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Magnetic force between two currents going in the same direction

Sal shows how to determine the magnetic force between two currents going in the same direction. Created by Sal Khan.

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Video transcript

We've now learned that a current or a stream of moving charges can be affected by a magnetic field. And we've also learned that it can induce a magnetic field. So that begs the question, what is the effect of one current carrying wire on another current carrying wire? Let's do that. Let's draw my first current carrying wire in green. That's the first current carrying wire. And let's say that the current is-- it's in magenta-- and we'll call this current 1. And then I have another current carrying wire not too far away. And I will call that current I2. Now what else do we need to figure out? Oh, well, let me just tell you. Let's say that they are a radius of r apart. And I say radius because we learned in the last video that the magnetic field created by a current carrying wire is kind of a, you know, they're kind of these circular cylinders around the wire. So let's say the distance from this wire to that wire is r. That distance is r. And so my question to you is-- well, first, just before we break into the math-- what's going to happen? Well, we don't know the magnitudes of the currents or anything just yet. But what's going to happen? What will be the net effect on, let's say, this wire? Let's say for some reason this wire is fixed or we could say they're floating in space. Let's just focus on wire 2 for now. This is wire 2, this is wire 1. What's going to happen to wire 2? Well, let's think about it. Wire 1-- the current in it is going to generate a magnetic field. Now what's the shape of that magnetic field going to look like? Well, we could take our right hand, do that right hand wrap around rule. It's a little different than the cross product rule, although it's kind of a byproduct. So that's my right hand and I'm wrapping it around. So if I point my thumb in the direction of the current-- so that's the direction of the current, just like I did-- then the magnetic field goes in the direction of my fingers. So they're going to go around this wire. And so if I were to just draw the magnetic field where it intersects with this screen, on the right hand side it will go into the screen. So we'll just see the rear ends of the magnetic field line. And I'll draw it in the same color as the current, so you know it's being created by I1. So I1-- its effect keeps going out to infinity, although it gets much weaker as we learned. It's inversely proportional with r. But this is the field of I1. I can draw these-- I don't want to crowd my page up too much. And then on this side of I1, what happens? Well, on this side, you can see the fingers come back around. So it pops out when it intersects with your video monitor. So on this side, the vectors-- this is the top of an arrow, coming out at you. Fair enough. So I1, by going in this direction, is generating a magnetic field that, at least where I2 is concerned, that magnetic field is going into the page. So what was our formula? And this all came from the first formula we learned about, the effect of a magnetic field soon. on a moving charge. But what was the formula of the net magnetic force on a current carrying wire? It was the force-- I'll do it in blue-- it's a vector, has a magnitude and direction-- is equal to the current. Well, in this case, we want to know the force on this current, on current 2, right? Caused by this magnetic field, by magnetic field 1. So it will be equal to I2, the magnitude of this current, times L-- where L is-- because you can't just say, oh well, what is the effect on this wire? You have to know how much wire is under consideration. So let's say we have a length of wire. And then of course, if you know the length of wire and we knew its mass and we knew the force on it, we could figure out its acceleration in some directions. So let's say that this distance is L, and it's a vector. L goes in the same direction as the current. That's just the convention we're using. It makes things simple. So that's L. So the force on this wire, or at least the length L of this wire, is going to be equal to current 2 times L. We could call that even L2, just so that you know that it deals with wire 2. That's a vector quantity. I could make it a full arrow. Doesn't matter. It's just a notation. I've seen professors do it either way, I've seen it written either way, as well. Cross the magnetic field that it's in. What's the magnetic field that it's in? The magnetic field-- I'll do it in magenta, because it's the magnetic field created by current 1. So it's magnetic field 1, which is this magnetic field. So before going into the math, let's just figure out what direction is this net force going to be in? So here we say, well, the current is a scalar, so that's not going to affect the direction. What's the direction of L2? This is L2. I didn't label it L2 on the diagram. What's the direction of L2? Well, it's up. And then the direction of B1, the magnetic field created by current 1, is going into the page here. So here we just do the standard cross product. Let me see if I can pull this off. This is actually an easy one to draw. So I put my index finger in the direction of L2. And then I put my middle finger in the direction of the field. So my middle finger's going to point straight down into this page. My other fingers just do what they would naturally do. And then my thumb would go in the direction of the net force. This is just the cross product. You'll see teachers teach the cross product other ways, where they tell you to put your thumb in the direction of the field, and this and that, your palm-- those are all valid. They're just different variations of the same thing. I find this one easier to remember. Because when I take the cross product, index finger is the first term of the cross product. Middle finger is the second term of the cross product. Thumb is the direction of the cross product. So anyway, this is the direction of L2. The magnetic field, we already know, goes into the page. So my middle finger is going into the page. And my thumb is in the direction of the force on the magnetic field. So that's the direction of the force. So there you have it. If this current is moving in this direction and its field is-- we know from this wrap around rule that pops out here and it goes in here-- the effect that it has on this other wire is that where the current is going in the same direction, is that it will be attracted. So the net force you is going in that direction. We could say the force from 1 on 2. That's just my convention. Maybe other people would have written it the force given to 2 by 1. That's the force given by 1 to 2. That's how I'm writing it. Now what's going to be the force on current 1 from I2? Well, it's going to be the current-- well, it's going to be the force there. Well it's going to be the same thing. Let me draw I2's magnetic field. You do the wrap around rule, it's going to look the same. So I2, sure, on this side its field is going to be going into the page. But what's I2's field going to be doing here? It's going to be popping out. I just did the wrap around-- take this wrap around, wrap it around that wire. So that's the field of I2. So then we can write down that the force-- and let's take, I don't know, this is some distance. Let's call that L1. So the force from current 2 on wire 1 of length L1, from here to here, is equal to current 1 times L1-- which is a vector-- cross the magnetic field created by current 2. And so we can do the same cross product here. Put our index finger in the direction of L1. That's what you do with the first element of the cross product. And then you put your middle finger in the direction of B2. And then your thumb is going to tell you what the net force is going to be. So let me draw that. So let me draw my hand. And just so you know, before I do any of these, I actually look at my hand, just to make sure I'm drawing the right thing. So my index finger in the direction of I1, my middle finger-- sorry, my index finger in the direction of L1, which is the same as I1, and then my middle finger is going to do what the magnetic field is doing. So my middle finger is actually going to point straight up. And then my other fingers are just going to do what they do. And so now you're looking at the palm of my hand. And my thumb-- let me make sure I'm doing this correctly. Oh, no. I was drawing my left hand. See, that's an error. You don't want to draw your left hand when you're doing the right hand rule with cross products. So let me draw it down here. My index finger going in the direction of L1. My middle finger's popping straight up, because the magnetic field created by I2 is popping straight out of the page here. So my middle finger goes straight up and my other fingers do what they need to do. Looking at the palm. And then my thumb will go in that direction. So the cross product of L with B2 popping out of this page, the net force is going to be in this direction. So there's a little bit of symmetry here. This wire's going to be attracted towards that wire, and this wire's going to be attracted to that wire. They're both going to-- eventually if they were floating in space, they would slowly get closer and closer to each other and their radiuses would get closer and closer and they would accelerate to each other, at ever increasing rates actually. Anyway, I'm out of time. In the next video I'll do this same principle, but we'll do it with some numbers. See