WEBVTT

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Prof: All right,
I'm hoping that this interlude

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into optics will give you a
chance to catch up and be on top

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of the class,
okay?

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I take that to be a yes.

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So if you have to give an
elevator presentation of

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geometrical optics,
you're riding in the elevator

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with some big shot,
you get what,

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60 seconds and you have to tell
that person everything,

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what would you say?

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Any guess?

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Student:
>

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Prof: Right,
Principle of Least Time covers

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everything,
and I thought I would do that,

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because I'm sure you guys,
most of you,

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have seen this in one form or
another.

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And I had to do something
slightly different,

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and that was to show you that
it all came from Least Time.

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Well, I haven't shown the whole
thing yet.

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What I did show you is that
angle of incidence = angle of

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reflection.

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That came out.

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Snell's law, that came out.

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But I was about to do mirrors
and the case of mirrors,

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I said if you want a parallel
set of rays from infinity,

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to focus on one point,
then what do you need to do?

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First of all,
focusing itself is a very

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special case.

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Focusing means many light rays
going in different directions

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all end up in the same place.

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That means they all must
satisfy the Principle of Least

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Time.

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How can all of them be the
least time?

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Because they all take the same
time.

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Therefore there's more than one
way for a light beam to go from

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start to finish and that's when
you get focusing.

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This is the first example of
that, and the way it works is,

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you start with this guy here.

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First of all,
the rays are coming from

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infinity.

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If you're going to keep track
of how long that takes,

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you're going to get infinity
for everybody.

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Maybe infinity 1 for one and
infinity 2 for another one.

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And we know from the time we
were children that they're all

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the same, so infinity's not a
good reference point.

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So you draw the gate here,
a mental gate,

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and you say,
"I'm going to attrack

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these guys from this point on.

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No one's got any head start.

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Then I want to see how long any
of them takes."

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Well, this guy hits that wall,
I mean, this mirror here.

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The mirror is locally vertical,
so it will bounce right back,

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and this focal point is where I
want everyone to end up.

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So this ray does that,
loops around and comes back

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here.

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How much time does it have?

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Well, it has the time to go
here, plus the time to go an

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equal distance f on the other
side.

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That means this light ray which
began here has the same amount

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of time.

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But for this mirror here,
it's got the time to go there.

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But you want it to hit the
mirror and come here in the same

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time.

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That means it will happen,
only that distance and that

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distance are equal.

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And this must be true for any
ray parallel to the axis.

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That means your mirror is a
curve with the property that

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points on it have the same
distance from the focal point

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and from this line here.

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And there's a name for that
curve,

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and that is the parabola,
and satisfies the equation,

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y^(2) = 4xf,
where f is the focal

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length.

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I derived that for you.

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Then the next question is,
it's not enough to have--then I

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told you that if you want,
you can approximate this by a

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spherical mirror.

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You take a sphere or radius
R and you slice a portion

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of it.

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A sphere is not a parabola,
because sphere is going to--put

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a sphere here,
it's going to start deviating

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from the parabola.

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But you can fool this beam of
light if you don't go too far

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from the axis.

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In that region,
it coincides with the parabola.

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And you say, how big a parabola?

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Well, it looks like a parabola
whose focal length is related to

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the radius of that sphere in
this fashion,

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where f is R/2.

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That's called a spherical
mirror.

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So whenever you do this,
you should be aware of all the

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limitations.

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A parabolic mirror will focus a
parallel beam of light,

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no matter how wide it is,
measured from the axis,

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because I've cooked this up to
work for any height.

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If you approximate it with a
spherical mirror,

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then you have to ask yourself,
"How long will this beam

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of light think that the sphere
is a parabola?"

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Well, you can see,
as you start moving off,

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it's going to catch on,
so you cannot go too far off

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the axis.

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So that's an approximation.

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And for a spherical mirror,
if you send a parallel beam of

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light, if it's too thick in the
transverse direction,

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it won't focus here.

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It will focus for small
thickness, small deviation from

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the axis.

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But now our goal in making
mirrors is not only to form the

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image of an object at infinity.

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We want an object at a finite
distance.

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We want object at finite
distance to form an image and

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that's what we were talking
about.

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So let's go back to that
problem.

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Okay, I'm going to be writing
some equations today,

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and I want to request people in
the last row,

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if you cannot read something
you have to tell me if the font

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is small.

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Guys in the front row,
if you cannot read something

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because handwriting is bad,
you've got to tell me.

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People in the last row may not
know why they cannot read it.

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Maybe the handwriting,
maybe the font.

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But between the two of you,
you've got to keep me on the

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lookout for that,
okay?

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Because I did realize last
time, I was drawing more and

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more and more lines in the same
figure.

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It was hard to read.

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All right, so I'm going to
write some equations.

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I'm trying to plan my
blackboard strategy here,

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but first I will tell you what
I'm going to do.

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I mentioned that to you last
time.

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We want to make sure that when
you follow the rules of ray

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optics that you learned in high
school.

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What do you learn in high
school?

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You say, well,
take any object here.

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You draw a bunch of rays and
you know where they're going to

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intersect because the parallel
ray goes through the focal

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point.

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That's how it's defined.

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Here's another ray that's a
little interesting.

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Go through the focal point,
you'll come out parallel.

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How do I know that?

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I know that because if I run
this thing backwards,

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it's a parallel ray going to
the focal point and ending up

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here.

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If this path was the path of
least time, the reversed one

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should also be a path of least
time.

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So these two meet here.

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So we say that's where the
image is.

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But that's not enough.

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I told you two lines will
always meet.

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You've got to draw more lines,
and here is one more that I

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drew.

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The one that hits the center
here, that's i = r,

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very obvious incidence =
reflection, because the mirror

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is vertical here.

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I showed you that the third
line demands that

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h_1/u is
h_2/v.

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But that's already true from
these two rays.

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Remember, the two rays
intersecting here gave the

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condition 1/u 1/v is
1/f and they gave this

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condition.

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So the third ray which I drew
demands the same thing,

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so it's consistent.

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But if you really want to nail
this down,

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you want to be able to prove
that no matter what height you

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pick for the interception of the
mirror here,

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you will take the same time.

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Because I've only taken three
special values for the height

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y.

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One is y =
h_1,

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this beam.

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One is y =
h_2,

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or if you like,
-h_2,

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this beam.

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And y = 0,
the third one,

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and they all happen to agree.

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I want to take an arbitrary
y and hope that the

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answer will not depend on
y.

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The time it takes to travel for
any y should be the same.

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And that's where we will find
it is not exactly the same.

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Yes?

