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

8
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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,

10
00:00:42 --> 00:00:50
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,

18
00:01:48 --> 00:01:58
just to remind you,
I mean we have seen this guy a

19
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couple of times,
but if I give you a uniform

20
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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

22
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circles centered in the z-axis
and our vector field is just

23
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going to be tangent to each of
these circles.

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00:02:26 --> 00:02:35
And, if you look at it from
above, then you will have this

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00:02:35 --> 00:02:42
rotation vector field that we
have seen many times.

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00:02:42 --> 00:02:48
Typically, the velocity vector
for this would be minus yi plus

27
00:02:48 --> 00:02:53
yj times maybe a number that
represents how fast we are

28
00:02:53 --> 00:02:56
spinning,
the angular velocity in

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00:02:56 --> 00:03:01
gradients per second.
And then.

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00:03:01 --> 00:03:06
if you compute the curl of
this, you will end up with two

31
00:03:06 --> 00:03:09
omega times k.
Now, the other kinds of vector

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00:03:09 --> 00:03:12
fields we have seen physically
are force fields.

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00:03:12 --> 00:03:17
The question is what does the
curl of a force field mean?

34
00:03:17 --> 00:03:24
What can we say about that?
The interpretation is a little

35
00:03:24 --> 00:03:34
bit less obvious,
but let's try to get some idea

36
00:03:34 --> 00:03:42
of what it might be.
I want to remind you that if we

37
00:03:42 --> 00:03:47
have a solid in a force field,
we can measure the torque

38
00:03:47 --> 00:03:51
exerted by the force on the
solid.

39
00:03:51 --> 00:03:55
Maybe first I should remind you
about what torque is in space.

40
00:03:55 --> 00:03:59
Let's say that I have a piece
of solid with a mass,

41
00:03:59 --> 00:04:03
delta m for example,
and I have a force that is

42
00:04:03 --> 00:04:09
being exerted to it.
Let's say that maybe my force

43
00:04:09 --> 00:04:14
might be F times delta m.
If you think,

44
00:04:14 --> 00:04:16
for example,
a gravitational field.

45
00:04:16 --> 00:04:20
The gravitational force is
actually the gravitational field

46
00:04:20 --> 00:04:24
times the mass.
I mean you can forget delta m

47
00:04:24 --> 00:04:29
if you don't like it.
And let's say that the position

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00:04:29 --> 00:04:33
vector, which should be aiming
for the origin,

49
00:04:33 --> 00:04:37
R is here.
And now let's say that maybe

50
00:04:37 --> 00:04:42
this guy is at the end of some
arm or some metal thing and I

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00:04:42 --> 00:04:46
want to hold it in place.
The force is going to exert a

52
00:04:46 --> 00:04:50
torque relative to the origin
that will try to measure how

53
00:04:50 --> 00:04:54
much I am trying to swing this
guy around the origin.

54
00:04:54 --> 00:04:57
And, consequently,
how much effort I have to exert

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00:04:57 --> 00:05:02
if I want to actually maintain
its place by just holding it at

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00:05:02 --> 00:05:12
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

59
00:05:31 --> 00:05:33
is the rotation effects of the
force.

60
00:05:33 --> 00:05:39
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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00:05:50 --> 00:05:58
velocity should be angular
acceleration which is related to

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00:05:58 --> 00:06:04
the torque per unit mass.
To just remind you, 

65
00:06:04 --> 00:06:06
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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00:06:21 --> 00:06:28
which is the derivative of
velocity.

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00:06:28 --> 00:06:35
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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00:06:49 --> 00:06:53
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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00:07:00 --> 00:07:06
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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00:07:30 --> 00:07:35
operation,
which from a velocity field

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00:07:35 --> 00:07:41
gives the angular velocity of
its rotation effects,

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00:07:41 --> 00:07:46
then you see that the curl of
an acceleration field gives the

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00:07:46 --> 00:07:49
angular acceleration in the
rotation part of the

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00:07:49 --> 00:07:53
acceleration effects.
And, therefore,

85
00:07:53 --> 00:07:59
the curl of a force field
measures the torque per unit

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00:07:59 --> 00:08:04
moment of inertia.
It measures how much torque its

87
00:08:04 --> 00:08:08
force field exerts on a small
test solid placed in it.

