Question Video: Identifying the Current Configuration That Generates a Given Magnetic Field | Nagwa Question Video: Identifying the Current Configuration That Generates a Given Magnetic Field | Nagwa

Question Video: Identifying the Current Configuration That Generates a Given Magnetic Field Physics • Third Year of Secondary School

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Several horizontal pairs of parallel conducting wires are stacked vertically. The magnitude of the current in each wire is the same. A cross section of the resultant magnetic field due to the currents is shown in the diagram. Which of the configurations of current directions shown would produce the resultant magnetic field?

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Video Transcript

Several horizontal pairs of parallel conducting wires are stacked vertically. The magnitude of the current in each wire is the same. A cross section of the resultant magnetic field due to the currents is shown in the diagram. Which of the configurations of current directions shown would produce the resultant magnetic field?

In our diagram, we see this cross section of the overall magnetic field formed by current-carrying pairs of wires. We’re told that these pairs of wires are parallel to one another. We can imagine then that each pair — this is one pair, here’s another pair, and so on — each one of these pairs consists of two wires, which could carry current into or out of the screen as we see them. Our four answer options show us possibilities for the direction of current through each wire. We know that a symbol like this represents current pointing into the screen, while a symbol like this represents current pointing out of the screen.

We want to pick which of these four possible answers shows us the current directions that would generate this magnetic field cross section we see here. Let’s begin by clearing some space on screen and recalling a bit about the direction of the magnetic field formed by current-carrying wires. If we have a long straight wire carrying a current 𝐼 in this direction, then by pointing the thumb on our right hand in the direction of that current, the fingers on that hand then curl closed in the direction of the magnetic field around the wire. In this case then, the magnetic field would point like this.

We can use the same right-hand rule to figure out the direction of the magnetic field formed by a current pointing into the screen, as well as one that travels in a wire where the current points out of the screen. If we point the thumb on our right hand into the screen as we see it, then our fingers on that hand curl closed in this direction, clockwise as we see it. If, however, we point the thumb on our right hand out towards us in accordance with the direction of this current, then the only way the fingers on that hand can curl closed is in a counterclockwise direction.

We know that the magnetic field created by each one of these current-carrying wires actually extends infinitely far away from that wire. That means if we were to pick a point right here, say, equidistant between the two wires, then the wire on the left would still create a magnetic field at this point. At this location, it would point straight downward, while the wire on the right would also create a magnetic field at this point. And at this location, it too would point downward. Then, that magnetic field then at this point would be indicated by this arrow in blue. It’s the vector sum of the two contributing magnetic fields.

What we’re finding here is that when we have a pair of parallel conducting wires and that those wires carry currents pointed in opposite directions, the magnetic field at a point midway between these wires doesn’t cancel to zero. Rather, the fields combine and add to one another. On the other hand, if we did have a pair of wires that have current traveling in the same direction, say, in this case out of the screen towards us, then by our right-hand rule, each of these wires would create a magnetic field that points counterclockwise around itself so that at the midpoint between these two wires, the field created due to this wire on the left would point upward like this, while the field due to the wire on the right would point downward in the opposite direction. If the magnitude of current carried by these two wires was the same, as it is in our scenario, then the net magnetic field midway between these wires is zero.

Knowing this, let’s look back at this original diagram. And we notice that the magnetic field in between the pairs of current-carrying wires is actually quite strong. We know that because of the high density of magnetic field lines in this region. This means that our pairs of current-carrying wires cannot be pairs such that the magnetic field between them is zero. We saw that that is the case when both wires in the pair carry current pointing out of the screen. Therefore, we can look at answer option II, which shows us many such pairs, and realize that this configuration would give us a magnetic field of zero in the middle of these pairs of wires. That though is not what we see in our diagram, so we can cross out answer choice II.

Now that we know that when we have a pair of wires like this, the magnetic fields they create actually combine and strengthen one another at the midpoint between them, let’s see what happens when we stack this pair with an oppositely arranged pair of wires. Once again using the right-hand rule to determine the direction of the magnetic fields formed around these wires, if for this wire, we point our thumb out of the screen towards us, we note that it creates a counterclockwise magnetic field, while for this current-carrying wire, with the current pointing into the screen, the current generates a magnetic field that points in the clockwise direction.

Once again, let’s consider the net magnetic field at a point midway between these two wires. The magnetic field from the wire on the left points upward at this location like this, and the magnetic field from the wire on the right also points in that same direction. And if the currents in these two wires have the same magnitudes, then the magnetic fields at this point have the same magnitudes too. What we find then is the net magnetic field points like this. Notice though that this opposes the net magnetic field direction from the pair of wires just above this lower pair.

If we were to keep this pattern going, adding more and more alternating pairs of current direction, we would continue to see this oppositional effect. With this pattern, what essentially happens is that each pair of pairs stifles, or diminishes, the magnetic field in between the pairs of wires. Now, let’s say we make a different arrangement. Imagine that we take this pair of wires here and this pair of wires here and we reverse them left to right. Before we do that though, notice that these two pairs that we’ve identified created a net upward pointing magnetic field between them. If we reverse the direction of current in each of these pairs of wires, now the wire on the left carries current into the screen and the wire on the right carries current out of it.

What that means is that now when we consider a point midway between these pairs, the net magnetic field points downward rather than upward. Notice what happens then to the magnetic field in between these pairs of wires. The field from each pair of wires combines with the field from other pairs. What this yields is a very strong magnetic field in between the pairs of wires like we see in our diagram. This arrangement of pairs of wires, where the current direction in all the wires on one side is the same and it’s opposite the direction of current in all the wires on the other side, matches answer option III.

Note that we’ve already in a sense analyzed answer options I and IV and found that they create situations where the net magnetic field in between the pairs is not as strong as it could be. Option III then will be our answer. This is the configuration of current direction in these pairs of wires that could give rise to the situation we see in our diagram. As a side note, notice that we don’t know the direction of the magnetic field lines in our diagram. Therefore, it could be that the current directions are exactly as shown in answer choice III, or they could be exactly the opposite. Either way, the magnetic field strength in between these pairs of current-carrying wires would be strengthened.

The best option among our possible answers though is answer choice III.

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