WEBVTT

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Prof: Before we start
today's lecture,

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I just wanted to explain to you
that at the end of the class,

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I have to run out and catch a
plane.

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So if you have to discuss any
other administrative matters,

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you should do it now,
because you won't find me after

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class.

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I couldn't take this weather.

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You know where I'm going?

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Seattle.

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That's the story of my life.

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So nothing else?

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Yes?

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Student:  Is there going
to be an equation sheet for the

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midterm and if so,
is it possible to get it?

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Prof: You will get the
equation sheet.

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Do you want it now?

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No.

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With the exam?

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Yeah.

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I don't give it out before,
but you will have all the

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reasonable equations on that
sheet.

00:56.110 --> 00:58.810
You know the reason for that,
right?

00:58.810 --> 01:03.660
The reason is that some of you
may be going into medicine and

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you control the anesthesia on
me, one of these days in the

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future.

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If I held back the equation
sheet, I know you will hold back

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the painkillers.

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It's just making a deal,
the way things are done

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nowadays.

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I made a deal with you guys.

01:22.860 --> 01:26.520
I couldn't sleep last night,
because today is when we're

01:26.519 --> 01:30.179
going to solve Maxwell's
equations and get the waves.

01:30.180 --> 01:31.630
I just couldn't wait to get to
work.

01:31.629 --> 01:32.019
It's great.

01:32.019 --> 01:36.529
No matter how many times I talk
about it, I just find it so

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amazing.

01:37.310 --> 01:40.720
So here are these great Maxwell
equations.

01:40.720 --> 01:41.840
I'm going to write them down.

01:41.840 --> 02:23.200


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Those are the equations.

02:26.090 --> 02:29.400
The first one tells you that if
you draw any surface and

02:29.397 --> 02:31.887
integrate
B over that surface,

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namely, you're counting the net
number of lines coming out,

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you're going to get 0.

02:37.348 --> 02:40.498
That's because lines begin and
end with charges and there are

02:40.496 --> 02:41.646
no magnetic charges.

02:41.650 --> 02:43.750
If you look at any magnetic
field problem,

02:43.747 --> 02:46.047
the lines don't have a
beginning or an end.

02:46.050 --> 02:48.200
They end on themselves.

02:48.199 --> 02:51.789
So if you put any surface
there, whatever goes in has to

02:51.791 --> 02:52.511
come out.

02:52.508 --> 02:55.018
That's the statement of no
magnetic charges.

02:55.020 --> 02:57.190
This one says,
yes, there are things that

02:57.193 --> 02:59.913
manufacture electric field lines
called charges.

02:59.910 --> 03:01.720
They make them and they eat
them.

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Depending on how many got in
the volume, the net will decide

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the flux out of that volume.

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This one, without that,
used to say electric field is a

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conservative field.

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It defines a potential and so
on.

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But then we found in the time
dependent problem,

03:17.543 --> 03:20.943
E⋅dl,
we said rate of change of the

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magnetic field.

03:22.020 --> 03:25.740
So changing magnetic field
sustains an electric field.

03:25.740 --> 03:29.230
And this one says a changing
electric field can produce a

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magnetic field.

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In addition,
a current can also produce a

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magnetic field.

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This is what we have to begin
with.

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Now I'm going to eventually
focus on free space.

03:43.628 --> 03:49.948
Free space means we're nowhere
near charges and currents.

03:49.949 --> 03:52.399
And the equations become more
symmetric.

03:52.400 --> 03:55.240
This has got no surface
integral, that has got no

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surface integral.

03:56.370 --> 03:59.730
The line integral of this guy
is essentially the rate of

03:59.730 --> 04:01.260
change of flux of that.

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And the line integral of that
guy is - the rate of change of

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flux of this.

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If I'm going to get a wave
equation, the triumphant moment

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is when I do this and do that
and out comes the wave equation.

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But if at that moment you have
not seen the wave equation,

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you're not going to get the
thrill.

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So I have to know,
how many people remember the

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wave equation from the first
part of the course?

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Okay, so I think I have to
remind you of what that is.

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It's a bit like explaining a
joke, but still,

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I have to do that,
because otherwise we won't have

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that climactic moment.

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So let me tell you one example
of a wave equation.

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People have seen them before.

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Here is the simplest example.

04:50.980 --> 04:55.000
You take a string,
you tie it between two fixed

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points and it has a tension
T.

04:58.399 --> 05:02.269
Tension is the force with which
the two ends are being pulled.

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Then you distort it in some
form and you release it,

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and you see what happens.

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Now let's measure x as
the distance from the left end

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and y as the height at
that time t.

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The wave equation tells you an
equation governing the behavior

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of y.

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It tells you how y
varies.

05:31.778 --> 05:35.088
Now if you want to do that,
you may think you have to

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invent new stuff,
but you don't.

05:37.250 --> 05:40.010
Everything comes from Newton's
laws.

05:40.009 --> 05:43.789
That's to me the amazing part,
that you have these equations,

05:43.786 --> 05:46.616
then of course the Lorentz
force equations.

05:46.620 --> 05:49.370
Then you've got Newton's laws
and that's all we have up to

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this point in the course.

05:50.670 --> 05:53.940
So there should be no need to
invoke anything beyond Newton's

05:53.942 --> 05:56.292
laws to find out the fate of
this string.

05:56.290 --> 05:59.000
And namely, we are going to
apply F = ma,

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but not to the whole string at
one time,

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because a is the
acceleration and whose

06:04.036 --> 06:06.046
acceleration are we talking
about?

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Piece of string here,
piece of string there?

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It can all be moving at
different rates.

06:10.490 --> 06:14.360
So you pick a tiny portion,
which I'm going to blow up for

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your benefit.

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That's a piece of string.

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Its interval,
it lies under region of length

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dx.

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In other words,
you take a dx here and

06:25.910 --> 06:27.630
look at that string.

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It is being pulled like that
with a tension T and it's

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being pulled like this with a
tension T here.

06:37.019 --> 06:40.859
That's the meaning of tension,
pulled at both ends.

06:40.860 --> 06:43.650
If this was a perfectly
straight segment,

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there'll be no net force on it,
because these forces will

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cancel.

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But if it's slightly curved,
then this line and that line

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are not quite in the same
direction and the forces don't

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have to cancel.

06:58.769 --> 07:03.029
So let's take that force and
write its vertical part.

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If this angle is
θ,

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T times
sinθ is the force

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here.

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-T times (because
T is pointing the

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opposite way)--
let me call this

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θ_1 and
θ_2 is

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the angle you have here.

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Do you understand?

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The vertical force is restoring
the strength to where it was is

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what I'm focused on.

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So you resolve that force into
a vertical part,

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up and this one has the
vertical part down.

07:43.199 --> 07:45.519
Now if θ_1 =
θ_2 they

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cancel and that's what I just
said.

07:46.980 --> 07:48.570
But θ_1
and θ_2

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are generally not equal,
because if I go to the top of

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the string,
for example,

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this end is pulling this way,
that end is pulling that way.

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There's a net force down.

07:58.420 --> 08:02.530
So you have a net force because
this angle or the tangent to the

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string is changing with
position.

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That's why they don't cancel.

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Now the approximation I'm going
to make is that theta is small,

08:10.910 --> 08:14.250
measured in radians,
and I remind you that

08:14.250 --> 08:19.060
sinθ is essentially
θ - very small

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corrections.

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And cosine θ is 1
− θ^(2)/2

08:24.975 --> 08:26.085
corrections.

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And tanθ,
which is the ratio of these

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guys, is θ other
corrections.

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For a very small
θ,

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you forget that,
you forget that.

08:37.730 --> 08:42.530
You keep things of order theta
in every equation.

08:42.529 --> 08:44.499
So for small angles,
cosine is just 1,

08:44.500 --> 08:46.740
and sinθ is
θ.

08:46.740 --> 08:48.970
If you do that,
and sinθ and

08:48.970 --> 08:52.590
tanθ being equal,
I can write it as

08:52.590 --> 08:59.190
tanθ_1 -
tanθ_2.

08:59.190 --> 09:01.660
But that's in the
approximation,

09:01.660 --> 09:05.090
where tan and sine are all
roughly equal.

09:05.090 --> 09:11.550
That is really T times
dy/dx at the point x

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dx −
dy/dx at the point

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x.

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Are you with me?

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dy/dx from calculus,
you remember,

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is the slope.

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It's a partial derivative.

09:23.590 --> 09:25.800
Why is it a partial derivative
now?

09:25.799 --> 09:28.949
Can you tell me?

09:28.950 --> 09:29.800
Yes?

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Student:  It's all
dependent on time.

09:30.519 --> 09:33.089
Prof: It can depend on
time and I'm doing F = ma

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at one instant in time.

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So I don't want to take this
guy's slope now and that guy's

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slope an hour later.

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I want them at the same time,
at a fixed time.

09:42.309 --> 09:46.489
Now what do I do with this?

09:46.490 --> 09:52.300
Have I taught you guys how to
handle a difference like this?

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There's one legacy I want to
leave behind is what to do when

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confronted with a difference
like this.

09:58.789 --> 10:03.489
You have any idea what to do?

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Pardon me?

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Student:  Multiply by
one.

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Prof: Multiply by one.

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Then what do you do with that?

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I'm asking you to evaluate the
difference between this guy at

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x dx - the same thing at
x.

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What is it equal to
approximately for small

10:25.695 --> 10:27.855
dx?

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Let me ask you another question.

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Suppose I have f of x
dx and I subtract from it

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f of x,
what is it equal to?

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Student:
>

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Prof: If I divide by
Δx and take the

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limit, it becomes a derivative.

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If I don't divide by
Δx,

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this is then df/dx
times Δx.

10:58.450 --> 11:02.490
There may be other corrections,
but when Δx goes

11:02.485 --> 11:05.085
to 0, you may make that
approximation.

11:05.090 --> 11:06.950
That is the meaning of the rate
of change.

11:06.950 --> 11:08.770
If I tell you,
here's a function and that's

11:08.773 --> 11:10.993
the same function somewhere to
the right, what's the

11:10.988 --> 11:11.638
difference?

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That's what the role of the
derivative is,

11:13.777 --> 11:15.547
to tell you how much it
changes.

11:15.548 --> 11:17.688
This slope changes from here to
there,

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therefore it's a derivative of
something which is already a

11:20.767 --> 11:24.207
derivative,
so it's d^(2)y/dx^(2)

11:24.214 --> 11:26.634
times Δx.

11:26.629 --> 11:29.459
In other words,
the f I'm using here is itself

11:29.455 --> 11:32.465
dy/dx and the
df/dx is then the

11:32.467 --> 11:33.847
second derivative.

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So that's the force.

11:36.649 --> 11:43.189
Now that force we must equate
to ma.

11:43.190 --> 11:44.240
That's the easy part.

11:44.240 --> 11:48.330
What is the mass of this
section, this portion of string?

11:48.330 --> 11:54.220
If the mass per unit length is
μ, that's the mass of that

11:54.221 --> 11:55.221
section.

11:55.220 --> 11:57.310
Then what's the acceleration.

11:57.308 --> 12:00.608
Acceleration is the rate at
which y is going up and down.

12:00.610 --> 12:05.520
That is d^(2)y/dt^(2).

12:05.519 --> 12:14.919
Canceling the Δx,
we get this wave equation which

12:14.916 --> 12:23.096
says d^(2)y/dx^(2) -
μ over T

12:23.095 --> 12:29.655
d^(2)y/dt^(2) is 0,
and that's the origin of the

12:29.662 --> 12:30.292
wave equation.

12:30.288 --> 12:34.528
Any time you get something like
this, you call it the wave

12:34.530 --> 12:35.350
equation.

