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Let's try to discuss a bit how
things relate to physics.

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There are two main things I
want to discuss.

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One of them is what curl says
about force fields and,

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in particular,a nice
consequence of that concerning

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gravitational attraction.
More about curl.

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If we have a velocity field,
then we have seen that the curl

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measures the rotation affects.
More precisely curl v measures

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twice the angular velocity,
or maybe I should say the

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angular velocity vector because
it also includes the axis of

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rotation.
I should say maybe for the

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rotation part of a motion.
For example,

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just to remind you,
I mean we have seen this guy a

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couple of times,
but if I give you a uniform

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rotation motion about the z,
axes.

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That is a vector field in which
the trajectories are going to be

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circles centered in the z-axis
and our vector field is just

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going to be tangent to each of
these circles.

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And, if you look at it from
above, then you will have this

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rotation vector field that we
have seen many times.

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Typically, the velocity vector
for this would be minus yi plus

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yj times maybe a number that
represents how fast we are

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spinning,
the angular velocity in

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gradients per second.
And then.

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if you compute the curl of
this, you will end up with two

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omega times k.
Now, the other kinds of vector

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fields we have seen physically
are force fields.

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The question is what does the
curl of a force field mean?

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What can we say about that?
The interpretation is a little

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bit less obvious,
but let's try to get some idea

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of what it might be.
I want to remind you that if we

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have a solid in a force field,
we can measure the torque

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exerted by the force on the
solid.

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Maybe first I should remind you
about what torque is in space.

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Let's say that I have a piece
of solid with a mass,

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delta m for example,
and I have a force that is

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being exerted to it.
Let's say that maybe my force

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might be F times delta m.
If you think,

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for example,
a gravitational field.

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The gravitational force is
actually the gravitational field

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times the mass.
I mean you can forget delta m

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if you don't like it.
And let's say that the position

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vector, which should be aiming
for the origin,

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R is here.
And now let's say that maybe

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this guy is at the end of some
arm or some metal thing and I

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want to hold it in place.
The force is going to exert a

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torque relative to the origin
that will try to measure how

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much I am trying to swing this
guy around the origin.

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And, consequently,
how much effort I have to exert

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if I want to actually maintain
its place by just holding it at

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the end of the stick here.
So the torque is now a vector,

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which is just the cross-product
of a position vector with a

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force.
What the torque measures again

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is the rotation effects of the
force.

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And if you remember the
principle that the derivative of

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velocity,
which is acceleration,

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is force divided by mass then
the derivative of angular

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velocity should be angular
acceleration which is related to

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the torque per unit mass.
To just remind you,

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if I look at translation
motions,

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say I am just looking at the
point mass so there are no

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rotation effects then force
divided by mass is acceleration,

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which is the derivative of
velocity.

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And so what I am claiming is
that for rotation effects we

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have a similar law,
which maybe you have seen in

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8.01.
Well, it is one of the

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important things of solid
mechanics, which is the torque

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of a force divided by the moment
of inertia.

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I am cheating a little bit here.
If you can see how I am

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cheating then I am sure you know
how to state it correctly.

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And if you don't see how I am
cheating then let's just ignore

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the details.
[LAUGHTER]

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Is angular acceleration.
And angular acceleration is the

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derivative of angular velocity.
If I think of curl as an

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operation,
which from a velocity field

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gives the angular velocity of
its rotation effects,

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then you see that the curl of
an acceleration field gives the

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angular acceleration in the
rotation part of the

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acceleration effects.
And, therefore,

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the curl of a force field
measures the torque per unit

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moment of inertia.
It measures how much torque its

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force field exerts on a small
test solid placed in it.

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If you have a small solid
somewhere, the curl will just

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measure how much your solid
starts spinning if you leave it

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in this force field.
In particular,

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a force field with no curl is a
force field that does not

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generate any rotation motion.
That means if you put an object

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in there that is completely
immobile and you leave it in

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that force field,
well, of course it might

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accelerate in some direction but
it won't start spinning.

