### Video Transcript

In this video, we’re talking about
the force that acts on conducting wires, which themselves are in magnetic
fields. As we learn about this magnetic
force, we’ll see which direction it acts in as well as its magnitude or
strength.

There’s one important thing to get
clear as we begin this discussion. In an earlier lesson, we saw that
when a wire is carrying current, that moving charge creates a magnetic field around
the wire. And it’s true such a field is
created by the charge moving down the wire. But this field, the magnetic field
created by the wire, is not the one we’re talking about in this case.

For our purposes in this lesson,
we’re considering an external magnetic field, one that has nothing to do with the
field of the wire. This external field might well be
supplied by a permanent magnet. Say we had a permanent magnet like
this that surrounded the wire so that the magnetic field it created pointed from the
north pole to the south pole, perpendicular to the direction of current flow in the
wire.

One of the fascinating things about
this setup is that this wire will experience a magnetic force. Let’s think about this for a
moment, going back to basics, electric charge. We know that electric charge is the
source of electric fields and that electric charge in motion is the source of
magnetic fields. If we consider the charges in our
current-carrying wire, that’s really what they are. They’re electric charges in
motion. Therefore, they create a magnetic
field around themselves, like we saw just a moment ago.

But now what we’ve done is created
an external magnetic field, the one created by the magnet, and put the wire in
it. When that happens, it’s a bit like
we’ve taken our moving charge and put it in a field like this. Under these circumstances,
depending on the amount of charge, the strength of the field, and the speed with
which the charge moves, this charge will experience a magnetic force.

Now before we go further, it will
be helpful to get a bit of notation in place. Up till now, we’ve talked about
magnetic fields and we’ve drawn in magnetic field lines. But we never labeled them to
indicate what they are. For historical reasons, the letter
capital 𝐵 has come to be used to indicate a magnetic field. So when we write a capital 𝐵 by a
bunch of field lines like we have here, that indicates that this is a magnetic field
we’re referring to. By the way, this symbol capital 𝐵
is also used in calculations to indicate the strength of a magnetic field. Okay, so that’s our first bit of
notation, that magnetic field can be written with a capital 𝐵.

Next, let’s say that our charge
here has a charge value of 𝑄. And we’ll say further that it’s
moving with a speed we’ll call 𝑣. We said earlier that, given this
setup, given a moving charge in an external magnetic field, that the charge would
experience a magnetic force. When the motion of the charge is in
a direction perpendicular to the external magnetic field, like it is now, we can
write that magnetic force this way. 𝐹 sub 𝐵, the magnetic force on
this charge, is equal to 𝑄, the charge value, multiplied by its speed, 𝑣,
multiplied by the strength of the field it’s in.

Based on the variables involved
here, a specific charge, and the specific speed on that charge, we can tell that
this relationship applies to individual electric charges. That doesn’t seem offhand to be of
much help when we’re talking about a current-carrying wire with many charges in a
magnetic field. But actually, the connection
between these two situations, where we have an individual charge compared to many
chargers in a current, is actually fairly close.

Consider the right-hand side of
this force equation, 𝑄 times 𝑣 times 𝐵. Let’s consider the middle term in
this expression, the speed. We know that, in general, speed is
equal to a distance divided by a time. We can say then that, effectively,
this speed, 𝑣, is equal to some distance, 𝐷, divided by some amount of time,
𝑡.

Now if we then substitute in 𝐷
over 𝑡 for 𝑣 in this expression, we can then make a bit of a modification by
taking this time, 𝑡, and bringing it outside the parentheses. That leaves us with a charge
divided by a time multiplied by a distance multiplied by a magnetic field. At this point though, we can recall
what charge divided by time is. That’s the definition for
current.

Current, 𝐼, is defined as the
amount of charge, 𝑄, that passes a point in a circuit over some amount of time,
𝑡. That means we can replace 𝑄
divided by 𝑡 in this expression by capital 𝐼, current. We can then consider this 𝐷 term,
which is a distance in this expression. That distance refers to the length
of current, so to speak, which is exposed to the magnetic field, 𝐵.

