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Let's explore the repercussions
of this

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equation some more.

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So what was the equation?

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It was that the force of a
magnetic field on a moving

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charged particle is equal to the
charge-- that's not what I

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wanted to do-- is equal to the
charge of the particle-- and

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that's just a scalar quantity--
times the

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velocity-- the cross product
of the velocity of the

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particle-- with the
magnetic field.

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Now, isn't the velocity vector
just the same thing as the

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distance vector divided
by time?

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So the velocity vector is equal
to-- let's call the

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distance that the electron
travels, l.

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Distance divided by time.

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So we could rewrite that, that
the force vector is equal to

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the charge times-- and I'm doing
this on purpose-- 1 over

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time, right?

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Times the distance vector
taken-- you take the cross

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product with the
magnetic field.

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All I did is I rewrote velocity
as per time times

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distance, or distance
per time.

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And this is a scalar quantity,
at least for our purposes,

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time only has a magnitude.

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Maybe we could call
it change in time.

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But it doesn't have
a direction.

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We're not going at
an angle in time.

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So we could take the scalar
quantity out.

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It doesn't affect this
vector cross product.

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So what we get left with is,
force is equal to charge per

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time times-- and this is just a
regular times, because this

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is just a number, it's not a
vector-- times the cross

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product of the distance vector
and the magnetic field.

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And what is charge per time?

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Coulombs per second?

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Well, that's just current.

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So we get that force is equal to
current times the distance

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that the current is flowing
along, taken-- and you take

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the cross product of that
with the magnetic field.

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And sometimes this is written as
a capital L because it's a

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vector and all that, but we
started with a lower case l,

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so we'll stay with
the lower case l.

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So let's see if we can apply
this formula, which is really

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the same thing as this.

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We just took the division by
time and took it out of

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velocity so we get distance.

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And we took it and we divided
the coulombs, or we took the

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charge divided by that.

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So we took charge divided by
time, or charge per unit of

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time, you get current.

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So this is really just another
way of writing this.

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It's not even a new formula.

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You could almost prove
it to yourself if

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you ever forget it.

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But let's see if we can use this
to figure out the effect

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that a magnetic field has on
a current carrying wire.

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So let me-- actually, I probably
want to put this up

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at the top, just so that I have
space to draw a current

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carrying wire.

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So let me rewrite it in green.

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So you're familiar with the
formula in all colors.

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So now our new derivation is
that the force of a magnetic

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field on a current carrying wire
is equal to the current

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in the wire-- and that's just a
scalar quantity, although it

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could be positive or negative
depending on the direction.

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Well, current is always a
positive number, but if this

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current is going in the opposite
direction as our

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distance vector, then it
might be negative.

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But I wouldn't worry
about that for now.

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Let's just assume this is a
current in the direction of

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the distance vector.

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So it's a scalar quantity
current times our distance

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vector l, or maybe the length
of the conductor.

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You take the cross product
of l with the

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magnetic field vector.

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So let's see if we
can apply that.

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Let's say that we have a wire.

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Actually, let's do the magnetic
field first. I've

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been doing a lot of magnetic
fields that

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pop out of the screen.

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Let's do a magnetic field that
goes into the screen.

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And those are even
easier to draw.

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They're just x's.

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Now, why is it an x?

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Because you're looking at the
rear end of an arrow.

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That's why it's an x.

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And that's why a circle with a
dot means a field or a vector

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coming out of the window.

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Because if an arrow was shot at
you, all you would see is

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the tip of the arrow
with maybe a little

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circle around it.

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But anyway, this shows us a
vector going into the screen.

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So this is our magnetic field.

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That is B.

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I don't know, let's
assign some value.

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Let's say that the magnitude
of B is equal to 1 tesla.

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And let's say I have a
wire going through

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that magnetic field.

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Let's say the wire is going
along or it's in the plane of

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your computer monitor.

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Let me just draw a wire going
through the magnetic field.

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And my question to you-- let
me tell you a little bit of

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information about this wire.

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Let's say the wire is
carrying a current.

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So I is going in
that direction.

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And it is carrying a current
of-- I'm just making up

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numbers-- 5 amperes, or
5 coulombs per second.

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My question to you is, what is
the net force of this magnetic

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field on a section
of this wire?

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And let's make this section of
the wire, I don't know, let's

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say it's a 2 meter
section of wire.

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So obviously the more wire you
have, the more charged moving

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particles you'll have. So the
larger a section you have, the

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more of a force you'll have on
that longer piece of wire.

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So we have to pick our length.

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So we want to know, what is
the force of the magnetic

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field on this section of wire?

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From here to here.

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So let's just go to
this formula.

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The force is equal
to the current.

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So that's 5 amperes.

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And remember, just from what we
learned about electricity,

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the current is the direction
that notional positive charges

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would travel in, and
suits us fine.

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Because when we did the first
equation, we cared about the

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direction a positive
charge would go in.

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And if it was an electron or a
negative charge, we would put

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a negative sign there.

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So that works fine.

