Video Transcript
In this video, our topic is
electromagnetic interactions in conducting loops. These interactions can lead to
phenomena that we may not expect. For example, we see in our sketch
that by passing a permanent magnet through the loops of a conducting coil, current
is generated in that coil. And the direction of the current
changes as we change the motion of the magnet through the loops. In this lesson, we’ll learn how
this happens, and we’ll also study the relative directions of changing magnetic
fields and changing currents.
As we get started, let’s consider
this simplified setup. Say that we’ve got a permanent
magnet here with a north and a south pole. And then a loop of conducting wire
is below the magnet with an ammeter, a device for measuring current, in it. If we keep our magnet and our loop
of wire stationary, then we know that the current readout on our ammeter will be
zero; there’s no current in this loop. But still there’s more going on
here than we can see.
Recall that any permanent magnet
creates a magnetic field around itself. If we could see those field lines,
they might look like this. And because these field lines
always point from the north pole of a magnet toward the south pole, we know that the
directions associated with them look like this. With these field lines visible, we
can see that some of them are passing through our circular conducting loop. Seeing magnetic field lines pass
through some area like this can remind us of magnetic flux.
Written as a symbol, we typically
express magnetic flux using Φ sub 𝑚. And we know it’s equal to the
strength of a magnetic field experienced by some cross-sectional area 𝐴. Looking back to our permanent
magnet and conducting loop, we can see that here we have a cross-sectional area and
we also have a magnetic field 𝐵 generated by our magnet. All this tells us that at this
moment in time, even before our magnet begins to move with respect to our loop,
there is some nonzero amount of magnetic flux linked to this loop.
But nonetheless, as we see our
ammeter showing us, there’s still no current induced in the loop. So this is our setup. And now let’s say we change it by
letting our magnet fall downward through the loop. When we do this, the first thing we
may notice as the magnet begins to fall is that the current 𝑖 on our ammeter moves
off of zero. In other words, as the magnet is in
motion, there is some induced current in our loop. A second thing we can take note of
is that the magnetic flux through our loop is changing. It’s not that the area of our loop
is changing. That’s staying constant. But what is varying in time is the
magnetic field that passes through the loop. As our magnet falls, the strength
of the vertical component of the magnetic field passing through our loop varies.
So then here’s what we know so far
in terms of magnetic flux and induced current. If we consider this quantity here,
which is a change in magnetic flux divided by a change in time, if we recall back to
a moment ago where our permanent magnet was stationary above the loop, at that
instant the change in magnetic flux over time through the loop was zero. And corresponding to this, we saw
that no current was induced in our conductor. But on the other hand, when we
dropped our permanent magnet so that it was in motion through our loop, then in that
case the change in magnetic flux per unit time was not zero through the loop. And when that happened, current was
indeed induced.
It turns out that these two
findings are true in general, and they’re described mathematically by a law called
Faraday’s law. Now interestingly, Faraday’s law
doesn’t explicitly mention current, but it does describe emf. And if we think about it, emf, or
voltage, is a necessary precursor to current. No charge will flow in a loop, no
current will exist in it, without an emf across it. So Faraday’s law describes emf. And it says it’s equal to negative
a constant, and we’ll talk about that constant in a moment, multiplied by the time
rate of change of magnetic flux.
To better understand this equation,
let’s consider this factor right here, ΔΦ sub 𝑚. We’ve seen that when ΔΦ sub 𝑚 is
zero, that means no emf and therefore no current is induced, and that when it’s
nonzero, some current is induced, and therefore so is emf.
But to understand what that means
for there to be a change in magnetic flux, let’s go back to our definition of that
term here. If we decide to modify this
expression so that we’re looking not at magnetic flux but a change in that flux,
then that means we would insert this symbol Δ, representing change, in front of both
sides of the equation. In order for ΔΦ sub 𝑚 to not be
zero then, there needs to be either some change in the magnetic field or some change
in the area that field is passing through or a change in both.
Just to consider a quick example,
say that we had a uniform magnetic field that was directed into the screen and that
in a plane perpendicular to that field direction we had a circular conducting
loop. Now, if our field was constant in
time — it didn’t get stronger or weaker or change direction — and if the area of our
conducting loop that was exposed to the field also didn’t change in time — in other
words, the ring didn’t grow or shrink or rotate — then we would have both a constant
magnetic field and a constant area exposed to that field. In this case, the change in
magnetic flux would then be zero. And therefore, no emf would be
induced in our loop, and so no current would travel through it.
But then we could imagine a
different scenario. Let’s say that over some amount of
time, and we can call that Δ𝑡, our magnetic field strength increased. So even though the area of our loop
exposed to the field hasn’t changed, 𝐵 has; it’s gone up. This means that ΔΦ sub 𝑚, the
change in magnetic flux, is not equal to zero. And therefore, this change in
magnetic flux will lead to an emf induced in the loop and then a current. We start to see then how important
these Δ symbols are in Faraday’s law. It’s not enough for there to simply
be some amount of magnetic flux. That amount needs to change over
time in order for an emf to be induced.
