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## Physics library

### Unit 13: Lesson 1

Magnets and Magnetic Force# Magnetic force on a current carrying wire

Sal derives the formula F=ILB to determine the force on a current carrying wire. Created by Sal Khan.

## Video transcript

Let's explore the repercussions
of this equation some more. So what was the equation? It was that the force of a
magnetic field on a moving charged particle is equal to the
charge-- that's not what I wanted to do-- is equal to the
charge of the particle-- and that's just a scalar quantity--
times the velocity-- the cross product
of the velocity of the particle-- with the
magnetic field. Now, isn't the velocity vector
just the same thing as the distance vector divided
by time? So the velocity vector is equal
to-- let's call the distance that the electron
travels, l. Distance divided by time. So we could rewrite that, that
the force vector is equal to the charge times-- and I'm doing
this on purpose-- 1 over time, right? Times the distance vector
taken-- you take the cross product with the
magnetic field. All I did is I rewrote velocity
as per time times distance, or distance
per time. And this is a scalar quantity,
at least for our purposes, time only has a magnitude. Maybe we could call
it change in time. But it doesn't have
a direction. We're not going at
an angle in time. So we could take the scalar
quantity out. It doesn't affect this
vector cross product. So what we get left with is,
force is equal to charge per time times-- and this is just a
regular times, because this is just a number, it's not a
vector-- times the cross product of the distance vector
and the magnetic field. And what is charge per time? Coulombs per second? Well, that's just current. So we get that force is equal to
current times the distance that the current is flowing
along, taken-- and you take the cross product of that
with the magnetic field. And sometimes this is written as
a capital L because it's a vector and all that, but we
started with a lower case l, so we'll stay with
the lower case l. So let's see if we can apply
this formula, which is really the same thing as this. We just took the division by
time and took it out of velocity so we get distance. And we took it and we divided
the coulombs, or we took the charge divided by that. So we took charge divided by
time, or charge per unit of time, you get current. So this is really just another
way of writing this. It's not even a new formula. You could almost prove
it to yourself if you ever forget it. But let's see if we can use this
to figure out the effect that a magnetic field has on
a current carrying wire. So let me-- actually, I probably
want to put this up at the top, just so that I have
space to draw a current carrying wire. So let me rewrite it in green. So you're familiar with the
formula in all colors. So now our new derivation is
that the force of a magnetic field on a current carrying wire
is equal to the current in the wire-- and that's just a
scalar quantity, although it could be positive or negative
depending on the direction. Well, current is always a
positive number, but if this current is going in the opposite
direction as our distance vector, then it
might be negative. But I wouldn't worry
about that for now. Let's just assume this is a
current in the direction of the distance vector. So it's a scalar quantity
current times our distance vector l, or maybe the length
of the conductor. You take the cross product
of l with the magnetic field vector. So let's see if we
can apply that. Let's say that we have a wire. Actually, let's do the magnetic
field first. I've been doing a lot of magnetic
fields that pop out of the screen. Let's do a magnetic field that
goes into the screen. And those are even
easier to draw. They're just x's. Now, why is it an x? Because you're looking at the
rear end of an arrow. That's why it's an x. And that's why a circle with a
dot means a field or a vector coming out of the window. Because if an arrow was shot at
you, all you would see is the tip of the arrow
with maybe a little circle around it. But anyway, this shows us a
vector going into the screen. So this is our magnetic field. That is B. I don't know, let's
assign some value. Let's say that the magnitude
of B is equal to 1 tesla. And let's say I have a
wire going through that magnetic field. Let's say the wire is going
along or it's in the plane of your computer monitor. Let me just draw a wire going
through the magnetic field. And my question to you-- let
me tell you a little bit of information about this wire. Let's say the wire is
carrying a current. So I is going in
that direction. And it is carrying a current
of-- I'm just making up numbers-- 5 amperes, or
5 coulombs per second. My question to you is, what is
the net force of this magnetic field on a section
of this wire? And let's make this section of
the wire, I don't know, let's say it's a 2 meter
section of wire. So obviously the more wire you
have, the more charged moving particles you'll have. So the
larger a section you have, the more of a force you'll have on
that longer piece of wire. So we have to pick our length. So we want to know, what is
the force of the magnetic field on this section of wire? From here to here. So let's just go to
this formula. The force is equal
to the current. So that's 5 amperes. And remember, just from what we
learned about electricity, the current is the direction
that notional positive charges would travel in, and
suits us fine. Because when we did the first
equation, we cared about the direction a positive
charge would go in. And if it was an electron or a
negative charge, we would put a negative sign there. So that works fine. But if you ever have to
visualize things as they maybe are in reality, but when you
talk about electrons it's hard to say that they really are
reality, because they're almost more an idea
than an object. But it's always good to remember
that when the current is flowing in this direction,
that would be true. Because if they were positive
charges moving, but we know it's a negative charge moving
in the opposite direction. Or you can think of it
as, maybe, holes. Well, I don't want
to get into that. But anyway, the current-- you
could visualize it if you like as positive charges going
in this direction. So the current is going
this direction. So you could view this
distance vector that we care about. Its magnitude is 2 meters. Because that's the length
of wire in question. And its direction is the
direction of the current. So let me-- this is l. Sometimes I get a little carried
away on tangents. So that is l. It's 2 meters in
that direction. I is 5 amperes. And we already figured
out that the magnetic field is 1 tesla. So what's this going
to be equal to? So the force is going to be
equal to-- we're using all SI units, so we don't have to
convert anything-- 5 amperes times 2 meters in
that direction. I won't specify right now,
let's just say that's a magnitude of l. Actually, let me write it. Well, 2 meters times the
magnetic field, 1 tesla. And so when you take a cross
product of something, this is just a reminder. l cross B. That's equal to the magnitude of
l times the magnitude of B times the sine of the angle
between them times some unit directional vector that
we figure out with the right hand rule. So we already did
the magnitude of the distance vector. That was 2 meters. We did the magnitude of
the magnetic field. And what's the sine of the
angle between them? Well, if the magnetic field is
going into the screen, if it's going straight into the screen,
you could imagine a bunch of arrows shooting
into the screen. Those are the vectors. While our distance vector, or
this l is in the screen, they actually are perpendicular,
in 3 dimensions. So this angle is 90 degrees. So this actually
just becomes 1. So in terms of the magnitude,
we're done. The l cross B magnitude
is 2 times 1 tesla. And then we multiply that
times the current. And then we actually have the
magnitude of the force. The magnitude of this force is
going to be equal to 5 amperes times 2 meters times 1 tesla. Which is equal to 10 newtons. And then the only question left
is, what is the direction of the force that the magnetic
field is exerting? And this is where we break
out the right hand rule. And it's no different. You could just imagine one of
the positive particles moving in that direction, and just
use the right hand rule. So let's take our hand out. And if we-- let me
draw a hand. A right hand. So this is my right hand. If I have my thumb sticking
out like that. So the l is going to
be my index finger. The first thing in the
cross product. And then the B is the
magnetic field. That's going into the screen. So you can't see it. All you can take my word for it
is that my middle finger is pointed downwards into the
screen and then my other fingers are just doing
something else. And there you have it. Your thumb is actually the
direction of the force. Your index finger is the
direction of-- we'll say l for these purposes. And then the magnetic field is
going into it, so you can't see my middle finger but it's
pointing downwards. I could draw a little x there,
to show it's going downwards. And then the force is what
my thumb is doing. So the force on this wire, or
at least on that section of wire, is going to be
perpendicular to the direction of the current. And that direction is going
to be a 10 newton force. Anyway, I've run out of time.