### Video Transcript

In this video, we’re talking about
the torque on a current-carrying rectangular loop of wire in a magnetic field. We’re going to learn why a torque
acts on such a current-carrying wire, how to calculate its magnitude, and also how
to determine something called the magnetic dipole moment for such a current-carrying
loop. We can get started on this topic by
thinking at first of just a rectangularly shaped wire. So here’s one side, here’s a second
side, a third side, and then a fourth side. And then we’ll say that this dotted
line right here represents an axis that goes right through the center of the
rectangle.

Now what if, perpendicular to this
axis, we set up a uniform magnetic field? We can call that field strength
𝐵. Even with our rectangular wire in
this uniform magnetic field, nothing would happen unless we put current in the
wire. If we do this, things would start
to change because now we have electrical charge in motion in a magnetic field. Those charges when they’re moving
in certain directions relative to the field will experience a force. And since the charges that make up
this electrical current are in the wire, the wire itself will then experience a
force.

A bit earlier, we identified the
four sides of our rectangular wire. We said that here was side one,
here was side two, and then three and then four. As electric charge moves along
through these four different straight segments of wire, only two of these sides will
experience a force, side one and side three. That comes down to the direction of
the moving charge relative to the external magnetic field. On sides one and three, there will
be a net magnetic force that acts along that whole segment of wire. And those two forces, the one on
side one and the one on side three, act in opposite directions.

Considering that magnetic force as
it acts on side one, let’s say that it points in this direction, upward. That means that the corresponding
force on side three will point the opposite way, downward. Given these two forces acting on
opposite sides of our rectangular wire, we can see what’s going to happen. These forces combine to create a
torque on the wire, which tends to make it rotate around this axis we drew through
its center. Now, the symbol to represent this
torque is the Greek letter 𝜏. And what we’d like to do is write a
mathematical relationship for this torque. It turns out that in a case like
the one we’re looking at here, this torque depends on several factors.

First, it depends on the strength
of the magnetic field that our current-carrying wire is in. The stronger that field is, the
more torque the wire experiences. It also depends on the magnitude of
the current in the wire. The more current, the more moving
charge, and therefore the greater the force sides one and three of our wire will
experience and so the greater the torque. Something else the torque depends
on is the cross sectional area, we’ve called it 𝐴, of our loop. And along with this, we’ll want to
consider the possibility that a rectangular coil of wire has more than one loop.

The way we’ve drawn it right here,
it has only one. But in general, there could be any
number of turns in this coil. There could be some integer value
we just call 𝑁. So the torque on this rectangular
current-carrying wire depends on 𝐵, the magnetic field, the current 𝐼, the cross
sectional area 𝐴, and the number of turns in our coil. And for all four of these factors,
the bigger they get, the greater the torque. That tells us that all four will be
in the numerator of our equation for torque.

Our equation now is almost
complete, but there’s one other factor we’ll want to add here. Remember, we noted earlier that
this coil in this field would experience a torque, and that torque would tend to
rotate the wire about this axis. When that happens, it will make the
overall orientation of our current-carrying loop change. And this change then leads to a
change in the torque it experiences. We can get a better sense for
what’s going on here by looking at our setup from a different perspective. If we watch the edge of our
rectangular coil, as the coil turns, we would see it start out horizontally in the
magnetic field, but then turn like this, then like this, and so forth as we continue
to rotate clockwise as we’re looking at the coil.

Well, the angle between our coil
and the external field that it’s in affects the torque that the coil
experiences. The way we measure that angle is by
considering the plane in which the coil lies. We represent that by our cross
sectional area 𝐴, and we picture a vector that’s perpendicular to that plane. To be clear about what that looks
like, if our coil looked like this and on relative to the magnetic field, then that
vector perpendicular to the cross sectional area of the rectangular loop would look
like this pink one. Once we have this vector, which is
normal to the area of our loop or perpendicular to it, then we measure the angle in
between this vector and the direction of the external magnetic field. If we call that angle 𝜃, then we
can enter, over on our equation for torque, the last factor we need to complete the
equation.

When 𝜃 is the angle between the
external magnetic field that our coil is in and the vector that points perpendicular
to the area of our coil, then the overall torque that our rectangular coil
experiences is 𝐵 times 𝐼 times 𝐴 times 𝑁 times the sin of 𝜃. Let’s consider now how this factor,
sin 𝜃, affects the torque on our coil. Say that our coil of wire was
oriented like this with respect to the magnetic field. In this case, the vector
perpendicular to the area of the coil would point in this direction. And we can see that this is
perpendicular to the external magnetic field. In this case then, 𝜃 would be 90
degrees, and we know that the sin of 90 degrees is one. It’s the maximum value the sine
function attains.

