WEBVTT

00:01.450 --> 00:03.660
Prof: All right,
welcome back.

00:03.660 --> 00:07.360
We're going to do a brand new
topic.

00:07.360 --> 00:09.720
Well actually,
a brand old topic,

00:09.722 --> 00:14.292
because it's about light,
but I'm going to go backwards

00:14.288 --> 00:18.208
in time,
because just before the break,

00:18.212 --> 00:21.992
we had this finish with a
flourish,

00:21.990 --> 00:24.300
Maxwell's theory of light.

00:24.300 --> 00:28.570
We took Ampere's law and Lenz's
law and Faraday's law and all

00:28.569 --> 00:32.149
kinds of stuff,
put them together and out came

00:32.151 --> 00:36.301
the news: the view that
electromagnetic waves can exist

00:36.295 --> 00:39.495
on their own,
travel away from charges and

00:39.501 --> 00:40.181
currents.

00:40.180 --> 00:43.300
And they travel at a speed
which happened to coincide with

00:43.304 --> 00:45.894
the speed of light,
and people conjectured,

00:45.890 --> 00:48.330
quite correctly,
that light was an

00:48.326 --> 00:50.556
electromagnetic phenomenon.

00:50.560 --> 00:53.990
And it was an oscillatory
phenomenon,

00:53.990 --> 00:58.940
but what's oscillating is not a
piece of wire or some water on a

00:58.941 --> 01:01.411
lake,
but what's oscillating is the

01:01.412 --> 01:02.432
electric field.

01:02.429 --> 01:03.929
It's oscillating in strength.

01:03.929 --> 01:06.069
The field is not jumping up and
down.

01:06.069 --> 01:08.279
The field is a condition at a
certain point,

01:08.280 --> 01:11.570
you sit at a certain point,
sometimes the field points up,

01:11.569 --> 01:13.279
sometimes the field points
down, it's strong,

01:13.280 --> 01:16.390
it's weak, and you can measure
it by putting a test charge.

01:16.390 --> 01:21.490
It's that condition in space
that travels from source to some

01:21.489 --> 01:22.679
other place.

01:22.680 --> 01:27.980
Now that point of view came
near the second half of the

01:27.983 --> 01:32.113
nineteenth century,
and it came after many,

01:32.108 --> 01:35.348
many years of studying light.

01:35.349 --> 01:39.819
And what I'm going to do is to
tell you two different ways in

01:39.816 --> 01:43.236
which you can go away from the
theory of light,

01:43.241 --> 01:45.401
of electromagnetic waves.

01:45.400 --> 01:53.360
One is, when the wavelength of
light is much smaller than your

01:53.358 --> 01:57.648
scale of observation,
namely, you're looking at a

01:57.646 --> 02:00.126
situation where you're thinking
in terms of centimeters and

02:00.129 --> 02:03.219
meters and so on,
and the wavelength of light is

02:03.221 --> 02:07.661
10^(-8) centimeters,
then light behaves in a much

02:07.656 --> 02:08.966
simpler way.

02:08.968 --> 02:12.958
You can forget about the waves,
then you get this theory of

02:12.955 --> 02:17.005
light in which you have what's
called geometrical optics.

02:17.008 --> 02:21.698
Geometrical optics is just
light going in a straight line

02:21.699 --> 02:25.719
from start to finish,
from source to your eye.

02:25.720 --> 02:29.670
So if you take Maxwell theory
and apply it to a situation

02:29.669 --> 02:32.349
where the wavelength is very
small,

02:32.348 --> 02:35.558
then you get this approximation
that I'm going to discuss for a

02:35.556 --> 02:36.276
while today.

02:36.280 --> 02:38.630
But when you say very small,
you always have to ask,

02:38.627 --> 02:40.327
"Small compared to
what?"

02:40.330 --> 02:41.310
Do you understand that?

02:41.310 --> 02:44.610
When someone says wavelength is
small, as it is,

02:44.614 --> 02:46.024
it has no meaning.

02:46.020 --> 02:48.920
I can pick units in which the
wavelength is million or 1

02:48.917 --> 02:49.547
millionth.

02:49.550 --> 02:50.700
That has no meaning.

02:50.699 --> 02:53.249
Small and large can be changed
by change of units.

02:53.250 --> 02:57.700
What we really mean is the
following: suppose I have a

02:57.696 --> 03:01.216
screen and there's a hole in the
screen.

03:01.218 --> 03:04.508
And behind the screen,
there is a source of light.

03:04.508 --> 03:10.038
Then I put another screen here
and the light goes through that

03:10.043 --> 03:15.763
and forms an image which you can
obtain just by drawing straight

03:15.758 --> 03:18.568
lines from start to finish.

03:18.568 --> 03:24.288
So you illuminate a region
which is the same shape as what

03:24.293 --> 03:25.803
you had here.

03:25.800 --> 03:29.380
That's what makes people think
of using ray optics.

03:29.378 --> 03:31.518
Ray optics, the light rays come
out,

03:31.520 --> 03:33.810
they're blocked by the screen
except near the hole,

03:33.810 --> 03:37.070
except inside the hole,
and the light escapes through

03:37.074 --> 03:40.974
that hole and fans out and forms
an image of the same shape.

03:40.970 --> 03:45.200
If you made a blip here in the
hole, you'll get a blip in the

03:45.197 --> 03:48.647
image, because it will simply
follow the shape.

03:48.650 --> 03:52.220
Now I can tell you what I mean
by, the wavelength is small or

03:52.223 --> 03:52.703
large.

03:52.699 --> 03:56.169
It's going to be small or large
compared to the size of this

03:56.170 --> 03:56.760
opening.

03:56.758 --> 03:59.138
If λ,
the wavelength,

03:59.139 --> 04:03.719
is much less than d,
which is the size of that

04:03.723 --> 04:07.163
opening here,
then you have this simpler

04:07.162 --> 04:09.192
geometric optics.

04:09.189 --> 04:15.329
That's the approximation.

04:15.330 --> 04:17.880
It's like saying,
if you have understood

04:17.877 --> 04:20.357
Einstein's relativistic
kinematics,

04:20.360 --> 04:22.750
if you go to small
velocities--again,

04:22.750 --> 04:24.170
one must say,
"Small compared to

04:24.166 --> 04:24.676
what?"

04:24.680 --> 04:27.040
The answer is small compared to
the speed of light--

04:27.040 --> 04:29.200
you get another kind of
mechanics called Newtonian

04:29.197 --> 04:31.467
mechanics,
which is simpler and was

04:31.471 --> 04:32.751
discovered first.

04:32.750 --> 04:36.170
Likewise, geometrical optics is
simpler than the real thing and

04:36.173 --> 04:37.723
it was discovered earlier.

04:37.720 --> 04:42.400
You will note that this picture
is incomplete if λ

04:42.403 --> 04:46.253
becomes comparable to the size
of that hole.

04:46.250 --> 04:50.380
Then you will find out that if
you put a very tiny hole in a

04:50.379 --> 04:52.169
screen,
and it would be very tiny

04:52.165 --> 04:55.805
compared to wavelength,
the waves have then spread out

04:55.805 --> 05:00.915
and formed something much bigger
than the geometric shadow.

05:00.920 --> 05:03.340
Then you will have to realize
that drawing straight lines

05:03.343 --> 05:03.953
won't do it.

05:03.949 --> 05:07.399
In other words,
if I show you a side view,

05:07.399 --> 05:10.779
you would think,
if you had a source,

05:10.778 --> 05:13.718
you'll form an image of that
dimension on the screen,

05:13.720 --> 05:17.110
but actually it will spread out
much more.

05:17.110 --> 05:19.570
And the smaller the hole,
the more the light will fan

05:19.574 --> 05:19.864
out.

05:19.860 --> 05:23.890
You're not going to get that
from geometrical optics.

05:23.889 --> 05:26.599
But in order to realize that,
you will have to deal with

05:26.596 --> 05:29.546
apertures small comparable to
the wavelength of light and the

05:29.548 --> 05:32.058
wavelength of visible light is
5,000 angstrom,

05:32.060 --> 05:33.270
which is very small.

05:33.269 --> 05:36.419
So it wasn't known for a while.

05:36.420 --> 05:39.320
Now you might say,
okay, if you want,

05:39.319 --> 05:42.899
you can go back in time,
but you should probably start

05:42.896 --> 05:46.806
with this and build your way to
electromagnetic theory a la

05:46.810 --> 05:47.620
Maxwell.

05:47.620 --> 05:50.680
Well, it turns out Maxwell
isn't right either,

05:50.675 --> 05:53.315
and to see where Maxwell's
theory fails,

05:53.322 --> 05:56.992
you will have to take light of
very low intensity.

05:56.990 --> 05:59.750
Remember, intensity is the
square of the electric field,

05:59.745 --> 06:00.945
or the magnetic field.

06:00.949 --> 06:02.639
They're all proportional.

06:02.639 --> 06:05.879
If light becomes really dim,
you might think the electric

06:05.882 --> 06:08.432
field is going to be smaller and
smaller,

06:08.430 --> 06:10.770
because E^(2) or
B^(2) is a measure of

06:10.771 --> 06:13.601
intensity,
but something else happens when

06:13.598 --> 06:15.538
light becomes really week.

06:15.540 --> 06:18.630
You realize that the light
energy is not coming to you

06:18.629 --> 06:22.009
continuously like a wave would,
but in discrete packets.

06:22.009 --> 06:24.649
These are called photons.

06:24.649 --> 06:27.419
But you won't be aware of
photons if the light is very

06:27.418 --> 06:30.498
intense, because there are so
many of them coming at you.

06:30.500 --> 06:32.550
It's like saying,
if you look at water waves,

06:32.545 --> 06:34.865
you see this nice surface and
you're looking at the

06:34.870 --> 06:36.870
description of that surface
undulating.

06:36.870 --> 06:38.820
You have a wave equation for
that.

06:38.819 --> 06:41.429
But if you really look deep
down, it's made of water

06:41.425 --> 06:44.485
molecules, but you don't see
them and you don't need that for

06:44.490 --> 06:45.870
describing ocean waves.

06:45.870 --> 06:49.570
But on some deeper level,
water is not a continuous body.

06:49.569 --> 06:50.949
It's discrete,
made of molecules.

06:50.949 --> 06:52.879
Likewise, light is not
continuous.

06:52.879 --> 06:56.969
It's made up of little
particles called photons.

06:56.970 --> 06:59.090
And that we will talk about
later.

06:59.089 --> 07:01.039
So you understand the picture
now?

07:01.040 --> 07:04.880
You're going back in time to
the ancient theory of light,

07:04.882 --> 07:06.532
then we did Maxwell's.

07:06.528 --> 07:08.678
I won't stop there,
because I have already done it,

07:08.680 --> 07:10.810
and we'll go onto the new
theory of light,

07:10.810 --> 07:13.290
which involves photons,
and that's part of what's

07:13.285 --> 07:17.685
called quantum mechanics,
so we'll certainly talk about

07:17.687 --> 07:18.297
that.

07:18.300 --> 07:20.610
So what did people know about
light?

07:20.610 --> 07:24.120
Well, they had an intuitive
feeling that something bright or

07:24.120 --> 07:26.740
shiny emits some light and you
can see it.

07:26.740 --> 07:29.560
It seemed to travel in a
straight line,

07:29.555 --> 07:33.705
and for the longest time,
people did not know how fast it

07:33.706 --> 07:34.666
traveled.

07:34.670 --> 07:37.010
It looked like it traveled
instantaneously,

07:37.007 --> 07:40.067
because you couldn't measure
the delay of light in daily

07:40.069 --> 07:40.569
life.

07:40.569 --> 07:43.939
You can measure the delay of
sound, but not the delay of

07:43.939 --> 07:44.429
light.

07:44.430 --> 07:46.690
They knew it travels in a
straight line,

07:46.689 --> 07:48.949
unlike some,
because if I close my face,

07:48.947 --> 07:51.437
you can hear me,
but you cannot see me.

07:51.440 --> 07:53.740
So sound waves can get around
an obstacle, but not light

07:53.735 --> 07:54.065
waves.

07:54.069 --> 07:57.019
That one, everybody knew.

07:57.019 --> 07:59.129
But they did not know how fast
it travels.

07:59.129 --> 08:01.129
Sound, they knew,
travels at a finite speed,

08:01.125 --> 08:03.625
because you go to the mountain
and start yelling at the

08:03.629 --> 08:04.929
mountain, it yells back.

08:04.930 --> 08:08.990
You can even time it and find
the velocity of sound.

08:08.990 --> 08:13.330
So Galileo tried to find the
velocity of light by asking one

08:13.329 --> 08:17.229
of his buddies to go stand on
top of one mountain,

08:17.230 --> 08:21.010
and he's going to stand on top
of the other mountain with a

08:21.009 --> 08:22.769
lantern which is blocked.

08:22.769 --> 08:25.469
Then he's going to open the
lantern and the minute his

08:25.473 --> 08:28.423
friend saw this light,
he was supposed to signal back

08:28.415 --> 08:31.065
with another light signal,
come back to Galileo.

08:31.069 --> 08:33.789
Meanwhile, he was timing it
with his pulse.

08:33.788 --> 08:38.198
Then he was going to--well,
that's the only clock you had

08:38.200 --> 08:39.460
in those days.

