Video: Force on Conducting Wires in Magnetic Fields

In this lesson, 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.

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Video Transcript

In this video, we’re talking about the force that acts on conducting wires, which themselves are in magnetic fields. As we learn about this magnetic force, we’ll see which direction it acts in as well as its magnitude or strength.

There’s one important thing to get clear as we begin this discussion. In an earlier lesson, we saw that when a wire is carrying current, that moving charge creates a magnetic field around the wire. And it’s true such a field is created by the charge moving down the wire. But this field, the magnetic field created by the wire, is not the one we’re talking about in this case.

For our purposes in this lesson, we’re considering an external magnetic field, one that has nothing to do with the field of the wire. This external field might well be supplied by a permanent magnet. Say we had a permanent magnet like this that surrounded the wire so that the magnetic field it created pointed from the north pole to the south pole, perpendicular to the direction of current flow in the wire.

One of the fascinating things about this setup is that this wire will experience a magnetic force. Let’s think about this for a moment, going back to basics, electric charge. We know that electric charge is the source of electric fields and that electric charge in motion is the source of magnetic fields. If we consider the charges in our current-carrying wire, that’s really what they are. They’re electric charges in motion. Therefore, they create a magnetic field around themselves, like we saw just a moment ago.

But now what we’ve done is created an external magnetic field, the one created by the magnet, and put the wire in it. When that happens, it’s a bit like we’ve taken our moving charge and put it in a field like this. Under these circumstances, depending on the amount of charge, the strength of the field, and the speed with which the charge moves, this charge will experience a magnetic force.

Now before we go further, it will be helpful to get a bit of notation in place. Up till now, we’ve talked about magnetic fields and we’ve drawn in magnetic field lines. But we never labeled them to indicate what they are. For historical reasons, the letter capital 𝐵 has come to be used to indicate a magnetic field. So when we write a capital 𝐵 by a bunch of field lines like we have here, that indicates that this is a magnetic field we’re referring to. By the way, this symbol capital 𝐵 is also used in calculations to indicate the strength of a magnetic field. Okay, so that’s our first bit of notation, that magnetic field can be written with a capital 𝐵.

Next, let’s say that our charge here has a charge value of 𝑄. And we’ll say further that it’s moving with a speed we’ll call 𝑣. We said earlier that, given this setup, given a moving charge in an external magnetic field, that the charge would experience a magnetic force. When the motion of the charge is in a direction perpendicular to the external magnetic field, like it is now, we can write that magnetic force this way. 𝐹 sub 𝐵, the magnetic force on this charge, is equal to 𝑄, the charge value, multiplied by its speed, 𝑣, multiplied by the strength of the field it’s in.

Based on the variables involved here, a specific charge, and the specific speed on that charge, we can tell that this relationship applies to individual electric charges. That doesn’t seem offhand to be of much help when we’re talking about a current-carrying wire with many charges in a magnetic field. But actually, the connection between these two situations, where we have an individual charge compared to many chargers in a current, is actually fairly close.

Consider the right-hand side of this force equation, 𝑄 times 𝑣 times 𝐵. Let’s consider the middle term in this expression, the speed. We know that, in general, speed is equal to a distance divided by a time. We can say then that, effectively, this speed, 𝑣, is equal to some distance, 𝐷, divided by some amount of time, 𝑡.

Now if we then substitute in 𝐷 over 𝑡 for 𝑣 in this expression, we can then make a bit of a modification by taking this time, 𝑡, and bringing it outside the parentheses. That leaves us with a charge divided by a time multiplied by a distance multiplied by a magnetic field. At this point though, we can recall what charge divided by time is. That’s the definition for current.

Current, 𝐼, is defined as the amount of charge, 𝑄, that passes a point in a circuit over some amount of time, 𝑡. That means we can replace 𝑄 divided by 𝑡 in this expression by capital 𝐼, current. We can then consider this 𝐷 term, which is a distance in this expression. That distance refers to the length of current, so to speak, which is exposed to the magnetic field, 𝐵.

If we were to highlight that in our sketch, that would be the length of wire which lies between the poles of this magnet. We can leave this distance as 𝐷. But since it refers to a length of wire, we may as well refer to it that way, as capital 𝐿.

At this point, let’s recall that this expression that we have here, which we haven’t altered in any meaningful way, is equal to the magnetic force acting on, in this case, this current 𝐼. What we’ve done then is developed equations for the magnetic force on an individual charge, 𝑄, in a magnetic field, 𝐵, as well as the magnetic force on a length 𝐿 of current 𝐼 carrying wire, also in a magnetic field 𝐵.

For the rest of our discussion, we’re gonna focus on the second of these two expressions. Magnetic force is equal to the magnetic field strength multiplied by the length of wire that’s in the field multiplied by the current running through the wire. We can see that, so far, what we’ve solved for is the magnitude of the magnetic force that acts on a current-carrying wire in a magnetic field. But, of course, that’s only half of the story because this force being a vector has a direction as well. So what direction will the magnetic force on this current-carrying wire in our magnetic field that we can now label capital 𝐵 be?

To figure that out, it’s helpful to know this. The magnetic force on a charged particle is always perpendicular to the motion of the charge as well as the field that the charge is in. Here’s what that means. Say we have a positive electric charge and it’s moving in this direction. Say further that that charge is in a magnetic field and that field points this way, perpendicular to the motion of the charge.

What we’ve just said is that the magnetic force acting on this charge must be perpendicular to both of these vectors. There are only two different ways then that it could point. It could point like this as a force vector or it could point the opposite way like this. The point is that force is perpendicular or at a 90-degree angle to both the magnetic field as well as the charge motion.

