### Video Transcript

In this video, weāre talking about
force on conducting wires in magnetic fields. The fact that putting a conducting
wire in a magnetic field can make it experience a force may come as a surprise. But as weāll see in this lesson,
there are circumstances where, indeed, this takes place. We can start out by considering
just what are the conditions needed for such a force to exist.

Letās start here. Say that we have a length of
conducting wire but no current is in the wire. Under these conditions, we know
that no magnetic or electric force will be experienced by the wire. But then say that we take a couple
of bar magnets. And we put them near this wire so
the north pole of one bar magnet faces the south pole of the other. By doing that, we set up a magnetic
field that runs from the north pole to the south pole.

Now, by the way, thatās not the
only part of the magnetic field created by these two bar magnets. But for now, this is the only part
of that field that weāll focus on. So we now have a conducting wire,
still with no current in it, thatās in the presence of a magnetic field. Under these conditions, our wire
still experiences no force, even the part of the wire thatās surrounded by this
magnetic field.

Finally, say weāre able to start
the flow of charge through this conducting wire. It now has a current. With all these pieces in place,
now, the section of the current-carrying wire thatās in the magnetic field would
begin to experience a magnetic force. But importantly, the
current-carrying wire only experiences that force when it is surrounded by this
field. Thereās no force on the wire here
or here or anywhere else outside of this magnetic field.

Now, letās say that the width of
these bar magnets and therefore also the width of the magnetic field that our
current-carrying wire is in we call capital šæ. In that case, we can say that a
length šæ of our current-carrying wire is exposed to this magnetic field and
therefore experiences a force. The magnitude of that force, we can
call it š¹ sub šµ, depends on three things. First, it depends on the strength
of the magnetic field that the wire is in. We can call that field strength šµ
in general. Second, it depends on the magnitude
of the current in the wire. The greater that current is, the
greater the magnetic force experienced.

And lastly, it depends on how much
of the wire, how great a length, is exposed to the magnetic field šµ. If šæ is the length of wire within
the field, then the overall magnetic force is equal to šµ times š¼ times šæ. So to reiterate, all we need to do
to create a magnetic force on a wire is to send some current through that wire and
put the current within a magnetic field. Now, thereās one important caveat
to all this. This equation weāve just written
down is true, but it requires a certain orientation between the magnetic field šµ
and the current š¼.

Notice that, in this diagram, our
current points to the right, while our magnetic field points up. So then, thereās a 90-degree angle
between these two quantities. So long as the current and the
magnetic field are perpendicular to one another, this equation applies. But if theyāre not, if the magnetic
field and the current are at some other angle to one another, then itās no longer
valid. In particular, if we had a magnetic
field that was pointed in parallel with the current in our wire, then that segment
of wire in the field would actually experience no magnetic force.

So letās expand on this
relationship a bit. We can say that this equation is
true if šµ is perpendicular to the current š¼. But on the other hand, if šµ is
parallel to š¼, then the magnetic force on that stretch of wire is zero. Thatās one important thing to keep
in mind regarding this equation. And another is that sometimes weāll
see it expressed in a slightly different way. For instance, we might see it
written as š¼ times šæ times šµ, current times length in the magnetic field times
magnetic field strength. Mathematically, these two
expressions mean the same thing, but they look a bit different. So itās helpful to know that really
they are the same.

Now, to this point, weāve
considered a wire with charge moving in it left to right. But what if we had an identical
conducting wire with the same magnitude of current in it, but this current pointed
in the opposite direction? The question is would the magnetic
force experienced by the second wire be the same as the magnetic force experienced
by the first? This is where we need to remember
that force is a vector quantity. That is, it has magnitude as well
as direction to it. Because these two identical
conducting wires had the same magnitude of current in them and theyāre exposed to
the same strength of a magnetic field and the same length of wire is exposed to that
field. Then the magnitude of the force
experienced by the two wires is the same, but the direction is not.

The reason we know this is thanks
to a rule called Flemingās left-hand rule. Through this rule, we use the thumb
and fingers on our left hand to figure out the direction of the magnetic force
acting on a current-carrying wire. Hereās how Flemingās left-hand rule
works. We start out by pointing our index
finger in the direction that the magnetic field šµ points. In our example, thatās upward to
the top of the screen. Then keeping that finger pointed
that same direction, we point our second finger, thatās our middle finger, in the
direction the current š¼ points.

When we do this, itās important to
use the direction of conventional current. In other words, the direction
positive charge would move in the wire if it did move. And then lastly, keeping our index
and middle fingers in these positions, we point the thumb on our left hand so that
itās perpendicular at 90 degrees to both of them. In this situation, that would be
pointing out of the screen towards us. That direction of our thumb is the
direction that the magnetic force acts on this current-carrying wire in a magnetic
field.

What weāve done here is applied
Flemingās left-hand rule to this current-carrying wire, the one carrying current
left to right. And weāve seen that the force
direction on this wire, where the wire is within the magnetic field, is out of the
screen at us. But if we then consider the other
current-carrying wire with charge running right to left, now our left-hand rule
works differently because instead of current pointing to the right, now, it points
the opposite way. So we twist our left hand around so
our index finger still points up in the same direction as the magnetic field
previously. But now, our middle finger points
to the left. Thatās to match the direction of
conventional current in the second wire.

And then, if we point the thumb on
our left hand perpendicular to both our index and middle fingers, this is now the
direction of the magnetic force on this second wire. Our thumb in this case points into
the page, away from us. So then going to this second wire
in our diagram, we see that the magnetic force direction on this is into the page
rather than out of the page.