08:39.460 --> 08:42.510
I think the two mountains were
like 20 miles apart or

08:42.506 --> 08:45.606
something, so it's not
impossible that with the pulse,

08:45.614 --> 08:47.904
you can probably find the
velocity.

08:47.899 --> 08:51.379
So you've got some answer,
but I think he realized very

08:51.384 --> 08:55.134
quickly that that answer just
measured the reaction time of

08:55.125 --> 08:56.605
him and his friend.

08:56.610 --> 08:58.960
Do you know how you will
realize that,

08:58.961 --> 09:02.711
that it's the reaction time and
not the propagation time?

09:02.710 --> 09:04.580
How will you find out?

09:04.580 --> 09:06.190
You all had a good laugh at
Mr. G.

09:06.190 --> 09:08.140
but what will you do?

09:08.139 --> 09:10.419
How will you know that it's
really--yes?

09:10.418 --> 09:11.618
Student:  Vary the
distance between them.

09:11.620 --> 09:12.500
Prof: You vary the
distance.

09:12.500 --> 09:13.820
If he and his friend,
instead of being on two

09:13.820 --> 09:15.690
different mountains,
are in the same room,

09:15.688 --> 09:18.888
and they do the experiment and
they get the same delay,

09:18.889 --> 09:20.999
then they know that it's
nothing to do with the travel

09:21.000 --> 09:21.280
time.

09:21.278 --> 09:24.798
It's just how long it takes
them to react.

09:24.798 --> 09:28.288
So the real serious measurement
of light,

09:28.288 --> 09:30.288
you can't ask yourself,
"How am I going to measure

09:30.287 --> 09:31.727
it if it's traveling that
fast?"

09:31.730 --> 09:34.430
First of all,
you don't even know if it takes

09:34.426 --> 09:35.526
any finite time.

09:35.529 --> 09:38.259
It's possible to imagine that
if you turn something on,

09:38.261 --> 09:39.681
you can see it right away.

09:39.679 --> 09:41.259
It looks very natural.

09:41.259 --> 09:45.079
So the fact that it could take
a finite time was a hypothesis,

09:45.081 --> 09:47.711
but to measure it,
if it's going very fast,

09:47.712 --> 09:49.532
you need a long distance.

09:49.529 --> 09:52.219
Even the distance equal to the
circumference of the earth is

09:52.221 --> 09:54.381
not enough,
because it takes one seventh of

09:54.375 --> 09:56.965
a second for a light signal to
go around the earth,

09:56.970 --> 10:00.100
if it could be made to go in a
circle.

10:00.100 --> 10:01.640
So that's too fast.

10:01.639 --> 10:06.609
So the idea of finding the
velocity of light,

10:06.611 --> 10:11.021
the first correct way,
came from Roemer,

10:11.019 --> 10:13.279
I think in 1676.

10:13.278 --> 10:17.278
He did the following very
clever experiment:

10:17.282 --> 10:21.102
here's the sun,
let's say, and here is the

10:21.096 --> 10:23.886
earth and here is Jupiter.

10:23.889 --> 10:27.319
It's got one of these moons
called Io.

10:27.320 --> 10:30.030
And the moon goes round and
round Jupiter,

10:30.032 --> 10:33.672
and we know from Newtonian
physics that it will go in an

10:33.671 --> 10:36.121
orbit with a certain time
period.

10:36.120 --> 10:39.810
If the earth was stationary,
this would go round and round

10:39.808 --> 10:41.038
with some period.

10:41.038 --> 10:44.458
I've forgot what it is,
an hour and something,

10:44.455 --> 10:46.575
to go once around Jupiter.

10:46.580 --> 10:49.490
So what you do is you record
the pulse.

10:49.490 --> 10:52.110
Let's say you wait until it's
hidden behind Jupiter,

10:52.106 --> 10:54.156
or it comes right in front of
Jupiter.

10:54.158 --> 10:57.568
Pick any one key event in its
orbit, then wait for the next

10:57.568 --> 11:00.858
pulse, and wait for the next
pulse, and wait for the next

11:00.861 --> 11:01.451
pulse.

11:01.450 --> 11:03.010
You understand what I mean by
pulse?

11:03.009 --> 11:06.209
You can see it all the time,
but wait till it comes to a

11:06.211 --> 11:08.831
particular location in its
orbit, then repeat,

11:08.832 --> 11:09.942
then time them.

11:09.940 --> 11:12.110
So that should be one hour and
something.

11:12.110 --> 11:15.450
Let's say one hour exactly.

11:15.450 --> 11:20.420
But you notice that as the
earth begins its journey around

11:20.423 --> 11:24.443
the sun, it takes longer and
longer and longer,

11:24.436 --> 11:28.446
the pulses get spaced apart a
little more.

11:28.450 --> 11:32.200
And he found out that if you go
to this situation when the earth

11:32.203 --> 11:34.833
is here,
Jupiter hasn't moved very much

11:34.832 --> 11:40.542
in this time,
it takes about 22 minutes more.

11:40.538 --> 11:44.758
Namely, this pulse,
with respect to the anticipated

11:44.764 --> 11:47.304
time, is 22 minutes delayed.

11:47.299 --> 11:47.779
Do you understand?

11:47.779 --> 11:51.809
The delay is continuous,
but take the case now and take

11:51.812 --> 11:55.662
the case six months later,
and the pulse should have come

11:55.658 --> 11:57.618
right there if it was not
moving,

11:57.620 --> 12:01.230
but it comes 22 minutes later.

12:01.230 --> 12:05.720
And he attributed that to the
fact that it takes light time to

12:05.722 --> 12:08.392
travel,
and it takes an extra time of

12:08.393 --> 12:12.263
traveling the whole diameter of
the orbit around the sun.

12:12.259 --> 12:15.529
And that's the 22 minutes.

12:15.528 --> 12:16.898
And how do you know you are
right?

12:16.899 --> 12:20.689
Well, you know you are right,
because as you start going back

12:20.690 --> 12:24.160
now, the remaining six months,
the pulses get closer and

12:24.164 --> 12:24.864
closer.

12:24.860 --> 12:26.560
So this delay is clearly due to
the motion.

12:26.559 --> 12:27.809
Yes?

12:27.808 --> 12:29.848
Student:  How could they
still see it

12:29.850 --> 12:31.180
>?

12:31.178 --> 12:34.708
Prof: Well,
it's not all in the same plane.

12:34.710 --> 12:39.210
So you can try to see it,
even if it's not the...

12:39.210 --> 12:43.990
If you can see Jupiter at
night, which you do,

12:43.985 --> 12:49.075
then you'll be able to see the
satellite also.

12:49.080 --> 12:54.660
All right, so he calculated
based on that timing and this

12:54.663 --> 12:57.213
distance,
which was known to some

12:57.205 --> 13:00.145
accuracy at that time,
a velocity of light,

13:00.153 --> 13:03.323
that was roughly 2/3 the
correct answer.

13:03.320 --> 13:07.530
The correct answer is what,
3�10^(8) meters per second.

13:07.528 --> 13:10.638
He got 2�10^(8) or roughly
that much.

13:10.639 --> 13:12.289
That was quite an achievement.

13:12.288 --> 13:17.038
I mean, it's off by some 50
percent, but till then,

13:17.035 --> 13:19.025
people had no clue.

13:19.028 --> 13:21.368
Also he used the best data he
had,

13:21.370 --> 13:23.180
but the travel time was not
really--

13:23.178 --> 13:24.908
the delay was not really 22
minutes,

13:24.908 --> 13:28.908
but maybe 13 or 14 minutes and
he didn't have the exact size of

13:28.914 --> 13:31.504
the orbit of the earth around
the sun.

13:31.500 --> 13:34.840
But it was quite an
achievement, take a number that

13:34.836 --> 13:38.766
could have been infinity and to
nail it to within 50 percent

13:38.772 --> 13:39.642
accuracy.

13:39.639 --> 13:41.589
Then after that,
people started doing laboratory

13:41.585 --> 13:43.525
experiments to measure the
velocity of light.

13:43.529 --> 13:45.469
I don't want to go into that.

13:45.470 --> 13:52.540
Everybody has something to say
about velocity of light.

13:52.539 --> 13:53.929
That's not the main thesis.

13:53.928 --> 13:57.748
The main thesis is to tell you
that what people had figured out

13:57.751 --> 14:00.531
by the seventeenth century is
that it travels,

14:00.527 --> 14:02.867
and it travels at a certain
speed.

14:02.870 --> 14:07.660
Now you guys have learned
geometrical optics in high

14:07.657 --> 14:09.157
school, right?

14:09.159 --> 14:09.869
Everybody?

14:09.870 --> 14:15.910
Who has not seen geometrical
optics, lenses and mirrors?

14:15.909 --> 14:17.349
You've not seen?

14:17.350 --> 14:20.520
Okay, that's all right.

14:20.519 --> 14:23.829
But I will tell you,
I'll remind you what the other

14:23.832 --> 14:24.962
guys have seen.

14:24.960 --> 14:27.960
I'm going to show you another
way to think about it.

14:27.960 --> 14:34.410
First thing they teach you is
if light hits a mirror,

14:34.407 --> 14:41.597
it bounces off in such a way
that the angle of incidence is

14:41.597 --> 14:45.067
the angle of reflection.

14:45.070 --> 14:48.690
Second thing they will teach
you is that if light travels

14:48.687 --> 14:51.077
from one medium to another
medium,

14:51.080 --> 14:56.830
say this is air and say this is
glass,

14:56.830 --> 15:00.440
then the first thing to note is
that the velocity of light,

15:00.440 --> 15:04.030
c, is the velocity in
vacuum.

15:04.028 --> 15:08.028
When light travels through a
medium, that's not the velocity.

15:08.028 --> 15:12.148
The velocity is altered by a
factor called n,

15:12.147 --> 15:17.227
which is bigger than 1 or equal
to 1, and n is called the

15:17.232 --> 15:20.142
refractive index of that medium.

15:20.139 --> 15:22.339
I think glass is like 1.33.

15:22.340 --> 15:26.380
Every medium has a refractive
index and the effective velocity

15:26.380 --> 15:29.430
of light is slowed by this
factor, n.

15:29.428 --> 15:31.478
So let us say,
this medium,

15:31.475 --> 15:34.855
let's not call it air,
n_1,

15:34.856 --> 15:38.786
this is refractive index
n_2.

15:38.788 --> 15:43.488
If a beam of light comes like
this and hits this interface,

15:43.494 --> 15:45.364
it won't go straight.

15:45.360 --> 15:49.510
It will generally deviate from
its original direction,

15:49.509 --> 15:53.509
and if you call this the theta
incident,

15:53.509 --> 15:57.329
and you call this the theta
refracted,

15:57.330 --> 16:00.240
then there is a law,
called Snell's law,

16:00.240 --> 16:02.450
which says n_1
sinθ

16:02.450 --> 16:04.710
_1 is
n_2 times

16:04.711 --> 16:06.421
sinθ_2.

16:06.418 --> 16:08.748
And theta is measured--in fact,
let me call it

16:08.750 --> 16:11.860
θ_1 and
θ_2.

16:11.860 --> 16:17.960
This is called Snell's law.

16:17.960 --> 16:19.440
Look, the way to think of the
law is,

16:19.440 --> 16:21.980
if n_2 is
bigger, then sinθ

16:21.977 --> 16:24.407
will be smaller,
so this angle would be smaller.

16:24.408 --> 16:28.008
So when light goes from a rare
medium to a dense medium,

16:28.009 --> 16:30.829
it will go even closer to the
perpendicular,

16:30.825 --> 16:32.195
or to the normal.

16:32.200 --> 16:35.460
And if you run the ray
backwards, from the dense medium

16:35.456 --> 16:38.886
to the rare medium at some
angle, it will go away from the

16:38.893 --> 16:40.163
normal even more.

16:40.158 --> 16:43.618
That was done and that was
measured and all that stuff.

16:43.620 --> 16:46.980
Then you can look at more
things.

16:46.980 --> 16:53.780
You look at mirrors,
parabolic mirrors,

16:53.779 --> 16:56.899
where you know if a light ray
comes like that,

16:56.899 --> 17:00.389
parallel to the axis,
it goes through what's called a

17:00.389 --> 17:01.329
focal point.

17:01.330 --> 17:04.300
Every parallel ray goes through
the focal point,

17:04.296 --> 17:07.006
so you can use it to focus the
light ray.

17:07.009 --> 17:07.739
That's what you see.

17:07.740 --> 17:11.310
Whenever you have these
antennas, your own satellite

17:11.308 --> 17:15.438
dish, here's the dish in which
the rays come and they're all

17:15.440 --> 17:17.330
focused onto one point.

17:17.328 --> 17:20.858
That's where you put your probe
that picks up the signal from

17:20.857 --> 17:21.797
the satellite.

17:21.798 --> 17:25.998
It's a way to focus all the
light into one place,

17:26.002 --> 17:30.992
so it's a property of these
concave mirrors that they will

17:30.993 --> 17:34.763
focus all the light at the focal
point.

17:34.759 --> 17:36.689
Then you learn other stuff.

17:36.690 --> 17:40.440
If you don't have the object at
infinity sending parallel rays,

17:40.444 --> 17:43.054
if you have an object here,
what happens?