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Student:  What are
u and v?

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Prof: u is the
distance of the object measured

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from here,
and v is the distance of

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the image measured from there.

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I don't know what the book
calls them.

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I don't know which book you
guys are using,

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d_i and
d_o and

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i and o.

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There's a lot of different
symbols you can use.

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So you're free to use any
symbol you want in your problem

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set, as long as you explain to
the TA what you're using.

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Unfortunately,
the only thing everyone agrees

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on is that f stands for
focal length.

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That's universal;
these are random.

08:57.850 --> 09:01.090
All right, so do you understand
what I'm going to do?

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Because it's going to take some
time and it's not worth doing it

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unless you know why we are doing
it.

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We're trying to verify that the
time it takes to go from here to

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the mirror and over here is the
same for every conceivable ray

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for any height y.

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And I'm telling you,
so what would you like to show?

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You would like to show the
following.

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You take this here.

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You want to hit that mirror at
some height y,

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you make it bigger.

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So these are all drawn as very
big,

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so that you can follow them,
but the understanding is,

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they are not really that big
compared to the dimensions of

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the mirror.

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So it goes there and it does
something, and it comes here.

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So this is the point
(x,y) on the parabola

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that it hits.

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d_1 is the
distance--

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I'm going to use Pythagoras'
Theorem to calculate it--

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is that distance,
and d_2 is

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that distance.

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So d_1 depends
on y and

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d_2 depends on
y and when I add them up,

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that's got to be independent of
y.

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That's got to be the same for
all y.

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In particular,
it's got to equal the value

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when y = 0,
= d_1(0)

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d_2(0).

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That's my reference signal.

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That's the one that goes here,
hits the center of that and

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comes here.

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You guys with me now?

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It should be independent of
y, therefore it should be

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equal to the value at any
y I like.

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I'm going to pick the y
= 0 as my reference.

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You find the distance
d_1 d_2

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when y is 0,
you move up and down,

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calculate it,
and it shouldn't depend on

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y.

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So when I do the calculation,
what do I really hope will

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happen?

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If I plot--if I make a plot of
d_1 d_2

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as a function of y,
I'm expecting the answer to be

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flat,
independent of y.

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And this is the reference
point, y = 0.

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But when you do the
calculation, first of all,

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if the graph of
d_1 d_2

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looked like this,
that's no good,

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because that means if you take
y positive,

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you take a longer time than
y negative.

11:52.658 --> 11:55.708
Obviously that y and that y are
not paths of least time,

11:55.712 --> 11:59.102
because one is clearly taking
less time than the other one.

11:59.100 --> 12:03.300
So you don't want anything
growing linearly in y,

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so it's got to be quadratic,
at best.

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But then even if it's
quadratic, you go far out,

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you can see that the answer
depends on y.

12:13.259 --> 12:15.599
So what I will show you is that
the graph,

12:15.600 --> 12:17.660
if you plot it as a function of
y,

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if you plot d_1
d_2 as the

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function of y,
it's got some constant,

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which is the time for y
= 0 some coefficient Ay

12:30.610 --> 12:34.160
some By^(2)
Cy^(3).

12:34.158 --> 12:36.928
These are all possible things
that can happen.

12:36.928 --> 12:41.208
If you want perfect focusing
for any value of y,

12:41.206 --> 12:45.716
all of these guys have to
vanish, because it cannot depend

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on y.

12:46.990 --> 12:53.230
But we will find out it's the
first two guys that vanish and

12:53.231 --> 12:57.571
there is no guarantee about the
others.

12:57.570 --> 13:00.230
But of course,
from dimension analysis,

13:00.230 --> 13:02.690
when you have a y and a
y^(2) and a y^(3),

13:02.690 --> 13:04.830
for them to all have the same
dimensions,

13:04.830 --> 13:08.040
they will be divided by some
numbers.

13:08.038 --> 13:09.968
So this y will be
divided by let's say u,

13:09.965 --> 13:11.245
this will be divided by
u^(2),

13:11.248 --> 13:12.708
this will be divided by
u^(3).

13:12.710 --> 13:18.050
It's an expansion in a small
parameter, and only when the

13:18.053 --> 13:21.493
parameter is small will this
work.

13:21.490 --> 13:25.610
So you have to notice two
things: in a parabolic mirror an

13:25.605 --> 13:28.775
object at infinity will form a
perfect image,

13:28.782 --> 13:31.312
no matter how wide the beam is.

13:31.308 --> 13:34.998
But an object at a finite
distance will not form a perfect

13:34.999 --> 13:37.199
image if the object is too tall.

13:37.200 --> 13:40.860
If the height y becomes too
much, then you don't get a good

13:40.855 --> 13:41.355
image.

13:41.360 --> 13:44.450
But you get a pretty good
image, because the possible

13:44.445 --> 13:47.585
dependence on y and
y^(2) will be 0,

13:47.590 --> 13:50.830
but not to arbitrary powers of
y.

13:50.830 --> 13:51.440
Do you understand?

13:51.440 --> 13:56.320
If you have a graph that's a
power series in y, if it

13:56.322 --> 14:00.462
begins with y^(2) it will
look like this.

14:00.460 --> 14:03.730
If the first non-zero term is
y to the fourth,

14:03.726 --> 14:07.116
it will look very flat,
but eventually it will move.

14:07.120 --> 14:10.070
The flatter this is,
the further out you can pretend

14:10.066 --> 14:11.796
nothing depends on y.

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So we want to see how far we
can go.

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That's the purpose.

14:14.970 --> 14:20.200
It will also tell you the
limits of a real mirror for a

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real object.

14:21.548 --> 14:26.428
Any questions on calculation
I'm about to do?

14:26.428 --> 14:31.008
Okay, before I do the
calculation, you're going to

14:31.009 --> 14:33.999
need the following
approximation:

14:34.001 --> 14:38.771
u^(2) θ^(2) to the
power ½.

14:38.769 --> 14:43.259
You're going to get such
combinations all the time.

14:43.259 --> 14:47.429
You can write that as u
times 1 θ^(2) over

14:49.549 --> 14:50.739
That's exact.

14:50.740 --> 14:51.470
Yes?

14:51.470 --> 14:56.870
Student:
>

14:56.870 --> 15:00.240
Prof: Oh,
R in the top?

15:00.240 --> 15:03.530
R/2.

15:03.528 --> 15:08.358
R stands for the radius
of the sphere from which you cut

15:08.364 --> 15:09.694
out that piece.

15:09.690 --> 15:11.300
Everybody understands that?