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While, if you put it in there
spinning already in some

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direction, it should continue to
spin in the same way.

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Of course, maybe there will be
friction and things like that

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which will slow it down but this
force field is not responsible

00:08:58.000 --> 00:09:05.000
for it.
The cool consequence of this is

00:09:05.000 --> 00:09:14.000
if a force field F derives from
a potential -- That is what we

00:09:14.000 --> 00:09:20.000
have seen about conservative
forces.

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Our main concern so far has
been to say if we have a

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conservative force field it
means that the work of a force

00:09:26.000 --> 00:09:29.000
is the change in the energy.
And, in particular,

00:09:29.000 --> 00:09:32.000
we cannot get energy for free
out of it.

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And the change in the potential
energy is going to be the change

00:09:36.000 --> 00:09:40.000
in kinetic energy.
You have conservation of energy

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principles.
There is another thing that we

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know now because if a force
derives from a potential then

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that means its curl is zero.
That is the criterion we have

00:09:53.000 --> 00:09:58.000
seen for a vector field to
derive from a potential.

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And if the curl is zero then it
means that this force does not

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generate any rotation effects.
For example,

00:10:23.000 --> 00:10:27.000
if you try to understand where
the earth comes from,

00:10:27.000 --> 00:10:32.000
well, the earth is spinning on
itself as it goes around the

00:10:32.000 --> 00:10:35.000
sun.
And you might wonder where that

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comes from.
Is that causes by gravitational

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attraction?
And the answer is no.

00:10:40.000 --> 00:10:44.000
Gravitational attraction in
itself cannot cause the earth to

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start spinning faster or slower,
at least if you assume the

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earth to be a solid,
which actually is false.

00:10:52.000 --> 00:10:57.000
I mean basically the reason why
the earth is spinning is because

00:10:57.000 --> 00:11:01.000
it was formed spinning.
It didn't start spinning

00:11:01.000 --> 00:11:03.000
because of gravitational
effects.

00:11:03.000 --> 00:11:08.000
And that is a rather deep
purely mathematical consequence

00:11:08.000 --> 00:11:12.000
of understanding gravitation in
this way.

00:11:12.000 --> 00:11:16.000
It is quite spectacular that
just by abstract thinking we got

00:11:16.000 --> 00:11:17.000
there.
What is the truth?

00:11:17.000 --> 00:11:21.000
Well, the truth is the earth,
the moon and everything is

00:11:21.000 --> 00:11:24.000
slightly deformable.
And so there is deformation,

00:11:24.000 --> 00:11:26.000
friction effects,
tidal effects and so on.

00:11:26.000 --> 00:11:29.000
And these actually cause
rotations to get slightly

00:11:29.000 --> 00:11:32.000
synchronized with each other.
For example,

00:11:32.000 --> 00:11:36.000
if you want to explain why the
moon is always showing the same

00:11:36.000 --> 00:11:39.000
face to the earth,
why the rotation of a moon on

00:11:39.000 --> 00:11:43.000
itself is synchronized with its
revolution around the earth,

00:11:43.000 --> 00:11:47.000
which is actually explained by
friction effects over time and

00:11:47.000 --> 00:11:50.000
the gravitational attraction of
the earth and the moon.

00:11:50.000 --> 00:11:59.000
There is something there,
but if you took perfectly

00:11:59.000 --> 00:12:09.000
rigid, solid bodies then
gravitation would never cause

00:12:09.000 --> 00:12:15.000
any rotation effects.
Of course that tells us that we

00:12:15.000 --> 00:12:20.000
do not know how to answer the
question of why is the earth

00:12:20.000 --> 00:12:22.000
spinning.
That will be left for another

00:12:22.000 --> 00:12:31.000
physics class.
I don't have a good answer to

00:12:31.000 --> 00:12:35.000
that.
That was kind of 8.01-ish.

00:12:35.000 --> 00:12:40.000
Let me now move forward to 8.02
stuff.