We may wonder then which of these two directions the force actually acts and how we figure that out. The answer comes down to an application of what we can call a right-hand rule. This name has the advantage of clearly showing us what hand we’ll use to figure this direction out. The tricky part is that there are actually several right-hand rules, some of which you may remember from previous lessons. It can be tricky to keep them all straight when they all go by the same name, right-hand rule. The important thing though is not so much the name as it is remembering which version to apply given the circumstances.

In this situation, we have a vector, which is the current, flowing through the wire, as well as a vector, which is the magnetic field created by the magnet. Let’s say then that this blue arrow we’ve drawn is the direction of the current flowing through the wire, which is a vector, and that the pink vector is the magnetic field created by our permanent magnet. And just to make things a bit more clear, let’s rotate this whole setup so that the current actually goes the way it is in our sketch.

Okay, so we have current moving along in this direction. And then at 90 degrees to that, we have the magnetic field created by the magnet, 𝐵. We’ve said that, in this situation, when there’s charge moving through an external magnetic field, there’s a force that acts on that charge and that that force is perpendicular to both the field and the flow of the charge.

Based on this diagram then, we can see that that force will either be straight up or it will be straight down. And here’s where our rule comes in “handy,” get it? The first thing we do here is we take our right hand and we point the fingers of our hand in the direction of the current flow. Note that this is conventional current, that is, the motion of positive charge through this wire.

Next, what we do is we curl our fingers until they point in the direction of the magnetic field, 𝐵. Then, finally, we point our thumb up. And the direction that our thumb points is the direction of the magnetic force on this current-carrying wire. Based on this right-hand rule then, we can see that the force on this particular current-carrying wire would indeed be up. That is the direction of the magnetic force vector.

This right-hand rule may take a bit of practice to get right. And that’s just fine. As you go along and use it, remember one important thing. In order for this rule to properly indicate the direction of magnetic force, it must be the case that the current and the magnetic field are perpendicular, at 90 degrees, to one another. If they’re not — say that they were running in the same directions or even in opposite directions — if the current and magnetic field relate to one another this way in terms of their direction, then there’s no net magnetic force on these charged particles. Under those conditions, the magnetic force on them is zero. Let’s take some time now to get a bit of practice with these ideas of the force on a current-carrying wire in a magnetic field.

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

We can see that this here is our wire. We’re told that it carries a current of four amperes and that it’s one meter long. We’re told that this current-carrying wire is in an external magnetic field, which is oriented at 90 degrees to the wire. Under these conditions, the wire is subjected to a magnetic force we can call 𝐹, which is given as 0.2 newtons.

Knowing all this, we want to solve for the strength of the magnetic field, what we’ve called 𝐵 in our sketch. To figure this out, we can recall a mathematical relationship between wire length, current, magnetic field, and force. The magnetic force on a current-carrying wire is equal to the magnetic field strength that the wire is in multiplied by the length of the wire times the current that’s running through it.

In our case, it’s not the magnetic force we want to solve for, but the magnetic field. And we can do that by rearranging this equation. If we divide both sides by the current times the length of the wire, we get this result. 𝐵 is equal to 𝐹 sub 𝐵 over 𝐼 times 𝐿. Looking at the information in our problem statement, we’re told 𝐹 sub 𝐵. That’s 0.2 newtons. We’re also told the current, 𝐼, in the wire of four amps and the length of the wire of one meters. Our next step then is to substitute these values into this equation. 0.2 newtons divided by four amps times one meter equals 0.05 teslas, where a tesla is the unit of magnetic field. It’s often abbreviated just using a capital T. 0.05 teslas then is the strength of the magnetic field in this situation. Now let’s look at a second example.

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?

Taking a look at our diagram, we see this wire, marked out in pink, with current running left to right. We see this wire is placed within a uniform magnetic field called 𝐵. And this field also points from left to right. Given the strength of the magnetic field and the magnitude of the current, we want to solve for the direction of the magnetic force acting on the wire due to the field.

What may first come to mind is the right-hand rule we used to help us figure out this force direction. Using our right hand, we might start out by pointing our fingers in the direction of the current as this rule calls for. But then when we seek to curl our fingers in the direction of the magnetic field, we notice something interesting. The magnetic field is in the same direction as the current. They point the same way.

It’s at this point we must be very careful to remember a condition of this right-hand rule. And that is that this rule will only be certain to give us the direction of the force on a current-carrying wire when the current and the magnetic field the wire is in are perpendicular to one another, at 90 degrees. And, in fact, in the special case when current and magnetic field are in the same direction or even when they’re 180 degrees opposed, in these two instances where they’re parallel or antiparallel to one another, the magnetic force on the wire is zero.

Looking back at our diagram, we find that that’s the case in this scenario. Our current and our field are moving in the same direction. They’re parallel, and therefore the force on the wire is zero. This isn’t the most common scenario, but we have seen that, in this case, it did come up. Because the current flowing in the wire and the magnetic field the wire is in are parallel, there is no force acting on the wire.

Let’s take a moment now to summarize what we’ve learned about force on conducting wires in magnetic fields. For starters, we learned in this lesson that when a current-carrying wire is perpendicular to an external magnetic field, that wire experiences a force. That force magnitude is equal to the magnitude of the field strength, 𝐵, multiplied by the length of the wire in the field multiplied by the current running through the wire.

We learned that the direction of that force is given by what’s called the right-hand rule. This version of that rule has us take our right hand and point our fingers in the direction of the conventional current flow. If we then curl our fingers in the direction of the magnetic field, assuming it’s perpendicular to the current, then in that case our thumb points in the direction of the magnetic force on the wire.

Lastly, we noticed then when the current and the external magnetic field are parallel to one another, like in this case, or antiparallel, like in this case, then the force acting on the wire is zero.

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