12:35.350 --> 12:42.440
I'm going to write this as
d^(2)y/dx^(2) - 1 over

12:42.436 --> 12:47.586
v^(2) d^(2)y/dt^(2 )= 0
where v is

12:47.590 --> 12:51.070
√(T/μ).

12:51.070 --> 12:54.480
I use the symbol v,
because this will turn out to

12:54.482 --> 12:57.092
be the velocity of waves on that
string.

12:57.090 --> 13:00.160
Namely, if you pluck the string
and make a little bump here,

13:00.162 --> 13:03.232
let it go, the bump will move
and v is going to be the speed

13:03.234 --> 13:04.384
at which it moves.

13:04.379 --> 13:08.439
The speed of waves is
determined by the tension and

13:08.443 --> 13:10.643
the mass per unit length.

13:10.639 --> 13:14.499
Now I want to show you that
this really is the velocity of

13:14.495 --> 13:15.235
the wave.

13:15.240 --> 13:17.010
How do we know that's the
velocity?

13:17.009 --> 13:19.919
It's got dimensions of
velocity, but maybe it's not the

13:19.917 --> 13:21.207
velocity of this wave.

13:21.210 --> 13:23.080
Maybe it's something else.

13:23.080 --> 13:26.690
I want to show you that
v stands for the velocity

13:26.690 --> 13:28.660
of the waves in this medium.

13:28.658 --> 13:33.038
The way you do that is you ask
yourself, what is the nature of

13:33.039 --> 13:35.409
the solutions to this equation?

13:35.408 --> 13:41.198
If y(x,t) satisfied the
equation, what can you say about

13:41.196 --> 13:43.246
its functional form?

13:43.250 --> 13:46.180
What functions do you think
will play a role?

13:46.178 --> 13:53.378
Does anybody have any idea what
kind of functions may play a

13:53.384 --> 13:54.244
role?

13:54.240 --> 13:55.380
Student:  Sine function?

13:55.379 --> 13:57.029
Prof: Sines maybe.

13:57.029 --> 13:59.259
With single oscillators,
when you have something

13:59.255 --> 14:01.475
something is d/dt
squared, you had cosine

14:01.481 --> 14:02.431
ωt.

14:02.428 --> 14:04.958
Whatever you can do for t
you can do for x.

14:04.960 --> 14:06.560
You can have a cosine in
x.

14:06.558 --> 14:09.288
So you might think it is all
sines and cosines and

14:09.293 --> 14:10.133
exponentials.

14:10.129 --> 14:14.419
All that is true,
but it turns out the range of

14:14.424 --> 14:17.884
solutions is much bigger than
that.

14:17.879 --> 14:20.699
I'm going to write down for you
the most generation solution to

14:20.702 --> 14:22.932
the wave equation,
namely what's the constraint it

14:22.933 --> 14:24.213
imposes on the function?

14:24.210 --> 14:31.910
The answer is this - y
can be any function you want of

14:31.907 --> 14:33.957
x - vt.

14:33.960 --> 14:37.540
I don't care what function it
is.

14:37.538 --> 14:42.768
So if I call z as a
single variable x - vt,

14:42.774 --> 14:47.734
y can be a function of
this single combination,

14:47.726 --> 14:49.406
x - vt.

14:49.408 --> 14:51.218
In other words,
y can depend on x

14:51.221 --> 14:53.271
and y can depend on
t. If it depends on

14:53.269 --> 14:55.119
x and t in an
arbitrary way,

14:55.120 --> 14:57.320
of course it won't satisfy this
equation.

14:57.320 --> 15:00.950
But I'm telling you that if it
depends on x and t

15:00.952 --> 15:03.532
only through this combination,
x - vt,

15:03.532 --> 15:05.762
it will satisfy the wave
equation.

15:05.759 --> 15:07.379
In other words,
here's the function.

15:07.379 --> 15:11.299
You don't even have to
think, x - vt squared

15:11.299 --> 15:14.749
over some number
x_0^(2).

15:14.750 --> 15:17.650
That's a completely good
solution to the wave equation.

15:17.649 --> 15:21.959
I guarantee if you took this,
it would satisfy the wave

15:21.958 --> 15:26.028
equation, because it's a
function only of x - vt.

15:26.028 --> 15:30.188
A function that's not a good
function is y = e^(

15:30.186 --> 15:33.176
−x2) −
v^(2)t^(2).

15:33.178 --> 15:35.558
That's not a function of x -
vt.

15:35.558 --> 15:42.338
That won't satisfy the wave
equation.

15:42.340 --> 15:46.860
So why is that true is what I
want to show you.

15:46.860 --> 15:51.710
Let us take dy/dx.

15:51.710 --> 15:58.350
Remember, y depends only
on this combination z,

15:58.351 --> 16:04.331
therefore it equals df/dz
times dz/dx.

16:04.330 --> 16:09.320
But z is x - vt,
dz/dx is 1,

16:09.318 --> 16:12.228
so it's just df/dz.

16:12.230 --> 16:14.880
Now let me rush through this
and do it one more time.

16:14.879 --> 16:16.569
Everything will go the same way.

16:16.570 --> 16:21.250
You will get this.

16:21.250 --> 16:27.690
But how about dy/dt?

16:27.690 --> 16:33.110
If you want dy/dt,
again you take df/dz

16:33.107 --> 16:35.927
times dz/dt.

16:35.928 --> 16:40.648
But that gives you--this should
be written as a

16:40.647 --> 16:46.387
partial--dz/dt is
df/dz times -v.

16:46.389 --> 16:47.889
Can you see that?

16:47.889 --> 16:50.219
By the chain rule,
you take any function you have,

16:50.221 --> 16:52.461
first differentiate with
respect to z.

16:52.460 --> 16:54.910
Then differentiate z
with respect to x and

16:54.908 --> 16:56.108
with respect to t.

16:56.110 --> 16:57.630
If you do an x
derivative, you get 1.

16:57.629 --> 16:59.559
If you do a t
derivative,

16:59.562 --> 17:00.812
you get -v.

17:00.808 --> 17:03.728
Now if I do it twice--again,
I don't want to spend too much

17:03.732 --> 17:05.952
time--if I do it twice,
I'm going to skip the

17:05.951 --> 17:07.061
intermediate step.

17:07.058 --> 17:12.058
I'll get a -v and a
−v^(2).

17:12.058 --> 17:15.878
It follows then that if I took
1/v^(2) times that,

17:15.880 --> 17:17.450
can you see that now?

17:17.450 --> 17:27.460
1/v^(2) d^(2)y/dt^(2) is
the same as v^(2)y

17:27.458 --> 17:29.958
dx^(2).

17:29.960 --> 17:32.160
Therefore to satisfy the wave
equation,

17:32.160 --> 17:35.720
it can be any function at all,
as long as it depends on

17:35.724 --> 17:39.494
x and t in the
combination x - vt.

17:39.490 --> 17:40.830
That is just a mathematical
fact.

17:40.828 --> 17:43.508
You just have to do the chain
rule, you will find that it's

17:43.506 --> 17:43.826
true.

17:43.828 --> 17:46.608
But more importantly,
I want you to understand what

17:46.612 --> 17:48.952
it tells you about the wave
propagation.

17:48.950 --> 17:51.610
Let me pick x_0
= 1 for convenience.

17:51.608 --> 17:55.888
Let me plot this guy at time
t = 0.

17:55.890 --> 18:00.620
This solution looks like
e to the -x

18:00.619 --> 18:06.219
squared, which is a function
like that, at t = 0.

18:06.220 --> 18:10.160
What does it look like at t
= 1 second?

18:10.160 --> 18:11.520
Think about it.

18:11.519 --> 18:15.489
t = 1 second looks like
e to the -x - v

18:15.493 --> 18:16.943
whole squared.

18:16.940 --> 18:22.390
That means it's the same bump,
shifted by an amount v.

18:22.390 --> 18:25.800
See this function x - v
squared behaves with respect to

18:25.796 --> 18:28.696
x = v the same as the
original function around

18:28.700 --> 18:29.650
x = 0.

18:29.650 --> 18:32.760
Whatever happens to this guy
here happens to that guy at

18:32.761 --> 18:34.631
distance v to the right.

18:34.630 --> 18:37.490
You can see that if you wait
t seconds,

18:37.492 --> 18:40.232
it would have moved a distance
vt.

18:40.230 --> 18:43.730
That's why you understand that
vt,

18:43.730 --> 18:45.900
v is the velocity of
propagation,

18:45.900 --> 18:49.050
because what this is telling
you is that if you start the

18:49.053 --> 18:52.153
system out with some
configuration at time t,

18:52.150 --> 18:55.070
t = 0,
later on the function looks

18:55.070 --> 18:58.720
like the same function
translated to the right at a

18:58.722 --> 19:00.332
velocity v.

19:00.328 --> 19:04.118
That means its profile,
whatever profile you have,

19:04.124 --> 19:06.374
is just moved to the right.

19:06.368 --> 19:16.238
Now can you think of another
family of solutions besides this

19:16.241 --> 19:17.231
one?

19:17.230 --> 19:21.720
Why should the wave move to the
right?

19:21.720 --> 19:22.610
Yes?

19:22.608 --> 19:25.098
Student:  It's x
vt.

19:25.098 --> 19:27.348
Prof: Do you understand
that if it was x vt,

19:27.348 --> 19:30.248
you go through this derivation,
you'd get a v^(2)

19:30.252 --> 19:32.762
instead of a
−v^(2) and that

19:32.758 --> 19:33.838
doesn't matter.

19:33.838 --> 19:37.128
So the answer is,
the most general solution to

19:37.126 --> 19:40.846
the wave equation is any
function you like of x -

19:40.853 --> 19:44.363
vt any function you like of
x vt.

19:44.358 --> 19:46.218
This will describe waves going
to the right;

19:46.220 --> 19:48.710
that will describe waves going
to the left, and you can

19:48.713 --> 19:49.503
superpose them.

19:49.500 --> 19:53.510
Because it's a linear equation,
you can add solutions.

19:53.509 --> 19:55.849
This is what I want you to know
about the wave equation.

19:55.848 --> 20:01.108
This is one example of how the
wave equation comes and what the

20:01.108 --> 20:03.568
meaning of the symbol v is.

20:03.568 --> 20:08.478
Now let me go to the old
Maxwell equations.

20:08.480 --> 20:11.910
Now I want to do them and ask
myself.

20:11.910 --> 20:16.470
I'm going to write down a
solution and I'm going to see if

20:16.474 --> 20:19.684
it obeys these four Maxwell
equations.

20:19.680 --> 20:23.430
If I write down a solution for
the electric field and the

20:23.429 --> 20:26.239
magnetic field,
they have to obey all those

20:26.241 --> 20:28.721
conditions, those four
equations.

20:28.720 --> 20:32.360
That means, let's take first
the case of working in a vacuum,

20:32.356 --> 20:35.506
when there is no current,
when there is no charge.

20:35.509 --> 20:37.239
So you want a surface integral
of

20:37.240 --> 20:43.750
E on any surface to be 0
and the surface integral of

20:43.751 --> 20:47.571
B on any surface to be 0.

20:47.568 --> 20:50.818
Look at the surface integral
first.

20:50.819 --> 20:51.339
Do you understand?

20:51.338 --> 20:53.468
This was never going to be
non-zero.

20:53.470 --> 20:56.150
This will be non-zero near
matter but we are far from any

20:56.150 --> 20:58.830
charges, so there is nothing to
enclose in the volume.

20:58.829 --> 21:01.079
There are no charges there.

21:01.079 --> 21:02.169
Look at those equations.