00:12:40.000 --> 00:12:54.000
I want to tell you things about
electric and magnetic fields.

00:12:54.000 --> 00:13:01.000
And, in fact,
something that is known as

00:13:01.000 --> 00:13:06.000
Maxwell's equations.
Just a quick poll.

00:13:06.000 --> 00:13:10.000
How many of you have been
taking 8.02 or something like

00:13:10.000 --> 00:13:13.000
that?
OK. That is not very many.

00:13:13.000 --> 00:13:15.000
For most of you this is a
preview.

00:13:15.000 --> 00:13:18.000
If you have been taking 8.02,
have you seen Maxwell's

00:13:18.000 --> 00:13:20.000
equations, at least part of
them?

00:13:20.000 --> 00:13:22.000
Yeah.
OK.

00:13:22.000 --> 00:13:23.000
Then I am sure,
in that case,

00:13:23.000 --> 00:13:25.000
you know better than me what I
am going to talk about because I

00:13:25.000 --> 00:13:30.000
am not a physicist.
But just in case.

00:13:30.000 --> 00:13:35.000
Maxwell's equations govern how
electric and magnetic fields

00:13:35.000 --> 00:13:39.000
behave, how they are caused by
electric charges and their

00:13:39.000 --> 00:13:41.000
motions.
And, in particular,

00:13:41.000 --> 00:13:45.000
they explain a lot of things
such as how electric devices

00:13:45.000 --> 00:13:49.000
work, but also how
electromagnetic waves propagate.

00:13:49.000 --> 00:13:54.000
In particular,
that explains light and all

00:13:54.000 --> 00:13:58.000
sorts of waves.
It is thanks to them,

00:13:58.000 --> 00:14:02.000
you know, your cell phone,
laptops and things like that

00:14:02.000 --> 00:14:06.000
work.
Anyway.

00:14:06.000 --> 00:14:11.000
Hopefully most of you know that
the electric field is a vector

00:14:11.000 --> 00:14:14.000
field that basically tells you
what kind of force will be

00:14:14.000 --> 00:14:18.000
exerted on a charged particle
that you put in it.

00:14:18.000 --> 00:14:23.000
If you have a particle carrying
an electric charge then this

00:14:23.000 --> 00:14:27.000
vector field will tell you,
basically there will be an

00:14:27.000 --> 00:14:31.000
electric force which is the
charge times E that will be

00:14:31.000 --> 00:14:33.000
exerted on that particle.
And that is what is

00:14:33.000 --> 00:14:36.000
responsible, for example,
for the flow of electrons when

00:14:36.000 --> 00:14:41.000
you have a voltage difference.
Because classically this guy is

00:14:41.000 --> 00:14:45.000
a gradient of a potential.
And that potential is just

00:14:45.000 --> 00:14:50.000
electric voltage.
The magnetic field is a little

00:14:50.000 --> 00:14:55.000
bit harder to think about if you
have never seen it in physics,

00:14:55.000 --> 00:15:00.000
but it is what is causing,
for example,

00:15:00.000 --> 00:15:04.000
magnets to work.
Well, basically it is a force

00:15:04.000 --> 00:15:09.000
that is also expressed in terms
of a vector field usually called

00:15:09.000 --> 00:15:12.000
B.
Some people call it H but I am

00:15:12.000 --> 00:15:15.000
going to use B.
And that force tends to cause

00:15:15.000 --> 00:15:20.000
it, if you have a moving charged
particle, to deflect its

00:15:20.000 --> 00:15:24.000
trajectory and start rotating in
a magnetic field.

00:15:24.000 --> 00:15:32.000
What it does is not quite as
easy as what an electric field

00:15:32.000 --> 00:15:35.000
does.
Just to give you formulas,

00:15:35.000 --> 00:15:39.000
the force caused by the
electric field is the charge

00:15:39.000 --> 00:15:43.000
times the electric field.
And the force caused by the

00:15:43.000 --> 00:15:47.000
magnetic field,
I am never sure about the sign.

