1

concept

## Magnetic Force on Current-Carrying Wire

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Hey, guys. So in this figure, we're gonna talk about forces acting on current carrying wires, Meaning if you have a wire and charges moving through it, it has a current. And if that wire is inside of a magnetic field, it will feel a force. Let's check it out. All right. So remember, charges can move in space, and we've done this. If you have a little charge moving through space and it walks into a magnetic field, it will feel a magnetic force. Well, charges can also move inside of a wire. And if you have charges inside of a wire, you have a current. You can think of a wire as simply a, um ah, way to sort of restrict the path of charges that they're not going crazy directions. They have to move along the direction of the wire. But in a way, ah, wire is simply charges moving just the same. So remember, if charges air moving in space, they will be producing a new magnetic fields. And we did this. The equation is mu, not Q v sign of data over four pi R square, right, Just like how a charge moving freely through space will produce a new magnetic fields away from itself. A charge moving in a wire. In other words, if you have a current carrying wire that is also going to produce a new magnetic field, Okay, the equation for that is mu not I divided by two pi r so look somewhat similar to this, but it's not exactly the same. We're not gonna talk about this now. We're gonna talk about this later, but I want to make the point that they produce a charge. They produce the fields away. Um, the Y will produce the fields away from itself. Remember that charges not only produce a field away from themselves, but they're also going to feel a force if they are in the presence of an existing magnetic field. And this is a really important theme of magnetism is that moving charges produce a field, and if they are in an existing field, they will also feel a force. And just like how this happens for charges in space, this is also going to happen for charges moving in a wire. Okay, so if a charge is moving through space, it will feel a force given by Q V. The sign of data right s O. For example, if you have a magnetic field this way and a charge moving this way, you can use your right hand rule and figure out that the force will be the force will be into the field. Okay. Similarly, if a charge is moving, not through free space, but it's moving inside of a wire. It is a current, and that wire will will feel a force as well. So charge moving in an existing field, fields of force and the wire in an existing field also feels the force because it's the same thing just and here it has to be moving. And here it has to have a currents. It's a current carrying wire, right? Just like here, current carrying wire. And that force is going to be given by B I hell sign of theta. Okay, And this is the big equation that we're going to talk about. I wanted to sort of do a big summary of all the different situations, but this is what we're gonna talk about for the next few videos. Now, most textbooks and professors write this equation of different order. I like to write it this way just because there's so much crap to room. In this chapter, this spells bill, which is easier to remember than some other combination. Okay, so F equals built sign of data. I has a direction which is direction of the wire and B has a direction and data is that three angle between those two directions. The directions of the forces will be given always by the right hand rule. Now remember, there's sort of this exception that if you have a negative charge, if you have a negative charge that's moving freely, then you're going to use the left hand rule. But in currents because of sign convention, you are always going to use the right hand rules at the right hand rule. Every time When you have a current, you don't have a negative current right. Eso you're always going to use the right hand rule no matter what. Alright, eso The last thing we'll talk about here is that this force will cause the wire to bend slightly. Okay, so it causes the wire to bend slightly. So let's say here we have a wire here and the current is up And here we have a wire between these two magnets and the current is moving down. Now, you may remember that magnetic fields