21:02.170 --> 21:05.220
I give you a function and I
say, "Tell me if it

21:05.224 --> 21:07.324
satisfies Maxwell's
equations,"

21:07.321 --> 21:08.941
what do you have to do?

21:08.940 --> 21:14.620
Operationally,
what is it that you have to do?

21:14.618 --> 21:24.558
Any ideas on what you must do
at this point to test?

21:24.559 --> 21:26.079
Any ideas?

21:26.078 --> 21:33.268
What do you have to see of a
potential solution?

21:33.269 --> 21:34.509
Come on, this is not the New
York subway.

21:34.509 --> 21:36.629
You can make eye contact.

21:36.630 --> 21:38.250
What is this?

21:38.250 --> 21:41.270
You have no idea what you may
have to do?

21:41.269 --> 21:45.369
What if I give you a function
and say, "See if it

21:45.372 --> 21:47.932
satisfies the wave
equation,"

21:47.926 --> 21:49.936
what will you do then?

21:49.940 --> 21:51.330
Take the function and then what?

21:51.329 --> 21:52.169
Yes?

21:52.170 --> 21:53.500
Student:  Differentiate
it and see--

21:53.500 --> 21:54.140
Prof: Yes!

21:54.140 --> 21:56.170
Take x derivative,
y derivative and all

21:56.165 --> 21:57.175
that, second derivative.

21:57.180 --> 21:59.230
If it vanishes,
you'll say it satisfies it,

21:59.229 --> 21:59.619
right?

21:59.618 --> 22:01.428
That's what it means to satisfy
it.

22:01.430 --> 22:05.050
If I give you fields
E and B that

22:05.053 --> 22:10.013
depends on space and time and I
want you to verify if it

22:10.010 --> 22:13.440
satisfies this,
what do you have to do,

22:13.435 --> 22:15.595
is what I'm asking?

22:15.598 --> 22:17.698
Student:  Come up with a
situation?

22:17.700 --> 22:18.880
Prof: No,
it must be true in every

22:18.875 --> 22:19.225
situation.

22:19.230 --> 22:22.070
In other words,
what's the surface on which

22:22.074 --> 22:23.704
this integral is done?

22:23.700 --> 22:26.850
On what surface do we do the
integral?

22:26.849 --> 22:27.509
Pardon me?

22:27.509 --> 22:28.569
Student:  Closed.

22:28.569 --> 22:29.589
Prof: Closed.

22:29.588 --> 22:31.208
But beyond closed,
anything else?

22:31.210 --> 22:31.980
Where is it located?

22:31.980 --> 22:36.480
How big is it?

22:36.480 --> 22:37.080
Pardon me?

22:37.078 --> 22:41.018
Student:
>

22:41.019 --> 22:42.259
Prof: I didn't hear that.

22:42.259 --> 22:43.209
Student:  It's
everywhere.

22:43.210 --> 22:46.040
Prof: It could be
anywhere and it could have any

22:46.041 --> 22:46.411
size.

22:46.410 --> 22:47.920
It could have any shape.

22:47.920 --> 22:52.030
And it must be true for all
those surfaces.

22:52.029 --> 22:54.389
So if I just give you a field,
E (x,y,z and

22:54.387 --> 22:56.167
t), it's a lot of work,
right?

22:56.170 --> 22:58.150
Draw all possible surfaces.

22:58.150 --> 23:01.470
On them, you can take the
surface, divide it into patches,

23:01.471 --> 23:04.851
do the surface integral patch
by patch, and you better keep

23:04.853 --> 23:05.673
getting 0.

23:05.670 --> 23:08.980
When you're done with that,
then you take the magnetic

23:08.980 --> 23:09.480
field.

23:09.480 --> 23:12.710
Then the other two Maxwells
tell you, take a loop,

23:12.712 --> 23:16.872
any loop, go round that and do
the line integral of B.

23:16.868 --> 23:19.528
That should be the flux
crossing it, the electric flux

23:19.534 --> 23:20.494
or magnetic flux.

23:20.490 --> 23:23.660
And it should be done for every
loop and for every surface.

23:23.660 --> 23:26.350
You realize that looks
impossible.

23:26.348 --> 23:30.908
But there is a quicker way to
verify all of these equations,

23:30.910 --> 23:37.980
and the quick way is the
following - if these equations

23:37.976 --> 23:42.946
are true for a tiny loop,
in this case,

23:42.951 --> 23:48.601
true for a tiny volume or a
tiny surface enclosing a tiny

23:48.599 --> 23:54.299
volume,
then it's true for big ones.

23:54.298 --> 23:59.458
And likewise,
if true for a tiny loop,

23:59.459 --> 24:04.479
it is true for any arbitrary
loop.

24:04.480 --> 24:06.850
In other words,
if I can show that at any

24:06.846 --> 24:10.556
arbitrary point,
if I pick an infinitesimal loop

24:10.561 --> 24:14.241
or infinitesimal surface,
and these guys satisfy these

24:14.244 --> 24:16.184
equations on an infinitesimal
thing,

24:16.180 --> 24:19.520
it's going to satisfy it on a
macroscopic thing.

24:19.519 --> 24:21.529
That's what we want to
understand.

24:21.529 --> 24:24.199
How does that come about?

24:24.200 --> 24:29.170
It comes about due to a very
beautiful way in which we define

24:29.166 --> 24:31.316
the sum of two surfaces.

24:31.319 --> 24:35.769
So here is a surface.

24:35.769 --> 24:37.859
I'm going to take everything to
be a cube, but it's not

24:37.855 --> 24:38.315
important.

24:38.319 --> 24:40.079
Here's a surface.

24:40.078 --> 24:41.778
Let's call it
S_1.

24:41.779 --> 24:43.679
And it's got its outward normal.

24:43.680 --> 24:47.160
That's the definition of
surface integral is you draw the

24:47.157 --> 24:49.457
normals pointing outwards and
you take

24:49.457 --> 24:52.867
E⋅dA
on every face of the cube

24:52.873 --> 24:53.933
and add it.

24:53.930 --> 24:58.940
The area vector is defined to
be outward pointing normal to

24:58.936 --> 25:00.056
every face.

25:00.059 --> 25:01.889
So you take the little cube.

25:01.890 --> 25:05.400
E may be pointing this
way there or that way there,

25:05.396 --> 25:09.086
but you take the dot product of
the area and you add them.

25:09.089 --> 25:16.059
Then I take a second cube here.

25:16.059 --> 25:21.939
Let's see.

25:21.940 --> 25:24.260
It's got its own outward normal.

25:24.259 --> 25:27.919
This is surface 2.

25:27.920 --> 25:33.290
Now imagine gluing this guy to
this guy, so it looks like this.

25:33.289 --> 25:40.459
You glue this common face.

25:40.460 --> 25:46.330
Bring that surface over and
glue them with that face common

25:46.327 --> 25:47.337
to both.

25:47.338 --> 25:51.588
Then I claim that the surface
integral on this one the surface

25:51.586 --> 25:55.686
integral on that one is the
surface integral on the union of

25:55.693 --> 25:56.533
the two.

25:56.529 --> 26:02.359
In other words,
let's see, take one cube,

26:02.355 --> 26:09.345
take another cube,
and that surface integral will

26:09.345 --> 26:14.875
be the same as on a longer
object.

26:14.880 --> 26:18.980
I don't know what this is
called, something over that.

26:18.980 --> 26:24.440
The two cubes used to be
sharing this.

26:24.440 --> 26:25.580
Do you understand why?

26:25.578 --> 26:28.878
Let's understand why the
integral on this guy,

26:28.882 --> 26:32.402
which is, if you want,
two cubes joined to form a

26:32.404 --> 26:34.024
rectangular solid.

26:34.019 --> 26:38.249
That surface integral is the
sum of these two surface

26:38.252 --> 26:39.232
integrals.

26:39.230 --> 26:43.440
Think for a minute about why
that is true.

26:43.440 --> 26:46.260
If you compare them,
piece by piece,

26:46.256 --> 26:48.666
this face matches that face.

26:48.670 --> 26:50.380
This face matches that half.

26:50.380 --> 26:51.940
This one matches that half.

26:51.940 --> 26:54.670
But something is missing in
this that's present here,

26:54.671 --> 26:55.461
do you agree?

26:55.460 --> 26:59.230
What is it we don't have?

26:59.230 --> 27:00.070
What's missing in the equal
union?

27:00.069 --> 27:00.579
Yes?

27:00.578 --> 27:01.868
Student:  Two sides
equal _________.

27:01.868 --> 27:05.318
Prof: The two common
faces are missing in the big

27:05.317 --> 27:07.447
surface, but it does not matter.

27:07.450 --> 27:10.420
The two common faces do not
matter because when you do the

27:10.420 --> 27:13.490
surface on the first one,
the outward normal will point

27:13.490 --> 27:15.100
out of that,
whereas if you do it on the

27:15.099 --> 27:17.149
second one,
the outward normal will point

27:17.151 --> 27:18.291
into the other one.

27:18.288 --> 27:22.488
When you glue them together,
the common face is opposite

27:22.492 --> 27:23.642
orientations.

27:23.640 --> 27:27.600
Consequently,
the surface integral on the big

27:27.598 --> 27:33.268
solid is equal to the sum of the
surface in integrals in the two

27:33.269 --> 27:34.799
small solids.

27:34.799 --> 27:35.449
Do you follow that?

27:35.450 --> 27:36.390
That's the way it works.

27:36.390 --> 27:40.160
Then you can go on adding more
and more pieces to this,

27:40.161 --> 27:43.651
right, and you can build like
Legos and arbitrarily

27:43.653 --> 27:45.123
complicated blob.

27:45.118 --> 27:47.428
Remember, these are all very,
very tiny volumes,

27:47.432 --> 27:49.992
so you can build them up to
look like a big thing.

27:49.990 --> 27:50.930
It may look like a pyramid.

27:50.930 --> 27:53.910
If you look very closely,
you'll have the steps like in a

27:53.910 --> 27:56.200
real pyramid,
but the steps here can be made

27:56.199 --> 27:57.369
arbitrarily small.

27:57.368 --> 28:00.718
Therefore if the surface
integral was 0 on every little

28:00.721 --> 28:03.021
piece that made up the big
object,

28:03.019 --> 28:04.719
then it's going to be 0 on the
big object,

28:04.720 --> 28:08.590
because the integral on the big
object is the integral of the

28:08.589 --> 28:11.299
tiny pieces that make up the big
object.

28:11.298 --> 28:14.978
This is a very profound idea,
because the big object has

28:14.983 --> 28:17.533
fewer surfaces than the small
ones,

28:17.528 --> 28:21.558
because when you cut it,
you create two new surfaces,

28:21.558 --> 28:24.028
but they don't contribute
between the two of them,

28:24.029 --> 28:26.839
because they cancel.

28:26.838 --> 28:29.978
And in particular,
if the surface integral had not

28:29.979 --> 28:32.949
been 0,
but because of matter it was

28:32.953 --> 28:37.403
equal to charge inside volume 1,
and this right hand side of

28:37.402 --> 28:39.912
this one was the charge inside
volume 2,

28:39.910 --> 28:43.250
you can see that the surface
integral on the bigger surface

28:43.249 --> 28:45.379
will be the sum of the two
charges.

28:45.380 --> 28:47.890
So anything you're trying to
prove about surface integrals,

28:47.890 --> 28:50.070
even if the right hand side is
not 0,

28:50.068 --> 28:52.978
if it is true in a tiny cube,
true in a union of two cubes

28:52.984 --> 28:55.854
and then in a union of any
number of these tiny guys,

28:55.849 --> 28:59.079
therefore on any surface.

28:59.079 --> 29:00.459
That's the thing to remember.

