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.