Everything else being equal then,
our coil will experience a maximum torque due to the magnetic field it’s in when
it’s oriented to that field this way. But then, what about this? What if our coil has rotated to
this position? At this point, the vector
perpendicular to the area of the coil would point this way. And we can see this points in the
same direction as the external field. These two vectors are parallel, and
so 𝜃 is zero degrees. And then, the sin of zero degrees
is zero. So when our coil is oriented this
way, it experiences no torque. So if our coil is perpendicular to
the field like this, it experiences zero torque. And if it’s parallel to the
magnetic field like this, 𝜃 is 90 degrees and the coil experiences a maximum
torque.

Now, let’s focus for a moment on
this case where our coil experiences that torque maximum. What we can do is write a specific
version of our torque equation for this maximum value, we’ll call it 𝜏 sub 𝑚. And the only difference between
this equation and our original general equation for torque is that now we’re
assuming that 𝜃 is 90 degrees. So the sin of 𝜃 is one. If we consider this maximum torque
our coil can experience and the strength of the field that causes that torque, we
can identify what’s called the magnetic dipole moment of our current-carrying
loop. This term, magnetic dipole moment,
refers to the tendency of an object to interact with an external magnetic field.

So given our external field, we’ve
called that capital 𝐵, the more torque our coil experiences, the more it interacts
with that field we can say. And the magnetic dipole moment
measures the extent of that interaction. If we represent magnetic dipole
moment symbolically using 𝑚 sub 𝑑, mathematically, it’s equal to this ratio, the
maximum torque that our current-carrying coil can experience divided by the strength
of the field that the coil is in. This shows more clearly what we
mean by saying the magnetic dipole moment measures the response of an object to the
magnetic field it’s in. Given a magnetic field strength 𝐵,
the more torque our object experiences, the greater its magnetic dipole moment.

Just as a side note, because 𝜏 sub
𝑚 down here is equal to 𝐵 times 𝐼 times 𝐴 times 𝑁, that means that specifically
for a current-carrying rectangular loop of wire, we can also write the magnetic
dipole moment as 𝐼 times 𝐴 times 𝑁. This shows us that if we were to
keep everything the same in our scenario, but increase the current 𝐼, then we would
also raise the magnetic dipole moment of our coil. Or, likewise, if we kept everything
the same but increased the cross sectional area of our coil or increased the number
of turns in it, those are also ways that we could increase the coil’s magnetic
dipole moment. Knowing all this about the torque
on a current-carrying rectangular loop of wire in a magnetic field, let’s get some
practice now with these ideas.

The diagram shows a rectangular
loop of current-carrying wire between the poles of a magnet. The longer sides of the loop are
initially parallel to the magnetic field, and the shorter sides of the loop are
initially perpendicular to the magnetic field. The loop then rotates through 90
degrees so that all its sides are perpendicular to the magnetic field. Which of the lines on the graph
correctly represents the change in the torque acting on the loop as the angle its
longest sides make with the magnetic field direction varies from zero degrees to 90
degrees?

Okay, in our scenario, we have a
rectangular current-carrying loop of wire that’s shown in our diagram here. And our loop, we can see, is
positioned between the poles of a permanent magnet, which means it’s exposed to a
uniform magnetic field. Because a rectangular loop carries
current and is in a magnetic field, it experiences a torque about its axis of
rotation. This torque leads to the rotation
of the coil that we see here in a clockwise direction. We’re told that, initially, our
coil is oriented like this, where the longer sides — that is, the side of the front
and the back we could say of the coil — are parallel to the magnetic field and the
shorter sides are perpendicular to it.

But then, under the influence of
torque, our coil rotates 90 degrees so that in this position, all four of the sides
of the coil are perpendicular to the magnetic field. And we know that that field points
from the north pole of our magnet to the south pole, so left to right as we’ve drawn
it here. The other part of our diagram is
this graph right here. Looking at it carefully, we see
that on the vertical axis, the torque acting on a rectangular coil is plotted
against the angle of orientation of that coil to the magnetic field. That angle varies from zero
degrees, which is the position of our coil relative to the field when it starts out
in this horizontal plane, all the way up to 90 degrees, which is the coil’s position
relative to the field once it’s rotated.

On this graph, we see these lines
of different colors. There’s a red curve, a yellow
curve, a blue one, and a green one. What we want to do here is identify
which of these four curves, which color, correctly represents the change in the
torque that acts on a rectangular current-carrying loop as this angle that its
longest sides make with the magnetic field direction varies from zero to 90
degrees. In other words, which of the four
lines on our graph correctly shows the torque acting on our coil as it changes from
this position here to this position? Notice that that change in position
is defined on our graph by a change in this angle called 𝜃. 𝜃 goes from zero up to 90
degrees.