17:43.048 --> 17:45.798
Well, you have to do other
constructions.

17:45.798 --> 17:48.258
If you have an object here,
for example,

17:48.259 --> 17:51.099
you want to know what image
will be formed.

17:51.098 --> 17:54.488
You draw a parallel line and
that goes through the focal

17:54.487 --> 17:54.977
point.

17:54.980 --> 17:57.010
You draw a line through the
focal point.

17:57.009 --> 18:01.449
That comes out parallel,
and where they meet is your

18:01.452 --> 18:02.152
image.

18:02.150 --> 18:03.790
And this is called
h_1,

18:03.785 --> 18:05.375
this is called
h_2.

18:05.380 --> 18:08.550
That distance is called
u, that's called

18:08.546 --> 18:12.606
v, and you have this
result, 1/u 1/v is

18:12.606 --> 18:13.636
1/f.

18:13.640 --> 18:17.520
By the way, there is no
universal agreement on what to

18:17.517 --> 18:19.197
call these distances.

18:19.200 --> 18:21.580
Some people call it i
and o for image and

18:21.582 --> 18:21.982
object.

18:21.980 --> 18:23.980
When I was growing up,
they called it u and

18:23.980 --> 18:24.430
v.

18:24.430 --> 18:31.340
I don't care what you want to
call it, but this is the law.

18:31.339 --> 18:35.589
Then you've got lenses.

18:35.588 --> 18:41.258
This is a piece of glass and it
has the property that when you

18:41.259 --> 18:46.459
shine parallel light from one
side, it all focuses on the

18:46.464 --> 18:47.864
other side.

18:47.859 --> 18:53.589
That's called the focal length.

18:53.588 --> 18:57.018
And if you have an object here,
it will go and form an image on

18:57.022 --> 19:00.122
the other side,
which will be upside down and

19:00.115 --> 19:02.715
that also obeys the same
equation,

19:02.720 --> 19:06.310
except u is the distance
of the object and v is

19:06.305 --> 19:09.945
the distance of the image and
f is the focal length.

19:09.950 --> 19:12.650
So there's a whole bunch of
things you learn.

19:12.650 --> 19:15.690
That's all I want you to know.

19:15.690 --> 19:20.170
Then there are some tricky
issues you must have seen

19:20.165 --> 19:24.015
yourself,
that if you got a lens that's

19:24.017 --> 19:29.747
not concave but convex,
like this, and if you shine

19:29.750 --> 19:33.470
light on that guy,
what will happen?

19:33.470 --> 19:37.630
This parallel ray of light,
you can sort of imagine,

19:37.631 --> 19:39.591
will go off like that.

19:39.588 --> 19:43.758
In fact, the way it will go off
is as if it came from some point

19:43.762 --> 19:45.422
called the focal point.

19:45.420 --> 19:47.730
In other words,
these rays of light in this

19:47.732 --> 19:50.382
mirror, instead of really
focusing at some point,

19:50.376 --> 19:52.466
seem to come from the focal
point.

19:52.470 --> 19:56.970
And if you draw a ray of light
here, since that is a vertical

19:56.967 --> 20:00.187
part of the mirror,
you use i = r.

20:00.190 --> 20:02.070
That will go off at i =r.

20:02.068 --> 20:06.138
That ray of light when seen by
person here will seem to come

20:06.144 --> 20:10.154
from there, and if you join
them, you get an image here.

20:10.150 --> 20:13.610
That's the virtual image,
in the sense that this is a

20:13.608 --> 20:17.468
concave mirror--convex mirror
like the one you have in your

20:17.467 --> 20:17.997
car.

20:18.000 --> 20:21.680
And if forms a reduced image of
the object.

20:21.680 --> 20:24.870
Okay, so this is the scene from
Jurassic Park.

20:24.868 --> 20:27.608
That's the dinosaur,
and there's the Jurassic--

20:27.608 --> 20:28.928
I mean, the image of that,
and it says,

20:28.930 --> 20:31.270
"Objects may be bigger
than what they appear in the

20:31.270 --> 20:31.910
mirror."

20:31.910 --> 20:34.940
That's what it's all about,
because one of these mirrors

20:34.942 --> 20:37.372
will make an image,
but it's called a virtual

20:37.367 --> 20:37.917
image.

20:37.920 --> 20:39.320
In other words,
if you put a screen there,

20:39.323 --> 20:40.183
you won't see anything.

20:40.180 --> 20:42.020
It's on the other side of the
mirror.

20:42.019 --> 20:44.399
Here, if you put a screen,
if you put a candle here and

20:44.395 --> 20:46.505
put a screen here,
you will see a bright image of

20:46.509 --> 20:47.169
the candle.

20:47.170 --> 20:49.830
So this is a real image,
and that's a virtual image.

20:49.828 --> 20:51.858
The way you do these
calculations,

20:51.861 --> 20:54.941
you use the same formula,
except f will be a

20:54.938 --> 20:56.168
negative number.

20:56.170 --> 20:59.060
Instead of really focusing,
it anti-focuses,

20:59.061 --> 21:01.151
so the focal point,
if you want,

21:01.145 --> 21:03.765
is on the wrong side of the
mirror.

21:03.769 --> 21:07.149
You'll get all the right
answers if you use a negative

21:07.147 --> 21:07.847
f.

21:07.848 --> 21:11.328
So your textbook will have many
examples of how to solve these

21:11.332 --> 21:13.162
problems, very simple algebra.

21:13.160 --> 21:16.340
But what I want to do,
since many of you have seen

21:16.339 --> 21:19.289
this,
and to make it interesting for

21:19.292 --> 21:22.992
you,
is to show you there is a

21:22.991 --> 21:28.161
single unifying principle,
just one principle,

21:28.163 --> 21:31.483
from which I can derive all
these laws.

21:31.480 --> 21:34.790
All the things I mentioned,
this is why I didn't stop and

21:34.791 --> 21:37.451
go into detail,
every single one of them comes

21:37.454 --> 21:39.234
from one single principle.

21:39.230 --> 21:44.450
Anybody know what that
principle might be?

21:44.450 --> 21:46.010
Have you heard of anything?

21:46.009 --> 21:47.049
Yes?

21:47.048 --> 21:47.808
Student:  I don't know
how to pronounce the name.

21:47.809 --> 21:49.059
It starts with an "H."

21:49.058 --> 21:50.468
Prof: You mean Huygens'
principle?

21:50.470 --> 21:51.210
Student:  Yes.

21:51.210 --> 21:53.470
Prof: No,
that's a different guy.

21:53.470 --> 21:58.030
This is the famous Fermat,
who had this theorem with prime

21:58.029 --> 21:58.829
numbers.

21:58.828 --> 22:03.818
His principle says light will
go from start to finish in a

22:03.817 --> 22:07.577
path that takes the least amount
of time.

22:07.579 --> 22:09.519
That's the path it will take.

22:09.519 --> 22:11.439
That's the Principle of Least
Time.

22:11.440 --> 22:27.040


22:27.038 --> 22:30.308
Now we find a lot of pleasure
when we can derive many,

22:30.308 --> 22:34.198
many things from a single
principle and you will see then,

22:34.200 --> 22:37.450
all the stuff I wrote,
I can deduce from this one

22:37.448 --> 22:40.628
principle and that's what I want
to do today.

22:40.630 --> 22:42.070
So you don't have to carry all
that baggage.

22:42.069 --> 22:43.809
You can derive everything.

22:43.809 --> 22:45.919
So let's see how it goes.

22:45.920 --> 22:49.260
So first let's say I am here
and you are here,

22:49.260 --> 22:50.970
you send me a signal.

22:50.970 --> 22:52.770
What's the path it will take?

22:52.769 --> 22:54.749
Where is the path of least time?

22:54.750 --> 22:57.650
And everybody knows that's a
straight line.

22:57.650 --> 22:59.770
No point going any other way.

22:59.769 --> 23:03.529
So that tells you first,
light travels in straight lines

23:03.526 --> 23:05.776
when there's no other obstacle.

23:05.778 --> 23:11.198
The next possibility is,
I want the light--let me do

23:11.196 --> 23:17.566
this right because I'm going to
really draw some pictures.

23:17.568 --> 23:23.788
I want the light to hit the
mirror and then come to me,

23:23.787 --> 23:26.317
so it's like a race.

23:26.319 --> 23:27.639
You are here.

23:27.640 --> 23:31.610
You've got to touch the wall
and go to the finish line.

23:31.609 --> 23:33.309
Whoever gets there first wins.

23:33.309 --> 23:37.209
That's the path light will take.

23:37.210 --> 23:40.260
Now there are different
attitudes you can have.

23:40.259 --> 23:43.409
First is, you can start
wandering like this.

23:43.410 --> 23:46.860
You know that person's a loser,
because that's not the way to

23:46.861 --> 23:48.071
optimize your time.

23:48.068 --> 23:50.738
So we don't even listen to that
person.

23:50.740 --> 23:52.860
There are other reasonable
people who may have a different

23:52.858 --> 23:53.118
view.

23:53.118 --> 23:55.698
One person may say,
"Look, he told me to touch

23:55.699 --> 23:58.639
the wall, so I'm going to get
that out of my way first.

23:58.640 --> 24:01.430
Then I'm going to go
there."

24:01.430 --> 24:03.250
Fine, that's a possibility.

24:03.250 --> 24:05.840
Another person can say,
"Well, let me touch the

24:05.836 --> 24:08.676
wall right in front of this
person, then run over to meet

24:08.678 --> 24:09.438
the person.

24:09.440 --> 24:11.270
That's another
possibility."

24:11.269 --> 24:13.619
So there are different options
open to you.

24:13.618 --> 24:18.188
And we've got to find from all
these possible paths the one of

24:18.185 --> 24:19.155
least time.

24:19.160 --> 24:20.800
That's the goal.

24:20.798 --> 24:23.488
Now I already said,
when you look at paths,

24:23.492 --> 24:27.022
the path to the mirror has got
to be a straight line.

24:27.019 --> 24:30.839
You gain nothing by wiggling
around.

24:30.838 --> 24:34.198
And the path back from the
mirror to the receiving point

24:34.199 --> 24:37.499
should also be a straight line,
because the winner lies

24:37.498 --> 24:38.718
somewhere there.

24:38.720 --> 24:42.310
Anybody who doesn't follow a
straight line in free space is

24:42.305 --> 24:43.475
not going to win.

24:43.480 --> 24:47.030
So the only freedom you have,
the only thing you want to ask,

24:47.030 --> 24:49.750
is the following:
"Where should I hit that

24:49.752 --> 24:50.762
mirror?"

24:50.759 --> 24:51.559
right?

24:51.558 --> 24:56.778
So let's call that point where
you hit as x.

24:56.779 --> 25:01.059
Let's say the distance between
these two points is L.

25:01.058 --> 25:04.258
This is at some height
h_1,

25:04.258 --> 25:07.678
this is at some height
h_2.

25:07.680 --> 25:11.590
So what I will do,
is I will simply calculate the

25:11.589 --> 25:16.559
time, then find the x for
which the time is minimum.

25:16.558 --> 25:18.878
So what's the time taken for
the first segment?

25:18.880 --> 25:21.440
So let's find the total time.

25:21.440 --> 25:23.660
It will be the distance,
d_1,

25:23.663 --> 25:25.643
divided by the velocity of
light distance

25:25.642 --> 25:28.512
d_2 divided by
the velocity of light.

25:28.509 --> 25:31.239
d_1,
you can see,

25:31.240 --> 25:36.010
is h_1^(2)
x^(2),

25:36.009 --> 25:43.999
divided by c
h_2^(2) L -

25:43.998 --> 25:49.908
x squared divided by
c,

25:49.910 --> 25:53.950
just from Pythagoras' theorem,
right?

25:53.950 --> 25:58.810
It's d_1/c
d_2/c.

25:58.808 --> 26:00.818
So let's multiply everything by
c.

26:00.819 --> 26:02.449
That doesn't matter.

26:02.450 --> 26:05.870
Whether you minimize cT
or T, it doesn't make any

26:05.872 --> 26:06.592
difference.

26:06.588 --> 26:11.158
So let's take d/dx of
this whole expression and equate

26:11.157 --> 26:11.917
it to 0.

26:11.920 --> 26:15.320
That's how we find the minimum
of anything.

26:15.318 --> 26:18.568
So let me take d/dx of
the first term.

26:18.568 --> 26:23.608
That is x divided by square
root of h_1^(2)

26:23.607 --> 26:24.887
x^(2).

26:24.890 --> 26:25.770
You understand?

26:31.851 --> 26:33.901
inside,
that's the 2x.

26:38.400 --> 26:41.480
This is the d/dx of the
first term.

26:41.480 --> 26:44.570
The d/dx of the second
term will look pretty much the

26:44.573 --> 26:44.943
same.

26:44.940 --> 26:47.930
It will look like L - x
divided by

26:47.926 --> 26:51.956
h_2^(2)  (L
− x)^(2),

26:51.960 --> 26:53.620
but when you take the
derivative of (L −

26:53.624 --> 26:55.614
x)^(2),
you get a 2 times L - x

26:55.605 --> 26:58.765
and another - sign from
differentiating that guy,

26:58.769 --> 27:00.169
so you will get that.

