WEBVTT
00:01.630 --> 00:05.260
Prof: Before we start
today's lecture,
00:05.260 --> 00:10.130
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.
00:33.640 --> 00:35.780
So nothing else?
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Yes?
00:36.870 --> 00:39.570
Student: Is there going
to be an equation sheet for the
00:39.568 --> 00:41.568
midterm and if so,
is it possible to get it?
00:41.570 --> 00:42.560
Prof: You will get the
equation sheet.
00:42.560 --> 00:43.460
Do you want it now?
00:43.460 --> 00:44.110
No.
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With the exam?
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Yeah.
00:46.720 --> 00:52.210
I don't give it out before,
but you will have all the
00:52.206 --> 00:56.106
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
01:08.260 --> 01:09.150
future.
01:09.150 --> 01:14.310
If I held back the equation
sheet, I know you will hold back
01:14.313 --> 01:15.893
the painkillers.
01:15.890 --> 01:18.540
It's just making a deal,
the way things are done
01:18.539 --> 01:19.159
nowadays.
01:19.159 --> 01:22.859
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
01:36.532 --> 01:37.312
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
02:23.199 --> 02:26.089
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,
02:31.894 --> 02:35.714
namely, you're counting the net
number of lines coming out,
02:35.705 --> 02:37.345
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.
03:01.718 --> 03:05.938
Depending on how many got in
the volume, the net will decide
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the flux out of that volume.
03:08.080 --> 03:11.270
This one, without that,
used to say electric field is a
03:11.269 --> 03:12.509
conservative field.
03:12.508 --> 03:14.878
It defines a potential and so
on.
03:14.878 --> 03:17.538
But then we found in the time
dependent problem,
03:17.543 --> 03:20.943
E⋅dl,
we said rate of change of the
03:20.943 --> 03:22.023
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
03:29.229 --> 03:30.289
magnetic field.
03:30.288 --> 03:32.798
In addition,
a current can also produce a
03:32.801 --> 03:33.871
magnetic field.
03:33.870 --> 03:39.090
This is what we have to begin
with.
03:39.090 --> 03:43.630
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
03:55.244 --> 03:56.374
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.
04:01.258 --> 04:08.558
And the line integral of that
guy is - the rate of change of
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flux of this.
04:10.408 --> 04:14.428
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.
04:18.949 --> 04:23.249
But if at that moment you have
not seen the wave equation,
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you're not going to get the
thrill.
04:26.050 --> 04:29.180
So I have to know,
how many people remember the
04:29.177 --> 04:32.577
wave equation from the first
part of the course?
04:32.579 --> 04:36.429
Okay, so I think I have to
remind you of what that is.
04:36.430 --> 04:39.180
It's a bit like explaining a
joke, but still,
04:39.177 --> 04:42.237
I have to do that,
because otherwise we won't have
04:42.235 --> 04:43.855
that climactic moment.
04:43.860 --> 04:47.390
So let me tell you one example
of a wave equation.
04:47.389 --> 04:49.029
People have seen them before.
04:49.029 --> 04:50.979
Here is the simplest example.
04:50.980 --> 04:55.000
You take a string,
you tie it between two fixed
04:54.995 --> 04:58.395
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.
05:02.269 --> 05:06.849
Then you distort it in some
form and you release it,
05:06.853 --> 05:09.283
and you see what happens.
05:09.278 --> 05:16.478
Now let's measure x as
the distance from the left end
05:16.478 --> 05:22.698
and y as the height at
that time t.
05:22.699 --> 05:27.429
The wave equation tells you an
equation governing the behavior
05:27.432 --> 05:28.522
of y.
05:28.519 --> 05:31.779
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
05:35.086 --> 05:37.246
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
05:49.367 --> 05:50.667
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,
05:59.000 --> 06:01.520
but not to the whole string at
one time,
06:01.519 --> 06:04.039
because a is the
acceleration and whose
06:04.036 --> 06:06.046
acceleration are we talking
about?
06:06.050 --> 06:07.890
Piece of string here,
piece of string there?
06:07.889 --> 06:10.489
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
06:14.360 --> 06:15.380
your benefit.
06:15.379 --> 06:17.619
That's a piece of string.
06:17.620 --> 06:21.410
Its interval,
it lies under region of length
06:21.411 --> 06:22.471
dx.
06:22.470 --> 06:25.910
In other words,
you take a dx here and
06:25.910 --> 06:27.630
look at that string.