88
00:08:08 --> 00:08:12
If you have a small solid
somewhere, the curl will just

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00:08:12 --> 00:08:16
measure how much your solid
starts spinning if you leave it

90
00:08:16 --> 00:08:18
in this force field.
In particular,

91
00:08:18 --> 00:08:22
a force field with no curl is a
force field that does not

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00:08:22 --> 00:08:26
generate any rotation motion.
That means if you put an object

93
00:08:26 --> 00:08:29
in there that is completely
immobile and you leave it in

94
00:08:29 --> 00:08:31
that force field,
well, of course it might

95
00:08:31 --> 00:08:35
accelerate in some direction but
it won't start spinning.

96
00:08:35 --> 00:08:38
While, if you put it in there
spinning already in some

97
00:08:38 --> 00:08:41
direction, it should continue to
spin in the same way.

98
00:08:41 --> 00:08:50
Of course, maybe there will be
friction and things like that

99
00:08:50 --> 00:08:58
which will slow it down but this
force field is not responsible

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00:08:58 --> 00:09:05
for it.
The cool consequence of this is

101
00:09:05 --> 00:09:14
if a force field F derives from
a potential -- That is what we

102
00:09:14 --> 00:09:20
have seen about conservative
forces.

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00:09:20 --> 00:09:23
Our main concern so far has
been to say if we have a

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00:09:23 --> 00:09:26
conservative force field it
means that the work of a force

105
00:09:26 --> 00:09:29
is the change in the energy.
And, in particular,

106
00:09:29 --> 00:09:32
we cannot get energy for free
out of it.

107
00:09:32 --> 00:09:36
And the change in the potential
energy is going to be the change

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00:09:36 --> 00:09:40
in kinetic energy.
You have conservation of energy

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00:09:40 --> 00:09:43
principles.
There is another thing that we

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00:09:43 --> 00:09:48
know now because if a force
derives from a potential then

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00:09:48 --> 00:09:53
that means its curl is zero.
That is the criterion we have

112
00:09:53 --> 00:09:58
seen for a vector field to
derive from a potential.

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00:09:58 --> 00:10:14
And if the curl is zero then it
means that this force does not

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00:10:14 --> 00:10:23
generate any rotation effects.
For example,

115
00:10:23 --> 00:10:27
if you try to understand where
the earth comes from,

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00:10:27 --> 00:10:32
well, the earth is spinning on
itself as it goes around the

117
00:10:32 --> 00:10:35
sun.
And you might wonder where that

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00:10:35 --> 00:10:37
comes from.
Is that causes by gravitational

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00:10:37 --> 00:10:40
attraction?
And the answer is no.

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00:10:40 --> 00:10:44
Gravitational attraction in
itself cannot cause the earth to

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00:10:44 --> 00:10:47
start spinning faster or slower,
at least if you assume the

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00:10:47 --> 00:10:52
earth to be a solid,
which actually is false.

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00:10:52 --> 00:10:57
I mean basically the reason why
the earth is spinning is because

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00:10:57 --> 00:11:01
it was formed spinning.
It didn't start spinning

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00:11:01 --> 00:11:03
because of gravitational
effects.

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00:11:03 --> 00:11:08
And that is a rather deep
purely mathematical consequence

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00:11:08 --> 00:11:12
of understanding gravitation in
this way.

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00:11:12 --> 00:11:16
It is quite spectacular that
just by abstract thinking we got

129
00:11:16 --> 00:11:17
there.
What is the truth?