27:00.170 --> 27:06.080
And that's what should = 0.

27:06.078 --> 27:09.408
Therefore the point x has to
satisfy this condition,

27:09.414 --> 27:12.624
but what is x over
d_1?

27:12.618 --> 27:16.518
This is just x over
d_1 = L - x

27:16.519 --> 27:18.709
over d_2.

27:18.710 --> 27:21.340
So here is x and here is
L - x.

27:21.338 --> 27:24.938
So x over
d_1 is cosine

27:24.938 --> 27:28.538
of this angle and that is cosine
of that angle,

27:28.537 --> 27:29.317
right?

27:29.318 --> 27:31.088
I don't know what you want to
call it?

27:31.088 --> 27:36.138
Let's say it's cosine α
= cosine β.

27:36.140 --> 27:42.930
That means α = β.

27:42.930 --> 27:45.420
Or if you like,
90 - α

27:45.415 --> 27:48.175
is 90 - β
and 90 - α

27:48.178 --> 27:53.698
is what one normally calls the
angle of incidence and this is

27:53.702 --> 27:57.112
called the angle of reflection.

27:57.108 --> 28:02.458
So you get
θ_i =

28:02.463 --> 28:06.563
θ_r.

28:06.558 --> 28:08.858
Now it's something everybody
should be following,

28:08.862 --> 28:11.262
because if you don't follow,
you should stop me.

28:11.259 --> 28:15.579
But it's very interesting that
i = r is the way for

28:15.577 --> 28:20.277
light to go from here to there
after touching the mirror in the

28:20.275 --> 28:22.165
least amount of time.

28:22.170 --> 28:25.520
So this is the first victory
for the Principle of Least Time.

28:25.519 --> 28:33.339
It reproduces this result.

28:33.338 --> 28:39.038
Now I'm going to reproduce a
second result.

28:39.038 --> 28:43.588
That's when light changes the
medium.

28:43.589 --> 28:46.129
So here it is.

28:46.130 --> 28:48.260
Now the challenge is different.

28:48.259 --> 28:52.639
So here is h_1
in a medium with a refractive

28:52.642 --> 28:56.732
index n_1, and
you want to go there in a

28:56.732 --> 28:58.402
medium,
refractive index

28:58.401 --> 29:03.921
n_2,
and the distance between these

29:03.920 --> 29:06.840
points is L.

29:06.838 --> 29:12.978
So imagine you are the light
ray and this is the beach and

29:12.983 --> 29:15.143
this is the ocean.

29:15.140 --> 29:20.550
You are the lifeguard and here
is the person asking for help.

29:20.548 --> 29:24.788
Now how do you get there in the
least amount of time?

29:24.788 --> 29:28.348
One point of view is to say,
"Look, let's go in a

29:28.353 --> 29:31.313
straight line,
because we have learned that's

29:31.310 --> 29:32.790
always good."

29:32.788 --> 29:36.188
But it may not be always good,
because maybe you want to spend

29:36.192 --> 29:39.092
less time in the water,
because you are slower in the

29:39.092 --> 29:39.652
water.

29:39.650 --> 29:41.190
One point of view is to say,
"Look,

29:41.190 --> 29:44.510
let's go as far as we can in
the land,

29:44.509 --> 29:47.869
and then minimize the swimming
time because we can swim slower

29:47.867 --> 29:49.187
than we can run."

29:49.190 --> 29:50.610
That's a possibility.

29:50.608 --> 29:52.428
Or you can draw all kinds of
possibilities.

29:52.430 --> 29:55.540
So we're going to find one that
has the least amount of time.

29:55.538 --> 30:01.198
If this happens to be the
answer, it should turn out in

30:01.196 --> 30:06.956
the end, so once again,
let's assume that we do that.

30:06.960 --> 30:14.140
And let this be at a distance
x from the left.

30:14.140 --> 30:15.680
Now what do you want to
minimize?

30:15.680 --> 30:18.670
Again, you want to minimize the
travel time, T.

30:18.670 --> 30:23.170
That's going to be
h_1^(2) x^(2)

30:23.170 --> 30:27.160
divided by
n_1c.

30:27.160 --> 30:31.710
That's the only subtlety,
because the time--I'm sorry,

30:31.713 --> 30:36.443
not n_1c--time
n_1.

30:36.440 --> 30:37.600
It's c over
n_1.

30:37.598 --> 30:39.828
You want to divide by the
velocity in the medium.

30:39.828 --> 30:41.708
Velocity is c divided by
n_1.

30:41.710 --> 30:44.400
Also you should know,
it's going to take longer in

30:44.404 --> 30:47.154
anything but a vacuum,
so n_1_

30:47.154 --> 30:48.754
should come on top.

30:48.750 --> 30:52.920
Then you have the other term,
that is,

30:52.917 --> 30:59.107
h_2^(2) L -
x squared divided by

30:59.113 --> 31:03.623
c times
n_2.

31:03.618 --> 31:06.948
And this will tell you what to
do.

31:06.950 --> 31:11.050
So you see, I'm teaching you a
lot of practical things in this

31:11.045 --> 31:11.645
course.

31:11.650 --> 31:14.800
I taught you,
if you're in a tsunami

31:14.804 --> 31:18.504
situation, remember what you
should do?

31:18.500 --> 31:22.630
You should calculate the
gradient and go along the

31:22.632 --> 31:23.562
gradient.

31:23.558 --> 31:26.028
On the other hand,
if it's a volcano and you

31:26.029 --> 31:29.189
calculate the gradient,
you go opposite the gradient.

31:29.190 --> 31:30.450
So this is one more thing.

31:30.450 --> 31:34.090
If you want to rescue somebody,
you've got to go towards the

31:34.090 --> 31:37.610
water in such an angle that this
function is minimized.

31:37.608 --> 31:41.308
So I suggest we calculate it
and keep the answer ready,

31:41.308 --> 31:43.278
because if you really want to
be a lifeguard,

31:43.279 --> 31:45.999
what you should do is swim and
measure your speed,

31:46.000 --> 31:47.990
run and measure your speed.

31:47.990 --> 31:50.180
It's the ratio of those two
speeds, n_1 to

31:50.180 --> 31:52.180
n_2 ,
that will tell you where to hit

31:52.181 --> 31:52.711
the water.

31:52.710 --> 31:55.250
Okay, so we're going to do that
now.

31:55.250 --> 31:57.950
So I take d/dx of all of
these things.

31:57.950 --> 31:58.780
What's the difference?

31:58.779 --> 32:01.849
It looks the same,
except you've got an

32:01.846 --> 32:04.586
n_1 everywhere,
right?

32:04.588 --> 32:11.288
h_1^(2) x^(2
)should = (L −

32:11.288 --> 32:16.958
x)n_2/h
_2^(2) (L −

32:16.957 --> 32:18.887
x)^(2).

32:18.890 --> 32:20.530
So what is x over this?

32:20.529 --> 32:27.159
This is the x.

32:27.160 --> 32:32.150
So n_1x over
that distance = cosine of this

32:32.152 --> 32:32.832
angle.

32:32.829 --> 32:34.159
You understand?

32:34.160 --> 32:39.120
That's the sine of that angle.

32:39.118 --> 32:41.728
x over that is cosine
of this angle or the sine of

32:41.729 --> 32:43.889
that angle,
since people like to write it

32:43.892 --> 32:48.312
in terms of sine,
you get this result,

32:48.310 --> 32:53.960
n_2
sinθ

32:53.958 --> 32:57.378
_2.

32:57.380 --> 32:58.720
So this is Snell's law.

32:58.720 --> 33:01.550
It also comes from the
principle of least time.

33:01.548 --> 33:04.138
Each one of them has got
interesting consequences.

33:04.140 --> 33:07.560
I don't have time to do it,
but you can imagine some of the

33:07.559 --> 33:11.229
consequences are,
if you are in the bottom of a

33:11.229 --> 33:13.769
lake,
and you look outside,

33:13.769 --> 33:18.119
the light rays go like that,
because lake is dense,

33:18.119 --> 33:19.629
air is not so dense.

33:19.630 --> 33:23.590
That means you can see stuff
right up to the horizon by

33:23.588 --> 33:26.668
taking an angle,
so that this comes exactly

33:26.670 --> 33:27.330
here.

33:27.328 --> 33:28.588
So if you're a fish and you
look out,

33:28.588 --> 33:30.548
you can see right up to the
surface of the lake without

33:30.546 --> 33:31.846
going to the surface of the
lake,

33:31.848 --> 33:33.988
because all the light,
right up to the surface,

33:33.990 --> 33:37.360
bends and comes into you.

33:37.358 --> 33:41.208
Or if you've got a flashlight
and you're sending a signal,

33:41.210 --> 33:42.870
maybe hoping somebody there
will see it,

33:42.868 --> 33:45.838
actually it will bend and
somebody at this angle will see

33:45.844 --> 33:46.114
it.

33:46.108 --> 33:49.058
And if your angle x is a
certain critical angle,

33:49.060 --> 33:52.070
your flashlight will go to the
surface, and beyond that,

33:52.067 --> 33:54.087
it will just get fully
reflected.

33:54.088 --> 33:56.958
It won't be able to go to the
other side.

33:56.960 --> 34:00.760
So another useful thing to
know, if you're going to be

34:00.759 --> 34:05.279
under water, you're lying there,
you've got some concrete blocks

34:05.278 --> 34:06.998
you're dealing with.

34:07.000 --> 34:08.800
Meanwhile you're trying to send
a signal.

34:08.800 --> 34:11.130
What angle should you send it?

34:11.130 --> 34:13.240
You've got to remember that
it's not going to go in a

34:13.239 --> 34:13.889
straight line.

34:13.889 --> 34:19.709
These are all useful lessons
from 201.

34:19.710 --> 34:23.930
All right, so now I'm going to
do the third thing.

34:23.929 --> 34:28.119
The third thing is very
interesting, which is the

34:28.119 --> 34:32.139
following: we say,
take the path of least time,

34:32.135 --> 34:33.005
right?

34:33.010 --> 34:37.420
Now there is a problem that
occurs when there's more than

34:37.420 --> 34:39.390
one path of least time.

34:39.389 --> 34:41.589
That's what we're going to talk
about now.

34:41.590 --> 34:42.900
What if there's more than one
answer?

34:42.900 --> 34:46.820
I'm going to give that to you,
so here it is.

34:46.820 --> 34:55.260
Take an elliptical room,
Oval Office.

34:55.260 --> 34:59.390
You stand here,
at one of the focal points,

34:59.385 --> 35:05.375
and you want to send a signal
to the person in the other focal

35:05.376 --> 35:07.926
point, a light signal.

35:07.929 --> 35:10.579
You know what you have to do.

35:10.579 --> 35:15.179
That portion of the mirror is
like horizontal mirror,

35:15.181 --> 35:15.891
right?

35:15.889 --> 35:18.089
So it's like the tangent to the
horizontal,

35:18.090 --> 35:21.090
so it's very clear that if you
send it like that,

35:21.090 --> 35:25.050
it will end up here,
because it will obey i =

35:25.050 --> 35:27.140
r,
and you can see from similar

35:27.139 --> 35:29.439
triangles,
that distance and that distance

35:29.440 --> 35:33.200
are equal,
and therefore they are similar

35:33.204 --> 35:34.344
triangles.

35:34.340 --> 35:36.060
Are you will me?

35:36.059 --> 35:37.909
This is the angle at which you
should send it.

35:37.909 --> 35:40.479
If you send it to this
midpoint, by symmetry,

35:40.476 --> 35:42.806
it will come to the other focal
point.

35:42.809 --> 35:47.619
Okay, so now imagine that this
is not a mirror,

35:47.619 --> 35:52.009
but some steel walls and you
have a gun.

35:52.010 --> 35:54.450
You've got one bullet left.

35:54.449 --> 35:58.419
You are here and your enemy is
here.

35:58.420 --> 36:03.070
Now what direction will you
fire in?

36:03.070 --> 36:04.370
Pardon me?

36:04.369 --> 36:05.779
Student:  At him.

36:05.780 --> 36:06.410
Prof: Right.

36:06.409 --> 36:08.369
So you can fire-- very good.

36:08.369 --> 36:10.259
See, this is why I forgot.

36:10.260 --> 36:14.520
So that's a little steel plate.

36:14.519 --> 36:16.389
Now what will you do?

36:16.389 --> 36:19.219
That's like asking the light
how to go from A to B without

36:19.219 --> 36:21.549
hitting the mirror,
I agree, that's the shortest

36:21.552 --> 36:22.002
time.

36:22.000 --> 36:24.400
But if there's something
blocking you,

36:24.398 --> 36:27.638
then you know the other
person's at the other focal

36:27.641 --> 36:28.291
point.

36:28.289 --> 36:32.859
Now which direction should you
aim?

36:32.860 --> 36:34.270
You know the answer.

36:34.269 --> 36:35.029
I gave it, right?

36:35.030 --> 36:38.130
Give me an answer then.

36:38.130 --> 36:39.240
Yes?

36:39.239 --> 36:42.849
Student:  At that point
in the wall.

36:42.849 --> 36:44.419
Prof: At that point in
the wall.

36:44.420 --> 36:50.380
But it turns out,
you can aim anywhere you like.

