Question Video: Motion of Straight Conductors in Uniform Magnetic Fields | Nagwa Question Video: Motion of Straight Conductors in Uniform Magnetic Fields | Nagwa

Question Video: Motion of Straight Conductors in Uniform Magnetic Fields Physics • Third Year of Secondary School

A 7.2 cm long conducting rod moves through a 36 mT uniform magnetic field, as shown in the diagram. The rod travels at 4.5 cm/s.

06:09

Video Transcript

A 7.2-centimeter-long conducting rod moves through a 36-millitesla uniform magnetic field, as shown in the diagram. The rod travels at 4.5 centimeters per second.

Before we get to the first part of our question, let’s consider this diagram. In it, we see a conducting rod with two ends, the top end called 𝐴 and the bottom end called 𝐵, that’s in motion at a steady speed — we can call that speed 𝑣 — through a uniform magnetic field. Now, that speed is 4.5 centimeters per second. And the field that the conducting rod moves through, we can see, is directed into the screen. Let’s call the magnitude of this field 𝐵. And we’re told that it’s equal to 36 millitesla or, in other words, 36 times 10 to the negative third tesla.

Along with all this, we’re told the length of this conducting rod. We’ll call that length 𝑙. And we can see it’s given as 7.2 centimeters. So our conducting rod of this length is moving with this speed through a magnetic field of this strength. Knowing this, let’s now look at the first part of our question.

What is the magnitude of the potential difference across the rod?

Well, it may surprise us that there’s any potential difference, but indeed there is. We know this based on the fact that the emf, or the potential difference, induced in a conducting rod is equal to the rod’s length times the strength of the magnetic field it moves in times its speed, so long as the rod is moving perpendicular to the magnetic field. We see in this case that that is indeed happening, that the rod is moving to the right, while the magnetic field is pointed 90 degrees to that into the screen.

So to calculate the potential difference across the rod, we’ll use the length, magnetic field, and speed values given to us. The rod’s length is 7.2 centimeters. The magnetic field strength is 36 times 10 to the negative third tesla. And it’s moving with a speed of 4.5 centimeters every second.

Before we calculate this potential difference though, let’s convert the distances in these two values from centimeters into meters. If we do that, then we’ll have all of the units in this whole expression in SI base units. Recalling that 100 centimeters is equal to one meter, in both of these cases, we’re gonna be shifting decimal points two spots to the left. Expressed this way, the length of our rod is 0.072 meters. And it’s moving with a speed of 0.045 meters per second.

When we multiply these three quantities together, to two significant figures, we find a result of 1.2 times 10 to the negative fourth volts. That’s the magnitude of the potential difference across the rod. Now, let’s look at the second part of our question.

This part asks, which end of the rod has a higher potential?

Looking at the rod, we see it has these two ends, end 𝐴 and end 𝐵. And as we think about how to answer this question, we can recall that positive electrical charges, by definition, have a higher electrical potential than negative electric charges do. This means that another way to answer this question of which end has higher potential is to figure out which end of the rod has more positive electrical charge.

Now, if we assume that our conducting rod is an electrically neutral object, then we would say it has the same number of positive charges as negative charges. In order for one end of the rod to have a higher potential than the other, there needs to be some sort of charge separation. That is, something has to cause one type of electrical charge to move to one end of the rod and the other type to move to the opposite end.

And indeed, there is such a force involved here. The magnitude of that force is given by this expression here. It says that if we have a charge 𝑞 moving with a speed 𝑣 perpendicularly to a magnetic field of strength 𝐵, then the force magnitude that charge experiences is equal to 𝑞 times 𝑣 times 𝐵.

However, in our case, it’s not the magnitude we’re interested in exactly, but rather the direction. We want to know which way charges are pushed in this conducting rod. Because this rod is a conducting material, that means that all throughout it has mobile electrons. Unlike the positive charges in this conducting rod, these electrons are free to move all throughout the conductor. And they do so in response to forces, specifically this magnetic force.

To figure out the direction in which this force acts on negative charges in our conductor, we can recall a right-hand rule for determining this. The first step involved is to figure out the direction of 𝑞 times 𝑣, where 𝑣 we take to be the velocity vector of our rod and 𝑞 is the value of our charge of interest, in our case an electron.

Now, we’re only concerned about the direction of 𝑞 times 𝑣. So if we look at our diagram, we can see that 𝑣 points to the right. That’s the way the rod overall is moving. But this isn’t the direction of 𝑞 times 𝑣 because recall that 𝑞 is the charge of an electron and, therefore, a negative value. This means that 𝑞 times 𝑣 instead of pointing to the right will actually point the opposite way, to the left.

And now that we know this, we take our right hand and we point our fingers in the direction of 𝑞 times 𝑣. As we said, that’s to the left. Our next step is to figure out the direction of the magnetic field 𝐵. That’s a bit easier. We know from our diagram that 𝐵 points into the screen. So now what we do is we curl the fingers on our right hand in that direction. Once we’ve done that, pointed our fingers into the screen, we then point our thumb perpendicular to both of the directions that our fingers have pointed. And when we do that, our thumb points in the direction of the force that acts on these charges, specifically the negative charges, the electrons, in our conducting rod.

So then as our rod moves along, the mobile negative charges in it will experience a force towards the lower end. And because they are mobile, they’ll actually collect down there at end 𝐵. This means that at the opposite end of the rod, end 𝐴, all the negative charges that left will leave an abundance of positive charge. And now that we have charge separation across the ends of our conductor, we can answer this question of which end of the rod has a higher potential. Because positive charges have a higher electrical potential than negative ones, we can say the end with the most positive charge has the higher electrical potential. And that, as we see, is end 𝐴.

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