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00:11:17 --> 00:11:21
Well, the truth is the earth,
the moon and everything is

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00:11:21 --> 00:11:24
slightly deformable.
And so there is deformation,

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00:11:24 --> 00:11:26
friction effects,
tidal effects and so on.

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00:11:26 --> 00:11:29
And these actually cause
rotations to get slightly

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00:11:29 --> 00:11:32
synchronized with each other.
For example, 

135
00:11:32 --> 00:11:36
if you want to explain why the
moon is always showing the same

136
00:11:36 --> 00:11:39
face to the earth,
why the rotation of a moon on

137
00:11:39 --> 00:11:43
itself is synchronized with its
revolution around the earth,

138
00:11:43 --> 00:11:47
which is actually explained by
friction effects over time and

139
00:11:47 --> 00:11:50
the gravitational attraction of
the earth and the moon.

140
00:11:50 --> 00:11:59
There is something there,
but if you took perfectly

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00:11:59 --> 00:12:09
rigid, solid bodies then
gravitation would never cause

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00:12:09 --> 00:12:15
any rotation effects.
Of course that tells us that we

143
00:12:15 --> 00:12:20
do not know how to answer the
question of why is the earth

144
00:12:20 --> 00:12:22
spinning.
That will be left for another

145
00:12:22 --> 00:12:31
physics class.
I don't have a good answer to

146
00:12:31 --> 00:12:35
that.
That was kind of 8.01-ish.

147
00:12:35 --> 00:12:40
Let me now move forward to 8.02
stuff.

148
00:12:40 --> 00:12:54
I want to tell you things about
electric and magnetic fields.

149
00:12:54 --> 00:13:01
And, in fact,
something that is known as

150
00:13:01 --> 00:13:06
Maxwell's equations.
Just a quick poll.

151
00:13:06 --> 00:13:10
How many of you have been
taking 8.02 or something like

152
00:13:10 --> 00:13:13
that?
OK. That is not very many.

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00:13:13 --> 00:13:15
For most of you this is a
preview.

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00:13:15 --> 00:13:18
If you have been taking 8.02,
have you seen Maxwell's

155
00:13:18 --> 00:13:20
equations, at least part of
them?

156
00:13:20 --> 00:13:22
Yeah.
OK.

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00:13:22 --> 00:13:23
Then I am sure,
in that case,

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00:13:23 --> 00:13:25
you know better than me what I
am going to talk about because I

159
00:13:25 --> 00:13:30
am not a physicist.
But just in case.

160
00:13:30 --> 00:13:35
Maxwell's equations govern how
electric and magnetic fields

161
00:13:35 --> 00:13:39
behave, how they are caused by
electric charges and their

162
00:13:39 --> 00:13:41
motions.
And, in particular,

163
00:13:41 --> 00:13:45
they explain a lot of things
such as how electric devices

164
00:13:45 --> 00:13:49
work, but also how
electromagnetic waves propagate.

165
00:13:49 --> 00:13:54
In particular,
that explains light and all

166
00:13:54 --> 00:13:58
sorts of waves.
It is thanks to them,

167
00:13:58 --> 00:14:02
you know, your cell phone,
laptops and things like that

168
00:14:02 --> 00:14:06
work.
Anyway.

169
00:14:06 --> 00:14:11
Hopefully most of you know that
the electric field is a vector

170
00:14:11 --> 00:14:14
field that basically tells you
what kind of force will be

171
00:14:14 --> 00:14:18
exerted on a charged particle
that you put in it.