15:11.298 --> 15:17.128
We had a parabola here,
and I'm going to approximate it

15:17.128 --> 15:23.818
by taking a sphere of some
radius R and that sphere,

15:23.820 --> 15:26.940
if you look at this portion
here, it looks like a parabola.

15:26.940 --> 15:28.630
You can ask,
"What parabola does it

15:28.631 --> 15:29.501
approximate?"

15:29.500 --> 15:34.070
I showed you that near the
origin, it looks like a parabola

15:34.067 --> 15:35.877
whose f = R/2.

15:35.879 --> 15:38.239
So if you take a sphere of
radius R and you slice a

15:38.235 --> 15:40.625
portion off it and you paint it
with silver so it becomes a

15:40.633 --> 15:43.083
mirror,
its focal length will be

15:43.082 --> 15:44.062
R/2.

15:44.058 --> 15:46.268
But it won't be a perfect
parabolic mirror.

15:46.269 --> 15:48.439
It's an approximate parabolic
mirror.

15:48.440 --> 15:50.550
There are different levels of
approximation.

15:50.548 --> 15:54.618
A sphere will not focus a
parallel beam even if the beam

15:54.623 --> 15:55.663
is too wide.

15:55.658 --> 15:58.888
A parabola will focus the
parallel beam no matter how wide

15:58.893 --> 16:01.393
it is, provided the object is at
infinity.

16:01.389 --> 16:04.279
If the object comes to a finite
distance, I'm saying even the

16:04.278 --> 16:06.828
parabolic mirror will not give
you a perfect image.

16:06.830 --> 16:09.930
But if it's very short,
short compared to u and

16:09.927 --> 16:12.497
v and so on,
it will do a good job.

16:12.500 --> 16:14.120
So let's see how that happens.

16:14.120 --> 16:18.060
So I'm going to need this
approximation that looks like

16:18.057 --> 16:18.857
u.

16:18.860 --> 16:21.980
now this thing,
I'm going to write as u

16:21.977 --> 16:25.787
times 1 θ^(2) over
2u squared ...

16:25.788 --> 16:29.088
θ to the fourth,
u to the fourth,

16:29.089 --> 16:29.469
etc.

16:29.470 --> 16:32.480
I'm going to drop all this.

16:32.480 --> 16:36.970
This is the approximate
expansion, (1 x)^(n) is 1

16:36.970 --> 16:37.950
nx.

16:37.950 --> 16:44.610
So that becomes = u
θ^(2) over 2u.

16:44.610 --> 16:49.450
So let's bear that in mind.

16:49.450 --> 16:51.990
This is just a mathematical
preliminary I'm getting out of

16:51.994 --> 16:53.784
the way so we don't get stuck in
that.

16:53.779 --> 16:54.489
Yes?

16:54.490 --> 16:57.930
Student:
>

16:57.928 --> 17:02.908
Prof: It was squared
here, but when I multiplied by

17:02.910 --> 17:05.970
this u,
it got un-squared.

17:05.970 --> 17:08.710
So in this calculation,
I mentioned to you,

17:08.709 --> 17:11.969
some numbers are big and some
numbers are small.

17:11.970 --> 17:14.420
I said that near the end of the
class.

17:14.420 --> 17:21.770
u, v, f are treated as
big numbers.

17:21.769 --> 17:25.029
Compared to that,
the height h_1

17:25.034 --> 17:27.864
of the object,
height h_2 of

17:27.859 --> 17:31.029
the image and y, the
height at which the light hits

17:31.034 --> 17:34.434
the mirror,
they're called small.

17:34.430 --> 17:38.520
And the coordinate x on
that mirror where you hit it is

17:38.519 --> 17:40.599
proportional to y^(2).

17:40.599 --> 17:42.129
You remember that?

17:42.130 --> 17:46.830
So it's proportional to small
squared.

17:46.828 --> 17:52.208
Remember the parabola equation
is x = y^(2)/4f.

17:52.210 --> 17:54.520
In fact, let me write it
exactly.

17:54.519 --> 17:58.719
You can see that f is a
number that's called--you can

17:58.720 --> 18:02.490
call these order one numbers,
these are small numbers,

18:02.493 --> 18:05.203
these are smaller numbers
squared.

18:05.200 --> 18:07.710
So in all these calculations,
I think you have no trouble

18:07.711 --> 18:10.361
seeing why x is much
smaller than y, right?

18:10.358 --> 18:13.048
The shape of the parabola is,
if you climb a lot of height,

18:13.050 --> 18:15.190
you don't move a lot in the
x direction,

18:15.185 --> 18:16.435
because it's quadratic.

18:16.440 --> 18:19.210
And the square of a small
number is even smaller.

18:19.210 --> 18:21.680
So if y is small,
x, which is

18:21.678 --> 18:23.268
y^(2), is even smaller.

18:23.269 --> 18:26.889
So I want to do this,
because no matter what field of

18:26.894 --> 18:29.464
life you go into--
well, not every field,

18:29.455 --> 18:32.155
but in most fields where you
have to do some kind of

18:32.161 --> 18:34.411
calculation,
whether it's in economics or

18:34.414 --> 18:36.834
biology or whatever,
you should know how to do

18:36.827 --> 18:38.187
approximate calculations.

18:38.190 --> 18:42.460
I'm trying to show you how to
do a systematic approximate

18:42.461 --> 18:43.531
calculation.

18:43.529 --> 18:46.889
Most of the calculations people
do are approximate.

18:46.890 --> 18:50.030
In fact, every calculation ever
done by any physicist is

18:50.027 --> 18:52.877
approximate, because we're
neglecting something.

18:52.880 --> 18:56.200
You show me one result that you
swear is accurate,

18:56.196 --> 18:57.006
it is not.

18:57.009 --> 18:59.379
Everything has got corrections.

18:59.380 --> 19:02.670
So we should learn how to find
the big numbers in the problem

19:02.672 --> 19:05.582
and keep them and how to
organize our calculation,

19:05.578 --> 19:07.708
if you like,
as a series in small and

19:07.710 --> 19:09.900
smaller squared and smaller
cubed,

19:09.900 --> 19:12.890
and you keep a few terms.

19:12.890 --> 19:17.090
So let's calculate now the
distance, d_1

19:17.086 --> 19:19.106
and d_2.

19:19.108 --> 19:21.938
So we'll have to look at the
picture here.

19:21.940 --> 19:24.390
Let me just write down what I
get.

19:24.390 --> 19:32.960
d_1 d_2
is going to look like

19:32.963 --> 19:42.633
h_1 - y whole
squared u - x squared.

19:42.630 --> 19:50.320
That is d_1.
And d_2 will be

19:50.320 --> 19:57.130
h_2 y squared
v - x squared.