29:00.460 --> 29:04.150
What we will do then is not to
take arbitrary surfaces,

29:04.150 --> 29:07.240
but infinitesimal surfaces and
we'll prove for them that the

29:07.244 --> 29:10.504
Maxwell equations are satisfied
for the solution that I come up

29:10.497 --> 29:10.967
with.

29:10.970 --> 29:14.930
Then you're guaranteed that it
will work for an arbitrary

29:14.932 --> 29:15.642
surface.

29:15.640 --> 29:18.860
When you do line
integrals--this is for surface

29:18.858 --> 29:21.728
integral--line integral is even
easier.

29:21.730 --> 29:26.670
Suppose I'm taking the line
integral of some function around

29:26.665 --> 29:27.665
this loop.

29:27.670 --> 29:29.150
This is loop
L_1.

29:29.150 --> 29:34.510
I take a line integral of some
field on L_1.

29:34.509 --> 29:38.569
Then somebody wants to do a
line integral on

29:38.565 --> 29:42.805
L_2 that looks
like this,

29:42.808 --> 29:46.028
E⋅dl
on line 2.

29:46.029 --> 29:51.279
You agree that this that really
is the same as the line integral

29:51.276 --> 29:55.936
on the union of the two loops
where you delete the common

29:55.940 --> 29:56.690
edge.

29:56.690 --> 29:58.360
You have to understand this.

29:58.358 --> 30:00.978
I will not leave any child
behind.

30:00.980 --> 30:03.110
You have to know why this is
true.

30:03.108 --> 30:06.228
I'm taking a lot of time so you
know where it's coming from.

30:06.230 --> 30:08.940
If you compare an integral of
any field on the big rectangle

30:08.936 --> 30:12.356
compared to the two squares,
the only difference is that the

30:12.363 --> 30:15.623
two squares had these two sides,
but they were doing them in

30:15.623 --> 30:19.473
opposite directions,
so it does not matter.

30:19.470 --> 30:21.590
So if the line integral of
this, for example,

30:21.588 --> 30:23.758
was some flux coming out of
this one,

30:23.759 --> 30:26.239
and the line integral of that
was the rate of change of flux

30:26.238 --> 30:28.068
coming out of that one,
if it was true,

30:28.065 --> 30:30.655
then it's guaranteed that the
line integral on the big

30:30.663 --> 30:33.803
rectangle will be the sum of the
fluxes or the rate of the change

30:33.801 --> 30:35.471
of fluxes coming out of both.

30:35.470 --> 30:38.970
So any Maxwell equation,
if it's true for a tiny square,

30:38.971 --> 30:42.221
infinitesimal square,
will be also true for anything

30:42.219 --> 30:42.919
bigger.

30:42.920 --> 30:46.200
There's only one subtlety when
you do loops and that's the

30:46.202 --> 30:48.682
following - if you're living in
a plane,

30:48.680 --> 30:51.910
you can prove the result for
the plane in the plane of the

30:51.906 --> 30:55.156
blackboard,
because there no loop I cannot

30:55.155 --> 30:57.445
form by joining these guys.

30:57.450 --> 31:02.210
But we live in three dimensions
so that we may have a loop like

31:02.209 --> 31:05.049
this, floating in three
dimensions.

31:05.048 --> 31:08.608
Then what you have to do is,
you've got to find any surface

31:08.607 --> 31:11.367
with a loop as a boundary,
maybe this dome.

31:11.368 --> 31:14.658
Then the integral on--let me
hide this for you.

31:14.660 --> 31:18.080
This is the part behind;
this is the part you can see.

31:18.078 --> 31:22.618
That's the same as tiling it
into little squares and doing an

31:22.624 --> 31:25.734
integral on each one of them
like this.

31:25.730 --> 31:28.750
You have a dome of some big
building, you take little tiles

31:28.750 --> 31:30.210
and you tile the building.

31:30.210 --> 31:35.290
Then all the interior edges
cancel, and all that remains is

31:35.291 --> 31:39.851
the edge that is the edge of the
original surface.

31:39.848 --> 31:42.848
So even in 3D it's going to
work, but now you should be

31:42.847 --> 31:46.177
ready for loops that are not
lying in the xy plane.

31:46.180 --> 31:49.570
So you will have to prove it
for the three independent loops.

31:49.568 --> 31:52.448
You will have to prove it for
an infinitesimal loop in that

31:52.454 --> 31:55.244
plane, infinitesimal in that
plane, infinitesimal in that

31:55.241 --> 31:55.741
plane.

31:55.740 --> 31:58.820
If it's true for three
independent directions,

31:58.816 --> 32:02.846
then by combining those little
pieces, you can make yourself

32:02.849 --> 32:04.559
any surface you want.

32:04.558 --> 32:06.768
In other words,
I'm saying, given a rim,

32:06.773 --> 32:10.013
you can build any surface with
the rim as the boundary.

32:10.009 --> 32:13.569
If you can take the little flat
pieces in any orientation,

32:13.573 --> 32:16.643
and it's enough to have them in
the xy, yz,

32:16.636 --> 32:18.196
and zx planes.

32:18.200 --> 32:20.210
So the strategy that I'm going
to follow,

32:20.210 --> 32:23.350
this is something one can skip,
but I wanted you to know the

32:23.345 --> 32:25.055
details,
if you really want to know

32:25.055 --> 32:26.305
where everything comes from.

32:26.308 --> 32:29.348
I'm going to write down or
search for a solution to Maxwell

32:29.349 --> 32:29.979
equations.

32:29.980 --> 32:32.500
I'm going to make it have a
certain form.

32:32.500 --> 32:34.820
You remember how we do this
with equations.

32:34.818 --> 32:37.578
We assume a certain form,
stick it into the equation,

32:37.579 --> 32:39.809
play with some parameters till
it works.

32:39.808 --> 32:42.638
I'm going to make it work on
infinitesimal loops and

32:42.638 --> 32:45.578
infinitesimal cubes and that's
going to be enough,

32:45.578 --> 32:47.248
because if it works on this
tiny loop,

32:47.250 --> 32:49.340
it works in a big loop,
works in a tiny cube,

32:49.338 --> 32:54.478
works on a big cube or
arbitrary surface.

32:54.480 --> 32:59.850
So let us write down the
functional form that I'm going

32:59.854 --> 33:00.754
to use.

33:00.750 --> 33:08.340
The functional form that I'm
going to use looks like this.

33:08.338 --> 33:12.918
I'm going to take an electric
field that is entirely in the

33:12.924 --> 33:17.274
z direction and the
z field will depend on

33:17.273 --> 33:19.333
y and t.

33:19.328 --> 33:22.368
And I'm going to take a
magnetic field which is going to

33:22.373 --> 33:25.803
be in the x direction and
the B field is going to

33:25.804 --> 33:29.184
depend on the y and
t. So let me tell you how my

33:29.180 --> 33:30.730
axes are defined here.

33:30.730 --> 33:36.190
This is x, this is
y, this is z.

33:36.190 --> 33:39.390
The electric and magnetic
fields, the electric field will

33:39.390 --> 33:41.620
always point along the z
axis.

33:41.618 --> 33:46.318
The magnetic field will always
point along the x axis.

33:46.318 --> 33:50.338
They will not vary,
as you vary x or

33:50.336 --> 33:51.386
z.

33:51.390 --> 33:56.750
They'll vary only if you vary
y by assumption.

33:56.750 --> 33:58.280
In general, it can vary with
everything,

33:58.279 --> 34:02.779
but I'm trying as a modest goal
to find a simple solution which

34:02.781 --> 34:07.431
has a dependence on only two of
the four possible coordinates.

34:07.430 --> 34:10.750
It depends on only y and
t, rather than x,

34:10.753 --> 34:12.133
y, z and t.

34:12.130 --> 34:13.880
Let's get any solution.

34:13.880 --> 34:15.220
We cannot get every possible
one;

34:15.219 --> 34:18.429
let's get something and
something, I assume,

34:18.425 --> 34:19.615
has this form.

34:19.619 --> 34:23.209
So this is called a plane wave,
because if you take the plane

34:23.210 --> 34:25.480
y = 0,
E and B are

34:25.483 --> 34:27.103
constant on that plane.

34:27.099 --> 34:29.229
Because when y is fixed
or some value,

34:29.226 --> 34:32.026
E and B are not
changing as you vary x

34:32.027 --> 34:32.847
and z.

34:32.849 --> 34:36.059
So you should think of it as
plane after plane and on each

34:36.059 --> 34:38.199
plane, the field is doing
something.

34:38.199 --> 34:39.099
It's doing the same thing.

34:39.099 --> 34:43.409
If I draw a plane here,
that field is a constant.

34:43.409 --> 34:45.989
E is a constant and that
plane and B is a constant

34:45.992 --> 34:46.652
on that plane.

34:46.650 --> 34:49.840
On another plane,
it could be a different

34:49.842 --> 34:54.952
constant, but within the plane,
it varies from plane to plane.

34:54.949 --> 34:57.859
So that is not an axiom,
that's not a law.

34:57.860 --> 35:02.550
That's an assumed simplicity in
the function I'm looking at.

35:02.550 --> 35:06.040
There's no theorem that says
that every solution to Maxwell

35:06.041 --> 35:08.271
equations much obey this
condition.

35:08.268 --> 35:10.878
In general, these functions
will be functions of x,

35:10.878 --> 35:12.058
y, z and t.

35:12.059 --> 35:14.419
The trick will be superposition.

35:14.420 --> 35:16.830
If I can get a solution that
depends only y and

35:16.833 --> 35:18.473
t,
and you can get a solution that

35:18.465 --> 35:19.685
depends on z and
t,

35:19.690 --> 35:21.830
I can add them up and they will
still be a solution,

35:21.829 --> 35:23.739
because the wave equation is
linear.

35:23.739 --> 35:24.899
You can add solutions.

35:24.900 --> 35:27.870
Then our sum of the two
solutions will be a function of

35:27.869 --> 35:29.189
y and z.

35:29.190 --> 35:30.820
You can bring in the x
and so on.

35:30.820 --> 35:34.530
So you do the simplest one,
then you can add them.

35:34.530 --> 35:39.070
So here is my function and I
have to know what I can say

35:39.068 --> 35:41.378
about these two functions.

35:41.380 --> 35:42.760
You understand?

35:42.760 --> 35:45.010
In general, the electric field
has three components,

35:45.012 --> 35:46.872
the magnetic field has three
components.

35:46.869 --> 35:49.029
Each of them depends on four
quantities, x,

35:49.027 --> 35:50.177
y, z and t.

35:50.179 --> 35:51.439
Big mess.

35:51.440 --> 35:54.840
In our simplified solution,
the only unknown component of

35:54.842 --> 35:56.972
E is
E_z.

35:56.969 --> 35:58.929
I'm assuming there is no
E_x and there

35:58.934 --> 35:59.994
is no E_y.

35:59.989 --> 36:02.619
And there is no
B_y and there

36:02.623 --> 36:04.323
is no B_z.

36:04.320 --> 36:06.330
Now you might say,
"Why don't you go a little

36:06.329 --> 36:06.739
further?

36:06.739 --> 36:09.109
Kill the B also."

36:09.110 --> 36:09.790
You can try that.

36:09.789 --> 36:12.809
If you kill the B,
you will find the only solution

36:12.806 --> 36:14.526
is to get everything equals 0.

36:14.530 --> 36:17.260
So by trial and error,
we know this is the first time

36:17.262 --> 36:19.472
you can get something
interesting going.

36:19.469 --> 36:21.869
If you make it simpler than
that, you get nothing.

36:21.869 --> 36:26.489
You can make it more
complicated, but not simpler.

36:26.489 --> 36:30.849
So now, will this satisfy the
surface integral condition?

36:30.849 --> 36:32.689
Let's check that?