172
00:14:18 --> 00:14:23
If you have a particle carrying
an electric charge then this

173
00:14:23 --> 00:14:27
vector field will tell you,
basically there will be an

174
00:14:27 --> 00:14:31
electric force which is the
charge times E that will be

175
00:14:31 --> 00:14:33
exerted on that particle.
And that is what is

176
00:14:33 --> 00:14:36
responsible, for example,
for the flow of electrons when

177
00:14:36 --> 00:14:41
you have a voltage difference.
Because classically this guy is

178
00:14:41 --> 00:14:45
a gradient of a potential.
And that potential is just

179
00:14:45 --> 00:14:50
electric voltage.
The magnetic field is a little

180
00:14:50 --> 00:14:55
bit harder to think about if you
have never seen it in physics,

181
00:14:55 --> 00:15:00
but it is what is causing,
for example,

182
00:15:00 --> 00:15:04
magnets to work.
Well, basically it is a force

183
00:15:04 --> 00:15:09
that is also expressed in terms
of a vector field usually called

184
00:15:09 --> 00:15:12
B.
Some people call it H but I am

185
00:15:12 --> 00:15:15
going to use B.
And that force tends to cause

186
00:15:15 --> 00:15:20
it, if you have a moving charged
particle, to deflect its

187
00:15:20 --> 00:15:24
trajectory and start rotating in
a magnetic field.

188
00:15:24 --> 00:15:32
What it does is not quite as
easy as what an electric field

189
00:15:32 --> 00:15:35
does.
Just to give you formulas,

190
00:15:35 --> 00:15:39
the force caused by the
electric field is the charge

191
00:15:39 --> 00:15:43
times the electric field.
And the force caused by the

192
00:15:43 --> 00:15:47
magnetic field,
I am never sure about the sign.

193
00:15:47 --> 00:15:52
Is that the correct sign?
Good.

194
00:15:52 --> 00:15:56
Now, the question is we need to
understand how these fields

195
00:15:56 --> 00:16:00
themselves are caused by the
charged particles that are

196
00:16:00 --> 00:16:03
placed in them.
There are various laws in there

197
00:16:03 --> 00:16:11
that explain what is going on.
Let me focus today on the

198
00:16:11 --> 00:16:17
electric field.
Maxwell's equations actually

199
00:16:17 --> 00:16:22
tell you about div and curl of
these fields.

200
00:16:22 --> 00:16:27
Let's look at div and curl of
the electric field.

201
00:16:27 --> 00:16:37
The first equation is called
the Gauss-Coulomb law.

202
00:16:37 --> 00:16:47
And it says that the divergence
of the electric field is equal

203
00:16:47 --> 00:16:51
to,
so this is a just a physical

204
00:16:51 --> 00:16:54
constant,
and what it is equal to depends

205
00:16:54 --> 00:16:57
on what units you are using.
And this guy rho,

206
00:16:57 --> 00:17:01
well, it is not the same rho as
in spherical coordinates because

207
00:17:01 --> 00:17:06
physicists somehow pretended
they used that letter first.

208
00:17:06 --> 00:17:08
It is the electric charge
density.

209
00:17:08 --> 00:17:15
It is the amount of electric
charge per unit volume.

210
00:17:15 --> 00:17:20
What this tells you is that
divergence of E is caused by the

211
00:17:20 --> 00:17:23
presence of electric charge.
In particular,

212
00:17:23 --> 00:17:29
if you have an empty region of
space or a region where nothing

213
00:17:29 --> 00:17:34
has electrical charge then E has
divergence equal to zero.

214
00:17:34 --> 00:17:38
Now, that looks like a very
abstract strange equation.

215
00:17:38 --> 00:17:43
I mean it is a partial
differential equation satisfied

216
00:17:43 --> 00:17:49
by the electric field E.
And that is not very intuitive

217
00:17:49 --> 00:17:56
in any way.
What is actually more intuitive

218
00:17:56 --> 00:18:05
is what we get if we apply the
divergence theorem to this

219
00:18:05 --> 00:18:11
equation.
If I think now about any closed

220
00:18:11 --> 00:18:16
surface,
and I want to think about the

221
00:18:16 --> 00:18:21
flux of the electric field out
of that surface,

222
00:18:21 --> 00:18:24
we haven't really thought about
what the flux of a force field

223
00:18:24 --> 00:18:27
does.
And I don't want to get into

224
00:18:27 --> 00:18:31
that because there is no very
easy answer in general,

225
00:18:31 --> 00:18:35
but I am going to explain soon
how this can be useful

226
00:18:35 --> 00:18:38
sometimes.
Let's say that we want to find

227
00:18:38 --> 00:18:43
the flux of the electric field
out of a closed surface.