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But if you ever have to
visualize things as they maybe

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are in reality, but when you
talk about electrons it's hard

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to say that they really are
reality, because they're

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almost more an idea
than an object.

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But it's always good to remember
that when the current

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is flowing in this direction,
that would be true.

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Because if they were positive
charges moving, but we know

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it's a negative charge moving
in the opposite direction.

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Or you can think of it
as, maybe, holes.

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Well, I don't want
to get into that.

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But anyway, the current-- you
could visualize it if you like

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as positive charges going
in this direction.

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So the current is going
this direction.

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So you could view this
distance vector

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that we care about.

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Its magnitude is 2 meters.

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Because that's the length
of wire in question.

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And its direction is the
direction of the current.

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So let me-- this is l.

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Sometimes I get a little carried
away on tangents.

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So that is l.

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It's 2 meters in
that direction.

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I is 5 amperes.

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And we already figured
out that the

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magnetic field is 1 tesla.

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So what's this going
to be equal to?

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So the force is going to be
equal to-- we're using all SI

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units, so we don't have to
convert anything-- 5 amperes

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times 2 meters in
that direction.

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I won't specify right now,
let's just say that's a

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magnitude of l.

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Actually, let me write it.

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Well, 2 meters times the
magnetic field, 1 tesla.

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And so when you take a cross
product of something, this is

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just a reminder.

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l cross B.

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That's equal to the magnitude of
l times the magnitude of B

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times the sine of the angle
between them times some unit

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directional vector that
we figure out with

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the right hand rule.

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So we already did
the magnitude of

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the distance vector.

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That was 2 meters.

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We did the magnitude of
the magnetic field.

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And what's the sine of the
angle between them?

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Well, if the magnetic field is
going into the screen, if it's

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going straight into the screen,
you could imagine a

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bunch of arrows shooting
into the screen.

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Those are the vectors.

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While our distance vector, or
this l is in the screen, they

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actually are perpendicular,
in 3 dimensions.

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So this angle is 90 degrees.

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So this actually
just becomes 1.

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So in terms of the magnitude,
we're done.

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The l cross B magnitude
is 2 times 1 tesla.

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And then we multiply that
times the current.

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And then we actually have the
magnitude of the force.

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The magnitude of this force is
going to be equal to 5 amperes

190
00:09:23,900 --> 00:09:26,570
times 2 meters times 1 tesla.

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00:09:26,570 --> 00:09:30,090
Which is equal to 10 newtons.

192
00:09:30,090 --> 00:09:32,980
And then the only question left
is, what is the direction

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00:09:32,980 --> 00:09:35,400
of the force that the magnetic
field is exerting?

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And this is where we break
out the right hand rule.

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And it's no different.

196
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You could just imagine one of
the positive particles moving

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in that direction, and just
use the right hand rule.

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00:09:46,950 --> 00:09:50,450
So let's take our hand out.

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00:09:50,450 --> 00:09:52,670
And if we-- let me
draw a hand.

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00:09:52,670 --> 00:09:54,840
A right hand.

201
00:09:54,840 --> 00:09:58,400
So this is my right hand.

202
00:09:58,400 --> 00:10:00,500
If I have my thumb sticking
out like that.

203
00:10:00,500 --> 00:10:03,550
So the l is going to
be my index finger.

204
00:10:03,550 --> 00:10:05,910
The first thing in the
cross product.

205
00:10:05,910 --> 00:10:08,610
And then the B is the
magnetic field.

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00:10:08,610 --> 00:10:09,980
That's going into the screen.

207
00:10:09,980 --> 00:10:11,210
So you can't see it.

208
00:10:11,210 --> 00:10:13,710
All you can take my word for it
is that my middle finger is

209
00:10:13,710 --> 00:10:16,390
pointed downwards into the
screen and then my other

210
00:10:16,390 --> 00:10:18,700
fingers are just doing
something else.

211
00:10:18,700 --> 00:10:20,100
And there you have it.

212
00:10:20,100 --> 00:10:22,760
Your thumb is actually the
direction of the force.

213
00:10:22,760 --> 00:10:25,450


214
00:10:25,450 --> 00:10:31,150
Your index finger is the
direction of-- we'll say l for

215
00:10:31,150 --> 00:10:32,180
these purposes.

216
00:10:32,180 --> 00:10:34,430
And then the magnetic field is
going into it, so you can't

217
00:10:34,430 --> 00:10:36,250
see my middle finger but it's
pointing downwards.

218
00:10:36,250 --> 00:10:38,780
I could draw a little x there,
to show it's going downwards.

219
00:10:38,780 --> 00:10:41,100
And then the force is what
my thumb is doing.

220
00:10:41,100 --> 00:10:44,400
So the force on this wire, or
at least on that section of

221
00:10:44,400 --> 00:10:47,630
wire, is going to be
perpendicular to the direction

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00:10:47,630 --> 00:10:48,830
of the current.

223
00:10:48,830 --> 00:10:52,290
And that direction is going
to be a 10 newton force.

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00:10:52,290 --> 00:00:00,000
Anyway, I've run out of time.