19:57.130 --> 20:01.490
Can you see in that figure in
the right that the right angle

20:01.491 --> 20:06.151
triangle--maybe I'll bring it
down one more time to show you.

20:06.150 --> 20:09.260
In fact, this is what we really
need, so let's keep it here.

20:09.259 --> 20:12.639
This whole thing is
h_1. This

20:12.637 --> 20:16.137
height is y,
so this side is

20:16.144 --> 20:22.114
h_1 - y,
and this side here is u

20:22.108 --> 20:24.178
- x,
because u brings you

20:24.183 --> 20:27.653
from here to the mirror,
but this point is to the right

20:27.646 --> 20:29.536
by an amount x.

20:29.539 --> 20:31.199
That's this right triangle.

20:31.200 --> 20:34.380
Then there's a second right
angled triangle here to find

20:34.380 --> 20:35.710
d_2.

20:35.710 --> 20:40.060
That height is h_2
y and that distance is again

20:40.063 --> 20:41.103
v - x.

20:41.098 --> 20:47.588
This is the point v and
this is the point u.

20:47.588 --> 20:54.338
And I want that to be same as
when the beam goes to this

20:54.337 --> 20:56.667
dotted line here.

20:56.670 --> 20:58.410
That's the y = 0 answer.

20:58.410 --> 21:00.610
That's very easy to do
Pythagoras on that one.

21:00.608 --> 21:03.568
This dotted square is
h_1^(2) u^(2)

21:03.574 --> 21:07.224
under root,
and this dotted square =

21:07.223 --> 21:11.733
h_2^(2) v^(2)
under root.

21:11.730 --> 21:25.680
So this whole thing has got to
= h_1^(2) u^(2)

21:25.682 --> 21:32.902
h_2^(2) v^(2).

21:32.900 --> 21:36.930
So the right hand side does not
depend on y.

21:36.930 --> 21:39.600
Left hand side depends on
y.

21:39.598 --> 21:43.178
In the best of all worlds,
somehow all the y

21:43.178 --> 21:47.038
dependents will magically
cancel, but it's not going to

21:47.041 --> 21:49.691
happen, I can tell you right
now.

21:49.690 --> 21:53.630
What you will find is that the
term linear in y will

21:53.633 --> 21:57.783
cancel, the term quadratic and
y can be made to cancel,

21:57.779 --> 21:59.819
and that's all you can do.

21:59.818 --> 22:02.078
So let's now look at these two
things.

22:02.078 --> 22:03.368
Let me look at the right hand
side.

22:03.369 --> 22:04.719
It's very easy to do this one.

22:04.720 --> 22:06.540
I only paved the way for this.

22:06.538 --> 22:09.608
Remember, h_1
is a small number,

22:09.606 --> 22:11.456
u is the big number.

22:11.460 --> 22:14.510
When you do this one,
you will get u

22:14.509 --> 22:17.639
h_1^(2) over
2u.

22:17.640 --> 22:18.930
You see that?

22:18.930 --> 22:22.370
h^(2) is like
θ^(2) in my

22:22.367 --> 22:23.147
formula.

22:23.150 --> 22:28.460
And this will look like v
h_2^(2) over

22:28.460 --> 22:29.640
2v.

22:29.640 --> 22:33.970
So the formula I'm using is
u^(2 ) θ^(2) to

22:33.972 --> 22:38.232
the½ = u θ^(2)
over 2u terms

22:38.227 --> 22:41.927
involving θ
to the fourth and so on,

22:41.930 --> 22:46.680
which I'm not keeping.

22:46.680 --> 22:49.170
So the right hand side is very
easy to calculate.

22:49.170 --> 22:49.900
It's that.

22:49.900 --> 22:52.020
So let's look at the left hand
side now.

22:52.019 --> 22:55.299
So let me open out everything
in the bracket here.

22:55.298 --> 23:11.028
I get h_1^(2)
y^(2) - 2h_1y u^(2) -

23:11.025 --> 23:15.435
2ux x^(2).

23:15.440 --> 23:23.030
Now the second guy is
h_2^(2) y^(2 )

23:23.027 --> 23:29.347
2h_2y v^(2) - 2vx
x^(2).

23:29.348 --> 23:35.338
That's got to equal the u v
h_1^(2 )over 2u

23:35.335 --> 23:39.095
h_2^(2) over
2v.

23:39.098 --> 23:42.908
Remember, the right hand side
is not really--I've put a dot to

23:42.909 --> 23:44.969
tell you it's an approximation.

23:44.970 --> 23:46.730
This square root goes on
forever.

23:46.730 --> 23:50.410
I've kept the leading,
first leading term.

23:50.410 --> 23:55.820
The first leading term happens
to be quadratic in h.

23:55.818 --> 23:59.898
Quadratic in the small
quantity, so to be consistent,

23:59.904 --> 24:05.014
I should look at the left hand
side, also to the same accuracy.

24:05.009 --> 24:08.169
So let's look at the left hand
side here.

24:08.170 --> 24:13.800
This is quadratic in the small
guy, quadratic in the small guy,

24:13.800 --> 24:16.620
small times small,
a big square,

24:16.616 --> 24:20.156
big times small,
and small squared.

24:20.160 --> 24:22.400
But you've got to be a little
careful.

24:22.400 --> 24:25.110
x is not small;
x is small squared.

24:25.109 --> 24:25.789
Remember that?

24:25.788 --> 24:27.298
x is proportional to
y^(2).

24:27.298 --> 24:29.398
This is proportional to
y to the fourth.

24:29.400 --> 24:32.050
You cannot keep a term to
fourth order in the small

24:32.054 --> 24:35.354
quantity on the left hand side
if you don't keep a similar term

24:35.346 --> 24:36.776
on the right hand side.

24:36.779 --> 24:40.549
So you systematically drop
things which are quartic in the

24:40.549 --> 24:41.079
small.

24:41.078 --> 24:45.458
So even though x makes
it, x^(2) doesn't make it

24:45.457 --> 24:47.177
in this approximation.

24:47.180 --> 24:47.900
Yes?

24:47.900 --> 24:52.380
Student:
>

24:52.380 --> 24:54.610
Prof: Yeah,
it can be anywhere it wants.

24:54.608 --> 24:56.468
You may think that because it's
under square root,

24:56.468 --> 24:58.288
when you take the root of it,
it may become x.

24:58.289 --> 25:00.009
Is that your worry?

25:00.009 --> 25:01.709
Is that what you're saying?

