna be 7.5 times 10 to the negative seven to the negative seven Tesla, tiny amounts of magnetic fields. Um And it's also asking for the direction the direction is easy. I'm going to grab the wire with my fingers curled in the direction of I and then my thumb points at myself, which means that this is the direction of the magnetic field and center is out of the page. Cool. So that's how loops and solid guards work. Let's go do some more problems.

Hey, guys. So let's check out this solenoid example. So here I wanna know how Maney turns. A So annoyed is going toe have How many terms is the variable big end insulin noise? Not to be confused with little in so big end is the number of terms 2 m long so annoyed, meaning the length of the solenoid, the sort of sideways length This l is m in order to produce a 20.4 t magnetic fields B equals 0.4 t when a three and current is ran through it. So when a current high equals three amp is ran through, it is their eyes. They're an equation that relates all these variables off course there is. And that's the equation for the magnetic fields through the center of a solenoid. B equals meal knots. I en over l we're looking for n so I could just move some stuff around and equals b l divided by divided by mu, Not I and B is 0.4 length is to me you not is four pi times 10 to the negative seven and the current is three amps. The current is three amps and If you multiply all of this, I have it here you're gonna get you're gonna get so I haven't rounded. Um, you're gonna get true times 10 to the fifth Turns. That's the value for end. Which means, by the way, that you're gonna have 200,000 turns. That's what you need to have to make this happen. Go. That's it for this one. Let's get going.

Hey, guys. So let's check out this example. So here you're you're tasked with designing a solid annoyed that produces a magnetic fields of this strength here. So B equals 0.3 Tesla at its center with a radius of four centimeters. So the radios is 0.4 m in a length of 30 or 50 centimeters or five meters, and we wanna know what is the minimum total length of 12 AMP. Wire. 12 AMP wire means that this wire is capable. Can can withstand currents of 12 amps or more. If you try. Ah, higher current. It's just probably gonna burn the wire or it's risky. What that means is that we're gonna use a I currents of 12 amps, and I want to know the minimum total length you should buy to construct this Illinois. So imagine going to the wire store and you gotta buy some some wire. How much total length Now remember, total length is different from length. Right length is just if you make a solid noi that looks something like this. This is length here, sort of like the side to side length, but the total length of wire is all these sir conferences here? Right, It's all of this. One way to think about this is if you get that solenoid that's all curled up and you pull all the way straight so that it doesn't doesn't curl anymore. What is a total length of wire you're gonna get if you did that. Okay. L is this which is given to us, but we wanna know total length, Total length. So I'm just gonna write total wire equals question Mark. And you may remember the equation for this, um, one circumference is two pi r right. Two pi r where r is the radius and were given that. But if you have any loops than the total wire is two pi r times and okay, so that's another equation that you need to know. Andi, that's what we're looking for here. Notice that I have are so that's good. And to empire constant. So that's good. But I don't have end. So before I can solve for this, I'm gonna have to calculate end and to find end. There's really only one of the equation that I can use, which is the magnetic field equation I'm given the magnetic field. So we might want to write the magnetic field equation. Be for a so annoyed is remember, Mu not I l over in. Whoops. It's actually end over. L don't get it twisted and over l or m you nuts? I little n because little n is and big and over l Okay, this is number of this is turns part meter. Okay, Just as a reminder. Cool. So I can I can find and using this equation very straightforward. So let's move some stuff out of the way and is gonna be bl divided by new knots high bl develop immunity. I b is 0. mu knots. Is four pi times 10 to the negative seven. And I is 12 amps. Okay. And if you do this, you get that end is 995 which some people would around that to 1000. But let's just say 995 turns or loops. Right? So there's 995 little winnings, and now we can plug this end into here so that the total wire is two pi The radius 0.4 and then end is 9 95. And this gives you this rounds toe Basically m of wire. Okay, so that's it for this one. Hopefully make sense. Let's get going.