36:50.380 --> 36:52.590
You will thank me when you use
that rule.

36:52.590 --> 36:58.410
In other words,
you can shoot any direction.

36:58.409 --> 36:59.109
See?

36:59.110 --> 37:01.960
This is the guy who took only
Physics 200.

37:01.960 --> 37:04.230
This person took Physics 201.

37:04.230 --> 37:06.930
That's what 201 gives you.

37:06.929 --> 37:07.959
Now that's amazing, right?

37:07.960 --> 37:10.710
I'm telling you,
shoot anywhere you want.

37:10.710 --> 37:12.830
You know that bullets are like
light.

37:12.829 --> 37:16.609
They follow angle of incidence
= angle of reflection.

37:16.610 --> 37:18.490
This beam obeys i = r.

37:18.489 --> 37:20.389
You can see by symmetry.

37:20.389 --> 37:23.099
How about this one I shot at
some random angle?

37:23.099 --> 37:26.619
The way to think of that is to
draw a tangential plane mirror

37:26.619 --> 37:27.089
there.

37:27.090 --> 37:29.900
As far as this beam is
concerned, the mirror could be

37:29.900 --> 37:30.280
flat.

37:30.280 --> 37:32.080
It doesn't know it's curving
away.

37:32.079 --> 37:35.719
That angle better be equal to
that angle.

37:35.719 --> 37:38.119
That's the way you should fire
it, because you find the

37:38.119 --> 37:39.779
tangent,
then draw the normal to the

37:39.784 --> 37:41.574
tangent,
and choose and angle so that

37:41.574 --> 37:44.464
that and that become equal,
and the bullet ends up here.

37:44.460 --> 37:46.660
But I'm saying you don't have
to do all that.

37:46.659 --> 37:49.979
You shoot anywhere you like,
you go crazy,

37:49.983 --> 37:54.123
shoot in any direction,
they will all end up on this

37:54.117 --> 37:55.007
person.

37:55.010 --> 38:00.070
So why is that?

38:00.070 --> 38:01.650
Pardon me?

38:01.650 --> 38:02.780
That's the definition of an
ellipse,

38:02.780 --> 38:07.010
but why does the definition--if
you follow the Principle of

38:07.009 --> 38:10.549
Least Time,
why should that also work,

38:10.548 --> 38:14.618
according to the Principle of
Least Time?

38:14.619 --> 38:18.099
I know this path is a path of
least time, because it obeys

38:18.101 --> 38:21.701
i = r with respect to
this mirror, so I know it's the

38:21.704 --> 38:23.114
path of least time.

38:23.110 --> 38:23.900
You agree?

38:23.900 --> 38:27.140
Student:
>

38:27.139 --> 38:28.009
Prof: That is correct.

38:28.010 --> 38:31.950
In other words,
the time it takes is really

38:31.949 --> 38:34.389
that length that length.

38:34.389 --> 38:38.489
But an ellipse is a figure that
is drawn keeping the sum of that

38:38.485 --> 38:40.885
distance to that distance
constant.

38:40.889 --> 38:42.139
That's how an ellipse is drawn.

38:42.139 --> 38:45.569
Take two thumbtacks and put
them in the paper and you take a

38:45.567 --> 38:48.467
string of some length,
and you stretch it out,

38:48.472 --> 38:52.112
grab your pencil and move it,
and you will draw the ellipse.

38:52.110 --> 38:54.430
So that distance is
r_1 and that

38:54.434 --> 38:56.184
distance is
r_2,

38:56.179 --> 38:59.899
r_1 r_2
= constant is what defines

38:59.902 --> 39:00.712
an ellipse.

39:00.710 --> 39:04.050
But the time taken is really
r_1 r_2

39:04.050 --> 39:05.550
divided by c.

39:05.550 --> 39:08.830
So if you were to design a
surface so that if you shot

39:08.827 --> 39:11.497
something one point,
it will all end up here,

39:11.503 --> 39:13.703
all the light from here will
focus here,

39:13.699 --> 39:17.239
you should build an ellipse,
and send the light from one

39:17.239 --> 39:18.139
focal point.

39:18.139 --> 39:21.399
Likewise, if you talk,
also the sound will come to

39:21.396 --> 39:22.656
that other point.

39:22.659 --> 39:25.629
Now sound waves behave more
like waves rather than

39:25.632 --> 39:27.822
geometrical optics,
but if it's high,

39:27.817 --> 39:29.817
long, short wavelength sound.

39:29.820 --> 39:33.320
Suppose you're talking to your
dog, then you can talk to the

39:33.320 --> 39:34.270
dog from here.

39:34.268 --> 39:38.178
At sufficiently high frequency,
the dog will hear it here.

39:38.179 --> 39:39.279
So it's a focusing effect.

39:39.280 --> 39:43.950
So the way focusing works,
is that there's more than one

39:43.947 --> 39:46.747
way to go from start to finish.

39:46.750 --> 39:48.790
But you are supposed to follow
the Principle of Least Time.

39:48.789 --> 39:52.229
That means all those paths take
the same time.

39:52.230 --> 39:53.640
That's the key.

39:53.639 --> 39:57.869
When you look at a mirror in
front of--an object in front of

39:57.873 --> 40:02.543
a mirror, there's only one path,
hit the mirror and bounce out.

40:02.539 --> 40:07.139
But if you have a geometry like
this one, curved,

40:07.144 --> 40:09.164
then it's not true.

40:09.159 --> 40:15.109
There is more than one way to
go from start to finish.

40:15.110 --> 40:25.140
Okay, so now let us ask how you
build a focusing mirror.

40:25.139 --> 40:26.659
Here's what we want to do.

40:26.659 --> 40:28.479
So this is not very practical.

40:28.480 --> 40:32.170
This is very useful,
but it's not what I'm talking

40:32.170 --> 40:36.610
about, because I didn't discuss
that in things you knew from

40:36.614 --> 40:37.824
high school.

40:37.820 --> 40:43.950
Here's what you knew from high
school, how to make a focusing

40:43.949 --> 40:44.869
mirror.

40:44.869 --> 40:48.609
So the deal is,
light's going to come from some

40:48.605 --> 40:52.495
object at infinity,
therefore it's coming in some

40:52.503 --> 40:56.243
parallel lines from a very
distant object.

40:56.239 --> 41:01.809
You want to put some mirror of
some shape so that every one of

41:01.813 --> 41:06.113
these guys will come to the same
focal point.

41:06.110 --> 41:07.630
You can ask,
"Can I even design such a

41:07.630 --> 41:07.920
thing?

41:07.920 --> 41:09.780
Is it possible?

41:09.780 --> 41:11.340
If so, what do I have to do?

41:11.340 --> 41:13.820
What's the shape of the
object?"

41:13.820 --> 41:15.050
So let's do the following.

41:15.050 --> 41:20.710
Let's take the ray that goes
along the axis of this thing.

41:20.710 --> 41:23.580
It goes here.

41:23.579 --> 41:27.579
It goes to that mirror,
hits the mirror,

41:27.577 --> 41:31.777
then it comes back a distance
f.

41:31.780 --> 41:35.150
So in the time it takes to go
from here to here,

41:35.150 --> 41:39.020
had it continued going,
it would have gone to this wall

41:39.023 --> 41:41.753
here, also at a distance
f.

41:41.750 --> 41:43.480
Do you agree?

41:43.480 --> 41:47.010
The time it takes for it to hit
the mirror and come to the focal

41:47.010 --> 41:49.980
point is the same as the time it
would have taken,

41:49.980 --> 41:54.790
but for the mirror,
to go the other side the same

41:54.786 --> 41:56.786
distance f.

41:56.789 --> 42:01.099
Okay, now I take a second ray
that's not on the symmetry axis,

42:01.097 --> 42:02.577
but above the axis.

42:02.579 --> 42:06.379
It comes here,
and having come here,

42:06.382 --> 42:11.712
if it has to take the same time
as the other beam,

42:11.708 --> 42:14.968
it's the time to go there.

42:14.969 --> 42:19.369
But you want it to instead come
here.

42:19.369 --> 42:22.249
So how will that happen?

42:22.250 --> 42:26.720
What will ensure that that
happens?

42:26.719 --> 42:31.749
Can you guys think of what
condition you have?

42:31.750 --> 42:33.450
Yes?

42:33.449 --> 42:36.079
Student:  The two
distances need to be the same.

42:36.079 --> 42:39.309
The distance from the mirror to
the--

42:39.309 --> 42:40.219
Prof: Do you understand
that?

42:40.219 --> 42:41.529
That's very important.

42:41.530 --> 42:46.600
That distance and that distance
have to be equal.

42:46.599 --> 42:49.129
Let's be very clear on why we
are doing that.

42:49.130 --> 42:50.720
See, these guys came from
infinity.

42:50.719 --> 42:52.589
They've been traveling in a
parallel line.

42:52.590 --> 42:56.100
Start with some plane here,
so that everybody is counted

42:56.103 --> 42:58.983
from now on and see how much
time you take.

42:58.980 --> 43:02.790
The ray from this center goes
to the mirror and goes an extra

43:02.793 --> 43:05.913
distance f,
because that's what it does.

43:05.909 --> 43:08.819
So that's the distance to which
any distance these rays would

43:08.824 --> 43:10.384
have gone, but for the mirror.

43:10.380 --> 43:13.350
That's how much time you have.

43:13.349 --> 43:16.329
So if you went there and you
want to turn around and come

43:16.333 --> 43:18.233
here,
that extra distance better be

43:18.228 --> 43:20.458
equal to the distance to go to
that plane,

43:20.460 --> 43:26.700
because that's the same time
for everything.

43:26.699 --> 43:30.419
So that's the condition of the
surface.

43:30.420 --> 43:35.100
It is a surface with a property
that its distance,

43:35.096 --> 43:40.056
any point on that curve,
has the same distance from a

43:40.057 --> 43:43.587
fixed point as from a fixed
line.

43:43.590 --> 43:47.740
If you can find that,
that's the surface you want.

43:47.739 --> 43:50.409
Now that happens to be a
parabola, but we'll derive that,

43:50.409 --> 43:52.459
but that's what you learn in
high school.

43:52.460 --> 43:57.840
A parabola is a curve which is
equidistant from a point and

43:57.840 --> 43:59.140
from a line.

43:59.139 --> 44:00.989
Distance to the point is very
clear.

44:00.989 --> 44:03.679
Distance to the line is
obtained by drawing a

44:03.681 --> 44:07.781
perpendicular and measuring that
distance, the shortest distance.

44:07.780 --> 44:09.880
So I'm just going to equate
these two, that's it.

44:09.880 --> 44:11.940
That will give me the equation
for this curve.

44:11.940 --> 44:14.930
So let this graph,
the shape of the mirror I'm

44:14.925 --> 44:17.775
trying to design,
let this be the origin.

44:17.780 --> 44:20.590
This is some point with
coordinate x and

44:20.594 --> 44:24.204
y, and y is some
function of x that I'm

44:24.204 --> 44:25.554
going to find out.

44:25.550 --> 44:26.120
That's the goal.

44:26.119 --> 44:29.489
What's the function y of
x that you want?

44:29.489 --> 44:35.549
So let's find out the different
distances.

44:35.550 --> 44:40.330
So what is that distance it has
to travel?

44:40.329 --> 44:43.769
You can see,
x is the coordinate of

44:43.768 --> 44:44.858
this point.

44:44.860 --> 44:48.950
It's got to do that and another
extra f on the other

44:48.945 --> 44:49.435
side.

44:49.440 --> 44:52.700
So going horizontally,
it's got to do an x here

44:52.699 --> 44:56.019
and an f there,
so it's got to do x f.

44:56.018 --> 45:01.508
That's the distance from the
mirror to this fictitious plane.

45:01.510 --> 45:03.240
That's the time they have.

45:03.239 --> 45:09.209
They all have the time to go to
this fictitious plane.

45:09.210 --> 45:12.820
So I'm going to equate that to
this distance.

45:12.820 --> 45:15.640
This one, you've got to use
Pythagoras' Theorem.

45:15.639 --> 45:20.139
So this height is y.

45:20.139 --> 45:26.029
So it's y^(2) (f −
x)^(2),

45:26.030 --> 45:29.970
because this side here is f
- x,

45:29.969 --> 45:32.639
because the whole distance is
f and that's x.

45:32.639 --> 45:33.199
You follow that?

45:33.199 --> 45:35.389
(x,y) is the coordinate
of this point.

45:35.389 --> 45:36.589
You drop a line down here.

45:36.590 --> 45:37.890
That distance is x.

45:37.889 --> 45:39.749
This is f - x and that's
y.

45:39.750 --> 45:40.680
That's that length.

45:40.679 --> 45:44.529
You want these two to be equal.

45:44.530 --> 45:46.520
So if you have a square root,
you know what you have to do.

45:46.519 --> 45:48.039
You've got to square both sides.

45:48.039 --> 45:55.939
You square both sides,
you get x^(2) f^(2)

45:55.936 --> 46:03.346
2xf = y^(2) (f −
x)^(2).

46:03.349 --> 46:11.719
So x^(2) cancels,
f^(2) cancels,

46:11.717 --> 46:18.967
then I get y^(2) =
4xf.

46:18.969 --> 46:21.879
That's the equation of the
parabola.