25:01.710 --> 25:04.230
Student:
>

25:04.230 --> 25:06.780
Prof: The point you're
making is, I see an x^(2)

25:06.776 --> 25:07.566
like that, right?

25:07.568 --> 25:08.648
But it's really not
x^(2);

25:08.652 --> 25:09.432
it's really x.

25:09.430 --> 25:11.340
That's certainly true.

25:11.338 --> 25:14.618
But you're adding a whole bunch
of numbers and then taking the

25:14.616 --> 25:15.796
square root of that.

25:15.798 --> 25:18.618
In that sum,
these numbers are going to be

25:18.621 --> 25:22.201
big compared to this one,
whether you take the square

25:22.199 --> 25:23.919
root r or not.

25:23.920 --> 25:27.820
So it is dominated by the terms
under the same square root.

25:27.819 --> 25:32.479
That's why you drop it.

25:32.480 --> 25:34.590
Okay, so this fellow is out.

25:34.589 --> 25:37.929
Here's what I want to do now.

25:37.930 --> 25:40.900
I'm going to write the first
thing.

25:40.900 --> 25:42.450
So let's decide what to call
what.

25:42.450 --> 25:44.010
This is my u^(2).

25:44.009 --> 25:48.399
The rest of these guys are my
θ^(2).

25:48.400 --> 25:49.780
You understand that?

25:49.779 --> 25:52.069
Unfortunately,
they're not next to each other,

25:52.067 --> 25:53.947
but all of that is
θ^(2),

25:53.948 --> 25:55.218
this is u^(2).

25:55.220 --> 26:04.870
Then the formula tells me,
this = u h_1^(2)

26:04.868 --> 26:12.588
y^(2) - 2h_1y - 2ux
over u v

26:12.587 --> 26:22.057
h_2^(2) y^(2)
2h_2y 2vx divided

26:22.061 --> 26:31.181
by v = u v
h_1^(2) over 2u

26:31.184 --> 26:39.084
h_2^(2) over
2v.

26:39.078 --> 26:45.068
This is what I want to this
order.

26:45.068 --> 26:47.998
Now I have an x
everywhere, but x is not

26:47.996 --> 26:48.766
independent.

26:48.769 --> 26:54.529
x = y^(2)/4f.

26:54.529 --> 26:57.439
So what is -2ux.

26:57.440 --> 27:07.470
-2ux will be
−y^(2) over

27:07.472 --> 27:13.542
2u over f.

27:13.539 --> 27:14.939
Yes?

27:14.940 --> 27:16.830
Student:  Should the
denominator for those be

27:16.834 --> 27:17.804
2u and 2v?

27:17.799 --> 27:20.959
Prof: Yes, thank you.

27:20.960 --> 27:27.300
Very good.

27:27.298 --> 27:32.988
Okay, therefore--so let's
compare everything.

27:32.990 --> 27:36.540
The big numbers have to cancel
the big numbers,

27:36.535 --> 27:39.305
otherwise you've got big
problems.

27:39.309 --> 27:40.639
You guys have a question?

27:40.640 --> 27:50.270
Student:
>

27:51.788 --> 27:55.478
but I've taken this
approximation that when you take

27:57.440 --> 27:58.680
Look at the top right.

27:58.680 --> 28:02.070
u^(2) θ^(2) to the
power ½

28:02.065 --> 28:05.145
is u θ^(2) over
2u.

28:05.150 --> 28:08.600
Okay?

28:08.598 --> 28:12.148
Yes, so I've done this a lot of
times, but I want you to follow

28:12.147 --> 28:15.117
this, because this is telling
you how to organize the

28:15.124 --> 28:16.044
calculation.

28:16.039 --> 28:17.189
This is how we organize.

28:17.190 --> 28:22.460
We identify quantities which
are quantities of importance,

28:22.460 --> 28:26.010
and then smaller and smaller
quantities and you balance the

28:26.006 --> 28:28.376
equation,
starting with the big numbers,

28:28.380 --> 28:30.130
then go to the smaller numbers.

28:30.130 --> 28:33.220
So the big numbers better match.

28:33.220 --> 28:38.090
So that tells you u v
matches u v.

28:38.089 --> 28:39.599
That's fine.

28:39.598 --> 28:44.158
Now take the terms proportional
to y.

28:44.160 --> 28:45.450
They have to separately match.

28:45.450 --> 28:49.200
When you have a function of
y, then if it functions

28:49.196 --> 28:52.676
equal on the two sides,
every power of y should

28:52.682 --> 28:53.342
match.

28:53.338 --> 28:56.408
Of course, we only have two
powers here and they should both

28:56.413 --> 28:56.833
match.

28:56.828 --> 28:59.968
So take the term linear in
y from both sides.

28:59.970 --> 29:02.490
Oh, let's get it to one more
stop.

29:02.490 --> 29:04.830
Do you see this
h_1

29:04.827 --> 29:07.767
^(2)/2u canceled this
one?

29:07.769 --> 29:10.729
And this h_2
^(2)/2v cancels that

29:10.731 --> 29:11.051
one.

29:11.049 --> 29:13.709
So I've got nothing left.

29:13.710 --> 29:16.680
So everything on the left hand
side should add up to 0,

29:16.676 --> 29:19.366
because there's nothing on the
right hand side.

29:19.368 --> 29:23.048
So what do I have to linear
order and y?

29:23.048 --> 29:27.548
This guy is linear and
y, this guy is linear and

29:27.545 --> 29:30.455
y and that's all there
is.

29:30.460 --> 29:36.030
So the coefficient of y
looks like y times

29:36.032 --> 29:41.912
-2h_1 over
2u 2h_2

29:41.911 --> 29:46.371
over 2v and that's got to
vanish.

29:46.368 --> 29:51.938
And that simply says
h_1/u is

29:51.942 --> 29:55.042
h_2/v.

29:55.038 --> 29:57.998
So this is what you get by
usual geometric optics

29:58.000 --> 29:58.740
arguments.

29:58.740 --> 30:01.630
I showed you last time with
similar triangles,

30:01.630 --> 30:04.760
but it comes from the Principle
of Least Time,

30:04.759 --> 30:08.429
by demanding that the function,
the time doesn't depend on y,

30:08.430 --> 30:11.260
to first order and y.

30:11.259 --> 30:13.889
Then I have to do the
y^(2) term,

30:13.892 --> 30:14.772
understand?

30:14.769 --> 30:19.219
You understand if you've got
numbers on the two sides,

30:19.220 --> 30:22.300
big numbers,
square of small numbers,

30:22.298 --> 30:25.378
linear and y,
quadratic and y,

30:25.380 --> 30:28.230
everything has to match,
order by order.

