Video: Electromagnetic Interactions in Conducting Loops

A loop of wire with radius 15 cm moves perpendicularly to a uniform 0.25 T magnetic field at a constant speed, as shown in the diagram. The motion takes 1.5 s to complete. Find the electromotive force induced in the loop.

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Video Transcript

A loop of wire with radius 15 centimeters moves perpendicularly to a uniform 0.25-tesla magnetic field at a constant speed, as shown in the diagram. The motion takes 1.5 seconds to complete. Find the electromotive force induced in the loop.

In our diagram, we see a uniform magnetic field pointed out of the screen at us. And moving perpendicularly to that is this loop of wire. We see its original position here and then its final position here. So this loop of wire moves like this at a constant speed. And the complete motion takes 1.5 seconds. Knowing all this, we want to solve for the electromotive force or emf induced in the loop.

To help us do that, we can recall Faraday’s law of electromagnetic induction. This law tells us that the emf induced in a conducting loop is proportional to the change in magnetic flux, ΔΦ sub 𝑚, through that loop divided by a change in time. We can recall further that magnetic flux in general, Φ sub 𝑚, is equal to a magnetic field strength through some area exposed to that field. In our application, this area would specifically be the cross-sectional area of some conducting loop.

So Faraday’s law tells us that we need to have a change in magnetic flux, in other words, a change in Φ sub 𝑚, in order for any emf to be induced in some conducting loop. And the way that happens, the way there is a nonzero change in magnetic flux, is if there is correspondingly a nonzero change either in the magnetic field strength or in the area exposed to that field. In other words, for ΔΦ sub 𝑚 not to be zero, either 𝐵 needs to change or 𝐴 or both 𝐵 and 𝐴. Knowing this, let’s look back at our scenario.

We have a loop of wire that moves perpendicularly to a uniform magnetic field. Now the fact that our magnetic field is uniform tells us that this magnetic field here in our equation for ΔΦ sub 𝑚 does not change. In our situation, we can say that Δ𝐵 is zero. That’s the meaning of the field being uniform. Nonetheless, we can still have a change in magnetic flux so long as the area exposed to our uniform field is changing in time. But then our problem statement tells us that our loop is moving perpendicularly to the field. This means that its initial area exposed to that field is this area here and its final area is equal to that initial area. Therefore, the area exposed to our magnetic field doesn’t change, which means that Δ𝐴 is equal to zero.

And if both Δ𝐴 and Δ𝐵 equal zero, as they do, then taken together, that implies that ΔΦ sub 𝑚, the change in magnetic flux, is also zero. And then since this is true, as we revisit Faraday’s law, we see that if ΔΦ sub 𝑚 in this equation equals zero, then so does the induced emf. And so because neither the magnetic field nor the area that field passes through changes, the electromotive force induced in this loop is zero volts.

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