We could say that this fraction
here, ΔΦ sub 𝑚 over Δ𝑡, is the most important part of Faraday’s law. But in order for this law to be
accurate, for the equation to be true, we also need this constant we mentioned
earlier and this negative sign.
Let’s first explain this constant
capital 𝑁. Remember that on our opening
screen, we had a coil of wire with a number of turns in it. This meant that when our permanent
magnet passed through the coil, it passed through each one of these loops. Each one of these individual turns
— we have one, two, three, four in our coil here — multiplies the emf that’s induced
in a coil when a magnet is passed through it.
Let’s say that this coil consists
of just one loop. And let’s say further that when a
magnet passes through this one loop, it induces an emf. And we’ll just call that emf 𝜀 sub
one. If we were to then double the
number of loops in our coil by adding one, then the emf induced when a magnet passed
through would be two 𝜀 one. And if we triple that number, we
would get three times the original emf and so on. That’s how the number of loops
multiplies the emf induced. The constant 𝑁 in our Faraday’s
law equation represents that number. However many loops we have, one or
seven or 1000, so long as the change in magnetic flux per unit time is the same
through each one, then we take that change and multiply it by 𝑁 to give us the
magnitude of the emf induced overall.
Now, the last part of Faraday’s law
to consider is this negative sign. It’s not always necessary to take
this minus sign into account, say, for example, if we wanted to solve just for the
magnitude of emf induced in some scenario. But the physical significance of
this sign is important.
Let’s consider again this situation
where we have a permanent magnet that’s being dropped through a conducting loop. As the magnet fell from its
original position, roughly here to here, we said that the magnetic flux through our
conducting loop changed, not because the area of that loop changed with respect to
the field — that stayed constant — but rather the strength of the magnetic field
increased over this time. And that’s because the north pole
of our magnet got closer and closer to the plane of a loop.
If we were to draw then the change
in magnetic field through this loop over this interval of time, we would see that
that change points downward. We could also say that the magnetic
field is increasing in the downward direction. This change in field leads to a
change in flux, which, when this change happens over some amount of time, leads to
an emf and, therefore, a current induced in this loop.
There’s something interesting about
that current though. The direction of the current
induced in the loop is such that it opposes the change in magnetic flux through the
loop. The way this happens, the way this
current opposes the change in magnetic flux, is by traveling in a direction such
that the current itself produces a magnetic field that points the opposite way. We could call this magnetic field
𝐵 induced, 𝐵 sub ind. This is the field created by the
current induced in the loop. And it’s always the case that
induced current resists a change in magnetic flux. This rule is called Lenz’s rule or
Lenz’s law.
The fact that induced current
creates a magnetic field that opposes a change in magnetic flux is the reason behind
the negative sign in Faraday’s law. The emf that’s induced in a
conducting loop due to a change in magnetic flux through it will drive current that
creates an induced magnetic field, we’ve called it 𝐵 sub ind, that works against or
opposes the change in magnetic flux originally experienced by the loop. We could say then that an induced
emf, and therefore an induced current, fights against change to the system. It tries to keep things the same by
counteracting changes the system experiences.
Now to be a bit more clear about
all these directions involved, let’s clear a bit of space on this diagram. And even though we’ve removed the
permanent magnet from our sketch, let’s say that it is actually still falling
through the loop. So, in other words, the change in
magnetic field, and therefore the change in magnetic flux experienced by the loop,
points down. As we mentioned, this will cause a
current to be induced in the loop, which generates a magnetic field, which opposes
this change Δ𝐵.
The question then is, which way
does current point in this loop in order to generate such a field 𝐵 sub ind? To figure this out, we can use
what’s called a right-hand rule. In this rule, we point the thumb of
our right hand in the direction of the induced magnetic field. Next, we curl our fingers
closed. And the direction of that curl
tells us which way current points in our loop to generate such a field, 𝐵
induced. In the case of our loop here, this
would indicate charge moving in this direction around the loop. And here, we could call this
current 𝐼 induced because it is induced by the change in magnetic flux due to the
magnet that is actually falling through this loop.
Knowing all this, let’s get some
practice now with these ideas through an example exercise.
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.
Let’s take a moment now to
summarize what we’ve learned about electromagnetic interactions in conducting
loops.
In this lesson, we saw that a
change in magnetic flux over time through a conducting loop induces an electromotive
force in that loop. This is described by Faraday’s law
of electromagnetic induction. We saw further that when emf is
induced, it generates a current which creates a magnetic field whose direction
opposes the change in magnetic flux originally experienced by the loop. So if we had a loop and a change in
magnetic flux directed downward through it, then the emf induced in the loop would
generate a current which creates a magnetic field in this opposite direction. We call this the induced magnetic
field. And we note that it’s caused by the
induced current.
And lastly, we saw that the
direction of this induced current is determined by what’s called a right-hand
rule. Using this rule, we point the thumb
on our right hand in the direction of the induced magnetic field, the one that
opposes the change in magnetic flux. And we then curl our fingers. And it’s the direction of this curl
that tells us the way induced current points. This is a summary of
electromagnetic interactions in conducting loops.