30:28.230 --> 30:29.770
So this is a linear and
y term.

30:29.769 --> 30:33.169
What is the quadratic and
y term?

30:33.170 --> 30:36.540
Quadratic and y,
look at the coefficients.

30:39.450 --> 30:41.840
You see that?

30:41.838 --> 30:45.108
And this one has got 1 over
2v, coefficient of

30:45.106 --> 30:46.046
y^(2).

30:46.048 --> 30:51.388
So I've got 1 over 2u 1
over 2v,

30:51.386 --> 30:57.076
but then I've got -2ux
and -2vx.

30:57.078 --> 31:03.488
So what is -2ux,
is −y^(2) u over

31:03.490 --> 31:10.270
2f divided by 2u -
2 y squared v by

31:10.270 --> 31:15.080
2f divided by 2v =
0.

31:15.078 --> 31:20.488
Student:
>

31:20.490 --> 31:21.820
Prof: Yes, I'm sorry.

31:21.819 --> 31:23.489
I'm not being consistent.

31:23.490 --> 31:27.110
So let me just call this the
coefficient of y squared.

31:27.109 --> 31:29.029
Do you agree?

31:29.028 --> 31:30.598
It's the coefficient of the
y^(2).

31:30.598 --> 31:33.468
If you want,
it's in front of everything,

31:33.472 --> 31:36.492
all of this is multiplying
y^(2).

31:36.490 --> 31:38.380
So this has to vanish.

31:38.380 --> 31:47.000
Okay, so cancel the 2's
everywhere, you get 1/u

31:47.003 --> 31:49.123
1/v.

31:49.119 --> 31:51.649
This cancels.

31:51.650 --> 31:56.460
Let me see.

31:56.460 --> 32:00.050
There is no 2 here,
because 2ux = y^(2)

32:00.048 --> 32:01.768
u over 2f.

32:01.769 --> 32:03.789
This 2 is not there, sorry.

32:03.789 --> 32:05.129
This is all you have.

32:05.130 --> 32:13.180
You get -1 over 2f -
another 1 over 2f = 0.

32:13.180 --> 32:16.130
Look guys, we know where we are
going, right?

32:16.130 --> 32:20.030
So it's going to happen.

32:20.028 --> 32:22.838
It's happened for the last 300
years and I'm not going to

32:22.836 --> 32:24.386
change it, so it will happen.

32:24.390 --> 32:27.050
So the 2's will all work out.

32:27.049 --> 32:28.769
What does this tell you?

32:28.769 --> 32:36.099
It tells you 1/u
1/v is 1/f.

32:36.098 --> 32:39.988
So you can ask,
what's the point of doing this

32:39.989 --> 32:40.939
exercise?

32:40.940 --> 32:46.730
This tells you that because of
the Principle of Least Time,

32:46.730 --> 32:49.460
the time it takes to go not
only for these two privileged

32:49.460 --> 32:51.450
rays,
but any ray you can think of

32:51.451 --> 32:54.141
drawing from here to the mirror
and back here.

32:54.140 --> 32:59.440
They all take the same time,
in the approximation in which

32:59.442 --> 33:02.232
the height over u,
for example,

33:02.233 --> 33:05.773
to the fourth power,
is neglected.

33:05.769 --> 33:09.399
Basically, anything that looks
like small over big to the

33:09.395 --> 33:12.435
fourth power and higher powers
are neglected.

33:12.440 --> 33:15.820
So it's not going to form a
perfect image for everything.

33:15.818 --> 33:19.138
It will form a perfect image in
a world in which you are able to

33:19.143 --> 33:21.573
keep the term to leading order
in u,

33:21.568 --> 33:23.548
to the order
h^(2)/u^(2),

33:23.549 --> 33:25.139
and stop.

33:25.140 --> 33:27.240
This is an approximate thing.

33:27.240 --> 33:30.420
So for it to successfully work,
you've got to keep your object

33:30.415 --> 33:33.175
and the image all to be much
smaller than u and

33:33.175 --> 33:34.525
v and f.

33:34.529 --> 33:35.979
That's the rule.

33:35.980 --> 33:39.440
If you take any parabolic
mirror and you have an object

33:39.442 --> 33:43.102
which is as tall as the focal
length, it won't form a good

33:43.097 --> 33:43.737
image.

33:43.740 --> 33:46.610
What it will mean,
what do you think it means not

33:46.605 --> 33:47.975
to form a good image?

33:47.980 --> 33:51.930
Rays will cross at different
points and you will get a

33:51.934 --> 33:53.134
blurred image.

33:53.130 --> 33:57.550
If you put a screen,
if you put a candle here,

33:57.548 --> 34:00.118
if that's a candle,
you can actually put a real

34:00.116 --> 34:03.016
screen here and there'll be a
glow on the screen,

34:03.019 --> 34:07.469
image of the candle upside down.

34:07.470 --> 34:11.660
It will be a very sharp image,
if the candle's not too tall.

34:11.659 --> 34:14.609
Once it gets taller,
image will get blurred,

34:14.605 --> 34:18.095
because the rays will not meet
at the same point.

34:18.099 --> 34:21.259
All right, I want to do one
other thing, then I'm going to

34:21.255 --> 34:23.465
stop with the Principle of Least
Time.

34:23.469 --> 34:26.439
That has to do with lenses.

34:26.440 --> 34:27.630
You saw how a mirror does a job.

34:27.630 --> 34:30.120
Let's take a lens.

34:30.119 --> 34:40.649
Let's say we have the following
lens.

34:40.650 --> 34:44.730
And I have an object which is
infinitesimally tall to begin

34:44.726 --> 34:45.216
with.

34:45.219 --> 34:56.469
And I wanted to form an image
somewhere on the other side.

34:56.469 --> 35:02.189
So you can ask yourself,
how can the light rays leaving

35:02.186 --> 35:08.116
this in different directions
arrive here and all take the

35:08.117 --> 35:09.597
same time?

35:09.599 --> 35:13.589
It's clear that this guy is the
shortest path.

35:13.590 --> 35:17.850
These are taking longer,
and yet they have to be

35:17.851 --> 35:21.571
competitive with the one in the
middle.

35:21.570 --> 35:23.390
So how do you think that
happens?

35:23.389 --> 35:28.389
How do you think they will take
the same time?

35:28.389 --> 35:29.339
Yes.

35:29.340 --> 35:35.370
Student:
>

35:35.369 --> 35:36.289
Prof: That's correct.