00:15:47.000 --> 00:15:52.000
Is that the correct sign?
Good.

00:15:52.000 --> 00:15:56.000
Now, the question is we need to
understand how these fields

00:15:56.000 --> 00:16:00.000
themselves are caused by the
charged particles that are

00:16:00.000 --> 00:16:03.000
placed in them.
There are various laws in there

00:16:03.000 --> 00:16:11.000
that explain what is going on.
Let me focus today on the

00:16:11.000 --> 00:16:17.000
electric field.
Maxwell's equations actually

00:16:17.000 --> 00:16:22.000
tell you about div and curl of
these fields.

00:16:22.000 --> 00:16:27.000
Let's look at div and curl of
the electric field.

00:16:27.000 --> 00:16:37.000
The first equation is called
the Gauss-Coulomb law.

00:16:37.000 --> 00:16:47.000
And it says that the divergence
of the electric field is equal

00:16:47.000 --> 00:16:51.000
to,
so this is a just a physical

00:16:51.000 --> 00:16:54.000
constant,
and what it is equal to depends

00:16:54.000 --> 00:16:57.000
on what units you are using.
And this guy rho,

00:16:57.000 --> 00:17:01.000
well, it is not the same rho as
in spherical coordinates because

00:17:01.000 --> 00:17:06.000
physicists somehow pretended
they used that letter first.

00:17:06.000 --> 00:17:08.000
It is the electric charge
density.

00:17:08.000 --> 00:17:15.000
It is the amount of electric
charge per unit volume.

00:17:15.000 --> 00:17:20.000
What this tells you is that
divergence of E is caused by the

00:17:20.000 --> 00:17:23.000
presence of electric charge.
In particular,

00:17:23.000 --> 00:17:29.000
if you have an empty region of
space or a region where nothing

00:17:29.000 --> 00:17:34.000
has electrical charge then E has
divergence equal to zero.

00:17:34.000 --> 00:17:38.000
Now, that looks like a very
abstract strange equation.

00:17:38.000 --> 00:17:43.000
I mean it is a partial
differential equation satisfied

00:17:43.000 --> 00:17:49.000
by the electric field E.
And that is not very intuitive

00:17:49.000 --> 00:17:56.000
in any way.
What is actually more intuitive

00:17:56.000 --> 00:18:05.000
is what we get if we apply the
divergence theorem to this

00:18:05.000 --> 00:18:11.000
equation.
If I think now about any closed

00:18:11.000 --> 00:18:16.000
surface,
and I want to think about the

00:18:16.000 --> 00:18:21.000
flux of the electric field out
of that surface,

00:18:21.000 --> 00:18:24.000
we haven't really thought about
what the flux of a force field

00:18:24.000 --> 00:18:27.000
does.
And I don't want to get into

00:18:27.000 --> 00:18:31.000
that because there is no very
easy answer in general,

00:18:31.000 --> 00:18:35.000
but I am going to explain soon
how this can be useful

00:18:35.000 --> 00:18:38.000
sometimes.
Let's say that we want to find

00:18:38.000 --> 00:18:43.000
the flux of the electric field
out of a closed surface.

00:18:43.000 --> 00:18:47.000
Then, by the divergence
theorem,

00:18:47.000 --> 00:18:53.000
that is equal to the triple
integral of a region inside of

00:18:53.000 --> 00:18:57.000
div E dV,
which is by the equation one

00:18:57.000 --> 00:19:00.000
over epsilon zero,
that is this constant,

00:19:00.000 --> 00:19:06.000
times the triple integral of
rho dV.

00:19:06.000 --> 00:19:09.000
But now, if I integrate the
charge density over the entire

00:19:09.000 --> 00:19:12.000
region,
then what I will get is

00:19:12.000 --> 00:19:17.000
actually the total amount of
electric charge inside the

00:19:17.000 --> 00:19:28.000
region.
That is the electric charge in

00:19:28.000 --> 00:19:31.000
D.
This one tells us,

00:19:31.000 --> 00:19:34.000
in a more concrete way,
how electric charges placed in

00:19:34.000 --> 00:19:38.000
here influence the electric
field around them.

