Magnetic Force Between Parallel Currents: Study with Video Lessons, Practice Problems & Examples

When two parallel currents flow in the same direction, they attract each other, while opposite currents repel. Each current generates a magnetic field, described by the equation B = μ0I2l2. The force between them can be calculated using F = μ0I1I2l2πr. Understanding these principles is crucial for applications in electromagnetism and electrical engineering.

Magnetic Force Between Parallel Currents

Hey guys. So in this video, we're going to talk about the magnetic force between 2 parallel currents. Let's check it out. Alright, so two things to remember. First, if you have a current-carrying wire, a wire that has currents, it will produce a magnetic field around itself. So you get a wire, it's got current, it produces a magnetic field around itself. But if you have a wire that has current and it sits on an existing magnetic field, someone else's magnetic field, it will feel a force. So, you produce your own but if you sit in someone else's, you also feel a force. Okay? And the equations for these, we've seen them before, are that the magnetic field that you produce is μ₀I / (2πr). And the force that you feel is BIL sin(θ). Bill sine of theta. Cool? Those two things are old news but they combine into an interesting conclusion here, which is if you have 2 parallel currents like here, you're going to have a mutual force between them. So let's check this out. So let's say you have a current i1 going up and a current i2 going up. We're going to put them in the same direction for the sake of this illustration. And what's going to happen is, a current will produce a magnetic field away from itself. And the direction of the magnetic field that is produced is given by the right-hand rule. So, here's my wire, I'm going to grab my wire, right? You always grab wires. And my current is going to go up like this, so my hand is going into the page over here, do this yourself. Right? So you can confirm. My hands go into the page on the right side and out of the page on the left. So that means that I one is going to look like this. Into the page, you're going to have a b1, and out of the page, you're going to have a b1 here due to that current. I2 is in the same direction, it's also going up. So your wire is going to be going, it's going to be coming out of the page, b2, the left of that wire, it's going to be going, it's going to be coming out of the page b2. What this means is that I one produces a b1 that is into the plane, over here. Let me do this in a different color. So there's going to be an into the plane b1 here and there's going to be an out of the plane b2 here. So this is really important, the 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 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 open. Okay? So here I have the magnetic field, let's do I 1 first. So over here, I 1, the magnetic field is coming out of the page and the current's 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 1, is going to be to the right, on wire 1. And if you do the same thing on the on the on the right wire, you're going to see that it's going to get pulled to the left. So fingers into the page, thumb up, and then my force is to the left. Do this yourself as well, the 3rd law of action, they are opposite to each other because of Newton's third law of action-reaction. Right? If 1 guy, if one is pulling on 2 then 2 must pull on 1 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 2 wires are going in the same direction, they will attract each other. But if they're going in opposite directions they will repel each other. And you may remember there are 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 it's backward. Opposites usually attract but if you have 2 parallel wires, it doesn't go in the same direction. It's going to be the magnetic field, that, 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 going to write l. Current I1, is this guy 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 b2i1l. Got it? If you're wire 1, you're feeling a force due to wire 2. Okay? And by the way the same thing happens if you are wire 2, f2 is going to be b1i2l. Okay? So now what I want to do is I want to replace, I want to replace b over here. I'm going to go off to the side and write, remember that b comes from here. So I'm actually just going to keep writing it here. So b is this, I'm going to replace this and it's going to be Now because it's b2, it's produced by wire 2, I have to use the current of wire 2 divided by 2π r2. Right? It's the current 2 and its distance 2. R 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 r1 is r2 is just r, which is just the distance between the two wires. So I don't really have to write r1, or r2. So that's B × I1L. Okay? So if you were to 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, it's the same thing for 2. It's the same exact equation. The force between them is going to be μ₀ × I1I2L / (2πr). This is the equation. Okay? You probably, for a lot of you, you don't actually need to know how to derive this equation, you can just use it. If your professor likes derivations, 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 into an example. One last point I want to make before we solve an example is 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 force divided by l. So all you gotta do is move this l over, right? And then this unit here, F over l, is μ₀×I1I2 / (2πr). So sometimes you might be asked for that. Cool. Let's do this example here. It says 2 horizontal wires 10 meters 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're going to see this is very very easy. What is the magnitude and the direction of the force exerted on the top wire and on the bottom wire? First of all, real quick, in terms of direction, notice that 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 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 <here, the Greek equation, f equals μ₀ × I1I2L / (2πr)>. And the numbers are 4π, μ is 4π times 10 to the negative 7. That's our μ₀. 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 2π. The distance is 0.5. Distance is 0.5 meters. So if you plug this monstrosity into your calculator, you'll get 2.4 times 10 to the negative 5th. This is a force, so it's measured in Newtons. Okay? This question is a little tricky in that it asks 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 a good one. Hopefully, it made sense. Let's keep 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?

• A. 3.6×10⁻⁵ N/m
• B. 3.0×10⁻⁷ N/m
• C. 9.0×10⁻⁷ N/m

Here’s what students ask on this topic:

What is the magnetic force between two parallel currents?

The magnetic force between two parallel currents can be calculated using the formula:

$F=\frac{\mu 0I{1}_{}I{2}_{}l}{2\pi r}$

where $\mu 0$ is the permeability of free space, $I{1}_{}$ and $I{2}_{}$ are the currents in the wires, $l$ is the length of the wires, and $r$ is the distance between the wires. If the currents flow in the same direction, the wires attract each other; if they flow in opposite directions, the wires repel each other.

How do you use the right-hand rule to determine the direction of the magnetic field around a current-carrying wire?

To use the right-hand rule to determine the direction of the magnetic field around a current-carrying wire, follow these steps:

1. Point your thumb in the direction of the current flow.

2. Curl your fingers around the wire.

The direction in which your fingers curl represents the direction of the magnetic field lines around the wire. For example, if the current flows upward, the magnetic field will circulate around the wire in a counterclockwise direction when viewed from above.

What happens to the magnetic force between two parallel wires if the currents are in opposite directions?

If the currents in two parallel wires are in opposite directions, the wires will repel each other. This is because the magnetic fields generated by each wire interact in such a way that the forces between them push the wires apart. The magnitude of the force can be calculated using the formula:

$F=\frac{\mu 0I{1}_{}I{2}_{}l}{2\pi r}$

where $\mu 0$ is the permeability of free space, $I{1}_{}$ and $I{2}_{}$ are the currents, $l$ is the length of the wires, and $r$ is the distance between them.

How do you calculate the force per unit length between two parallel current-carrying wires?

The force per unit length between two parallel current-carrying wires can be calculated using the formula:

$\frac{F}{l}=\frac{\mu 0I{1}_{}I{2}_{}}{2\pi r}$

where $\mu 0$ is the permeability of free space, $I{1}_{}$ and $I{2}_{}$ are the currents in the wires, and $r$ is the distance between the wires. This formula gives the force experienced per unit length of the wires.

Why do parallel currents in the same direction attract each other?

Parallel currents in the same direction attract each other due to the interaction of their magnetic fields. Each current-carrying wire generates a magnetic field that circles around it. When the currents flow in the same direction, the magnetic fields interact in such a way that the forces between the wires pull them together. This attraction can be explained using the right-hand rule and the principle that magnetic fields exert forces on moving charges (currents). The force can be calculated using the formula:

$F=\frac{\mu 0I{1}_{}I{2}_{}l}{2\pi r}$

where $\mu 0$ is the permeability of free space, $I{1}_{}$ and $I{2}_{}$ are the currents, $l$ is the length of the wires, and $r$ is the distance between them.