228
00:18:43 --> 00:18:47
Then, by the divergence
theorem,

229
00:18:47 --> 00:18:53
that is equal to the triple
integral of a region inside of

230
00:18:53 --> 00:18:57
div E dV,
which is by the equation one

231
00:18:57 --> 00:19:00
over epsilon zero,
that is this constant, 

232
00:19:00 --> 00:19:06
times the triple integral of
rho dV.

233
00:19:06 --> 00:19:09
But now, if I integrate the
charge density over the entire

234
00:19:09 --> 00:19:12
region,
then what I will get is

235
00:19:12 --> 00:19:17
actually the total amount of
electric charge inside the

236
00:19:17 --> 00:19:28
region.
That is the electric charge in

237
00:19:28 --> 00:19:31
D.
This one tells us,

238
00:19:31 --> 00:19:34
in a more concrete way,
how electric charges placed in

239
00:19:34 --> 00:19:38
here influence the electric
field around them.

240
00:19:38 --> 00:19:40
In particular,
one application of that is if

241
00:19:40 --> 00:19:43
you want to study capacitors.
Capacitors are these things

242
00:19:43 --> 00:19:46
that store energy by basically
you have two plates,

243
00:19:46 --> 00:19:49
one that contains positive
charge and a negative charge.

244
00:19:49 --> 00:19:52
Then you have a voltage between
these plates.

245
00:19:52 --> 00:19:57
And, basically,
that can provide electrical

246
00:19:57 --> 00:20:03
energy to power maybe an
electric circuit.

247
00:20:03 --> 00:20:06
That is not really a battery
because it doesn't store energy

248
00:20:06 --> 00:20:08
in large enough amounts.
But, for example,

249
00:20:08 --> 00:20:11
that is why when you switch
your favorite gadget off it

250
00:20:11 --> 00:20:14
doesn't actually go off
immediately but somehow you see

251
00:20:14 --> 00:20:18
things dimming progressively.
There is a capacitor in there.

252
00:20:18 --> 00:20:20
If you want to understand how
the voltage and the charge

253
00:20:20 --> 00:20:23
relate to each other,
the voltage is obtained by

254
00:20:23 --> 00:20:26
integrating the electric field
from one plate to the other

255
00:20:26 --> 00:20:29
plate.
And the charges in the plates

256
00:20:29 --> 00:20:34
are what causes the electric
field between the plates.

257
00:20:34 --> 00:20:37
That is how you can get the
relation between voltage and

258
00:20:37 --> 00:20:41
charge in these guys.
That is an example of

259
00:20:41 --> 00:20:44
application of that.
Now, of course,

260
00:20:44 --> 00:20:49
if you haven't seen any of this
then maybe it is a little bit

261
00:20:49 --> 00:20:54
esoteric, but that will tell you
part of what you will see in

262
00:20:54 --> 00:20:59
8.02.
Questions?

263
00:20:59 --> 00:21:07
I see some confused faces.
Well, don't worry.

264
00:21:07 --> 00:21:14
It will make sense some day.
[LAUGHTER]

265
00:21:14 --> 00:21:23
The next one I want to tell you
about is Faraday's law.

266
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In case you are confused,
Maxwell's equations,

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there are four equations in the
set of Maxwell's equations and

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00:21:29 --> 00:21:31
most of them don't carry
Maxwell's name.

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That is a quirky feature.
That one tells you about the

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curl of the electric field.
Now, depending on your

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knowledge,
you might start telling me that

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the curl of the electric field
has to be zero because it is the

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gradient of the electric
potential.

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00:21:52 --> 00:21:54
I told you this stuff about
voltage.

