Lesson Video: Force on Conducting Wires in Magnetic Fields Physics • 9th Grade

In this video, we will learn how to use the formula 𝐹 = 𝐵𝐼𝐿 to calculate the force experienced by a current-carrying wire that has been placed in a uniform magnetic field.

12:49

Video Transcript

In this video, we’re talking about force on conducting wires in magnetic fields. The fact that putting a conducting wire in a magnetic field can make it experience a force may come as a surprise. But as we’ll see in this lesson, there are circumstances where, indeed, this takes place. We can start out by considering just what are the conditions needed for such a force to exist.

Let’s start here. Say that we have a length of conducting wire but no current is in the wire. Under these conditions, we know that no magnetic or electric force will be experienced by the wire. But then say that we take a couple of bar magnets. And we put them near this wire so the north pole of one bar magnet faces the south pole of the other. By doing that, we set up a magnetic field that runs from the north pole to the south pole.

Now, by the way, that’s not the only part of the magnetic field created by these two bar magnets. But for now, this is the only part of that field that we’ll focus on. So we now have a conducting wire, still with no current in it, that’s in the presence of a magnetic field. Under these conditions, our wire still experiences no force, even the part of the wire that’s surrounded by this magnetic field.

Finally, say we’re able to start the flow of charge through this conducting wire. It now has a current. With all these pieces in place, now, the section of the current-carrying wire that’s in the magnetic field would begin to experience a magnetic force. But importantly, the current-carrying wire only experiences that force when it is surrounded by this field. There’s no force on the wire here or here or anywhere else outside of this magnetic field.

Now, let’s say that the width of these bar magnets and therefore also the width of the magnetic field that our current-carrying wire is in we call capital 𝐿. In that case, we can say that a length 𝐿 of our current-carrying wire is exposed to this magnetic field and therefore experiences a force. The magnitude of that force, we can call it 𝐹 sub 𝐵, depends on three things. First, it depends on the strength of the magnetic field that the wire is in. We can call that field strength 𝐵 in general. Second, it depends on the magnitude of the current in the wire. The greater that current is, the greater the magnetic force experienced.

And lastly, it depends on how much of the wire, how great a length, is exposed to the magnetic field 𝐵. If 𝐿 is the length of wire within the field, then the overall magnetic force is equal to 𝐵 times 𝐼 times 𝐿. So to reiterate, all we need to do to create a magnetic force on a wire is to send some current through that wire and put the current within a magnetic field. Now, there’s one important caveat to all this. This equation we’ve just written down is true, but it requires a certain orientation between the magnetic field 𝐵 and the current 𝐼.

Notice that, in this diagram, our current points to the right, while our magnetic field points up. So then, there’s a 90-degree angle between these two quantities. So long as the current and the magnetic field are perpendicular to one another, this equation applies. But if they’re not, if the magnetic field and the current are at some other angle to one another, then it’s no longer valid. In particular, if we had a magnetic field that was pointed in parallel with the current in our wire, then that segment of wire in the field would actually experience no magnetic force.

So let’s expand on this relationship a bit. We can say that this equation is true if 𝐵 is perpendicular to the current 𝐼. But on the other hand, if 𝐵 is parallel to 𝐼, then the magnetic force on that stretch of wire is zero. That’s one important thing to keep in mind regarding this equation. And another is that sometimes we’ll see it expressed in a slightly different way. For instance, we might see it written as 𝐼 times 𝐿 times 𝐵, current times length in the magnetic field times magnetic field strength. Mathematically, these two expressions mean the same thing, but they look a bit different. So it’s helpful to know that really they are the same.

Now, to this point, we’ve considered a wire with charge moving in it left to right. But what if we had an identical conducting wire with the same magnitude of current in it, but this current pointed in the opposite direction? The question is would the magnetic force experienced by the second wire be the same as the magnetic force experienced by the first? This is where we need to remember that force is a vector quantity. That is, it has magnitude as well as direction to it. Because these two identical conducting wires had the same magnitude of current in them and they’re exposed to the same strength of a magnetic field and the same length of wire is exposed to that field. Then the magnitude of the force experienced by the two wires is the same, but the direction is not.

The reason we know this is thanks to a rule called Fleming’s left-hand rule. Through this rule, we use the thumb and fingers on our left hand to figure out the direction of the magnetic force acting on a current-carrying wire. Here’s how Fleming’s left-hand rule works. We start out by pointing our index finger in the direction that the magnetic field 𝐵 points. In our example, that’s upward to the top of the screen. Then keeping that finger pointed that same direction, we point our second finger, that’s our middle finger, in the direction the current 𝐼 points.

When we do this, it’s important to use the direction of conventional current. In other words, the direction positive charge would move in the wire if it did move. And then lastly, keeping our index and middle fingers in these positions, we point the thumb on our left hand so that it’s perpendicular at 90 degrees to both of them. In this situation, that would be pointing out of the screen towards us. That direction of our thumb is the direction that the magnetic force acts on this current-carrying wire in a magnetic field.

What we’ve done here is applied Fleming’s left-hand rule to this current-carrying wire, the one carrying current left to right. And we’ve seen that the force direction on this wire, where the wire is within the magnetic field, is out of the screen at us. But if we then consider the other current-carrying wire with charge running right to left, now our left-hand rule works differently because instead of current pointing to the right, now, it points the opposite way. So we twist our left hand around so our index finger still points up in the same direction as the magnetic field previously. But now, our middle finger points to the left. That’s to match the direction of conventional current in the second wire.