36:32.690 --> 36:35.200
So what do I need to do for
that?

36:35.199 --> 36:39.019
I have to take a cube, right?

36:39.019 --> 36:40.999
Let me take the cube.

36:41.000 --> 36:43.330
It's infinitesimal,
but I'm just keeping it near

36:43.326 --> 36:43.966
the origin.

36:43.969 --> 36:47.259
It can be anywhere you want,
but I'm drawing it near the

36:47.260 --> 36:47.800
origin.

36:47.800 --> 36:51.670
It's a surface,
and it's got these outward

36:51.668 --> 36:53.178
going normals.

36:53.179 --> 37:00.919
And I must take E ⋅
surface area for every face.

37:00.920 --> 37:02.680
And I've got to get 0.

37:02.679 --> 37:09.569
That's the condition.

37:09.570 --> 37:11.110
So is that going to work or not?

37:11.110 --> 37:12.610
Let us see.

37:12.610 --> 37:16.860
There are six faces in this
cube, so I'm going to draw 1,2

37:16.858 --> 37:20.658
and 3 that I can see,
and -1, -2 and -3 refer to the

37:20.659 --> 37:22.969
faces on the opposite side.

37:22.969 --> 37:26.579
This is 3, that's -3.

37:26.579 --> 37:28.279
So I can only show you 1,2 and
3.

37:28.280 --> 37:33.220
Let's look at surface 1 and ask
what I get from the surface

37:33.219 --> 37:35.179
integral of E.

37:35.179 --> 37:38.699
Do you agree that E
points this way?

37:38.699 --> 37:41.029
So
E⋅dA

37:41.025 --> 37:44.075
is a non-zero contribution on
surface 1.

37:44.079 --> 37:48.849
But on surface -1,
E still points up,

37:48.849 --> 37:51.969
but dA points down,
because E doesn't vary

37:51.972 --> 37:54.302
from the upper face to the lower
face,

37:54.300 --> 37:56.060
because in going from upper to
lower,

37:56.059 --> 37:57.879
I'm varying the coordinate
z,

37:57.880 --> 38:00.910
but nothing depends on z.

38:00.909 --> 38:03.929
The same electric field is
sitting on the upper plane of

38:03.925 --> 38:06.445
this cube as on the lower plane
of the cube.

38:06.449 --> 38:10.759
Therefore the surface integrals
will cancel and give you 0,

38:10.764 --> 38:13.744
because the area vectors are
opposite.

38:13.739 --> 38:16.189
That is a simple statement,
that if you've got a constant

38:16.186 --> 38:17.886
electric field going through a
cube,

38:17.889 --> 38:21.469
the net flux will be 0 because
what's coming in on one side

38:21.469 --> 38:23.259
goes out of the other side.

38:23.260 --> 38:28.170
So that's the cancelation of 1
and -1 giving me 0.

38:28.170 --> 38:33.280
But there are other faces,
like 3 and -3.

38:33.280 --> 38:35.990
What surface integral will I
get from 3?

38:35.989 --> 38:39.779
The area vector looks like
this, the electric field looks

38:39.779 --> 38:42.079
like that, the dot product is 0.

38:42.079 --> 38:44.199
In other words,
the field lines are parallel to

38:44.195 --> 38:46.535
this face, so they are not going
to penetrate it.

38:46.539 --> 38:47.969
You're not going to get any
flux.

38:47.969 --> 38:50.689
Or the area vector is
perpendicular to the field

38:50.688 --> 38:51.208
vector.

38:51.210 --> 38:53.330
So on this face,
E is 0,

38:53.333 --> 38:55.673
on the opposite face is also 0.

38:55.670 --> 39:00.350
The same thing goes for 2 and
-2.

39:00.349 --> 39:03.609
If you go to face number 2
here, the electric field is

39:03.606 --> 39:06.616
pointing like that,
but there is no flux and there

39:06.617 --> 39:08.827
is no flux on the opposite face.

39:08.829 --> 39:13.049
So if you get 0,
either because the field is

39:13.045 --> 39:16.625
parallel to the face,
or if it's perpendicular,

39:16.626 --> 39:18.686
it has the same value on
opposite faces,

39:18.690 --> 39:22.000
with opposite pointing normals,
or opposite pointing area

39:22.001 --> 39:23.421
vectors and you get 0.

39:23.420 --> 39:27.490
That's how you get the surface
integral of E to be 0 on

39:27.494 --> 39:28.634
this tiny cube.

39:28.630 --> 39:31.400
But if it's 0 on a tiny cube,
it's 0 on anything you can

39:31.400 --> 39:32.710
build out of tiny cubes.

39:32.710 --> 39:35.880
That means 0 on any surface.

39:35.880 --> 39:38.650
If you repeat the calculation
for B,

39:38.652 --> 39:42.742
you'll get pretty much the same
logic, except that B now

39:42.744 --> 39:44.134
points like this.

39:44.130 --> 39:46.430
So on the top face,
B will have no flux

39:46.427 --> 39:48.367
because it's running along the
face.

39:48.369 --> 39:49.979
There is no flow through that.

39:49.980 --> 39:51.790
Top and bottom are 0 and 0.

39:51.789 --> 39:55.839
The only faces that matter are
2 and -2, because 2 is coming

39:55.840 --> 39:59.890
out of the board and B is
coming out of the board.

39:59.889 --> 40:02.329
But on the other face,
which you cannot see,

40:02.333 --> 40:05.403
the -2, B is still going
this way, but E,

40:05.403 --> 40:08.023
the area vector is going the
opposite way.

40:08.018 --> 40:10.578
The key to this is that
B does not vary.

40:10.579 --> 40:14.669
You see, if the flux was not
constant,

40:14.670 --> 40:17.970
if the field was not constant,
the fact that you've got two

40:17.972 --> 40:21.222
faces with opposite pointing
area vectors doesn't mean the

40:21.217 --> 40:22.127
answer is 0.

40:22.130 --> 40:23.440
Even though the area vectors
are opposite,

40:23.440 --> 40:25.620
the field strength would be
bigger on one face,

40:25.619 --> 40:28.279
smaller on the opposite face,
in which case they won't

40:28.277 --> 40:28.727
cancel.

40:28.730 --> 40:32.240
But the field is not varying in
the coordinate in which I've

40:32.240 --> 40:33.610
displaced the planes.

40:33.610 --> 40:39.570
That's the reason you get 0 for
both of those.

40:39.570 --> 40:41.430
All right.

40:41.429 --> 40:46.619
So far what I've shown you is
that the solution I have

40:46.619 --> 40:51.809
automatically satisfies 0
surface integral without any

40:51.809 --> 40:54.159
further assumptions.

40:54.159 --> 40:57.299
Of course, it's very important
that E did not vary with

40:57.295 --> 40:58.525
x and z.

40:58.530 --> 41:00.700
But with the assumed form,
I don't have to worry.

41:00.699 --> 41:03.819
I still have to only worry
about the other two Maxwell

41:03.815 --> 41:05.985
equations involving line
integrals.

41:05.989 --> 41:09.499
I've got to make sure that
works out.

41:09.500 --> 41:14.670
So let's see.

41:14.670 --> 41:18.270
So here I have to take loops
and I told you,

41:18.268 --> 41:21.108
when you take loops now,
you've got to take loops in

41:21.110 --> 41:24.000
that plane,
that plane, and that plane,

41:23.996 --> 41:28.826
because it takes three kinds of
little flat Lego pieces to form

41:28.831 --> 41:30.861
a curved surface in 3D.

41:30.860 --> 41:33.160
So let me take this loop first.

41:33.159 --> 41:41.409
I want to take an infinitesimal
loop that looks like this.

41:41.409 --> 41:44.599
I choose the orientation of
this so that if I do the right

41:44.603 --> 41:47.683
hand rule, the area vector is
pointing in the positive y

41:47.684 --> 41:48.474
direction.

41:48.469 --> 41:52.759
In other words,
this is a tiny loop of size

41:52.764 --> 41:55.734
Δy--let's see.

41:55.730 --> 42:01.540
Δx this way and
Δz that way.

42:01.539 --> 42:06.349
The area vector is coming out
like that.

42:06.349 --> 42:10.739
So I have to now look at the
line integral of

42:10.737 --> 42:15.317
E⋅dl
and equate it to

42:15.324 --> 42:21.014
-dΦ
_B /dt.

42:21.010 --> 42:25.610
I have to see,
is that true or false?

42:25.610 --> 42:30.690
Well, take this loop and look
at the line integral of E

42:30.686 --> 42:32.596
and see what happens.

42:32.599 --> 42:36.389
There is an E going up
this face.

42:36.389 --> 42:38.429
E is perpendicular to
that face , and E is

42:38.425 --> 42:39.495
perpendicular to that edge.

42:39.500 --> 42:41.500
E is anti-parallel to
this edge.

42:41.500 --> 42:42.630
You see that?

42:42.630 --> 42:46.830
If E is pointing up,
it cancels between these two.

42:46.829 --> 42:48.719
These two have no contribution,
because E is

42:48.722 --> 42:49.822
perpendicular to dl.

42:49.820 --> 42:52.000
But these two cancel.

42:52.000 --> 42:54.740
They cancel again because the
E that you have here

42:54.739 --> 42:57.579
going up is the same E
that you have here also going

42:57.577 --> 42:59.827
up,
but the dl is going in

42:59.831 --> 43:01.181
opposite directions.

43:01.179 --> 43:04.649
Therefore
E⋅dl

43:04.648 --> 43:08.028
around this tiny loop is
actually 0.

43:08.030 --> 43:09.720
You understand why it is 0?

43:09.719 --> 43:13.389
Cancelation between opposite
edges, and two edges that don't

43:13.393 --> 43:14.643
give you anything.

43:14.639 --> 43:18.079
So what we hope for is that on
the right hand side,

43:18.079 --> 43:21.919
there better not be any
magnetic flux coming out of this

43:21.920 --> 43:24.490
thing,
because otherwise right hand

43:24.494 --> 43:27.554
side will give you a non-zero
contribution.

43:27.550 --> 43:29.560
But luckily,
that is the case,

43:29.559 --> 43:32.679
because the magnetic field
looks like this.

43:32.679 --> 43:34.819
It's in the plane of this loop.

43:34.820 --> 43:39.830
The dot product of the area
vector, which is perpendicular

43:39.827 --> 43:45.097
to the loop and the B
field is 0, so that's also 0.

43:45.099 --> 43:50.269
So this is identically
satisfied on that loop.

43:50.268 --> 43:52.508
Now if you take line integral
of

43:52.505 --> 43:56.545
B⋅dl,
and that's supposed to be

43:56.545 --> 43:59.085
μ_0
ε_0

43:59.085 --> 44:02.445
dΦ/dt of
the electric flux,

44:02.449 --> 44:05.179
again, you should try to do the
exercise with me.

44:05.179 --> 44:07.499
Magnetic field,
I said, is going in the

44:07.498 --> 44:09.978
x direction,
so it has nothing to do with

44:09.978 --> 44:12.128
those two edges,
because they are perpendicular

44:12.130 --> 44:14.900
to it,
but it cancels between these

44:14.902 --> 44:15.332
two.

44:15.329 --> 44:17.539
It cancels because the edges
are going in the opposite

44:17.541 --> 44:19.961
direction, but B doesn't
vary from this edge to that

44:19.963 --> 44:20.343
edge.

44:20.340 --> 44:24.360
It varies only with y,
so again you get 0.

44:24.360 --> 44:26.850
And there is no electric flux
coming out of this surface,

44:26.851 --> 44:29.391
because electric field lines
also lie in the plane of that

44:29.389 --> 44:29.789
loop.

