Magnetic Force Between Parallel Currents

Hey guys in this video we're gonna talk about the magnetic force between two parallel currents, let's check it out.

Hey, so two things to remember first if you have a current carrying wire, a wire that has current, it will produce a magnetic field around itself. So you got a wire, its got current, it produces a magnetic field around itself, but if you have a wire that has current and it sits on existing magnetic field, someone else's magnetic field, it will field a force. So you produce your own but if you sit in someone else's, you also field force. Okay. And the equations for these, you've seen them before, are that the magnetic field that your produce is _not I divided by 2 Pi r and the force that you field is BILsin_. 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 gonna have to have a mutual force between them, so let's check this out. Let's see you have a current I1 going up and a current I2 going up where 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. And the direction of the magnetic field that distributes is given by the right hand rule. So here's my wire then grab my wire like you always grab wires and my current is gonna go up like this, so my hand is going into the page over here, do this yourself, right, so you can confirm. My hand is going into the page on the right side and out of the page on the left. So that means that I1 is gonna look like this, into the page your gonna have a B1 and out of the page you're gonna have a B1 here due to that current. I choose the same direction also going up to also gonna have a B2, you're gonna have a B2 due to the second wire, on the right going into the page and to the left of that wire

its gonna be coming out of the page. What this means is that I1 produces a B1 that is into the plane over here. Let me do this in a different color. So there's gonna be an into the plane B1 here and there's gonna be an out of the plane B2 here. So this is really important, wire 1 produces a field at wire 2 and wire 2 produces a field at wire 1. And because when you're sitting in someone else's field you field a force, both of these guys would field a force. Okay. The direction of the forces can also be given by the right hand rule but slightly different, you grab wires, right. but to find force you just keep your hand flat or open. Okay. So here I have the magnetic field, let's do I1 first, so over here I1, the magnetic field is coming out of the page and the current's up. And if you do this, please do this to yourself, you 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 right on wire 1. And if you do the same thing on the right wire you gonna see that its gonna get pulled to the left. So fingers into the page, thumb up and my force is to the left. Do this yourself as well and this is going to be the force on wire 2. 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, if 1 is pulling on 2 then 2 must pull on 1 as well. Okay. By the way in terms of direction, let's talk about directions since we just did that, one conclusion here is that with two wires are going on the same direction they will attract each other but if they are going opposite direction 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 you have that opposites repel so that means you cannot remember opposites attract here, or you might remember that its 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 wire 1, the force on wire 1 is going to be the magnetic field that is on wire 1 and then there is the current of wire 1 and the length. Okay. First of all the lengths will be the same, L1 equals L2 equals just L. So we're just gonna write L, current I1 is this current here and B on 1. So the magnetic field on 1 on wire 1 is actually the magnetic field produced by wire 2. So this is actually B2 I1 L. Got it. If you're wire 1 you field your force to wire 2. Okay. By the way the same thing happens if you're wire 2, F2 is gonna be B1 I2 L. Okay. So now what I wanna do, is I wanna replace B over here, I'm gonna go off to the side and write, remember that B comes from here so I'm actually just gonna keep writing in here. So B is this, I'm gonna replace this and its gonna be _not I, now because its B2, its produced by wire 2, I have to use the current of wire 2 divided by 2 Pi r2, right. Its the current 2 and its distance 2. r is the same thing her for both. Its the distance this way and its the distance this way. So we can say that the distances are the same, so r1 is r2 is just r which is just the distance between the two wires. So I normally have to write r1 or r2 so that B times I1 L. Okay. So if we're gonna organize this a little bit better, you can 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 1, its the same thing for 2, its the same exact equation. The force between them is gonna be _not, I'm gonna write this a little bit prettier, I1 I2 L divided by 2 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 prefer my observation, this is very simple, this is how you do it. The reason I wanted to show you, just so you're more comfortable with seeing these equations. But that's the final equation that you can just plug-in to an example. One last point I want to make before I saw an example is that sometimes you'd be asked for the force per length unit. force per length unit. So if you read this, it says force per means divided by unit length which is just L. So sometimes you're asked for 4, for F, force divided by L. So all you gonna do is move this L over, right, and then this unit here F over L is _not I1 I2 over 2 Pi r. So sometimes you might be asked for that. Cool. Lets do this example here says 2 horizontal wires 10 meter in length are parallel to each other, separated by 50 centimeters. Okay. So 2 meters in length like this, 10 meters in length, so L equals 10 meters and they're separated, this is little r, by 50 centimeters or 0.5 meters. The top wire has current 2 to the right, I1 equals 2, and the bottom wire has current 3 to the left, I2 equals 3. And you can see this is very very easy. What is the magnitude in the direction of the force exerted on the top wire and on the bottom wire. First of all group 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 2 is being pushed that way and wire I'm sorry, wire 1 is being pushed up and wire 2 is pushed down. Okay. So that's the direction of the force. So right away I know that the top wire would be pushed up, the force will be up and the bottom wire will have a force that is down, but we also wanna know the magnitude. The magnitude is easy you just plug into the equation, F equals rhoher, the good equation F equals _not I1 I2 L divided 2 Pi r. And the numbers are 4 Pi, _ is 4 Pi times 10 to the negative 7 that's our _not, that's a constant. The currents are 2 and 3, so I can just put 2 and 3. And the length of the wire is 10 meters, so I put a 10 over here divided by 2 Pi, the distance is 0.5, the distance is 0.5 meters. So if you plug this monstrosity on you calculator you'll get 2.4 times 10 to the negative 5, this is a force so its magnitude in Newtons. Okay. This question is a little bit tricky and that asked you for the force on the top and the bottom wire. It's the same force. Okay. So the top wire is 2.4 times 10 to the negative 5 up and the bottom wire is 2.4 times 10 to the negative 5 Newtons pointing down. Cool. That's it for this one. Hopefully made sense. Let's keep going.