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Well, that doesn't account for
the fact that sometimes you can

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create voltage out of nowhere
using magnetic fields.

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And, in fact,
you have a failure of

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conservativity of the electric
force if you have a magnetic

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field.
What this one says is the curl

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00:22:12 --> 00:22:17
of E is not zero but rather it
is the derivative of the

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00:22:17 --> 00:22:21
magnetic field with respect to
time.

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More precisely it tells you
that what you might have learned

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00:22:26 --> 00:22:31
about electric fields deriving
from electric potential becomes

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00:22:31 --> 00:22:35
false if you have a variable
magnetic field.

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00:22:35 --> 00:22:41
And just to tell you again that
is a strange partial

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00:22:41 --> 00:22:47
differential equation relating
these two vector fields.

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00:22:47 --> 00:22:51
To make sense of it one should
use Stokes' theorem.

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00:22:51 --> 00:22:56
If we apply Stokes' theorem to
compute the work done by the

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00:22:56 --> 00:23:00
electric field around a closed
curve,

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00:23:00 --> 00:23:04
that means you have a wire in
there and you want to find the

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00:23:04 --> 00:23:07
voltage along the wire.
Now there is a strange thing

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00:23:07 --> 00:23:10
because classically you would
say, well, if I just have a wire

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00:23:10 --> 00:23:13
with nothing in it there is no
voltage on it.

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00:23:13 --> 00:23:18
Well, a small change in plans.
If you actually have a varying

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00:23:18 --> 00:23:23
magnetic field that passes
through that wire then that will

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00:23:23 --> 00:23:31
actually generate voltage in it.
That is how a transformer works.

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00:23:31 --> 00:23:34
When you plug your laptop into
the wall circuit,

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00:23:34 --> 00:23:36
you don't actually feed it
directly 110 volts,

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00:23:36 --> 00:23:40
120 volts or whatever.
There is a transformer in there.

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00:23:40 --> 00:23:45
What the transformer does it
takes some input voltage and

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00:23:45 --> 00:23:49
passes that through basically a
loop of wire.

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00:23:49 --> 00:23:53
Not much seems to be happening.
But now you have another loops

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00:23:53 --> 00:23:56
of wire that is intertwined with
it.

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00:23:56 --> 00:23:59
Somehow the magnetic field
generated by it,

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00:23:59 --> 00:24:03
and it has to be a donating
current.

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00:24:03 --> 00:24:06
The donating current varies
over time in the first wire.

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00:24:06 --> 00:24:09
That generates a magnetic field
that varies over time,

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00:24:09 --> 00:24:13
so that causes 2B by 2t and
that causes curl of the electric

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00:24:13 --> 00:24:15
field.
And the curl of the electric

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00:24:15 --> 00:24:18
field will generate voltage
between these two guys.

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00:24:18 --> 00:24:21
And that is how a transformer
works.

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00:24:21 --> 00:24:25
It uses Stokes' theorem.
More precisely,

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00:24:25 --> 00:24:28
how do we find the voltage
between these two points?

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00:24:28 --> 00:24:32
Well, let's close the loop and
let's try to figure out the

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00:24:32 --> 00:24:37
voltage inside this loop.
To find a voltage along a

316
00:24:37 --> 00:24:42
closed curve places in a varying
magnetic field,

317
00:24:42 --> 00:24:47
we have to do the line integral
along a closed curve of the

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00:24:47 --> 00:24:51
electric field.
And you should think of this as

319
00:24:51 --> 00:24:54
the voltage generated in this
circuit.

320
00:24:54 --> 00:25:05
That will be the flux for this
surface bounded by the curve of

321
00:25:05 --> 00:25:11
curl E dot dS.
That is what Stokes' theorem

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00:25:11 --> 00:25:14
says.
And now if you combine that

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00:25:14 --> 00:25:21
with Faraday's law you end up
with the flux trough S of minus

324
00:25:21 --> 00:25:25
dB over dt.
And, of course, you could take, 

325
00:25:25 --> 00:25:27
if your loop doesn't move over
time,

326
00:25:27 --> 00:25:31
I mean there is a different
story if you start somehow

327
00:25:31 --> 00:25:34
taking your wire and somehow
moving it inside the field.