44:29.789 --> 44:31.949
They don't cross it.

44:31.949 --> 44:38.079
So those equations are also
satisfied.

44:38.079 --> 44:40.629
But I'm not done,
because I still have to

44:40.630 --> 44:44.140
consider loops in this plane and
loops in that plane.

44:44.139 --> 44:46.729
So far I've gotten no
conditions at all.

44:46.730 --> 44:49.730
What this means so far is that
there are no further

44:49.733 --> 44:53.583
restrictions on E_z
and B_x.

44:53.579 --> 44:54.969
They can be anything you like.

44:54.969 --> 44:59.279
But I'm going to get
restrictions by finally

44:59.277 --> 45:05.087
considering loops in this plane
and loops in that plane.

45:05.090 --> 45:11.940
So let's see how you get
conditions on one of them.

45:11.940 --> 45:15.190
This is x, y, z.

45:15.190 --> 45:22.780
So let's take a loop that looks
like this.

45:22.780 --> 45:25.920
I've chosen it so that with the
right hand rule,

45:25.916 --> 45:29.256
the area vector is
perpendicular and coming out the

45:29.255 --> 45:29.985
x axis.

45:29.989 --> 45:35.209
I remind you once again,
E looks like that,

45:35.208 --> 45:38.508
and B looks like this.

45:38.510 --> 45:44.140
So let me give the edges a
name, 1,2, 3,4.

45:44.139 --> 45:50.239
And let's take the condition
E⋅dl =

45:50.244 --> 45:54.014
- d/dt of the magnetic
flux.

45:54.010 --> 45:57.200
Now this loop has dimension
dy in this direction,

45:57.199 --> 45:58.939
dz in that direction.

45:58.940 --> 46:00.800
This is an infinitesimal loop.

46:00.800 --> 46:04.590
I've drawn it so you can see
it, but it's infinitesimal.

46:04.590 --> 46:09.860
Okay, so what do I get on the
right hand side?

46:09.860 --> 46:13.990
Right hand side says
−d/dt of the

46:13.994 --> 46:17.674
magnetic flux coming out of the
board.

46:17.670 --> 46:23.470
The magnetic flux is coming out
like this, right?

46:23.469 --> 46:25.719
It's coming out of the
blackboard, so there really is a

46:25.722 --> 46:26.392
magnetic flux.

46:26.389 --> 46:30.469
That magnetic flux is the
number I wrote down,

46:30.472 --> 46:36.372
B_x times the
loop area which is dy/dz.

46:36.369 --> 46:41.139
That gives me -
ΔyΔz

46:41.141 --> 46:44.371
B_x dt.

46:44.369 --> 46:49.319
That's the right hand side.

46:49.320 --> 46:53.120
This loop is so tiny,
I am approximating B by

46:53.119 --> 46:55.949
the value at the center if you
like.

46:55.949 --> 46:57.339
You might say,
"Look, I don't think

46:57.344 --> 46:58.674
B is a constant on the
loop.

46:58.670 --> 47:00.090
B is varying.

47:00.090 --> 47:01.820
Why do you take the value of
the center?"

47:01.820 --> 47:04.970
Well, if it varies,
the variation is proportional

47:04.967 --> 47:06.407
to Δx .

47:06.409 --> 47:08.379
I'm to Δy or
Δz,

47:08.380 --> 47:10.840
therefore that will be a term
proportional to Δy

47:10.835 --> 47:12.345
squared or Δz
squared.

47:12.349 --> 47:16.179
But we are going to keep things
to first order in Δy

47:16.182 --> 47:19.252
and Δz,
so it doesn't matter.

47:19.250 --> 47:21.550
Now how about the left hand
side?

47:21.550 --> 47:25.210
If you look at the left hand
side, I hope you'll try to do

47:25.213 --> 47:25.923
this one.

47:25.920 --> 47:30.170
If you go like that on edge 2,
you will get

47:30.170 --> 47:35.940
E_z times
Δz --I'm sorry.

47:35.940 --> 47:39.730
Let me write it as
Δz times

47:39.726 --> 47:45.646
E_z at the
point y Δy.

47:45.650 --> 47:50.160
And on this edge,
I will get - Δz

47:50.155 --> 47:56.195
times E_z
at the point y.

47:56.199 --> 47:58.399
In other words,
the electric field is

47:58.396 --> 48:01.016
everywhere, going up,
so it goes up here.

48:01.018 --> 48:04.798
E is parallel to it,
so that should be that times

48:04.802 --> 48:06.042
Δz.

48:06.039 --> 48:08.159
These two don't contribute,
because E and

48:08.157 --> 48:09.637
ΔL are
perpendicular,

48:09.643 --> 48:10.503
so forget that.

48:10.500 --> 48:12.280
But this one,
it goes in the opposite

48:12.284 --> 48:14.164
direction,
so I subtract it,

48:14.159 --> 48:17.919
but I bear in mind that this is
y Δy,

48:17.920 --> 48:19.890
but that's only y.

48:19.889 --> 48:22.529
So here I hope you guys will
know what to do.

48:22.530 --> 48:27.170
You will say that is then
roughly equal to

48:27.172 --> 48:32.722
dE_z/dy times
Δy.

48:32.719 --> 48:36.419
Therefore the line integral of
the electric field is

48:36.422 --> 48:40.632
proportional to the derivative
of dz with respect to

48:40.632 --> 48:43.612
y, times the area of the
loop.

48:43.610 --> 48:49.370
And the surface integral of the
flux change is also proportional

48:49.373 --> 48:52.583
to the area of the loop,
and you get

48:52.576 --> 48:55.226
dB_x/dt.

48:55.230 --> 49:02.380
So this is the final equation
you get by considering that

49:02.376 --> 49:03.266
loop.

49:03.268 --> 49:07.418
So what is remarkable is the
line integral and the surface

49:07.416 --> 49:10.686
integral are both proportional
to the area.

49:10.690 --> 49:13.480
The surface integral being
proportional to the area of the

49:13.483 --> 49:15.443
loop is obvious,
because it's the surface

49:15.443 --> 49:16.083
integral.

49:16.079 --> 49:18.479
Why is the line integral
proportional to the area?

49:18.480 --> 49:22.090
Because one part of the line
integral is the width of the

49:22.092 --> 49:23.502
loop,
Δz,

49:23.498 --> 49:26.608
other one comes in because the
extent to which they don't

49:26.612 --> 49:30.012
cancel is due to the derivative
of the field in the transverse

49:30.005 --> 49:30.835
direction.

49:30.840 --> 49:33.080
That brings you a
Δy.

49:33.079 --> 49:35.859
So if you cancel all of this,
you get the equation that I'm

49:35.864 --> 49:43.944
interested in,
which is very important -

49:43.936 --> 49:59.606
dE_z/dy =
-dB_x/dt.

49:59.610 --> 50:03.490
This came from looking at this
equation.

50:03.489 --> 50:06.079
The last one,
another loop equation,

50:06.083 --> 50:09.053
I'm going to go through
somewhat quickly,

50:09.045 --> 50:11.635
because it's going to be 0 = 0.

50:11.639 --> 50:15.969
Suppose I take the integral of
the magnetic field around this

50:15.965 --> 50:16.465
loop.

50:16.469 --> 50:18.509
Do you understand why it is 0?

50:18.510 --> 50:20.920
The magnetic field is coming
out of the blackboard.

50:20.920 --> 50:26.480
dl's are all lying in
the plane, so line integral of

50:26.476 --> 50:28.486
B is then 0.

50:28.489 --> 50:30.989
And that better be equal to
-μ_0

50:30.987 --> 50:33.007
ε_0
dΦ

50:33.007 --> 50:35.237
_electric
/dt.

50:35.239 --> 50:37.379
That is the case,
because there is no electric

50:37.382 --> 50:38.862
flux coming out of this face.

50:38.860 --> 50:40.740
The electric lines are in the
plane of the loop.

50:40.739 --> 50:43.659
There is no electric flux,
there is no rate of change,

50:43.657 --> 50:44.537
so it's 0 = 0.

50:44.539 --> 50:49.259
So from this loop,
I've managed to get one

50:49.255 --> 50:50.515
equation.

50:50.518 --> 50:53.428
So my plan now is,
I don't want to do yet another

50:53.425 --> 50:56.995
loop, because you've got the
idea, or it's not going to help

50:56.996 --> 50:58.566
to draw one more loop.

50:58.570 --> 51:03.290
But I would just say that if
you draw the last loop,

51:03.289 --> 51:05.789
which lies in what plane?

51:05.789 --> 51:10.439
Which lies in this plane.

51:10.440 --> 51:11.410
You will get one more equation.

51:11.409 --> 51:14.739
I'm just going to tell you what
it is.

51:14.739 --> 51:33.579
I got dE_z/dy
is -dB_x/dt.

51:33.579 --> 51:38.829
You'll get another equation,
dB_x/dy = -

51:38.826 --> 51:42.756
μ_0ε
_0dE

51:42.760 --> 51:45.010
_z/dt.

51:45.010 --> 51:47.490
So I confess,
that I've not derived this one,

51:47.485 --> 51:50.915
but it's going to involve just
drawing one more loop and doing

51:50.916 --> 51:51.306
it.

51:51.309 --> 51:54.279
Now it's really up to you guys
how much you want to do this,

51:54.275 --> 51:56.935
but you should at least have
some idea what we did.

51:56.940 --> 52:02.850
It turns out these are the only
restrictions I have to satisfy.

52:02.849 --> 52:07.339
If this is true of my field,
I'm done, because I've

52:07.338 --> 52:10.478
satisfied every Maxwell
equation.

52:10.480 --> 52:13.830
Every surface integral was 0 on
every tiny cube and therefore 0

52:13.831 --> 52:15.941
everywhere for E and
B.

52:15.940 --> 52:18.460
The line integrals,
some were identically

52:18.460 --> 52:20.290
satisfied, some were 0 = 0.

52:20.289 --> 52:24.849
There were some that led to non
trivial conditions and it's

52:24.847 --> 52:25.787
these two.

52:25.789 --> 52:28.419
Therefore I'm told that if the
fields that I pick,

52:28.420 --> 52:31.870
that depend on y and t,
have the property that the y

52:31.867 --> 52:35.787
derivative of this guy is the t
derivative of that guy,

52:35.789 --> 52:37.699
and the y derivative of
B_x is up to

52:37.695 --> 52:39.825
some constant,
the -t derivative of the other

52:39.827 --> 52:41.337
guy,
you're done.

52:41.340 --> 52:45.010
That's all you require of that
function.

52:45.010 --> 52:48.930
So let's do the following -
take this equation,

52:48.925 --> 52:52.155
take its y derivative on both
sides.

52:52.159 --> 52:54.739
IN other words,
I want to take y derivative of

52:54.737 --> 52:57.427
this and I want to take y
derivative of that.

52:57.429 --> 53:05.829
Then here you will get
d^(2)E _z/dy

53:05.826 --> 53:13.326
^(2) = -dB_x/dy
dt.

53:13.329 --> 53:15.519
Take the d/dz of this.

53:15.518 --> 53:19.088
Now you know partial
derivatives, you can take the

53:19.094 --> 53:23.034
derivatives in any order you
like, so let's write it as

53:23.034 --> 53:25.154
d/dt, d/dy.

53:25.150 --> 53:28.510
But dB_x/dy is
μ_0dE

53:28.507 --> 53:29.927
_z/dt.

53:29.929 --> 53:37.439
Another dt makes it
μ_0ε

53:37.440 --> 53:43.700
_0d^(2)E
_z/dt^(2).

53:43.699 --> 53:47.069
Take first the y derivative,
dB_x/dy,

53:47.074 --> 53:49.434
is that one single time
derivative.