00:19:38.000 --> 00:19:40.000
In particular,
one application of that is if

00:19:40.000 --> 00:19:43.000
you want to study capacitors.
Capacitors are these things

00:19:43.000 --> 00:19:46.000
that store energy by basically
you have two plates,

00:19:46.000 --> 00:19:49.000
one that contains positive
charge and a negative charge.

00:19:49.000 --> 00:19:52.000
Then you have a voltage between
these plates.

00:19:52.000 --> 00:19:57.000
And, basically,
that can provide electrical

00:19:57.000 --> 00:20:03.000
energy to power maybe an
electric circuit.

00:20:03.000 --> 00:20:06.000
That is not really a battery
because it doesn't store energy

00:20:06.000 --> 00:20:08.000
in large enough amounts.
But, for example,

00:20:08.000 --> 00:20:11.000
that is why when you switch
your favorite gadget off it

00:20:11.000 --> 00:20:14.000
doesn't actually go off
immediately but somehow you see

00:20:14.000 --> 00:20:18.000
things dimming progressively.
There is a capacitor in there.

00:20:18.000 --> 00:20:20.000
If you want to understand how
the voltage and the charge

00:20:20.000 --> 00:20:23.000
relate to each other,
the voltage is obtained by

00:20:23.000 --> 00:20:26.000
integrating the electric field
from one plate to the other

00:20:26.000 --> 00:20:29.000
plate.
And the charges in the plates

00:20:29.000 --> 00:20:34.000
are what causes the electric
field between the plates.

00:20:34.000 --> 00:20:37.000
That is how you can get the
relation between voltage and

00:20:37.000 --> 00:20:41.000
charge in these guys.
That is an example of

00:20:41.000 --> 00:20:44.000
application of that.
Now, of course,

00:20:44.000 --> 00:20:49.000
if you haven't seen any of this
then maybe it is a little bit

00:20:49.000 --> 00:20:54.000
esoteric, but that will tell you
part of what you will see in

00:20:54.000 --> 00:20:59.000
8.02.
Questions?

00:20:59.000 --> 00:21:07.000
I see some confused faces.
Well, don't worry.

00:21:07.000 --> 00:21:14.000
It will make sense some day.
[LAUGHTER]

00:21:14.000 --> 00:21:23.000
The next one I want to tell you
about is Faraday's law.

00:21:23.000 --> 00:21:25.000
In case you are confused,
Maxwell's equations,

00:21:25.000 --> 00:21:29.000
there are four equations in the
set of Maxwell's equations and

00:21:29.000 --> 00:21:31.000
most of them don't carry
Maxwell's name.

00:21:31.000 --> 00:21:40.000
That is a quirky feature.
That one tells you about the

00:21:40.000 --> 00:21:44.000
curl of the electric field.
Now, depending on your

00:21:44.000 --> 00:21:46.000
knowledge,
you might start telling me that

00:21:46.000 --> 00:21:50.000
the curl of the electric field
has to be zero because it is the

00:21:50.000 --> 00:21:52.000
gradient of the electric
potential.

00:21:52.000 --> 00:21:54.000
I told you this stuff about
voltage.

00:21:54.000 --> 00:21:58.000
Well, that doesn't account for
the fact that sometimes you can

00:21:58.000 --> 00:22:02.000
create voltage out of nowhere
using magnetic fields.

00:22:02.000 --> 00:22:05.000
And, in fact,
you have a failure of

00:22:05.000 --> 00:22:09.000
conservativity of the electric
force if you have a magnetic

00:22:09.000 --> 00:22:12.000
field.
What this one says is the curl

00:22:12.000 --> 00:22:17.000
of E is not zero but rather it
is the derivative of the

00:22:17.000 --> 00:22:21.000
magnetic field with respect to
time.