If we were to highlight that in our
sketch, that would be the length of wire which lies between the poles of this
magnet. We can leave this distance as
𝐷. But since it refers to a length of
wire, we may as well refer to it that way, as capital 𝐿.

At this point, let’s recall that
this expression that we have here, which we haven’t altered in any meaningful way,
is equal to the magnetic force acting on, in this case, this current 𝐼. What we’ve done then is developed
equations for the magnetic force on an individual charge, 𝑄, in a magnetic field,
𝐵, as well as the magnetic force on a length 𝐿 of current 𝐼 carrying wire, also
in a magnetic field 𝐵.

For the rest of our discussion,
we’re gonna focus on the second of these two expressions. Magnetic force is equal to the
magnetic field strength multiplied by the length of wire that’s in the field
multiplied by the current running through the wire. We can see that, so far, what we’ve
solved for is the magnitude of the magnetic force that acts on a current-carrying
wire in a magnetic field. But, of course, that’s only half of
the story because this force being a vector has a direction as well. So what direction will the magnetic
force on this current-carrying wire in our magnetic field that we can now label
capital 𝐵 be?

To figure that out, it’s helpful to
know this. The magnetic force on a charged
particle is always perpendicular to the motion of the charge as well as the field
that the charge is in. Here’s what that means. Say we have a positive electric
charge and it’s moving in this direction. Say further that that charge is in
a magnetic field and that field points this way, perpendicular to the motion of the
charge.

What we’ve just said is that the
magnetic force acting on this charge must be perpendicular to both of these
vectors. There are only two different ways
then that it could point. It could point like this as a force
vector or it could point the opposite way like this. The point is that force is
perpendicular or at a 90-degree angle to both the magnetic field as well as the
charge motion.

We may wonder then which of these
two directions the force actually acts and how we figure that out. The answer comes down to an
application of what we can call a right-hand rule. This name has the advantage of
clearly showing us what hand we’ll use to figure this direction out. The tricky part is that there are
actually several right-hand rules, some of which you may remember from previous
lessons. It can be tricky to keep them all
straight when they all go by the same name, right-hand rule. The important thing though is not
so much the name as it is remembering which version to apply given the
circumstances.

In this situation, we have a
vector, which is the current, flowing through the wire, as well as a vector, which
is the magnetic field created by the magnet. Let’s say then that this blue arrow
we’ve drawn is the direction of the current flowing through the wire, which is a
vector, and that the pink vector is the magnetic field created by our permanent
magnet. And just to make things a bit more
clear, let’s rotate this whole setup so that the current actually goes the way it is
in our sketch.

Okay, so we have current moving
along in this direction. And then at 90 degrees to that, we
have the magnetic field created by the magnet, 𝐵. We’ve said that, in this situation,
when there’s charge moving through an external magnetic field, there’s a force that
acts on that charge and that that force is perpendicular to both the field and the
flow of the charge.

Based on this diagram then, we can
see that that force will either be straight up or it will be straight down. And here’s where our rule comes in
“handy,” get it? The first thing we do here is we
take our right hand and we point the fingers of our hand in the direction of the
current flow. Note that this is conventional
current, that is, the motion of positive charge through this wire.

Next, what we do is we curl our
fingers until they point in the direction of the magnetic field, 𝐵. Then, finally, we point our thumb
up. And the direction that our thumb
points is the direction of the magnetic force on this current-carrying wire. Based on this right-hand rule then,
we can see that the force on this particular current-carrying wire would indeed be
up. That is the direction of the
magnetic force vector.

This right-hand rule may take a bit
of practice to get right. And that’s just fine. As you go along and use it,
remember one important thing. In order for this rule to properly
indicate the direction of magnetic force, it must be the case that the current and
the magnetic field are perpendicular, at 90 degrees, to one another. If they’re not — say that they were
running in the same directions or even in opposite directions — if the current and
magnetic field relate to one another this way in terms of their direction, then
there’s no net magnetic force on these charged particles. Under those conditions, the
magnetic force on them is zero.