go from positive to negative on the outside. So positive to negative. So it looks like this. But more importantly, we want to know what is the fields around here? Well, it goes from positive. I'm sorry. From north to south, from north to south, so it's gonna look like this. So as far as the wires concern, you have Byfield lines going this way, I guess. This way. Okay. So what is the direction of the magnetic force that this wire will experience? Well, right hand rule, right? And guess what current is the direction of the motion. So it's gonna be your thumb, and your B is going to be your fingers. So here I have might be is gonna go to the right, and I want my current to go up. So this is already in the direction that I want, right? This matches the picture. So if I do this notice that my poem is away from me, right? So please do this. It's away from you. Which means that it's into the page. So this force on the force of the wire will be into the page everywhere the wires being pushed into the page. Okay, we can't calculate this because it's with magnets, but we're just talking about the direction here you also have north to south. So the magnetic field look the same. And what do you think will be the direction of the force? Well, hopefully you're thinking, Well, if we're flipping the current, it must flip the force. And that's correct. So, right hand rule again. This is going to the right. But now my currents down. So the only way that I can keep these going to the right with my current going down if I do this right, and if I do this on your sort of, bring it close to your face, right? This is weird. But hopefully see that your palm is facing you, which means it's coming out of the page. So FB is coming out of the page everywhere so F b is out of page. So that's direction of the force. Let's do, um, example. Here there are two parts, so it says here a 2 m long wire. That's the length of the wire l equals two is passed through. A constant magnetic field has shown. So there's a little piece of wire here, and you might be wondering, How is their current on a little piece of wire that's floating? Don't worry about that. For now. I'll talk about that a little bit later. Just assume that this is possible, which it is. Okay, so there's a wire there. Somehow it's connected to a battery, but don't worry about it. So the piece of the wire there is 2 m. It sits through a constant magnetic fields. This magnetic field is going into the page by and you can tell by the little excess. And we wanna know for part A. If the wire experiences a force of three Newton's when it has a current of four, what is the strength of the wire? What is the strength of the field? So what is be So is there an equation that ties these guys together so that you can find me? And it's the equation we talked about earlier. F equals Bill Bill, Sign of Fada. Okay, we're looking for be B equals FB divided by I l sign of data. Remember, the angle is the theta is the angle between the B and the I. I is either to the right or to the left. We actually don't know, but it's either one of those in the direction of the field is into the page. So the current is going this way or this way. Right this way, this way, and the field is going into the page so they make 90 degrees with each other. It's either this, which is 90 degrees right here, or it's this right, which is also 90 degrees. So this angle here is 90 degrees. And by the way, generally speaking, if you don't see like a weird angle, it's probably 90 right? Um, so let's plug these numbers. That f is three. I is four. L is, too. This is 3/8, and that is 0.3 75 Tesla. That's part a part B has to do with the direction. It's as if the wide experiences a downward force. So if the wide experience down force what most the direction of the current be, so here's the wire, and it's experiencing of force F B. That's down right hand rule the forces down. So I want my palm to be down. And, um, I want my four fingers for field to be going into the page, which into the page is away from you, right? So I want my fingers like this and I want this force down. Notice what happens with my thumb, which is It's going left and that's it. That tells me that the currents must be going left so the direction of the current is left. That's all there is to this. Hopefully makes sense. Let's keep going.