53:49.429 --> 53:52.289
Take one more time derivative,
you get this.

53:52.289 --> 53:55.709
Now we get this wonderful--we
really have the wave equation

53:55.706 --> 53:58.836
now,
because I get d^(2)E

53:58.844 --> 54:05.244
_z/dy ^(2) -
μ_0ε

54:05.239 --> 54:11.279
_0d^(2)E
_z/dt^(2) = 0,

54:11.280 --> 54:17.500
which you recognize to be the
wave equation.

54:17.500 --> 54:22.820
That's what the wave equation
looks like in the top of the

54:22.817 --> 54:24.027
blackboard.

54:24.030 --> 54:27.340
But the things that are
oscillating now are not some

54:27.335 --> 54:27.915
string.

54:27.920 --> 54:31.910
It is really the electric field
oscillating.

54:31.909 --> 54:33.829
It's not a medium that's
oscillating.

54:33.829 --> 54:34.679
There's nothing there.

54:34.679 --> 54:37.109
This is all in vacuum.

54:37.110 --> 54:39.820
So let's find the velocity.

54:39.820 --> 54:47.280
v^(2) will be
1/μ_0ε

54:47.280 --> 54:51.160
_0,
right?

54:51.159 --> 54:57.169
Because this thing that comes
with our equation is the

54:57.166 --> 54:59.656
1/v^(2) term.

54:59.659 --> 55:01.809
Then you go back to your
electrostatics and

55:01.813 --> 55:04.533
magnetostatics and find out what
these numbers are.

55:04.530 --> 55:09.430
And I remind you that
¼Πε

55:09.431 --> 55:13.031
_0 is 9�10^(9)
and

55:13.027 --> 55:17.927
μ_0/4Π is
10^(-7).

55:17.929 --> 55:21.739
So let's write this as
4Π/μ_0

55:21.735 --> 55:25.535
times ¼Πε
_0,

55:25.539 --> 55:27.969
because we know what those guys
are.

55:27.969 --> 55:32.739
¼Πε
_0 is 9�10^(9).

55:32.739 --> 55:37.079
And 4Π/μ
_0_ is

55:37.081 --> 55:40.131
10^(7), so you get 9�10^(16).

55:40.130 --> 55:49.240
Or the velocity is 3�10^(8)
meters per second.

55:49.239 --> 55:53.959
So this was the big moment in
physics, when you suddenly

55:53.963 --> 55:59.033
realize that these things are
propagating at the velocity of

55:59.030 --> 55:59.890
light.

55:59.889 --> 56:02.549
So people knew the velocity of
light from other measurements.

56:02.550 --> 56:04.480
They had a fairly good idea
what it was.

56:04.480 --> 56:07.510
They know μ_0 by
doing experiments with currents.

56:07.510 --> 56:10.110
They knew ε_0
from Coulomb's law.

56:10.110 --> 56:13.120
It's one of the greatest
syntheses that you put them

56:13.119 --> 56:15.479
together and out comes an
explanation.

56:15.480 --> 56:19.160
Now this doesn't mean that
electromagnetic waves are the

56:19.159 --> 56:20.229
same as light.

56:20.230 --> 56:23.620
You agree that if you run next
to a buffalo at the same speed,

56:23.617 --> 56:24.947
you are not a buffalo.

56:24.949 --> 56:27.409
You just happen to have the
same speed.

56:27.409 --> 56:30.859
So that was a bit of a leap to
say it is really light.

56:30.860 --> 56:33.690
But it was also known that
whenever you have sparks in

56:33.690 --> 56:36.200
electrical circuits and so on,
you see light.

56:36.199 --> 56:38.779
So it took a little more than
that, but it's quite amazing,

56:38.784 --> 56:41.374
because gravity waves also
travel at the speed of light.

56:41.369 --> 56:44.179
You cannot assume that the
speed means the same phenomenon.

56:44.179 --> 56:45.669
But they were really right this
time.

56:45.670 --> 56:48.760
It really was the speed of
light.

56:48.760 --> 56:52.150
So we now have a new
understanding of what light it.

56:52.150 --> 56:57.120
Light is simply electromagnetic
waves traveling at the speed.

56:57.119 --> 57:01.559
It consists of electric and
magnetic fields.

57:01.559 --> 57:06.059
What we have is an example of a
simple wave,

57:06.059 --> 57:08.979
but one can show in general
that if you took the most

57:08.983 --> 57:11.573
general E and B
you can have,

57:11.570 --> 57:14.010
you will get similar wave
equations.

57:14.010 --> 57:17.090
You will get it for every
component of E and every

57:17.090 --> 57:18.410
component of B.

57:18.409 --> 57:21.879
So waves in three dimensions
will satisfy the general wave

57:21.880 --> 57:22.550
equation.

57:22.550 --> 57:25.530
But I'm not interested in the
most general case,

57:25.532 --> 57:29.662
because this is enough to show
you where everything comes from.

57:29.659 --> 57:31.839
So you have to think about how
wonderful this is,

57:31.840 --> 57:34.400
because you do experiments with
charges,

57:34.400 --> 57:36.690
with currents,
and you describe the

57:36.690 --> 57:39.050
phenomenology as best as you
can.

57:39.050 --> 57:45.440
Then Maxwell added that extra
term from logical consistency,

57:45.440 --> 57:47.890
by taking this capacitor and
drawing different surfaces,

57:47.889 --> 57:51.309
and realizing that unless you
added the second term,

57:51.309 --> 57:52.989
dΦ
_E /dt

57:52.989 --> 57:53.559
it didn't work.

57:53.559 --> 57:56.779
And without the second term,
you don't get the wave.

57:56.780 --> 58:00.770
It's only by adding the second
term and then fiddling with

58:00.773 --> 58:04.983
equations, he was able to
determine that waves can travel.

58:04.980 --> 58:08.010
So the reason that
electromagnetic waves travel in

58:08.010 --> 58:10.980
space without any charges is,
once you've got an E

58:10.983 --> 58:12.313
field or B field
somewhere,

58:12.309 --> 58:15.049
it cannot just disappear.

58:15.050 --> 58:16.610
It's like the LC circuit.

58:16.610 --> 58:20.220
If your capacitor is charged,
by the time it discharges,

58:20.215 --> 58:22.965
it has driven a current in the
inductor.

58:22.969 --> 58:25.479
Inductor is like a mass and
it's moving with velocity.

58:25.480 --> 58:28.100
Current is like velocity,
so it won't stop.

58:28.099 --> 58:30.869
So the current keeps going till
it charges the capacitor the

58:30.865 --> 58:31.565
opposite way.

58:31.570 --> 58:33.850
Then they go back and forth.

58:33.849 --> 58:36.489
They are the only 1 degree of
freedom, which is the charge in

58:36.485 --> 58:39.335
the capacitor or the current in
the circuit, which are related.

58:39.340 --> 58:42.580
Here E and B are
variables defined everywhere,

58:42.579 --> 58:45.709
but you cannot kill E
because the minute you try to

58:45.710 --> 58:49.070
destroy E,
you will produce a B.

58:49.070 --> 58:53.130
The minute you try to destroy
B, you'll produce an

58:53.132 --> 58:57.122
E, so they keep on
swapping energy and going back

58:57.123 --> 58:58.143
and forth.

58:58.139 --> 59:02.339
So it is self sustaining and
it's an oscillation if you like,

59:02.335 --> 59:05.195
but it's an oscillation over
all of space,

59:05.202 --> 59:08.212
and not over 1 or 2 degrees of
freedom.

59:08.210 --> 59:13.750
So I'm going to now write down
a specific form.

59:13.750 --> 59:20.450
The specific form I'm going to
write down looks like E = k

59:20.452 --> 59:23.042
times E_0.

59:23.039 --> 59:28.209
Now I'm going to pick a
particular function,

59:28.213 --> 59:33.633
ky - ωt,
and I'm going to take B to be I

59:33.626 --> 59:38.796
times B_0 sine ky -
ωt.

59:38.800 --> 59:45.640
Now I'm making a very special
function of y and t.

59:45.639 --> 59:47.829
Till now, I just said it's a
function of y and t.

59:47.829 --> 59:49.439
All I needed was it's a
function of y and t.

59:49.440 --> 59:51.900
That was enough to get me all
this.

59:51.900 --> 59:55.080
I'm going to find a relation
between these constants,

59:55.079 --> 59:57.089
E_0 and
B_0,

59:57.090 --> 1:00:02.620
by putting them into these two
Maxwell equations I had.

1:00:02.619 --> 1:00:04.799
That's all I want to do now.

1:00:04.800 --> 1:00:12.510
So one condition I had was
dE_z/dy was

1:00:12.510 --> 1:00:16.510
-dB_x/dt.

1:00:16.510 --> 1:00:20.110
Then I had
dB_x/dy is

1:00:20.110 --> 1:00:24.290
-μ_0ε
_0dE

1:00:24.293 --> 1:00:26.633
_z/dt.

1:00:26.630 --> 1:00:31.110
These are the two equations.

1:00:31.110 --> 1:00:35.740
I'm going to demand that these
particular functions obey these

1:00:35.744 --> 1:00:36.964
two equations.

1:00:36.960 --> 1:00:39.610
If they obey these two
equations, they will obey the

1:00:39.612 --> 1:00:41.742
wave equation,
because when I combine this

1:00:41.744 --> 1:00:44.244
equation with that,
I got the wave equation.

1:00:44.239 --> 1:00:45.329
Do you understand?

1:00:45.329 --> 1:00:48.729
I have an equation A and an
equation B, if I put one into

1:00:48.726 --> 1:00:50.966
the other, I got the wave
equation.

1:00:50.969 --> 1:00:53.749
But you really should satisfy
separately equation A and

1:00:53.746 --> 1:00:56.676
equation B, because it's not
enough to satisfy the one you

1:00:56.677 --> 1:00:58.527
get by shoving one in the other.

1:00:58.530 --> 1:01:00.480
They should be true
independently.

1:01:00.480 --> 1:01:06.730
So let's demand that this be
true and demand that be true.

1:01:06.730 --> 1:01:09.460
So what is
dE_z/dy?

1:01:09.460 --> 1:01:13.400
If I take the d/dy of
this, this is what I call

1:01:13.402 --> 1:01:16.222
E_z and this is
what I call

1:01:16.219 --> 1:01:17.979
B_x.

1:01:17.980 --> 1:01:20.420
dE_z/dy,
if you take the d/dy of

1:01:20.423 --> 1:01:22.433
that,
will give me

1:01:22.425 --> 1:01:29.005
kE_0cos(ky -
ωt). And we want that

1:01:29.009 --> 1:01:33.069
to be =
-dB_x/dt.

1:01:33.070 --> 1:01:35.410
So take the d/dt of this
guy.

1:01:35.409 --> 1:01:40.519
d/dt of this guy is
B_0 -

1:01:40.518 --> 1:01:44.988
ωcos(ky -
ωt).

1:01:44.989 --> 1:01:47.949
So these things cancel out,
then I get

1:01:47.954 --> 1:01:50.364
B_0ω.

1:01:50.360 --> 1:01:57.530
So this tells you that
E_0 = ω/k

1:01:57.530 --> 1:02:00.400
B_0.

1:02:00.400 --> 1:02:03.220
It tells you about the
magnitude of the E and

1:02:03.217 --> 1:02:05.757
B vectors,
what size they should bear in

1:02:05.757 --> 1:02:07.247
relation to each other.

1:02:07.250 --> 1:02:09.980
This is one condition.

1:02:09.980 --> 1:02:11.170
I'm almost done.

1:02:11.170 --> 1:02:16.650
So I want to take this
condition now.

1:02:16.650 --> 1:02:21.760
So dB_x/dy,
what is that equal to?