So to summarize this, the magnitude
of the magnetic force experienced by these current-carrying wires in the magnetic
field is the same. But since charge is moving in
opposite directions in these two wires, Flemingās left-hand rule tells us that the
force directions on the wires is opposite. One force, in this case, points
into the page and the other points out of the page. For this magnetic force on
current-carrying wires, the equation, š¹ sub šµ is equal to šµ times š¼ times šæ,
tells us that force magnitude and Flemingās left-hand rule tells us the force
direction. Knowing all this, letās now get a
bit of practice with these ideas through an example.

When positioned at 90 degrees to a
magnetic field, a wire of length one meter carrying a current of four amperes
experiences a force of 0.2 newtons. What is the strength of the
magnetic field?

All right, so in this example, we
have a length of wire. This length is one meter long. And weāre also told that the wire
carries a current of four amperes. And then, in addition to this, this
current-carrying wire is positioned at 90 degrees to a magnetic field. So we could draw magnetic field
lines that look like this, where these lines, as weāre told in the problem
statement, are perpendicular to our wire. For our purposes, whether the field
lines point straight up as weāve drawn them or if they pointed straight down doesnāt
make a difference so long as the field lines are at 90 degrees to our wire.

Weāre told that, in this magnetic
field, our wire experiences a force, and it has a magnitude of 0.2 newtons. Knowing all this, we want to solve
for the strength of the magnetic field that the current-carrying wire is in. We can call that magnetic field
strength capital šµ. And letās say that the force the
wire experiences in the field at 0.2 newtons is š¹ sub šµ. So we have all this information,
but we need some way to tie it all together. We can do this by recalling a
mathematical expression for the magnetic force experienced by a current-carrying
wire in a magnetic field. That force is equal to the strength
of the magnetic field šµ times the magnitude of the current running through the wire
multiplied by the length of wire exposed to the magnetic field.

Now, in our case, itās not š¹ sub
šµ we want to solve for, but rather the magnetic field strength šµ. So to rearrange and solve for that,
letās divide both sides of the equation by š¼ times šæ causing those two terms to
cancel out on the right-hand side. So then, the magnetic field is
equal to the magnetic force divided by the current times the length of the wire in
the field. We can now apply this relationship
using the numbers weāve been given. šµ is equal to 0.2 newtons divided
by four amperes times one meter.

And if we consider just the units
for a moment, a newton per ampere meter, thatās equal to whatās called a tesla,
symbolized capital š. This is the SI base unit of
magnetic field strength. So šµ is equal to 0.2 divided by
four times one tesla, which is equal to 0.05 teslas. Thatās the strength of the magnetic
field that the current-carrying wire is in.

Letās look now at a second example
exercise.

The diagram shows a section of wire
that has been positioned parallel to a uniform 0.1-tesla magnetic field. The wire carries a current of two
amperes. What is the direction of the force
acting on the wire due to the magnetic field?

Okay, in this diagram, we see this
section of wire here, which is parallel to a uniform magnetic field. The diagram also shows us that
charge in the wire moves left to right. Based on this, we want to solve for
the direction of the force that acts on this wire due to the magnetic field. Now, we have to be a bit careful
here. Seeing that we have a
current-carrying wire in a magnetic field, we might think of the mathematical
relation for the force on such a wire.

That equation says that this force
is equal to the strength of the magnetic field multiplied by the current magnitude
in the wire times the length of the wire exposed to the field. But thereās an important condition
thatās required in order for this equation to be valid. The condition is that the magnetic
field direction is perpendicular to the direction of current in the wire. In our case though, we can see that
condition is not met because the current and the magnetic field point parallel to
one another. When this is the case, this
equation here for magnetic force doesnāt apply. And, in fact, the magnetic force
experienced by this current-carrying segment of wire is zero when the current and
magnetic field are parallel. In other words, the force acting on
this section of wire is zero.

Now, weāre not asked about the
force magnitude itself, but rather its direction. We can see though that this
realization impacts our answer to the force direction. Thatās because a force of zero has
no direction to it. So our answer to this question can
simply be that there is no force acting on the wire. Since thereās no force, thereās no
direction. And this then is our answer.

Letās take a moment now to
summarize what weāve learned about force on conducting wires in magnetic
fields. Starting out, we saw that if a
current-carrying wire is placed in a magnetic field, it may experience a force. We say āmayā because if the
magnetic field that the current-carrying wire is in is perpendicular to the current,
then that magnetic force experienced by the wire is equal to the magnetic field
strength times the current magnitude times the length of the wire thatās in the
field.

But on the other hand, if the
magnetic field and the wire are arranged differently so that the magnetic field is
parallel to the current, then in that case thereās no force on the wire. So if we have a situation that
looks like this, where šµ and š¼ are perpendicular, then we do have a nonzero force
acting on that wire, šµ times š¼ times šæ. But if our scenario looks like
this, where šµ and š¼ are parallel, no magnetic force is exerted on the wire.

And lastly, when there is a force
on a current-carrying wire in a magnetic field, Flemingās left-hand rule shows us
how to find the direction of that force. We do it by pointing our index
finger in the direction of the magnetic field šµ, our middle finger in the direction
of conventional current in the wire, and then, perpendicular to both those
directions, we point our thumb. And our thumb gives the direction
of the resultant magnetic force on the wire in the field. This is a summary of force on
conducting wires in magnetic fields.