35:36.289 --> 35:39.999
The answer was,
it is true that this path is

35:40.001 --> 35:44.921
shorter than one that goes to
the top of the lens and goes

35:44.920 --> 35:48.460
down here, but this is through
glass.

35:48.460 --> 35:52.160
It's like saying I have a lake
here shaped like this,

35:52.164 --> 35:56.444
and I tell you to run so that
you go from here to here in the

35:56.440 --> 35:57.510
least time.

35:57.510 --> 36:00.080
It's not at all clear that
heading straight for the lake is

36:00.079 --> 36:02.339
the best thing to do,
because you've got to swim.

36:02.340 --> 36:05.580
This person doesn't have to
swim at all, but has to go

36:05.577 --> 36:06.187
further.

36:06.190 --> 36:10.960
You can imagine somehow cooking
things up so that they all take

36:10.958 --> 36:12.188
the same time.

36:12.190 --> 36:14.280
So I'm going to do it only for
the two extreme rays.

36:14.280 --> 36:17.000
Since I've done a lot of
algebra, I don't want to keep

36:17.003 --> 36:17.573
doing it.

36:17.570 --> 36:25.140
So let us say this height is
h, height of the lens,

36:25.135 --> 36:32.035
and this distance is u,
this distance is v.

36:32.039 --> 36:37.349
And the refractive index is n
of the lens.

36:37.349 --> 36:42.299
What that means is,
1 meter in air--I'm sorry.

36:42.300 --> 36:45.990
1 meter in glass is like
n meters in the air,

36:45.989 --> 36:49.029
you understand,
in terms of travel time.

36:49.030 --> 36:49.710
Everybody with me?

36:49.710 --> 36:52.350
That's what it means to say it
goes slower.

36:52.349 --> 36:55.329
So a meter of lens is worth
n meters of air.

36:55.329 --> 36:58.439
That's the consequence of the
velocity of light.

36:58.440 --> 37:01.480
We're trying to find equal
times, so when it travels here,

37:01.480 --> 37:04.230
it travels with a velocity
c/n,

37:04.230 --> 37:06.810
therefore the time it takes,
when you divide the distance by

37:06.809 --> 37:10.159
time,
will be n times as big.

37:10.159 --> 37:12.829
Okay, then there's one other
result I'm going to need,

37:12.829 --> 37:15.599
so I don't want to interrupt
you later,

37:15.599 --> 37:20.779
which is the following:
I'll take this glass here,

37:20.780 --> 37:24.650
this face of the lens,
and let's say it is coming from

37:24.648 --> 37:26.838
a sphere of radius R.

37:26.840 --> 37:33.600
So that's the center of the
sphere.

37:33.599 --> 37:40.069
And draw a normal here and call
that distance delta and call

37:40.068 --> 37:42.588
this height h.

37:42.590 --> 37:44.220
This is all exaggerated.

37:44.219 --> 37:48.289
In reality, delta's a very
small number.

37:48.289 --> 37:53.249
I'm going to need an
approximate formula for delta in

37:53.251 --> 37:56.401
terms of h and R.

37:56.400 --> 38:02.650
So since it's a sphere,
that distance is also R.

38:02.650 --> 38:03.830
Right?

38:03.829 --> 38:07.129
This is a sphere of radius
R from which I've cut out

38:07.132 --> 38:09.812
that portion,
which is this face of the lens.

38:09.809 --> 38:13.739
The lens ends here,
but the mental sphere can keep

38:13.737 --> 38:15.177
going like that.

38:15.179 --> 38:17.969
So what can you say about this
distance delta?

38:17.969 --> 38:19.309
You can say the following?

38:19.309 --> 38:27.929
delta = R - this side.

38:27.929 --> 38:31.589
So let's write the formula this
side, which I don't know,

38:31.590 --> 38:34.270
let's say is l,
is R - l.

38:34.268 --> 38:44.968
But we also know that l^(2)
h^(2 )= R^(2).

38:44.969 --> 38:52.349
Therefore l^(2) = R^(2)
− h^(2),

38:52.349 --> 38:57.919
or l is square root of
R^(2) −

38:57.920 --> 39:01.910
h^(2),
and by the same trick I did,

39:01.911 --> 39:05.741
it is R −
h^(2) over 2R.

39:05.739 --> 39:11.619
The same θ^(2) formula.

39:11.619 --> 39:16.389
So l is shorter than R
by an amount h^(2)/2R,

39:16.391 --> 39:19.191
so this delta =
h^(2)/2R.

39:19.190 --> 39:24.910
That's the formula I'm going to
invoke.

39:24.909 --> 39:28.319
You will see why I bother to
prove that in a minute.

39:28.320 --> 39:34.740
So you should now think about
what equation you will write

39:34.744 --> 39:36.214
down, okay?

39:36.210 --> 39:41.080
I'm going to give you a few
seconds to think about what two

39:41.083 --> 39:44.643
things should be equal,
for the condition that

39:44.639 --> 39:48.319
everything takes the same amount
of time to go from start to

39:48.315 --> 39:48.935
finish.

39:48.940 --> 39:52.490
So first I'm going to find the
time it takes to go on that

39:52.492 --> 39:53.492
straight line.

39:53.489 --> 39:56.299
By the way, since I'm going to
divide by velocity everywhere to

39:56.304 --> 39:58.364
get time,
distance is all you need,

39:58.358 --> 40:01.498
except here you've got to
remember 1 inch of glass is

40:01.498 --> 40:03.068
n inches of air.

40:03.070 --> 40:04.580
If you remember that,
you'll be fine.

40:04.579 --> 40:07.259
So let's see how long is that
path.

40:07.260 --> 40:15.790
That path = square root of
u^(2) h^(2) square root

40:15.786 --> 40:19.436
of v^(2) h^(2).

40:19.440 --> 40:23.060
By right angle,
this right angle triangle,

40:23.063 --> 40:27.573
that right angle triangle,
very easy to get that.

40:27.570 --> 40:33.400
And that's going to = the
effective length going straight

40:33.398 --> 40:34.438
through.

40:34.440 --> 40:37.760
So let's look at the effective
length in two portions.

40:37.760 --> 40:43.490
That's a segment here of size
delta and a segment there of

40:43.487 --> 40:45.697
size delta in glass.

40:45.699 --> 40:48.319
So let's look at the distance
in the air first.

40:48.320 --> 40:52.540
Distance in the air is u
- delta on the left v -

40:52.536 --> 40:53.986
delta on the right.

40:53.989 --> 40:55.899
That's the air distance.

40:55.900 --> 41:00.540
Now I've got to add to that 2
delta times n.

41:00.539 --> 41:02.029
That's what you have to
remember.