In our problem statement, we’re
told that 𝜃 represents the angle between the longest sides of our coil, those would
be the sides that initially are parallel with the magnetic field and then end up
perpendicular to it, and the magnetic field direction itself, which we’ve seen goes
from left to right between the north and south pole of our magnet. So the angle between this line
here, which is one of the longest sides of our coil, is zero degrees. That corresponds to this point on
our graph. And then, after our coil has turned
90 degrees, the angle between the longest side of the coil, which is now here, and
our external magnetic field is, we can see, 90 degrees. And that represents this data point
here on our curve.

To answer our question then, of
which of these four lines correctly represents the torque experienced by our loop as
it goes through this rotation, we’ll need to know how the torque on our rectangular
current-carrying wire varies with the angle 𝜃. To figure that out, we can recall a
general mathematical equation for the torque on just such a current-carrying
rectangular loop of wire in a magnetic field. That torque 𝜏 is equal to the
strength of the magnetic field the coil is in times the magnitude of the current in
it multiplied by its cross sectional area 𝐴. On our diagram, that area would be
this area we’re showing here times the number of turns in our rectangular coil all
multiplied by the sin of an angle we’ll call 𝜙.

Now it’s important to note that 𝜙,
this angle here, is not equal to the angle 𝜃 identified on our graph. The angles are different. But nonetheless, now that we have
this equation written out, we see how the torque on a rectangular current-carrying
wire in a magnetic field varies with the angular orientation of the coil. And that’s really what we need to
know to answer our question. So even though the torque 𝜏
depends on all these various factors, really we’re interested in the fact that it is
directly proportional to the sin of the angle we’ve called 𝜙. And now just what is this
angle?

Going back over to our diagram, if
we draw a vector that’s perpendicular to the cross sectional area of our loop, then
𝜙 is the angle between this vector, that we’ve drawn in blue, and the magnetic
field lines. And in this case, it’s worth
noticing that that angle is 90 degrees. So this is very important. It tells us that when the angle 𝜃
is zero degrees, the angle 𝜙 is equal to 90 degrees. So it’s true that 𝜃 and 𝜙 are not
the same, but this is how they correspond when 𝜃 is zero. And if we then consider the
orientation of our coil after it’s gone through its 90-degree rotation, in this
case, a vector drawn perpendicular to the cross sectional area of our coil would
look like this.

Once again, the angle 𝜙 is the
angle between this blue vector and the direction of the magnetic field. But now we can see these two
vectors are parallel. In other words, the angle between
them is zero degrees. And note that this corresponds to
the point where 𝜃, the angle between the longest side of our coil and the magnetic
field, is 90 degrees. So it’s a bit confusing, but when
𝜃 is zero degrees, 𝜙 is 90 degrees. And then when 𝜃 is 90 degrees, 𝜙
is zero degrees. We do all this because once we know
what 𝜙 is, we can take the sine of that angle. And then we’ll know how the torque
on our coil will vary over the course of this rotation, specifically, a rotation
from 𝜙 equals 90 degrees over to 𝜙 equals zero degrees.

In order to see what happens to the
sin of 𝜙, when we change 𝜙 this way, let’s recall the shape of the sine
function. If we plot the sin of an angle 𝜙
as 𝜙 varies from zero up to 360 degrees, we see that the graph reaches its maximum
value at an angle of 90 degrees. And then when the angle is zero,
back at the origin, the sine of that angle is zero. So as we transition from 𝜙 equals
90 to 𝜙 equals zero degrees, we’re covering this portion of our graph. And this tells us what kind of
curve to look for among our four candidates. It should be a line that starts out
at its maximum value when 𝜙 is equal to 90 degrees and then tails off to zero when
𝜙 goes to zero degrees.

Looking at our four curves then, we
see that just one of them starts out at the maximum value it attains over this range
of angles and then, over the angular change from 𝜙 equals 90 to 𝜙 equals zero
degrees, tails off to zero. That’s the red curve on our
graph. This is the only line that
consistently decreases whereas the other three options at some point increase in
value. And so this is the answer we’ll
give to our question. It’s the red curve that correctly
represents the change in torque acting on a loop as it rotates.

Let’s take a moment now to
summarize what we’ve learned about the torque on a current-carrying rectangular loop
of wire in a magnetic field. In this lesson, we saw that if we
have a current-carrying rectangular loop in a uniform magnetic field, then that loop
experiences an overall torque equal to the magnetic field strength times the current
magnitude in the coil times the cross sectional area of the coil multiplied by the
number of turns in the coil loop. And all of this is multiplied by
the sin of an angle we’ve called 𝜃, where 𝜃 is the measure of an angle between a
vector perpendicular to the cross sectional area of the coil and the magnetic field
lines.

Along with this, we learned the
term “magnetic dipole moment”, which indicates how strongly a coil such as this
interacts with an external magnetic field. The magnetic dipole moment 𝑚 sub
𝑑 is given by the maximum torque experienced by the coil divided by the strength of
the field it’s in. This is a summary of the torque on
a current-carrying rectangular loop of wire in a magnetic field.