00:22:21.000 --> 00:22:26.000
More precisely it tells you
that what you might have learned

00:22:26.000 --> 00:22:31.000
about electric fields deriving
from electric potential becomes

00:22:31.000 --> 00:22:35.000
false if you have a variable
magnetic field.

00:22:35.000 --> 00:22:41.000
And just to tell you again that
is a strange partial

00:22:41.000 --> 00:22:47.000
differential equation relating
these two vector fields.

00:22:47.000 --> 00:22:51.000
To make sense of it one should
use Stokes' theorem.

00:22:51.000 --> 00:22:56.000
If we apply Stokes' theorem to
compute the work done by the

00:22:56.000 --> 00:23:00.000
electric field around a closed
curve,

00:23:00.000 --> 00:23:04.000
that means you have a wire in
there and you want to find the

00:23:04.000 --> 00:23:07.000
voltage along the wire.
Now there is a strange thing

00:23:07.000 --> 00:23:10.000
because classically you would
say, well, if I just have a wire

00:23:10.000 --> 00:23:13.000
with nothing in it there is no
voltage on it.

00:23:13.000 --> 00:23:18.000
Well, a small change in plans.
If you actually have a varying

00:23:18.000 --> 00:23:23.000
magnetic field that passes
through that wire then that will

00:23:23.000 --> 00:23:31.000
actually generate voltage in it.
That is how a transformer works.

00:23:31.000 --> 00:23:34.000
When you plug your laptop into
the wall circuit,

00:23:34.000 --> 00:23:36.000
you don't actually feed it
directly 110 volts,

00:23:36.000 --> 00:23:40.000
120 volts or whatever.
There is a transformer in there.

00:23:40.000 --> 00:23:45.000
What the transformer does it
takes some input voltage and

00:23:45.000 --> 00:23:49.000
passes that through basically a
loop of wire.

00:23:49.000 --> 00:23:53.000
Not much seems to be happening.
But now you have another loops

00:23:53.000 --> 00:23:56.000
of wire that is intertwined with
it.

00:23:56.000 --> 00:23:59.000
Somehow the magnetic field
generated by it,

00:23:59.000 --> 00:24:03.000
and it has to be a donating
current.

00:24:03.000 --> 00:24:06.000
The donating current varies
over time in the first wire.

00:24:06.000 --> 00:24:09.000
That generates a magnetic field
that varies over time,

00:24:09.000 --> 00:24:13.000
so that causes 2B by 2t and
that causes curl of the electric

00:24:13.000 --> 00:24:15.000
field.
And the curl of the electric

00:24:15.000 --> 00:24:18.000
field will generate voltage
between these two guys.

00:24:18.000 --> 00:24:21.000
And that is how a transformer
works.

00:24:21.000 --> 00:24:25.000
It uses Stokes' theorem.
More precisely,

00:24:25.000 --> 00:24:28.000
how do we find the voltage
between these two points?

00:24:28.000 --> 00:24:32.000
Well, let's close the loop and
let's try to figure out the

00:24:32.000 --> 00:24:37.000
voltage inside this loop.
To find a voltage along a

00:24:37.000 --> 00:24:42.000
closed curve places in a varying
magnetic field,

00:24:42.000 --> 00:24:47.000
we have to do the line integral
along a closed curve of the

00:24:47.000 --> 00:24:51.000
electric field.
And you should think of this as

00:24:51.000 --> 00:24:54.000
the voltage generated in this
circuit.

00:24:54.000 --> 00:25:05.000
That will be the flux for this
surface bounded by the curve of

00:25:05.000 --> 00:25:11.000
curl E dot dS.
That is what Stokes' theorem

00:25:11.000 --> 00:25:14.000
says.
And now if you combine that

00:25:14.000 --> 00:25:21.000
with Faraday's law you end up
with the flux trough S of minus

00:25:21.000 --> 00:25:25.000
dB over dt.
And, of course, you could take,

00:25:25.000 --> 00:25:27.000
if your loop doesn't move over
time,

00:25:27.000 --> 00:25:31.000
I mean there is a different
story if you start somehow

00:25:31.000 --> 00:25:34.000
taking your wire and somehow
moving it inside the field.