Hey, guys. So let's check out this example. So here we have to wire loops that are concentric Lee arranged meaning concentric means one circle inside of the other, right, with a common middle eso It's shown below, and the inner wire has diameter four. Now, real quick in physics, remember, we almost never used diameter were almost always use radius. I'm gonna right away. Change this. Um, instead of writing diameter one, I'm gonna radius one and radius is half of the diameter. So that's 2 m in the clockwise currents of five. So that's the inner one here, which is blue, and it's got a clockwise currents of five amps. So I'm gonna put here five amps s o I one is five amps and radius one is m and then the red one is counter clockwise counterclockwise, which is this way, and it's got a current of seven amps in that direction. So I can write that I to is 7 a.m. and R. Two is the diameter, which is six. It's actually the radius, which is gonna be three half of the diameter. Okay, and we're looking for is the net the net magnetic field at the center. Remember what you have a current when you have a loop current. So you have a loop of wire with current going through it. It's gonna produce a magnetic field through the center of the ring, either in or out. Right. And we have two rings with the same common center. So both rings or both loops will be contributing. Um, to this here, which is why we're talking about the next magnetic fields because it's gonna be a contribution of both. It's gonna be a combination of both. So let's find those two numbers B one and B two, and the equation is mu knots. I divided by two big art where big R is the radius. And we have all of these numbers. Um, it's I won, since it's B one and it's our one since it's B one, right? So once go, go with once. So this is four pi times 10 to the negative seven and the current is a five, and the radius here is a true okay. And if you plug this into your calculator, you're gonna get that this is 15 7 or actually should say, um, 1.57 times, 10 to the negative. Six times 10 to the negative six. Okay. And if you do this with B two, it's very similar. Just the numbers are a little bit different. So instead of a five up here, you're gonna have a seven. And instead of a two over here, you're gonna have a three, okay? And if you do this, you get 1.67 times 10 times 10 to the negative six. Okay, now let's talk about direction. Let's talk about direction to find direction. I'm gonna use the right hand rule. So first, let's look at the blue inner circle. The blue inner circle is not going this direction, but it's actually going in this direction, right? It's going clockwise like this. If you do this, your thumb points away from you, which is into the page. Which means that the first one, the inner one, is going to go into the page and the other one is in the opposite direction, so it must go in the opposite direction. So this is going to be out of the page. And if you want to confirm, uh, if you want to confirm you could just use again your hand and grab the outer wire goes this way right this way. And look, my thumb is not pointing my face, which is out of the page and towards meat because these guys were going in different directions. We can't just add their magnitudes. In fact, we have to subtract. And the way to do this is you start with the bigger one and then you're gonna say, Hey, this guy is the bigger one. So it's the winner. This one wins, right? It's kinda like a tug of war ones pulling this way, the other ones pulling the other way. This one wins. So the next magnetic field is going to be a winner minus loser. So 1.67 times 10 to the negative six minus 1.57 times 10 to the negative six. This is actually just a matter of subtracting this minus this because it's got the same power of 10. So this is going to be 0.1 times 10 to the negative 60.1 Time center. Negative six. But we can multiply despite 10 and then we have to divide this by 10. We multiply this by 10 so we get one times instead of one. And if we multiply here, we have to divide here. So it's fair. So it's so we're not actually changing the number, and this divided by 10 is 1 10 to the negative seven. By the way, you could also have answered just 10 to negative seven. But that's that. So this is one times 10 to the negative seven. Tesla, uh, and in what direction? It's going out of the page because that was the winning direction of the two. Okay, so that's it's That's one way you could do it another way. You could have done this. You could have just assigned signs, and you could have said, Hey, um, into the page into the page is like this right away from you with my thumb and my fingers are currently in the clockwise direction. Clockwise is usually negative, so we can say that into the page is negative and out of the pages positive, right? So then you would have done this with numbers and you would have gotten the same results anyway cope so you can think of winner the big one minus loser, the smallest one and then the winner dictates the direction, the next direction. Or you could just assign positives and negatives on Do the math. That's it for this one. Let's get going.