Let’s take some time now to get a
bit of practice with these ideas of the force on a current-carrying wire in a
magnetic field.

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

We can see that this here is our
wire. We’re told that it carries a
current of four amperes and that it’s one meter long. We’re told that this
current-carrying wire is in an external magnetic field, which is oriented at 90
degrees to the wire. Under these conditions, the wire is
subjected to a magnetic force we can call 𝐹, which is given as 0.2 newtons.

Knowing all this, we want to solve
for the strength of the magnetic field, what we’ve called 𝐵 in our sketch. To figure this out, we can recall a
mathematical relationship between wire length, current, magnetic field, and
force. The magnetic force on a
current-carrying wire is equal to the magnetic field strength that the wire is in
multiplied by the length of the wire times the current that’s running through
it.

In our case, it’s not the magnetic
force we want to solve for, but the magnetic field. And we can do that by rearranging
this equation. If we divide both sides by the
current times the length of the wire, we get this result. 𝐵 is equal to 𝐹 sub 𝐵 over 𝐼
times 𝐿. Looking at the information in our
problem statement, we’re told 𝐹 sub 𝐵. That’s 0.2 newtons. We’re also told the current, 𝐼, in
the wire of four amps and the length of the wire of one meters. Our next step then is to substitute
these values into this equation. 0.2 newtons divided by four amps
times one meter equals 0.05 teslas, where a tesla is the unit of magnetic field. It’s often abbreviated just using a
capital T. 0.05 teslas then is the strength of
the magnetic field in this situation.

Now let’s look at a second
example.

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?

Taking a look at our diagram, we
see this wire, marked out in pink, with current running left to right. We see this wire is placed within a
uniform magnetic field called 𝐵. And this field also points from
left to right. Given the strength of the magnetic
field and the magnitude of the current, we want to solve for the direction of the
magnetic force acting on the wire due to the field.

What may first come to mind is the
right-hand rule we used to help us figure out this force direction. Using our right hand, we might
start out by pointing our fingers in the direction of the current as this rule calls
for. But then when we seek to curl our
fingers in the direction of the magnetic field, we notice something interesting. The magnetic field is in the same
direction as the current. They point the same way.

It’s at this point we must be very
careful to remember a condition of this right-hand rule. And that is that this rule will
only be certain to give us the direction of the force on a current-carrying wire
when the current and the magnetic field the wire is in are perpendicular to one
another, at 90 degrees. And, in fact, in the special case
when current and magnetic field are in the same direction or even when they’re 180
degrees opposed, in these two instances where they’re parallel or antiparallel to
one another, the magnetic force on the wire is zero.

Looking back at our diagram, we
find that that’s the case in this scenario. Our current and our field are
moving in the same direction. They’re parallel, and therefore the
force on the wire is zero. This isn’t the most common
scenario, but we have seen that, in this case, it did come up. Because the current flowing in the
wire and the magnetic field the wire is in are parallel, there is no force acting on
the wire.

Let’s take a moment now to
summarize what we’ve learned about force on conducting wires in magnetic fields. For starters, we learned in this
lesson that when a current-carrying wire is perpendicular to an external magnetic
field, that wire experiences a force. That force magnitude is equal to
the magnitude of the field strength, 𝐵, multiplied by the length of the wire in the
field multiplied by the current running through the wire.

We learned that the direction of
that force is given by what’s called the right-hand rule. This version of that rule has us
take our right hand and point our fingers in the direction of the conventional
current flow. If we then curl our fingers in the
direction of the magnetic field, assuming it’s perpendicular to the current, then in
that case our thumb points in the direction of the magnetic force on the wire.

Lastly, we noticed then when the
current and the external magnetic field are parallel to one another, like in this
case, or antiparallel, like in this case, then the force acting on the wire is
zero.