1:02:21.760 --> 1:02:27.110
B_0kcos(ky
- ωt) = -

1:02:27.112 --> 1:02:31.482
μ_0
ε_0

1:02:31.481 --> 1:02:37.381
dE_z/dt which will
be - ωE_0

1:02:37.380 --> 1:02:41.970
cos(ky -
ωt).

1:02:41.969 --> 1:02:44.859
Cosines cancel,
then I get a condition

1:02:44.864 --> 1:02:48.864
B_0k =
μ_0ε

1:02:48.856 --> 1:02:53.156
_0ω
times E_0.

1:02:53.159 --> 1:02:57.719
But that equals ω/c
squared times

1:02:57.724 --> 1:03:01.024
E_0,
because μ_0

1:03:01.018 --> 1:03:02.448
ε_0 is
1/c^(2).

1:03:02.449 --> 1:03:06.339
So I get a second condition
which I can write as

1:03:06.342 --> 1:03:10.402
E_0 =
c^(2)/ω

1:03:10.402 --> 1:03:12.972
times B_0.

1:03:12.969 --> 1:03:17.389
So these are the two conditions
in the end.

1:03:17.389 --> 1:03:20.319
If you want the function to
look like this traveling wave,

1:03:20.320 --> 1:03:24.100
with sine waves in it,
then the amplitude for E

1:03:24.097 --> 1:03:28.017
and the amplitude for B
have to satisfy these two

1:03:28.019 --> 1:03:29.089
conditions.

1:03:29.090 --> 1:03:31.380
But look at these two equations.

1:03:31.380 --> 1:03:32.800
They're both telling you
something about

1:03:32.800 --> 1:03:33.640
E_0.

1:03:33.639 --> 1:03:35.779
One says E_0
should be ω/k times

1:03:35.782 --> 1:03:36.592
B_0.

1:03:36.590 --> 1:03:38.040
The other says
E_0 should be

1:03:38.041 --> 1:03:39.691
c^(2)/ω times
B_0.

1:03:39.690 --> 1:03:44.720
That means ω/k
better equal c^(2)--I

1:03:44.717 --> 1:03:48.097
bet I dropped a k
somewhere.

1:03:48.099 --> 1:03:50.629
Student: You had
B_0k in the--

1:03:50.630 --> 1:03:52.880
Prof: Thank you, yes.

1:03:52.880 --> 1:03:59.920
So c^(2)k/ω.

1:03:59.920 --> 1:04:05.430
That tells me that
ω^(2)=

1:04:05.431 --> 1:04:11.111
k^(2)c^(2)ω
= kc or -.

1:04:11.110 --> 1:04:12.680
Or if you like,
k is

1:04:12.679 --> 1:04:14.009
�ω/c.

1:04:14.010 --> 1:04:17.290
It doesn't matter how you write
it, but you can see what that

1:04:17.291 --> 1:04:17.731
means.

1:04:17.730 --> 1:04:25.170
What this is telling you is
that if you take the function

1:04:25.168 --> 1:04:31.928
sine ky - ωt,
and if ω = kc,

1:04:31.929 --> 1:04:39.189
it becomes sine of ky -
kct.

1:04:39.190 --> 1:04:45.520
That becomes sine of k
times y - ct which is a

1:04:45.523 --> 1:04:48.533
function of y - ct.

1:04:48.530 --> 1:04:51.430
In other words,
we knew this was going to

1:04:51.425 --> 1:04:54.335
happen,
because the wave equation says

1:04:54.335 --> 1:04:58.335
the function should be a
function only of y - ct.

1:04:58.344 --> 1:05:01.484
That will happen if ω
= kc.

1:05:01.480 --> 1:05:02.010
You see that?

1:05:02.010 --> 1:05:04.340
If ω
= kc, you pull the k

1:05:04.342 --> 1:05:05.882
out, you get y - ct.

1:05:05.880 --> 1:05:10.500
Or you can have y ct,
but my solution was y -

1:05:10.500 --> 1:05:11.270
ct.

1:05:11.269 --> 1:05:12.109
I'm almost done.

1:05:12.110 --> 1:05:15.520
So the last thing I want to get
from this same equation,

1:05:15.518 --> 1:05:18.988
now that that condition is
satisfied, if ω =

1:05:18.987 --> 1:05:23.367
kc, I get
E_0 = c

1:05:23.373 --> 1:05:26.163
times B_0.

1:05:26.159 --> 1:05:27.609
So now I'm going to summarize
this.

1:05:27.610 --> 1:05:29.160
Don't worry about the details.

1:05:29.159 --> 1:05:35.209
I will tell you the part you
should know all the time.

1:05:35.210 --> 1:05:37.770
So I know this is somewhat
heavy.

1:05:37.768 --> 1:05:41.098
I had to do this calculation at
home to make sure I got all the

1:05:41.099 --> 1:05:41.959
- signs right.

1:05:41.960 --> 1:05:43.750
There's an orgy of - signs.

1:05:43.750 --> 1:05:45.250
I'm not that interested in that.

1:05:45.250 --> 1:05:48.070
I know that you guys,
unless you're going to major in

1:05:48.072 --> 1:05:51.492
physics and want to do it for a
lifetime, don't want to know all

1:05:51.490 --> 1:05:51.980
that.

1:05:51.980 --> 1:05:54.250
So what should you know?

1:05:54.250 --> 1:05:57.170
What you should know is that by
doing a few experiments,

1:05:57.166 --> 1:05:59.336
one wrote down these Maxwell
equations.

1:05:59.340 --> 1:06:02.300
You've got to understand a
little bit where everything came

1:06:02.302 --> 1:06:02.662
from.

1:06:02.659 --> 1:06:06.249
Then the whole class was
demonstrating that they implied

1:06:06.246 --> 1:06:10.156
some waves and the demonstration
was shown for proving it for

1:06:10.159 --> 1:06:13.489
infinitesimal loops and
infinitesimal cubes and then

1:06:13.487 --> 1:06:15.767
seeing what conditions I had.

1:06:15.768 --> 1:06:19.718
And I found that I could get a
set of functions that are

1:06:19.722 --> 1:06:23.392
dependent on y and t,
and that they really travel at

1:06:23.387 --> 1:06:25.037
the speed of light.

1:06:25.039 --> 1:06:26.259
So that was the bottom line.

1:06:26.260 --> 1:06:29.180
Actual derivatives and how
everything happened is really

1:06:29.179 --> 1:06:31.729
not something I expect you to
carry in your head,

1:06:31.730 --> 1:06:34.120
so don't let the exams ruin
that for you.

1:06:34.119 --> 1:06:35.089
I don't care.

1:06:35.090 --> 1:06:40.070
I will not quiz you on that
part of this derivation.

1:06:40.070 --> 1:06:43.140
But you must understand that
the structure of physics was

1:06:43.139 --> 1:06:46.259
such that it was an interplay
between some experiments and

1:06:46.264 --> 1:06:48.954
some purely theoretical
reasoning on the nature of

1:06:48.952 --> 1:06:49.832
equations.

1:06:49.829 --> 1:06:54.019
And you put them together and
what makes it worthwhile in the

1:06:54.021 --> 1:06:57.871
end is to get fantastic
productions like this that unify

1:06:57.865 --> 1:07:01.635
electricity and magnetism and
light into one shot.

1:07:01.639 --> 1:07:04.559
So the picture I get now,
if you look at all of this,

1:07:04.559 --> 1:07:10.789
the final answer is E
looks like k times some

1:07:10.791 --> 1:07:17.561
number E_0 sine
ky - ωt.

1:07:17.559 --> 1:07:22.729
But I want you to know that
ω = kc there.

1:07:22.730 --> 1:07:27.370
Then I get B =
IB_0

1:07:27.367 --> 1:07:31.527
also the same sine,
with the extra restriction that

1:07:31.534 --> 1:07:35.574
B_0 times
c = E_0.

1:07:35.570 --> 1:07:40.050
So here is what the
electromagnetic wave looks like.

1:07:40.050 --> 1:07:45.110
One guy, the electric field,
is always living in this plane.

1:07:45.110 --> 1:07:46.880
This is the E field.

1:07:46.880 --> 1:07:49.020
At one instant,
if you take a snapshot,

1:07:49.016 --> 1:07:51.486
that's what it will be doing,
going up there,

1:07:51.489 --> 1:07:53.569
coming down here,
going up there.

1:07:53.570 --> 1:07:56.180
That's E.

1:07:56.179 --> 1:08:04.629
B field looks like this,
lies horizontal here,

1:08:04.626 --> 1:08:09.496
goes like hat and comes out.

1:08:09.500 --> 1:08:13.470
So this is in that plane and
that is in the vertical plane.

1:08:13.469 --> 1:08:17.169
That's horizontal plane,
that's the vertical plane.

1:08:17.170 --> 1:08:20.120
So that is E and this is
B.

1:08:20.118 --> 1:08:23.708
And the point is,
E over B =

1:08:23.707 --> 1:08:24.667
C.

1:08:24.670 --> 1:08:27.410
In other words,
E is much bigger than

1:08:27.408 --> 1:08:28.108
B.

1:08:28.109 --> 1:08:32.319
The ratio of them is the
velocity of light.

1:08:32.319 --> 1:08:35.439
This is interesting,
because if you took the force

1:08:35.436 --> 1:08:39.256
on a charge, you remember is
q times E v

1:08:39.255 --> 1:08:40.205
x B.

1:08:40.210 --> 1:08:43.590
That means if you took an
electron and you left it in the

1:08:43.585 --> 1:08:46.835
electromagnetic field,
the field comes and hits you.

1:08:46.840 --> 1:08:48.660
If E field is
oscillating one way,

1:08:48.657 --> 1:08:51.017
the B field is
oscillating perpendicularly.

1:08:51.020 --> 1:08:54.280
The wave is traveling in a
direction perpendicular to both.

1:08:54.279 --> 1:08:55.909
If you want,
it's in the direction of

1:08:55.912 --> 1:08:58.222
E x B,
is the direction of propagation

1:08:58.224 --> 1:08:58.954
of the wave.

1:08:58.948 --> 1:09:01.868
If it hits an electron there,
the oscillating up and down

1:09:01.872 --> 1:09:04.852
electric field will make the
electron move up and down.

1:09:04.850 --> 1:09:07.360
It will also be feeling a
magnetic force,

1:09:07.359 --> 1:09:08.739
V x B.

1:09:08.738 --> 1:09:12.128
But notice that the size of
B is the size of E

1:09:12.134 --> 1:09:13.404
divided by c.

1:09:13.399 --> 1:09:16.719
So the electric force to
magnetic force ratio,

1:09:16.721 --> 1:09:20.711
or magnetic electric will be
the ratio of v over

1:09:20.710 --> 1:09:21.670
c.

1:09:21.670 --> 1:09:24.360
So for most velocities,
for electrons and circuits and

1:09:24.359 --> 1:09:27.099
so on, the velocity is much
tinier than the velocity of

1:09:27.101 --> 1:09:27.611
light.

1:09:27.609 --> 1:09:29.669
So when a radio wave hits your
antenna,

1:09:29.670 --> 1:09:32.560
gets the electric charges in
action,

1:09:32.560 --> 1:09:34.820
it's the electric field that
does most of the forcing,

1:09:34.819 --> 1:09:37.079
not the magnetic field.

1:09:37.078 --> 1:09:38.958
But in astrophysics and cosmic
ray physics,

1:09:38.960 --> 1:09:42.290
where particles can travel at
velocities comparable to that of

1:09:42.287 --> 1:09:44.977
light,
then these forces can become

1:09:44.978 --> 1:09:45.918
comparable.

1:09:45.920 --> 1:09:51.000