46:21.880 --> 46:26.750
You're probably used to drawing
parabolas that look like this,

46:26.751 --> 46:28.831
but it's the same thing.

46:28.829 --> 46:30.549
I've just turned it around.

46:30.550 --> 46:35.650
So if you go a distance y
here, then this x is

46:35.648 --> 46:38.068
quadratic in the y.

46:38.070 --> 46:41.160
So that's the equation,
that's the process by which you

46:41.155 --> 46:44.465
can design a mirror that will
focus light from infinity.

46:44.469 --> 46:47.419
So it can be done.

46:47.420 --> 46:49.710
Likewise, if you said in the
elliptical case,

46:49.710 --> 46:52.710
"Can I find a surface
inside which the distance will

46:52.708 --> 46:55.328
go from here to there after
touching the figure is

46:55.331 --> 46:57.581
independent of where I touch
it?"

46:57.579 --> 46:59.609
the equation you will get will
be an ellipse.

46:59.610 --> 47:01.220
That's more complicated to
derive.

47:01.219 --> 47:02.529
This is a lot easier to derive.

47:02.530 --> 47:04.640
This is the equation of the
parabola.

47:04.639 --> 47:09.009
So if you want to build a dish
that will really focus light,

47:09.010 --> 47:11.900
no matter how far,
how wide the beam is,

47:11.898 --> 47:13.378
this will do it.

47:13.380 --> 47:16.030
Parabolic mirrors is something
like what Hubble would use.

47:16.030 --> 47:18.870
Anybody would use parabolic
mirrors,

47:18.869 --> 47:23.889
but there is a cheap trick
people use if they cannot afford

47:23.887 --> 47:26.787
a parabola,
because it's very hard to

47:26.788 --> 47:29.218
design things in a parabolic
shape.

47:29.219 --> 47:32.319
Do you know what the simplest
solution is?

47:32.320 --> 47:33.850
It's a sphere.

47:33.849 --> 47:37.039
Now a sphere is not quite a
parabola,

47:37.039 --> 47:40.739
but you can imagine that if you
have a parabola like this,

47:40.739 --> 47:43.889
and have a sphere,
the sphere can sort of mimic

47:43.885 --> 47:46.275
the parabola up to some
distance.

47:46.280 --> 47:48.580
Then of course it will deviate.

47:48.579 --> 47:52.079
But if you promise that you'll
only take beams very near the

47:52.083 --> 47:54.773
axis,
then the two are just as good,

47:54.766 --> 47:59.106
except it's easier to make a
spherical mirror than to make a

47:59.108 --> 48:00.578
parabolic mirror.

48:00.579 --> 48:03.359
But if you've got the money,
parabola is what you want.

48:03.360 --> 48:06.550
Otherwise, this is the cheaper
solution.

48:06.550 --> 48:09.720
So let's ask ourselves the
following question:

48:09.721 --> 48:13.881
if I take a sphere of radius R
and I slice a part of it,

48:13.880 --> 48:16.900
okay, it's a hollow sphere and
I slice a part of it and I paint

48:16.902 --> 48:19.052
it with silver,
so I've got a mirror,

48:19.050 --> 48:21.510
this part of it,
what will be the focal length

48:21.505 --> 48:21.925
of that?

48:21.929 --> 48:25.679
That's what we are asking.

48:25.679 --> 48:31.649
I get that by saying the
following: so here is that

48:31.653 --> 48:32.733
sphere.

48:32.730 --> 48:35.430
It's got radius R,
and there is some point

48:35.431 --> 48:37.411
(x,y) on that sphere.

48:37.409 --> 48:43.529
Then if this is my origin,
the equation for a sphere will

48:43.530 --> 48:48.120
be (x −
R)^(2)  y^(2) =

48:48.121 --> 48:49.981
R^(2).

48:49.980 --> 48:52.670
See, normally is x^(2)
y^(2) = R^(2),

48:52.668 --> 48:53.098
right?

48:53.099 --> 48:55.829
That's when the origin is at
the center of the sphere.

48:55.829 --> 48:59.779
But the center of this sphere
is at a point x = R,

48:59.777 --> 49:03.657
so the equation in this
coordinate system will look like

49:03.655 --> 49:04.285
this.

49:04.289 --> 49:12.139
So you open everything out,
you get x^(2) R^(2) - 2xr

49:12.137 --> 49:18.837
y^(2) =
R^(2). You cancel the

49:18.844 --> 49:26.554
R^(2) and you get
y^(2) = 2xr x^(2).

49:26.550 --> 49:31.620
I want you to compare this
equation to the equation for a

49:31.621 --> 49:33.071
real parabola.

49:33.070 --> 49:34.680
They don't look the same.

49:34.679 --> 49:45.869
Yes?

49:45.869 --> 49:50.819
Student:
Yes, that's correct.

49:50.820 --> 49:53.090
Yes.

49:53.090 --> 49:56.440
Now the way to think about this
is that but for this

49:56.440 --> 50:00.370
−x^(2) term,
this equation looks like a

50:00.367 --> 50:03.357
parabola,
but if you compare the two,

50:03.360 --> 50:06.410
you find that 4xf =
2xR.

50:06.409 --> 50:09.939
That means R = 2f.

50:09.940 --> 50:13.530
Or, if you like,
the focal length is R/2.

50:13.530 --> 50:15.580
But we're not done yet,
because I just threw away the

50:15.577 --> 50:16.127
second term.

50:16.130 --> 50:19.610
I've got to give you a reason
for throwing the second term.

50:19.610 --> 50:22.580
So here is where you've got to
get used to the following notion

50:22.577 --> 50:23.867
of big and small numbers.

50:23.869 --> 50:28.789
Whenever you deal with a mirror
or a lens, things like u,

50:28.788 --> 50:33.288
v, f are all going to be
treated as big numbers.

50:33.289 --> 50:38.089
Things like y that take
you off the axis are going to be

50:38.088 --> 50:40.178
considered small numbers.

50:40.179 --> 50:43.809
Things like x are even
smaller.

50:43.809 --> 50:49.069
So the hierarchy is,
u, v,

50:49.072 --> 50:54.342
f, big,
y is small,

50:54.335 --> 50:59.265
x is small squared.

50:59.269 --> 51:00.879
You can see that already.

51:00.880 --> 51:04.240
Suppose someone tells you to
look at this equation.

51:04.239 --> 51:05.329
You can look at the two terms.

51:05.329 --> 51:08.339
One is x times R,
other is x times

51:08.336 --> 51:08.946
x.

51:08.949 --> 51:11.209
So x times R
beats x times x,

51:11.213 --> 51:13.763
because one is a small number,
one is a small number squared.

51:13.760 --> 51:15.900
So we're going to drop that
term.

51:15.900 --> 51:18.590
Then we get,
in that approximation,

51:18.585 --> 51:19.925
this condition.

51:19.929 --> 51:22.139
But that had to be such an
approximation,

51:22.139 --> 51:24.679
because a sphere can never
equal a parabola.

51:24.679 --> 51:28.129
It can look like a parabola
only for small deviations from

51:28.132 --> 51:28.802
the axis.

51:28.800 --> 51:32.450
This is cautioning you that if
your rays come too far off,

51:32.449 --> 51:35.069
like way over there,
then the x times

51:35.070 --> 51:38.910
R term will be comparable
to the x^(2) term and it

51:38.911 --> 51:41.351
will no longer look like a
parabola.

51:41.349 --> 51:42.469
So you should be very clear.

51:42.469 --> 51:46.029
When you make a real parabolic
mirror, it will focus rays,

51:46.029 --> 51:48.839
no matter how far they are from
the origin.

51:48.840 --> 51:51.870
If I take a spherical
approximation to it,

51:51.869 --> 51:56.379
it will work only if the rays
are very close to the center.

51:56.380 --> 51:59.050
So here's a spherical mirror.

51:59.050 --> 52:02.590
You cannot have the rays going
too far from here,

52:02.585 --> 52:06.115
in the scale of u,
v and f. Yes?

52:06.119 --> 52:09.909
Student:  With the first
equation, if f = 0,

52:09.907 --> 52:11.167
isn't it a plane?

52:11.170 --> 52:12.030
Prof: Yes.

52:12.030 --> 52:15.890
Student:  But the focal
point isn't at the origin,

52:15.887 --> 52:16.437
is it?

52:16.440 --> 52:19.990
Because the plane here has--
Prof: Yeah,

52:19.990 --> 52:21.220
what did you want to do for
that one?

52:21.219 --> 52:25.339
Student:  If you put
f = 0,

52:25.336 --> 52:28.936
so you have the focal length =
0.

52:28.940 --> 52:31.060
Prof: Plane mirror is
focal length = infinity.

52:31.059 --> 52:32.429
Student:  Infinity.

52:32.434 --> 52:34.024
Oh, it's the other way around.

52:34.018 --> 52:34.388
Okay.

52:34.389 --> 52:36.289
Prof: It's the fact
there is bending that's focusing

52:36.286 --> 52:36.446
it.

52:36.449 --> 52:39.659
As you straighten it out more
and more, it will just reflect

52:39.659 --> 52:40.909
it and go right back.

52:40.909 --> 52:42.079
And when will those lines meet?

52:42.079 --> 52:46.809
They'll meet at infinity.

52:46.809 --> 52:48.699
So I want to do one thing.

52:48.699 --> 52:50.389
I want to show you something.

52:50.389 --> 52:53.509
Here is a spherical mirror I've
cut out.

52:53.510 --> 52:57.130
I want to send a parallel beam.

52:57.130 --> 53:01.550
I've already shown you that
this should go through the focal

53:01.548 --> 53:04.918
point, because it's the path of
least time.

53:04.920 --> 53:07.910
But you can say,
"How do I know once again

53:07.911 --> 53:10.581
this is the same as i =
r?"

53:10.579 --> 53:13.689
Suppose you grew up on
incidence = reflection.

53:13.690 --> 53:16.000
I'm not going to reassure you
continuously.

53:16.000 --> 53:19.610
I'm going to do it one last
time, okay?

53:19.610 --> 53:20.850
Don't ever ask me again.

53:20.849 --> 53:22.139
It's the last time.

53:22.139 --> 53:24.869
I'm going to show you that
Principle of Least Time is the

53:24.869 --> 53:25.989
same as i = r.

53:25.989 --> 53:27.529
I don't want to do it over.

53:27.530 --> 53:30.180
Here's the last time we're
going to do this.

53:30.179 --> 53:31.659
So what's our question?

53:31.659 --> 53:35.269
Question is,
if I draw a tangent to that

53:35.271 --> 53:39.441
graph and drew the normal to
that tangent,

53:39.440 --> 53:44.450
right, we want to know that
that angle a will be

53:44.452 --> 53:47.332
equal to that angle b.

53:47.329 --> 53:49.489
That's what we want to show.

53:49.489 --> 53:51.629
But where will this go?

53:51.630 --> 53:54.730
If that's a circle,
if it's a spherical mirror,

53:54.728 --> 53:58.428
you draw a normal to the
tangent, it will go through the

53:58.434 --> 54:00.124
center of the sphere.

54:00.119 --> 54:01.059
You understand?

54:01.059 --> 54:07.159
And this one is supposed to go
to the focal point.

54:07.159 --> 54:12.629
Now let this height be h.

54:12.630 --> 54:14.920
Now let this angle is also
a.

54:14.920 --> 54:20.780
This figure is too small,
so let me draw you a bigger

54:20.782 --> 54:23.492
figure so you can see.

54:23.489 --> 54:29.149
This is sort of exaggerated so
you've got to be a little

54:29.148 --> 54:35.008
careful that the--let's call
that a and that's also

54:35.012 --> 54:36.352
a.

54:36.349 --> 54:38.669
Let's call this b.

54:38.670 --> 54:41.150
This is f and this is
R.

54:41.150 --> 54:42.260
I hope you understand why.

54:42.260 --> 54:44.630
That's the important part.

54:44.630 --> 54:48.060
This mirror is locally
tangential to that line,

54:48.059 --> 54:49.849
tangent to the circle.

54:49.849 --> 54:53.359
And the normal to the circle is
pointing towards the center.

54:53.360 --> 54:55.140
That's the way circles work.

54:55.139 --> 54:57.209
You draw a perpendicular from
the circumference,

54:57.213 --> 54:59.243
you hit the center,
but that's looking like the

54:59.244 --> 55:01.324
plane mirror for that particular
light ray.

55:01.320 --> 55:03.950
That light ray doesn't care if
you bend it somewhere else.

55:03.949 --> 55:06.689
As far as this ray is
concerned, you are a plane

55:06.686 --> 55:07.206
mirror.

55:07.210 --> 55:09.030
You want that angle to be equal
to that angle.

55:09.030 --> 55:12.030
That's what I want to show you.

55:12.030 --> 55:18.350
So let's take this height h
here, and you can notice that

55:18.347 --> 55:24.887
tan b = h/f and
tan a = h/R.

55:24.889 --> 55:26.029
Can you see that?

55:26.030 --> 55:29.450
That's b and a?

55:29.449 --> 55:34.639
For small angles,
for a small angle,

55:34.639 --> 55:38.359
I remind you again and again,
sinθ is roughly

55:38.364 --> 55:41.374
θ and
tanθ is roughly

55:41.371 --> 55:44.511
θ and
cosθ is roughly 1

55:44.507 --> 55:47.577
corrections of order
θ^(2).