2

example

## Find Force on Current-Carrying Wire at an Angle

8m

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Hey, guys. So in this example, we have a wire that sits on a magnetic field and it has a current, and we want to know what is the force on that wire in three different scenarios. So let's see. The wire has a length of 2 m. L equals to the magnetic field. Strength is three. B equals three, and it's directed in the negative y axis the magnetic field. And we wanna know what is the magnitude of the magnetic force, which, in this case, because it's in a wire, it's going to be given by Bill. Sign of data be I'll sign of data, by the way, were also given that the current is for okay, little Why a big guy doesn't matter. Current is four amps. That's a really ugly 44 amps. Cool. So let's get to it. So the magnetic fields I like to draw multiple lines, so let's just do too, is going down right right there. And the The wire is also or the current rather is flowing the negative y axis, which means that the wire goes down this way and then the current is going down in this direction. Here So the equation is F B B. I sign of data, and then these questions. The tricky part is the angle, because I'm given B i n l. So it's just plug and chug. Now the angle is going to be the angle between be and I they're both going down, so they're parallel to each other. So the angle is zero, and the sign of zero is zero the sign of which means there is no force here because they're going parallel to each other. Remember, encouraged to use the right hand rule always, Um, and this is a reminder you can think of. This is a reminder that you're supposed to have a 90 degree angle between your B and your eye, right, 90 degree angle between your being your eye. And here they're like that, and that's not it's supposed to be. In this situation, you have maximum force. If you have an angle like this, you have less than maximum force. But you still got some force. And as you keep going this way, the force and it goes from Max max, 90 degrees to smaller, smaller, smaller, and you get here and you get zero. Okay, So if the angle is zero because they're parallel or if the angles 1 80 which is anti parallel, opposite directions, there will be no force. So no force on this one here. What about here? So be again is down. And the current is in the positive X axis, which means the wire, this horizontal and the current is going in this direction. Theatrical between these two guys is 90 degrees. So FB is Bill sign of 90. Sign of nineties. Just one. So, really, we only have built so three eyes. Four l is too. So this is 24. Newton's very easy. What about the What about the direction of the force? What is the direction of the force? Well, right can rule because it's a wire B is going down. I is going to the right. So you gotta do this, right? You gotta do this. Okay. Away from me. Bees down. So my poem, even though it's pointing at your face, you gotta lose yourself, right? It's not my poem. It's your poem. Your poem is pointing away from you, which means it's going towards your page. So it's into the plane okay or into the page into the page. Make sure your master, your right hand rule into the pages direction. And the magnitude is 24. Okay, so this one's a little more complicated because it's got an angle on Dhere. We have the be this way. Um and we have a wire in the direction that makes 53 with the Y axis. So here is the positive y axis is up here. This makes 53. Now, this is a little tricky because it's ambiguous. It's not totally clear whether it's 53 with the y axis this way or 53 with the Y axis this way. But what you will see is that it actually doesn't matter when it comes to calculating this. Okay, so we're gonna think of this as two possibilities that the current could be going this way or it could be going that way. And then we're gonna calculate that. So the equation is f B equals bill sign of data, and the angle is the angle between the direction of the current and direction of be so B is pointing down. So if you want, what you can do is you can draw. Be over here. Okay. And what is the difference between these guys? So if you go counterclockwise here, this is 90. And then this here is a 37. So 37 plus 90 is 37 plus nineties, 1 27 degrees, 1 27 degrees. Or you can go another. Or you can go in this direction here, which would be negative. Negative 1 27 degrees. Or it could go all the way positive. And you can say that it's 90 plus another 90 plus which is 1 80 plus 53 1 80 plus 53 which is 2 30 three 2. 33. That I got that rights. Yeah. 2. 33 away from this. Um, and the sign off all these numbers will be the same. They may have different signs what might be a positive or negative? But here we're just looking for the magnitude of this thing. So you can pick your choosing. I'm gonna write that B equals B equals 34 eyes for Ellis to sign of 1 27 positive again. Whatever we get here, just think of this is an absolute value because we're just looking for the magnitude of this force on. But if you do this, you get that. The answer is, uh, uh, four is that a 14 can't read my own handwriting. Um, no. 19 point to the answer is 19. Newton's. Okay, so that's the force. And it would work whether you're going this way or this way. What about the direction? Right hand rule bees down. So let's do this. And I want this guy to be going in this direction, okay? I want this guy to be going this direction. So in case of in this case here where I won, right, if you weren't going that way, you'll be going down and you would have the I like this right? Which means my palm is full pointing towards me, which means it's away from the page. So, for in the case of I one, if it was going in that direction on, by the way, a question a question on your test would tell you exactly which one, right? I just wanted to talk about both cases here. Um, so in this case, you would be What do we say we was? We said it was out of the plane out of the page. What about for I to what if you were actually talking about this direction? Well, if you try to do beat down in this I hear you can't put this thumb all the way over here, right? Like not without breaking it. And don't do that, because now you're just you're just messing yourself up in and you're breaking the right hand roll anyway, right? Eso what can we do? Well, you have to do this right. You have to do this. Where now your fingers air still down following B and this guy is up like that. So this looks all kinds of weird, but my palm is away from me, which means it's going into into the page. So in this case, the direction of those two wires, um, actually made a difference in terms of different in terms of direction. Okay, doesn't make a difference in terms off of the force, but it does in terms of the not the magnitude, but it does make a difference for the direction. Cool. It's like this one. Let's get going

3

Problem

A 5-m current-carrying wire (red line) is ran through a 4 T magnetic field (blue lines), as shown. The angle shown is 30°. What must the magnitude and direction of the current in the wire be when it feels a 3 N force directed into the page?

A

0.17 A @ 30° below the +x-axis

B

0.17 A @ 30° above the −x-axis

C

0.30 A @ 30° below the +x-axis

D

0.30 A @ 30° above the −x-axis

E

0.49 A @ 30° below the +x-axis

F

0.49 A @ 30° above the −x-axis

G

3.25 A @ 30° below the +x-axis

H

3.25 A @ 30° above the −x-axis

Additional resources for Magnetic Force on Current-Carrying Wire

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