And then, if we point the thumb on our left hand perpendicular to both our index and middle fingers, this is now the direction of the magnetic force on this second wire. Our thumb in this case points into the page, away from us. So then going to this second wire in our diagram, we see that the magnetic force direction on this is into the page rather than out of the page.

So to summarize this, the magnitude of the magnetic force experienced by these current-carrying wires in the magnetic field is the same. But since charge is moving in opposite directions in these two wires, Fleming’s left-hand rule tells us that the force directions on the wires is opposite. One force, in this case, points into the page and the other points out of the page. For this magnetic force on current-carrying wires, the equation, 𝐹 sub 𝐵 is equal to 𝐵 times 𝐼 times 𝐿, tells us that force magnitude and Fleming’s left-hand rule tells us the force direction. Knowing all this, let’s now get a bit of practice with these ideas through an example.

When positioned at 90 degrees to a magnetic field, a wire of length one meter carrying a current of four amperes experiences a force of 0.2 newtons. What is the strength of the magnetic field?

All right, so in this example, we have a length of wire. This length is one meter long. And we’re also told that the wire carries a current of four amperes. And then, in addition to this, this current-carrying wire is positioned at 90 degrees to a magnetic field. So we could draw magnetic field lines that look like this, where these lines, as we’re told in the problem statement, are perpendicular to our wire. For our purposes, whether the field lines point straight up as we’ve drawn them or if they pointed straight down doesn’t make a difference so long as the field lines are at 90 degrees to our wire.

We’re told that, in this magnetic field, our wire experiences a force, and it has a magnitude of 0.2 newtons. Knowing all this, we want to solve for the strength of the magnetic field that the current-carrying wire is in. We can call that magnetic field strength capital 𝐵. And let’s say that the force the wire experiences in the field at 0.2 newtons is 𝐹 sub 𝐵. So we have all this information, but we need some way to tie it all together. We can do this by recalling a mathematical expression for the magnetic force experienced by a current-carrying wire in a magnetic field. That force is equal to the strength of the magnetic field 𝐵 times the magnitude of the current running through the wire multiplied by the length of wire exposed to the magnetic field.

Now, in our case, it’s not 𝐹 sub 𝐵 we want to solve for, but rather the magnetic field strength 𝐵. So to rearrange and solve for that, let’s divide both sides of the equation by 𝐼 times 𝐿 causing those two terms to cancel out on the right-hand side. So then, the magnetic field is equal to the magnetic force divided by the current times the length of the wire in the field. We can now apply this relationship using the numbers we’ve been given. 𝐵 is equal to 0.2 newtons divided by four amperes times one meter.

And if we consider just the units for a moment, a newton per ampere meter, that’s equal to what’s called a tesla, symbolized capital 𝑇. This is the SI base unit of magnetic field strength. So 𝐵 is equal to 0.2 divided by four times one tesla, which is equal to 0.05 teslas. That’s the strength of the magnetic field that the current-carrying wire is in.

Let’s look now at a second example exercise.

The diagram shows a section of wire that has been positioned parallel to a uniform 0.1-tesla magnetic field. The wire carries a current of two amperes. What is the direction of the force acting on the wire due to the magnetic field?

Okay, in this diagram, we see this section of wire here, which is parallel to a uniform magnetic field. The diagram also shows us that charge in the wire moves left to right. Based on this, we want to solve for the direction of the force that acts on this wire due to the magnetic field. Now, we have to be a bit careful here. Seeing that we have a current-carrying wire in a magnetic field, we might think of the mathematical relation for the force on such a wire.

That equation says that this force is equal to the strength of the magnetic field multiplied by the current magnitude in the wire times the length of the wire exposed to the field. But there’s an important condition that’s required in order for this equation to be valid. The condition is that the magnetic field direction is perpendicular to the direction of current in the wire. In our case though, we can see that condition is not met because the current and the magnetic field point parallel to one another. When this is the case, this equation here for magnetic force doesn’t apply. And, in fact, the magnetic force experienced by this current-carrying segment of wire is zero when the current and magnetic field are parallel. In other words, the force acting on this section of wire is zero.

Now, we’re not asked about the force magnitude itself, but rather its direction. We can see though that this realization impacts our answer to the force direction. That’s because a force of zero has no direction to it. So our answer to this question can simply be that there is no force acting on the wire. Since there’s no force, there’s no direction. And this then is our answer.

Let’s take a moment now to summarize what we’ve learned about force on conducting wires in magnetic fields. Starting out, we saw that if a current-carrying wire is placed in a magnetic field, it may experience a force. We say “may” because if the magnetic field that the current-carrying wire is in is perpendicular to the current, then that magnetic force experienced by the wire is equal to the magnetic field strength times the current magnitude times the length of the wire that’s in the field.

But on the other hand, if the magnetic field and the wire are arranged differently so that the magnetic field is parallel to the current, then in that case there’s no force on the wire. So if we have a situation that looks like this, where 𝐵 and 𝐼 are perpendicular, then we do have a nonzero force acting on that wire, 𝐵 times 𝐼 times 𝐿. But if our scenario looks like this, where 𝐵 and 𝐼 are parallel, no magnetic force is exerted on the wire.

And lastly, when there is a force on a current-carrying wire in a magnetic field, Fleming’s left-hand rule shows us how to find the direction of that force. We do it by pointing our index finger in the direction of the magnetic field 𝐵, our middle finger in the direction of conventional current in the wire, and then, perpendicular to both those directions, we point our thumb. And our thumb gives the direction of the resultant magnetic force on the wire in the field. This is a summary of force on conducting wires in magnetic fields.

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