Skip to main content
Pearson+ LogoPearson+ Logo
Start typing, then use the up and down arrows to select an option from the list.


Learn the toughest concepts covered in Physics with step-by-step video tutorials and practice problems by world-class tutors

29. Sources of Magnetic Field

Magnetic Force Between Parallel Currents


Magnetic Force Between Parallel Currents

Play a video:
Was this helpful?
Hey, guys, in this video, we're gonna talk about the magnetic force between two Palo currents. Let's check it out. All right, so two things to remember first, if you have a current carrying wire ah, wire that has current it will produce a magnetic fields around itself to get a wire. It's got current. It produces a magnetic field around itself. But if you have a wire that has currents and it sits on an existing magnetic fields, someone else's magnetic fields, it will feel a force, so you produce your own. But if you sit in someone else's, you also feel force. Okay, And the equations for these we've seen them before our that the magnetic field that you produce is mu knots. I divided by two pi r and the force that you feel is B I l sign of data bill. Sign of data. Cool. Those two things are old news, but they combine into an interesting conclusion here, which is, if you have two parallel currents like here, you're gonna have to have a mutual force between them. So let's check this out. So let's say you have a current I one going up in a current to going up. We're gonna put them in the same direction for the sake of this illustration. And what's gonna happen is a current will produce a magnetic field away from itself in the direction of magnetic field that is produced is given by the right hand rule. So here's my wire. I'm gonna grab my wire. Right. You always grab wires. Um, and my current is gonna go up like this. Eso my hand is going into the page over here. Do this yourself. Right. So you can confirm my hands going into the page on the right side and out of the page on the left. So that means that I want is gonna look like this into the page. You're gonna have a B one and out of the page. You're gonna have a B one here. Due to that current I two is in the same direction. It's also going up, So you're also gonna have a B two. You're gonna have a B two due to the second wire on the right, going into the page and to the left of that wire. It's gonna be going. It's gonna be coming out of the page. Beach. What this means is that I won produces a B one that is into the plane over here. Let me do this in a different color. So there's gonna be an into the plane, be one here, and there's gonna be an out of the plane Be to here, So this is really important. The why're one produces a field at wire to wire to produces a field at wire one. And because when you're sitting in someone else's field, you feel a force, both of these guys will feel a force. Okay? The direction of the forces can also be given by the right hand rule, but slightly different. You grab wires, right? You grab wires, but to find force, you just keep your hand flat or or open. Okay, so here I have the magnetic fields. Let's do I won first. So over here, I won. The magnetic field is coming out of the page and the currents up. And if you do this, please do this yourself. You'll see that the palm of your hand is pointing to the right. So that means that this guy, the force on one, is going to be to the rights on the wire one. And if you do the same thing on the right wire, you're gonna see that it's gonna get pulled to the left. So fingers into the page, thumb up, and then my force is to the left. Do this yourself is well, and this is going to be the force on wire to okay. It should make sense that they are opposite to each other because of Newton's third law of action reaction. Right? If one guy, um, if one is pulling on to, then two must pull on one as well. Okay, by the way, in terms of direction, let's talk about direction. Since we just did that. One conclusion here is that if two wires air going in the same direction, they will attract each other. But if they're going opposite directions, they will repel each other. And you may remember, there's a lot of instances in physics where opposites attract. This is not one of those cases. Okay, so here here you have that opposite repel. So that means you cannot remember opposites attract here. Or you might remember that it's backwards, right? Opposites usually attract, but if you have two parallel wires. It doesn't work that way. Let's calculate the magnitude of this force. So the force on any wire Let's look at Weir. Won the force on wire. One is going to be the magnetic fields. Um, that that is on wire one. And then then there is the current of wire one and the length. Okay, first of all, the length would be the same. L one equals l two equals just l So we're just gonna write l current? I one is this guy here and be on one. So the magnetic field on one on wire one is actually the magnetic fields produced by wire to So this is actually be to I one. L got it. If your wire one You're feeling of force due to wire too, By the way, the same thing happens. If you are wire to F two is gonna be be one I to Hello. Okay. So now what I wanna do is I want to replace I wanna replace, Be over here. I'm gonna go off to the side and right. Remember that be comes from here, so I'm actually just gonna keep writing here. So be is this. I'm gonna replace this and it's gonna be mu not I. Now, because it's be too. It's produced by wire to I have to use the current of wire two divided by two pi are too right. It's the current two, and its distance to our is the same thing here for both. It's the distance this way, and it's the distance this way so we can say that the distances are the same. So are one is our two is just our, which is just the distance between the two wires. So I don't really have to write are one or two. So that's be times I one elf. Okay, so if you wanna organize this a little bit better, you could just say that this equation is just the force between them. And by the way, this is a mutual force, okay? Because of action reaction. So even though I was solving for force on one, it's the same thing for two. It's the same exact equation. The force between them is gonna be immune knots. I'm gonna be right. It's it's a little prettier. I one i two l divided by two pi r. This is the equation. Okay? You probably for a lot of you. You don't actually need to know how to go. How to derive this equation. You can just use it If you're Professor likes derivations. This is very simple. This is how you do it. The reason I want to show you. Just so you're more comfortable with seeing these equations. But that's the final equation that you could just plug into an example. One last point I wanna make before we saw an example because that sometimes you'll be asked for the force per length units, force per length units. So if you read this, it says force per means divided by unit length, which is just l. So sometimes you ask for four for F force divided by l. So all you gotta do is move this l over right. And then this unit here f over l is new, not I one. I 2/2 pi r. So sometimes you might be asked for that. Cool. Let's do this example here says to horizontal wires 10 m in length are parallel to each other, separated by 50 centimeters. Okay, so 2 m in length like this, uh 10 m length. So l equals 10 m and they're separated. This is little are by 50 centimeters or 500.5 m. The top wire has current to to the rights. I one equals two and the bottom wire has current three to the left. I two equals three and you're gonna see this is very, very easy. What is the magnitude and the direction of the force exerted on the top wire in on the bottom wire. First of all, real quick in terms of direction. Notice that the currents are the currents are in opposite directions opposite currents, which means that they will repel. They will repel. So instead of them pulling on each other like this, they will actually push each other away. Which means that wire to is being pushed that way and wire. I'm sorry. Where one is being pushed up and wire to is pushed down. Okay, so that's the direction of the force. So right away, I know that the top wire will be pushed up, the force will be up and the bottom wire will have a force that is down. But we also want to know the magnitude. The magnitude is easy. You just plug into the equation. F equals Roger. Uh, the equation f equals mu notes. Um, I one i two l divided by two pi r and the numbers are four pi mu is four pi times to the negative seven. That's are you not that's a constant. The currents are two and three. So I could just put two and three and the length of the wire is 10 m. I put a 10 over here divided by two pi. The distance is 20.5 distances 0.5 m. So if you plug this monstrosity on your calculator, you'll get 2.4 times 10 to the negative fifth. This is a force. So it's measured in Newtons. Okay, this question is a little bit tricky in that and ask you for the force on the top of the bottom wire. It's the same force. Okay, so the top wire is 2.4 times 10 to the negative five up, and the bottom water is 2.4 times 10 to the negative. Fifth. Newton's pointing down cool. That's it for this one. Hopefully make sense. Let's get going

Two very long wires of unknown lengths are a parallel distance of 2 m from each other. If both wires have 3 A of current flowing through them in the same direction, what must the force per unit length on each wire be? 

BONUS:Is the mutual force between the wires attractive or repulsive?