55:47.579 --> 55:50.279
So we delete the tan.

55:50.280 --> 55:54.070
And if you use the fact that
R = 2f,

55:54.067 --> 55:58.007
it becomes h over
2f, you can see that

55:58.010 --> 56:00.020
a = b/2.

56:00.019 --> 56:05.729
So a = b/2.

56:05.730 --> 56:09.470
That means b is twice as
big as a,

56:09.469 --> 56:12.839
but in any triangle,
the external angle is sum of

56:12.842 --> 56:16.832
the internal opposite angles,
therefore if this is 2a

56:16.826 --> 56:21.946
and this guy is a,
that's also a.

56:21.949 --> 56:26.089
That means the angle of
incidence is the angle of

56:26.088 --> 56:27.208
reflection.

56:27.210 --> 56:31.600
So what I'm telling you is,
you can always go back to angle

56:31.601 --> 56:34.481
of incidence = angle of
reflection,

56:34.480 --> 56:37.520
but it's going to take more
work because you have to find

56:37.518 --> 56:38.278
the tangent.

56:38.280 --> 56:40.290
You've got to draw the normal
to the tangent.

56:40.289 --> 56:42.919
You've got to find the angle
with respect to that normal and

56:42.920 --> 56:44.170
equate it with that angle.

56:44.170 --> 56:47.990
You will find in fact every ray
goes through f.

56:47.989 --> 56:53.039
But it works only in the small
angle approximation.

56:53.039 --> 56:58.609
The small angle basically means
your object is not too tall

56:58.612 --> 57:02.362
compared to the radius of the
mirror.

57:02.360 --> 57:05.650
That's because if the object is
comparable, then the

57:05.650 --> 57:09.010
approximation I made that a
circle will approximate a

57:09.005 --> 57:11.065
parabola is no longer valid.

57:11.070 --> 57:13.170
So remember,
if you took a real parabola,

57:13.170 --> 57:15.690
if you have the stomach,
you can do the following

57:15.690 --> 57:16.530
calculation.

57:16.530 --> 57:19.440
Take a real parabola,
you will find angle of

57:19.440 --> 57:22.350
incidence is angle of reflection
exactly.

57:22.349 --> 57:24.229
Whenever you draw a line,
horizontal line,

57:24.230 --> 57:26.340
that hits the mirror,
comes to the focal point,

57:26.340 --> 57:28.600
if you find the local value of
the perpendicular,

57:28.599 --> 57:31.859
you'll find i = r,
no matter how far you go.

57:31.860 --> 57:33.900
But if you approximate it by a
spherical mirror,

57:33.900 --> 57:36.030
we have seen,
the spherical mirror is only an

57:36.025 --> 57:39.975
approximation to the parabola,
when you can drop the

57:39.981 --> 57:42.121
x^(2) term.

57:42.119 --> 57:47.579
Okay, so we have seen the
Principle of Least Time is able

57:47.577 --> 57:52.737
to give us i = r,
Snell's law and focusing of a

57:52.742 --> 57:54.012
parabola.

57:54.010 --> 57:57.530
Now I want to consider the
following thing:

57:57.525 --> 58:02.465
just because a parabola can
focus light ray at infinity to a

58:02.465 --> 58:07.735
focal point does not mean if you
put a finite object at a finite

58:07.739 --> 58:13.549
distance from it,
it will form a clear image.

58:13.550 --> 58:16.040
It's only an approximation.

58:16.039 --> 58:19.339
And I will show you what we
normally get from geometrical

58:19.338 --> 58:19.868
optics.

58:19.869 --> 58:22.379
I'll remind you what we know
from geometrical optics.

58:22.380 --> 58:24.520
Geometrical optics tells you
the following:

58:24.516 --> 58:27.406
if you have an object of height
h at a distance u from the

58:27.414 --> 58:30.674
mirror and you want to know
where the image will be formed,

58:30.670 --> 58:34.560
you first draw a horizontal
line whose fate you know.

58:34.559 --> 58:39.389
It has to go through the focal
point.

58:39.389 --> 58:43.289
The second one says you draw a
line through the focal point and

58:43.289 --> 58:45.429
it's got to come out horizontal.

58:45.429 --> 58:47.139
How do I know that?

58:47.139 --> 58:49.899
I know that if I run the ray
backwards,

58:49.900 --> 58:53.050
a horizontal ray will go
through the focal point,

58:53.050 --> 58:55.890
but if going backwards,
it's a good idea,

58:55.889 --> 58:59.949
namely least time,
it's also a good idea going

58:59.947 --> 59:00.847
forward.

59:00.849 --> 59:04.869
That's why we know that
parallel will go through focus.

59:04.869 --> 59:06.609
Through focus it will come out
parallel.

59:06.610 --> 59:10.970
So you join them and you've got
the image there.

59:10.969 --> 59:20.169
And that's at a distance
v at a height

59:20.170 --> 59:24.980
h_2.

59:24.980 --> 59:28.970
So I'm now going to use ideas
of geometrical optics,

59:28.969 --> 59:32.109
having shown you enough times
that least time and geometrical

59:32.112 --> 59:34.412
optics are equal,
to find the usual relation

59:34.409 --> 59:38.299
between u,
v and f.

59:38.300 --> 59:40.100
So how do we do that?

59:40.099 --> 59:45.959
We say, take that triangle with
angle alpha and that with angle

59:45.963 --> 59:50.033
alpha and draw a triangle like
this here.

59:50.030 --> 59:54.140
Then you equate tangent of this
angle to tangent of that angle.

59:54.139 --> 1:00:04.099
Then you find tanα
= h_1 divided

1:00:04.097 --> 1:00:08.987
by u - f,
that distance,

1:00:08.994 --> 1:00:13.734
= h_2/f.

1:00:13.730 --> 1:00:15.260
Actually, there's a tiny bit.

1:00:15.260 --> 1:00:16.290
It's not quite f.

1:00:16.289 --> 1:00:19.059
It's (f −
x)^(2), but x^(2) is

1:00:19.059 --> 1:00:22.679
negligible compared to f,
so we won't worry about that.

1:00:22.679 --> 1:00:26.339
You see this triangle here?

1:00:26.340 --> 1:00:27.860
That height is certainly
h_2.

1:00:27.860 --> 1:00:29.460
That length is not quite
f.

1:00:29.460 --> 1:00:32.320
f goes all the way to
the mirror, but if I drop a

1:00:32.324 --> 1:00:34.884
perpendicular here,
there's a tiny little x

1:00:34.876 --> 1:00:35.446
inside.

1:00:35.449 --> 1:00:36.589
I'm now showing you that
x.

1:00:36.590 --> 1:00:38.720
I've been neglecting it.

1:00:38.719 --> 1:00:41.129
In other words,
you really should put an f -

1:00:41.130 --> 1:00:43.910
x, but x is quadratic
in the small numbers,

1:00:43.907 --> 1:00:45.687
so we're not going to keep it.

1:00:45.690 --> 1:00:48.300
Then draw another similar
triangle.

1:00:48.300 --> 1:00:52.730
This angle β
is the same as this angle

1:00:52.730 --> 1:00:53.680
β.

1:00:53.679 --> 1:00:58.609
So let's say tan β
found two ways it's equal.

1:00:58.610 --> 1:01:01.730
This one, tan β
on the top,

1:01:01.730 --> 1:01:05.530
you can see is the
h_1/f

1:01:05.530 --> 1:01:10.140
and that's going to = the tan
β of this triangle,

1:01:10.139 --> 1:01:15.279
which will be
h_2 divided by

1:01:15.284 --> 1:01:17.044
v - f.

1:01:17.039 --> 1:01:18.609
These are the two conditions
you get.

1:01:18.610 --> 1:01:37.220


1:01:37.219 --> 1:01:39.919
Okay, I may want to draw bigger
pictures.

1:01:39.920 --> 1:01:42.410
Can you guys see this,
or there's no hope?

1:01:42.409 --> 1:01:44.429
Can you see in the last row?

1:01:44.429 --> 1:01:45.349
Cannot.

1:01:45.349 --> 1:01:48.699
You should tell me when you
cannot, because I'll be happy to

1:01:48.695 --> 1:01:49.315
fix that.

1:01:49.320 --> 1:01:51.340
So let me draw this bigger.

1:01:51.340 --> 1:01:54.010
The reason that I'm drawing
everything small is that I don't

1:01:54.010 --> 1:01:56.730
want the rays to go too far up
the axis, but I'm not going to

1:01:56.728 --> 1:01:57.678
worry about that.

1:01:57.679 --> 1:01:59.379
So let's make sure you can see
the rays.

1:01:59.380 --> 1:02:06.070
There's one guy who did that,
one which did that,

1:02:06.074 --> 1:02:07.474
correct?

1:02:07.469 --> 1:02:11.969
This is α
and that is α

1:02:11.971 --> 1:02:17.401
and this is β
and this is β`.

1:02:17.400 --> 1:02:19.680
That's all I've done now.

1:02:19.679 --> 1:02:23.979
So tan α
= h_1 divided

1:02:23.976 --> 1:02:26.896
by this side,
where u is the distance

1:02:26.902 --> 1:02:29.082
from the mirror,
you take away f,

1:02:29.077 --> 1:02:32.797
because that's f. That's
h_1 over u -

1:02:32.797 --> 1:02:33.327
f.

1:02:33.329 --> 1:02:36.689
It's the same as tan alpha
measured on this triangle.

1:02:36.690 --> 1:02:38.640
That tan α
is

1:02:38.641 --> 1:02:43.631
h_2/f -
a tiny portion,

1:02:43.630 --> 1:02:48.620
which is the x,
which I'm dropping.

1:02:48.619 --> 1:02:51.239
But similarly for β,
you have a similar rule.

1:02:51.239 --> 1:02:55.659
So here's all you can get out
of these two rays.

1:02:55.659 --> 1:03:04.509
So let's multiply this one and
this one and that one and that

1:03:04.505 --> 1:03:05.385
one.

1:03:05.389 --> 1:03:08.789
So I'll get
h_1h_2

1:03:08.791 --> 1:03:12.451
over u - f times v
- f.

1:03:12.449 --> 1:03:14.889
I'm going to cross multiply
like that.

1:03:14.889 --> 1:03:18.219
It =
h_1h_2

1:03:18.217 --> 1:03:20.637
over f^(2).

1:03:20.639 --> 1:03:26.349
That tells you,
u - f times v - f

1:03:26.353 --> 1:03:29.213
= f^(2).

1:03:29.210 --> 1:03:32.440
You may not have seen it this
way, but it's a very symmetric

1:03:32.438 --> 1:03:33.968
way to write the equation.

1:03:33.969 --> 1:03:37.349
If you measure all distances,
not from this point here but

1:03:37.353 --> 1:03:41.033
from the focal point,
then it says u - f times

1:03:41.034 --> 1:03:43.514
v - f equals
f^(2).

1:03:43.510 --> 1:03:45.670
But let's make contact with
what we all know.

1:03:45.670 --> 1:03:51.890
So let's open out the brackets,
so I get uv - uf - vf f

1:03:51.885 --> 1:03:55.045
squared = f^(2).

1:03:55.050 --> 1:03:57.670
You cancel that guy.

1:03:57.670 --> 1:04:04.680
Then you get uf vf =
uv.

1:04:04.679 --> 1:04:10.699
Now divide everything by
uvf.

1:04:10.699 --> 1:04:20.569
You divided everything by
uvf, you'll get 1/v

1:04:20.565 --> 1:04:23.485
1/u = 1/f.

1:04:23.489 --> 1:04:27.029
Anyway, this is derived by
standard geometrical optics,

1:04:27.030 --> 1:04:30.440
without going back to the
Principle of Least Time.

1:04:30.440 --> 1:04:31.330
So this is the result.

1:04:31.329 --> 1:04:33.599
But there's one more result,
because you've got two

1:04:33.596 --> 1:04:35.496
equations, you can learn one
more thing.

1:04:35.500 --> 1:04:38.600
You can ask yourself,
what's the ratio of the object

1:04:38.599 --> 1:04:40.119
size to the image size?

1:04:40.119 --> 1:04:43.659
You can say,
what is h_2/h

1:04:43.659 --> 1:04:45.429
_1?

1:04:45.429 --> 1:04:49.279
So
h_2/h_1

1:04:49.282 --> 1:04:54.222
is f divided by u
-f.

1:04:54.219 --> 1:04:55.759
Or let me write it another way,
it's easier.

1:04:55.760 --> 1:05:01.750
h_1/h_2
= (u −

1:05:01.746 --> 1:05:03.086
f)/f.

1:05:03.090 --> 1:05:07.710
That's equal to u/f - 1.

1:05:07.710 --> 1:05:13.010
u/f is u times
1/f and 1/f is

1:05:13.010 --> 1:05:16.230
1/u  1 over v - 1.

1:05:16.230 --> 1:05:21.300
And if you do that,
you'll find it's just u

1:05:21.295 --> 1:05:22.945
over v.

1:05:22.949 --> 1:05:26.469
So the ratio of the object size
to the image size is just the

1:05:26.465 --> 1:05:29.565
ratio of the object distance to
the image distance.