328
00:25:34 --> 00:25:37
But if you don't do that,
if it is the field that is

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00:25:37 --> 00:25:40
moving then you just can take
the dB by dt outside.

330
00:25:40 --> 00:25:48
But let's not bother.
Again, what this equation tells

331
00:25:48 --> 00:25:52
you is that if the magnetic
field changes over time then it

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00:25:52 --> 00:25:55
creates, just out of nowhere,
and electric field.

333
00:25:55 --> 00:26:09
And that electric field can be
used to power up things.

334
00:26:09 --> 00:26:11
I don't really claim that I
have given you enough details to

335
00:26:11 --> 00:26:15
understand how they work,
but basically these equations

336
00:26:15 --> 00:26:20
are the heart of understanding
how things like capacitors and

337
00:26:20 --> 00:26:23
transformers work.
And they also explain a lot of

338
00:26:23 --> 00:26:25
other things,
but I will leave that to your

339
00:26:25 --> 00:26:28
physics teachers.
Just for completeness,

340
00:26:28 --> 00:26:33
I will just give you the last
two equations in that.

341
00:26:33 --> 00:26:37
I am not even going to try to
explain them too much.

342
00:26:37 --> 00:26:42
One of them says that the
divergence of the magnetic field

343
00:26:42 --> 00:26:45
is zero,
which somehow is fortunate

344
00:26:45 --> 00:26:49
because otherwise you would run
into trouble trying to

345
00:26:49 --> 00:26:53
understand surface independence
when you apply Stokes' theorem

346
00:26:53 --> 00:26:58
in here.
And the last one tells you how

347
00:26:58 --> 00:27:04
the curl of the magnetic field
is caused by motion of charged

348
00:27:04 --> 00:27:09
particles.
In fact, let's say that the

349
00:27:09 --> 00:27:16
curl of B is given by this kind
of formula, well,

350
00:27:16 --> 00:27:23
J is what is called the vector
of current density.

351
00:27:23 --> 00:27:30
It measures the flow of
electrically charged particles.

352
00:27:30 --> 00:27:34
You get this guy when you start
taking charged particles,

353
00:27:34 --> 00:27:38
like electrons maybe,
and moving them around.

354
00:27:38 --> 00:27:40
And, of course, 
that is actually part of how

355
00:27:40 --> 00:27:44
transformers work because I have
told you running the AC through

356
00:27:44 --> 00:27:46
the first loop generates a
magnetic field.

357
00:27:46 --> 00:27:49
Well, how does it do that?
It is thanks to this equation.

358
00:27:49 --> 00:27:52
If you have a current passing
in the loop that causes a

359
00:27:52 --> 00:27:54
magnetic field and,
in turn, for the other equation

360
00:27:54 --> 00:27:59
that causes an electric field,
which in turn causes a current.

361
00:27:59 --> 00:28:08
It is all somehow intertwined
in a very intricate way and is

362
00:28:08 --> 00:28:15
really remarkable how well that
works in practice.

363
00:28:15 --> 00:28:17
I think that is basically all I
wanted to say about 8.02.

364
00:28:17 --> 00:28:23
I don't want to put your
physics teachers out of a job.

365
00:28:23 --> 00:28:24
[LAUGHTER]
If you haven't seen any of this

366
00:28:24 --> 00:28:26
before,
I understand that this is

367
00:28:26 --> 00:28:28
probably not detailed enough to
be really understandable,

368
00:28:28 --> 00:28:32
but hopefully it will make you
a bit curious about that and

369
00:28:32 --> 00:28:36
prompt you to take that class
someday and maybe even remember

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00:28:36 --> 00:28:39
how it relates to 18.02.
 

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