00:25:34.000 --> 00:25:37.000
But if you don't do that,
if it is the field that is

00:25:37.000 --> 00:25:40.000
moving then you just can take
the dB by dt outside.

00:25:40.000 --> 00:25:48.000
But let's not bother.
Again, what this equation tells

00:25:48.000 --> 00:25:52.000
you is that if the magnetic
field changes over time then it

00:25:52.000 --> 00:25:55.000
creates, just out of nowhere,
and electric field.

00:25:55.000 --> 00:26:09.000
And that electric field can be
used to power up things.

00:26:09.000 --> 00:26:11.000
I don't really claim that I
have given you enough details to

00:26:11.000 --> 00:26:15.000
understand how they work,
but basically these equations

00:26:15.000 --> 00:26:20.000
are the heart of understanding
how things like capacitors and

00:26:20.000 --> 00:26:23.000
transformers work.
And they also explain a lot of

00:26:23.000 --> 00:26:25.000
other things,
but I will leave that to your

00:26:25.000 --> 00:26:28.000
physics teachers.
Just for completeness,

00:26:28.000 --> 00:26:33.000
I will just give you the last
two equations in that.

00:26:33.000 --> 00:26:37.000
I am not even going to try to
explain them too much.

00:26:37.000 --> 00:26:42.000
One of them says that the
divergence of the magnetic field

00:26:42.000 --> 00:26:45.000
is zero,
which somehow is fortunate

00:26:45.000 --> 00:26:49.000
because otherwise you would run
into trouble trying to

00:26:49.000 --> 00:26:53.000
understand surface independence
when you apply Stokes' theorem

00:26:53.000 --> 00:26:58.000
in here.
And the last one tells you how

00:26:58.000 --> 00:27:04.000
the curl of the magnetic field
is caused by motion of charged

00:27:04.000 --> 00:27:09.000
particles.
In fact, let's say that the

00:27:09.000 --> 00:27:16.000
curl of B is given by this kind
of formula, well,

00:27:16.000 --> 00:27:23.000
J is what is called the vector
of current density.

00:27:23.000 --> 00:27:30.000
It measures the flow of
electrically charged particles.

00:27:30.000 --> 00:27:34.000
You get this guy when you start
taking charged particles,

00:27:34.000 --> 00:27:38.000
like electrons maybe,
and moving them around.

00:27:38.000 --> 00:27:40.000
And, of course,
that is actually part of how

00:27:40.000 --> 00:27:44.000
transformers work because I have
told you running the AC through

00:27:44.000 --> 00:27:46.000
the first loop generates a
magnetic field.

00:27:46.000 --> 00:27:49.000
Well, how does it do that?
It is thanks to this equation.

00:27:49.000 --> 00:27:52.000
If you have a current passing
in the loop that causes a

00:27:52.000 --> 00:27:54.000
magnetic field and,
in turn, for the other equation

00:27:54.000 --> 00:27:59.000
that causes an electric field,
which in turn causes a current.

00:27:59.000 --> 00:28:08.000
It is all somehow intertwined
in a very intricate way and is

00:28:08.000 --> 00:28:15.000
really remarkable how well that
works in practice.

00:28:15.000 --> 00:28:17.000
I think that is basically all I
wanted to say about 8.02.

00:28:17.000 --> 00:28:23.000
I don't want to put your
physics teachers out of a job.

00:28:23.000 --> 00:28:24.000
[LAUGHTER]
If you haven't seen any of this

00:28:24.000 --> 00:28:26.000
before,
I understand that this is

00:28:26.000 --> 00:28:28.000
probably not detailed enough to
be really understandable,

00:28:28.000 --> 00:28:32.000
but hopefully it will make you
a bit curious about that and

00:28:32.000 --> 00:28:36.000
prompt you to take that class
someday and maybe even remember

00:28:36.000 --> 00:28:39.000
how it relates to 18.02.