1:05:29.570 --> 1:05:33.720
And people sometimes define a
magnification M to be

1:05:33.722 --> 1:05:35.182
−u/v.

1:05:35.179 --> 1:05:38.629
What they mean by the - sign is
that if this comes out negative,

1:05:38.626 --> 1:05:40.976
the object is in the upside
down version.

1:05:40.980 --> 1:05:45.170
The image is an upside down
version of the object.

1:05:45.170 --> 1:05:47.490
In this problem u and
v are both positive,

1:05:47.487 --> 1:05:49.127
then M will come out
negative.

1:05:49.130 --> 1:05:54.220
It just means it is that much
bigger, but flipped upside down.

1:05:54.219 --> 1:05:56.759
In some other mirrors,
you will find v is

1:05:56.764 --> 1:05:58.934
negative because the image is
virtual.

1:05:58.929 --> 1:06:00.929
Then it will mean M is
positive.

1:06:00.929 --> 1:06:02.629
That means the object is
upright.

1:06:02.630 --> 1:06:06.840
That's when you look into the
mirror, the bathroom mirror,

1:06:06.842 --> 1:06:10.762
then your image has the same
orientation as your face,

1:06:10.760 --> 1:06:12.240
not upside down.

1:06:12.239 --> 1:06:16.979
Then M will be positive.

1:06:16.980 --> 1:06:20.810
Okay, now here's a question one
can ask.

1:06:20.809 --> 1:06:25.159
When you do geometric optics,
there's a question one can ask,

1:06:25.164 --> 1:06:28.724
which usually occurs about 30
years afterwards.

1:06:28.719 --> 1:06:30.049
I never asked that question.

1:06:30.050 --> 1:06:32.290
I kept doing all the problems.

1:06:32.289 --> 1:06:35.749
Then a few years ago,
when I started teaching this

1:06:35.753 --> 1:06:39.923
course, I began to ask myself,
you know, you can always draw

1:06:39.922 --> 1:06:42.682
two rays and they will always
meet.

1:06:42.679 --> 1:06:45.199
What if I draw another one?

1:06:45.199 --> 1:06:51.029
What if I draw one that goes
like this?

1:06:51.030 --> 1:06:52.000
How do I know it will come here?

1:06:52.000 --> 1:06:55.090
A lot of pictures show you that
coming here, but should it come

1:06:55.090 --> 1:06:55.440
here?

1:06:55.440 --> 1:06:59.320
Maybe it won't,
so let me check this thing.

1:06:59.320 --> 1:07:03.080
Let me make sure that if I have
a ray coming to the center of

1:07:03.077 --> 1:07:06.707
this, it will also end up where
the other two guys came.

1:07:06.710 --> 1:07:09.960
Well, if you do the center,
remember, it's angle of

1:07:09.960 --> 1:07:13.930
incidence = angle of reflection,
that means the tangent of the

1:07:13.925 --> 1:07:15.285
angles are equal.

1:07:15.289 --> 1:07:17.819
That means h_1/u
should be

1:07:17.815 --> 1:07:19.315
h_2/v.

1:07:19.320 --> 1:07:21.950
Luckily that happens to be
true, because

1:07:21.945 --> 1:07:25.915
h_1/h_2
is in fact u/v.

1:07:25.920 --> 1:07:31.260
Thereby you can show that this
ray, which hits the center of

1:07:31.260 --> 1:07:35.970
the mirror, will also come to
where this one came.

1:07:35.969 --> 1:07:38.779
But that's not enough,
because somebody can draw yet

1:07:38.777 --> 1:07:40.647
another one and yet another one.

1:07:40.650 --> 1:07:44.630
How do you know they will all
come to the same focal point?

1:07:44.630 --> 1:07:48.030
How will you know they're all
from the same image?

1:07:48.030 --> 1:07:50.340
Do you understand the question
now?

1:07:50.340 --> 1:07:53.620
You have to show that every ray
leaving that source hits the

1:07:53.621 --> 1:07:56.571
mirror and comes back to the
very same image point.

1:07:56.570 --> 1:07:59.880
And it's not enough to draw two
rays and show them meeting,

1:07:59.882 --> 1:08:01.942
because two rays will always
meet.

1:08:01.940 --> 1:08:05.010
Now I've drawn a third one and
shown you that it certainly

1:08:05.012 --> 1:08:07.712
comes to the right place,
but that's not enough.

1:08:07.710 --> 1:08:13.530
You can sort of argue that the
evidence is overwhelming it will

1:08:13.528 --> 1:08:17.428
come here,
because if you look at all the

1:08:17.434 --> 1:08:22.004
rays fanning out of this,
the one that went to the top

1:08:21.996 --> 1:08:22.836
came here.

1:08:22.840 --> 1:08:24.750
One that went to the bottom and
also came here,

1:08:24.750 --> 1:08:25.830
but that's pretty solid.

1:08:25.829 --> 1:08:27.459
This involves the focal point.

1:08:27.460 --> 1:08:29.870
One in the middle also came to
the same point.

1:08:29.868 --> 1:08:32.668
You can sort of say,
"Look, this end is good,

1:08:32.667 --> 1:08:35.407
that end is good,
point in the middle is good.

1:08:35.409 --> 1:08:37.389
What do you think will happen?

1:08:37.390 --> 1:08:38.050
We don't know.

1:08:38.050 --> 1:08:41.360
Sometimes it can happen that
there are three points which are

1:08:41.359 --> 1:08:43.399
good, but everything else is
wrong.

1:08:43.399 --> 1:08:46.559
So what is the way to nail this
thing?

1:08:46.560 --> 1:08:50.200
I'm not going to do it today,
but I want you to think about

1:08:50.195 --> 1:08:53.895
what calculation will satisfy
you that no matter where I hit

1:08:53.895 --> 1:08:56.925
the mirror,
I will get the same time.

1:08:56.930 --> 1:09:00.360
What do you think I have to do?

1:09:00.359 --> 1:09:08.589
What would you do?

1:09:08.590 --> 1:09:09.610
What do you want to check?

1:09:09.609 --> 1:09:10.449
Yes?

1:09:10.448 --> 1:09:14.088
Student:  You can make
the object infinitely small

1:09:14.086 --> 1:09:15.836
>

1:09:15.840 --> 1:09:17.720
Prof: Yes,
if you make the object

1:09:17.722 --> 1:09:19.902
infinitely small,
perhaps every ray will start

1:09:19.895 --> 1:09:20.905
looking parallel.

1:09:20.909 --> 1:09:21.829
That's correct.

1:09:21.828 --> 1:09:26.458
Student:  You can take
infinitely small pieces

1:09:26.462 --> 1:09:28.872
>

1:09:28.868 --> 1:09:31.548
Prof: No,
I think I'll explain what my

1:09:31.550 --> 1:09:32.830
question really is.

1:09:32.829 --> 1:09:34.129
Then you can think about it.

1:09:34.130 --> 1:09:36.040
Here is the mirror, right?

1:09:36.038 --> 1:09:39.088
I took an object here,
that's the focal point,

1:09:39.085 --> 1:09:42.665
and I draw some number of rays,
three of them in fact,

1:09:42.672 --> 1:09:43.352
right?

1:09:43.350 --> 1:09:51.560
This one, that one and one
through the center.

1:09:51.560 --> 1:09:54.230
They all came.

1:09:54.229 --> 1:09:57.909
If I want to show you that if I
took an arbitrary point at

1:09:57.912 --> 1:09:59.402
height y--yes.

1:09:59.399 --> 1:10:00.679
Student:  Create an
ellipse based off of the two

1:10:00.676 --> 1:10:00.886
points.

1:10:00.890 --> 1:10:01.530
Prof: Pardon me?

1:10:01.529 --> 1:10:03.899
Student:  Could you
create a function for an ellipse

1:10:03.902 --> 1:10:04.722
off of two points?

1:10:04.720 --> 1:10:05.290
Prof: Yes.

1:10:05.288 --> 1:10:10.848
What you have to do is to pick
a random generic point on that

1:10:10.854 --> 1:10:13.814
graph,
not a parabola,

1:10:13.805 --> 1:10:21.985
and ask how long it will take
light to go to that point and

1:10:21.985 --> 1:10:23.815
come here.

1:10:23.819 --> 1:10:25.449
So what do I want to show?

1:10:25.448 --> 1:10:29.678
Every ray of light hitting this
is going to end up here,

1:10:29.680 --> 1:10:30.450
correct?

1:10:30.448 --> 1:10:36.618
For every possible altitude,
all the way from 0 to the full

1:10:36.622 --> 1:10:37.582
height.

1:10:37.578 --> 1:10:40.248
Now in order to show that it
will come here,

1:10:40.252 --> 1:10:43.862
it also has to be a path of
least time, because you need to

1:10:43.858 --> 1:10:45.908
go in the path of least time.

1:10:45.908 --> 1:10:48.438
These three guys are obviously
path of least time,

1:10:48.439 --> 1:10:49.989
the three rays I showed you.

1:10:49.989 --> 1:10:51.649
I want you never to forget that.

1:10:51.649 --> 1:10:54.009
If three rays leave here and
they meet here,

1:10:54.012 --> 1:10:57.362
that means they take the same
time, because light travels in a

1:10:57.364 --> 1:10:58.634
path of least time.

1:10:58.630 --> 1:11:01.320
If three guys get there,
they all take the same time.

1:11:01.319 --> 1:11:06.019
But it's not enough to consider
that height, 0 height and that

1:11:06.015 --> 1:11:06.705
height.

1:11:06.710 --> 1:11:08.410
I only took a height
h_2,

1:11:08.408 --> 1:11:09.608
h_1 and 0.

1:11:09.609 --> 1:11:13.009
I want to take a generic height
y, and I want to

1:11:13.006 --> 1:11:16.526
calculate that distance,
divided by velocity of light.

1:11:16.529 --> 1:11:20.919
Just take that distance that
distance and show that the

1:11:20.921 --> 1:11:23.931
answer does not depend on
y.

1:11:23.930 --> 1:11:26.150
The answer does not depend on
y, then you vary any

1:11:26.146 --> 1:11:26.936
y you like.

1:11:26.939 --> 1:11:31.629
Then you get the same time.

1:11:31.630 --> 1:11:33.950
So I'm not going to do the
calculation, but I want you to

1:11:33.951 --> 1:11:35.901
think about what it is you want
to calculate.

1:11:35.899 --> 1:11:39.319
I'm going to set it up,
but then come back next time

1:11:39.323 --> 1:11:42.013
and do it, because it takes some
time.

1:11:42.010 --> 1:11:46.150
I'm going to exaggerate
everything so you can sort of

1:11:46.145 --> 1:11:49.005
see what we are trying to do
here.

1:11:49.010 --> 1:11:52.000
You want to go to that guy,
not at that height

1:11:52.002 --> 1:11:54.002
h_1,
sorry.

1:11:54.000 --> 1:11:58.310
I want to pick an arbitrary
height y,

1:11:58.305 --> 1:12:02.105
then I wanted to form an image
here.

1:12:02.109 --> 1:12:04.989
So that's at u,
that's at v,

1:12:04.990 --> 1:12:07.100
that's h_2.

1:12:07.100 --> 1:12:11.280
I want to find that distance
and I want to find that distance

1:12:11.277 --> 1:12:14.337
and add them up,
and the answer should be the

1:12:14.341 --> 1:12:16.501
same as any of the winners.

1:12:16.500 --> 1:12:21.360
The winner I want to take is
this guy, which came like this,

1:12:21.356 --> 1:12:24.646
which I know is one of the
least time.

1:12:24.649 --> 1:12:28.689
If you want,
that corresponds to y =

1:12:28.694 --> 1:12:29.084
0.

1:12:29.078 --> 1:12:32.918
So the time we are trying to
match is really

1:12:32.916 --> 1:12:36.126
h_1^(2)
u^(2)

1:12:36.128 --> 1:12:40.408
h_2^(2)
v^(2) divided by

1:12:40.412 --> 1:12:43.162
c,
but I'm not going to divide by

1:12:43.158 --> 1:12:43.598
c.

1:12:43.600 --> 1:12:46.320
Just imagine everywhere we're
dividing by c.

1:12:46.319 --> 1:12:50.469
The path length for this path
that goes to the middle of the

1:12:50.470 --> 1:12:53.930
mirror and comes out,
you can see from Pythagoras'

1:12:53.934 --> 1:12:56.724
Theorem's h_1^(2 )
u^(2) and

1:12:56.721 --> 1:12:58.941
h_2^(2 ) v^(2).

1:12:58.939 --> 1:13:02.329
And that's going to be equal to
this length

1:13:02.329 --> 1:13:05.799
d_1_
this length

1:13:05.801 --> 1:13:07.821
d_2.

1:13:07.819 --> 1:13:09.749
And they will depend on
y.

1:13:09.750 --> 1:13:12.080
d_1 and
d_2 will depend

1:13:12.078 --> 1:13:12.658
on y.

1:13:12.658 --> 1:13:15.788
And we want to expand it as a
function of y and make

1:13:15.789 --> 1:13:18.809
sure it doesn't vary with
y, and I'll tell you the

1:13:18.811 --> 1:13:20.001
details next time.

1:13:20.000 --> 1:13:25.000