06:27.629 --> 06:32.569
It is being pulled like that
with a tension T and it's
06:32.572 --> 06:37.022
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,
06:43.648 --> 06:47.548
there'll be no net force on it,
because these forces will
06:47.553 --> 06:48.323
cancel.
06:48.319 --> 06:52.819
But if it's slightly curved,
then this line and that line
06:52.821 --> 06:57.241
are not quite in the same
direction and the forces don't
06:57.242 --> 06:58.772
have to cancel.
06:58.769 --> 07:03.029
So let's take that force and
write its vertical part.
07:03.028 --> 07:06.858
If this angle is
θ,
07:06.860 --> 07:12.300
T times
sinθ is the force
07:12.297 --> 07:13.407
here.
07:13.410 --> 07:18.370
-T times (because
T is pointing the
07:18.365 --> 07:21.865
opposite way)--
let me call this
07:21.872 --> 07:28.332
θ_1 and
θ_2 is
07:28.333 --> 07:31.293
the angle you have here.
07:31.290 --> 07:32.210
Do you understand?
07:32.209 --> 07:36.479
The vertical force is restoring
the strength to where it was is
07:36.476 --> 07:37.986
what I'm focused on.
07:37.990 --> 07:40.620
So you resolve that force into
a vertical part,
07:40.622 --> 07:43.202
up and this one has the
vertical part down.
07:43.199 --> 07:45.519
Now if θ_1 =
θ_2 they
07:45.523 --> 07:46.983
cancel and that's what I just
said.
07:46.980 --> 07:48.570
But θ_1
and θ_2
07:48.565 --> 07:50.725
are generally not equal,
because if I go to the top of
07:50.730 --> 07:52.270
the string,
for example,
07:52.273 --> 07:56.113
this end is pulling this way,
that end is pulling that way.
07:56.110 --> 07:58.420
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
08:02.533 --> 08:04.823
string is changing with
position.
08:04.819 --> 08:06.929
That's why they don't cancel.
08:06.930 --> 08:10.910
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
08:19.055 --> 08:20.355
corrections.
08:20.360 --> 08:24.980
And cosine θ is 1
− θ^(2)/2
08:24.975 --> 08:26.085
corrections.
08:26.089 --> 08:29.679
And tanθ,
which is the ratio of these
08:29.682 --> 08:32.912
guys, is θ other
corrections.
08:32.908 --> 08:35.168
For a very small
θ,
08:35.173 --> 08:37.733
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
09:11.552 --> 09:15.832
dx −
dy/dx at the point
09:15.825 --> 09:17.245
x.
09:17.250 --> 09:17.620
Are you with me?
09:17.620 --> 09:20.260
dy/dx from calculus,
you remember,
09:20.259 --> 09:21.249
is the slope.
09:21.250 --> 09:23.590
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?
09:29.798 --> 09:30.518
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
09:33.091 --> 09:34.401
at one instant in time.
09:34.399 --> 09:36.729
So I don't want to take this
guy's slope now and that guy's
09:36.726 --> 09:37.606
slope an hour later.
09:37.610 --> 09:42.310
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?
09:52.298 --> 09:56.128
There's one legacy I want to
leave behind is what to do when
09:56.129 --> 09:58.789
confronted with a difference
like this.
09:58.789 --> 10:03.489
You have any idea what to do?
10:03.490 --> 10:04.320
Pardon me?
10:04.320 --> 10:07.350
Student: Multiply by
one.
10:07.350 --> 10:10.660
Prof: Multiply by one.
10:10.659 --> 10:13.009
Then what do you do with that?
10:13.009 --> 10:15.989
I'm asking you to evaluate the
difference between this guy at
10:15.985 --> 10:18.115
x dx - the same thing at
x.
10:18.120 --> 10:25.700
What is it equal to
approximately for small
10:25.695 --> 10:27.855
dx?
10:27.860 --> 10:29.470
Let me ask you another question.
10:29.470 --> 10:34.080
Suppose I have f of x
dx and I subtract from it
10:34.078 --> 10:37.458
f of x,
what is it equal to?
10:37.460 --> 10:48.430
Student:
>
10:48.428 --> 10:51.418
Prof: If I divide by
Δx and take the
10:51.424 --> 10:53.164
limit, it becomes a derivative.
10:53.158 --> 10:55.348
If I don't divide by
Δx,
10:55.345 --> 10:58.445
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?
11:11.639 --> 11:13.779
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,
11:17.690 --> 11:20.770
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.
11:33.850 --> 11:36.650
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 9ï¿½10^(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
ï¿½ω/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