41:02.030 --> 41:04.410
That's the only non trivial
part of this,

41:04.409 --> 41:06.429
is that this distance,
2 delta, is should be

41:06.434 --> 41:09.824
multiplied by n,
because you're traveling at a

41:09.820 --> 41:13.100
speed reduced by n in
terms of time.

41:13.099 --> 41:16.079
It's the distance factor
increase by n.

41:16.079 --> 41:29.679
So this becomes u v 2
delta times n - 1.

41:29.679 --> 41:30.619
That's the right hand side.

41:30.619 --> 41:35.699
The left hand side,
it's u h^(2) over

41:35.699 --> 41:40.659
2u v h^(2) over
2v.

41:40.659 --> 41:42.469
Again, in the same
approximation,

41:42.469 --> 41:44.339
in the quantity
h_2/u

41:44.338 --> 41:46.488
_2, or
h_2/v

41:46.487 --> 41:47.787
_2.

41:47.789 --> 41:51.279
There are higher powers of
h/v which I'm not keeping

41:51.284 --> 41:52.314
in either side.

41:52.309 --> 41:55.589
Even here, there are higher
powers of h/R that I'm

41:55.590 --> 41:56.410
not keeping.

41:56.409 --> 41:58.789
So get used to the notion, okay?

41:58.789 --> 42:01.019
u, v, f, R, big.

42:01.019 --> 42:03.109
h, y, etc.

42:03.110 --> 42:05.490
are small.

42:05.489 --> 42:12.719
So you cancel the u and
the v.

42:12.719 --> 42:14.559
Write the formula for delta.

42:14.559 --> 42:22.799
Delta = h^(2)/2r.

42:22.800 --> 42:26.380
I'm sorry, did I make a mistake
here?

42:26.380 --> 42:29.370
I guess not.

42:29.369 --> 42:37.179
Okay.

42:37.179 --> 42:46.569
So you compare the two sides,
you find 1/u 1/v.

42:46.570 --> 42:49.590
In other words,
drop these 2's and cancel the

42:49.586 --> 42:51.366
h^(2) everywhere.

42:51.369 --> 42:59.349
And 1/u 1/v = 2
times n - 1 over

42:59.351 --> 43:08.561
R, which I'm free to
write as 1/f if I like.

43:08.559 --> 43:10.469
In fact, that will be
1/f.

43:10.469 --> 43:13.489
How do I know that it's
1/f, can you tell me?

43:13.489 --> 43:24.799
Why should I call this quantity
that I get as 1/f?

43:24.800 --> 43:27.490
It is what I got,
but my question is,

43:27.489 --> 43:30.029
what allows me to identify
that?

43:30.030 --> 43:37.720
Student:
>

43:37.719 --> 43:41.609
Prof: Remember,
v is not the focal

43:41.608 --> 43:42.268
point.

43:42.268 --> 43:45.548
v is the location of the
image and u is the

43:45.547 --> 43:47.007
location of the object.

43:47.010 --> 43:48.780
But what do you want the focal
point to be?

43:48.780 --> 43:53.410
What's the definition of the
focal point?

43:53.409 --> 43:54.359
How is it defined?

43:54.360 --> 43:56.310
If you had a lens and I said,
"What's the focal

43:56.313 --> 43:56.853
point?"

43:56.849 --> 43:59.319
how will you find it?

43:59.320 --> 44:01.990
Haven't you guys played with
ants when you were kid with a

44:01.989 --> 44:02.879
magnifying glass?

44:02.880 --> 44:05.970
Come on, what's the focal point?

44:05.969 --> 44:09.209
Where a parallel set of rays
will converge on the ant.

44:09.210 --> 44:14.580
So if you send rays from
infinity, if u =

44:14.579 --> 44:19.949
infinity, v should be the
focal point.

44:19.949 --> 44:22.159
If u is infinity,
you drop this guy,

44:22.161 --> 44:24.271
then v,
the image should be at the

44:24.269 --> 44:27.009
focal point, therefore
1/v is 1/f.

44:27.010 --> 44:30.490
That's what allows you to
identify then this quantity as

44:30.487 --> 44:31.307
1/f.

44:31.309 --> 44:33.709
Because in the limit,
u goes to infinity,

44:33.706 --> 44:36.406
1/v should equal
1/f, and therefore this

44:36.411 --> 44:37.331
is what it is.

44:37.329 --> 44:40.129
But this is a calculation of
the focal length.

44:40.130 --> 44:40.930
See there are two things.

44:40.929 --> 44:43.979
One can define focal length
operationally by saying take a

44:43.983 --> 44:46.933
lens, send a parallel beam,
see where it all focuses.

44:46.929 --> 44:48.289
That's the focal length.

44:48.289 --> 44:51.789
But you can also calculate it
from first principles by saying,

44:51.789 --> 44:56.139
if I made a lens whose faces
are cut out of spheres,

44:56.139 --> 44:58.729
the two phases,
of radius of curvature

44:58.730 --> 45:01.090
R,
and the refractive index of the

45:01.094 --> 45:04.164
glass is n,
then I can actually calculate

45:04.157 --> 45:08.097
the focal length in terms of the
radius of curvature and

45:08.099 --> 45:08.959
n.

45:08.960 --> 45:12.070
So one is a derivation of the
focal length in terms of other

45:12.074 --> 45:15.084
given parameters,
whereas parallel rays focusing

45:15.076 --> 45:18.316
at f is the definition of
the focal point.

45:18.320 --> 45:20.530
It doesn't tell you how to
calculate it for a lens.

45:20.530 --> 45:21.910
But this will tell you.

45:21.909 --> 45:25.189
So if you've got a focal length
that is too big or too small,

45:25.190 --> 45:28.200
you can vary it by varying
either n or by varying

45:28.199 --> 45:28.909
R.

45:28.909 --> 45:32.039
You can take different
materials or you can take

45:32.043 --> 45:34.113
different radii of curvature.

45:34.110 --> 45:35.320
Question, yes?

45:35.320 --> 45:39.090
Student:  In that
drawing of the lens,

45:39.088 --> 45:41.058
what is the f?

45:41.059 --> 45:42.639
Prof: Which one?

45:42.639 --> 45:46.929
Student:  In the drawing
of the lens, what is the little

45:46.929 --> 45:47.689
f?

45:47.690 --> 45:48.690
Prof: This little
f?

45:48.690 --> 45:55.170
Student:
>

45:55.170 --> 45:56.590
Prof: You want to know
what f means?

45:56.590 --> 45:57.780
Student:  Yes.

45:57.780 --> 46:03.290
Prof: It means take this
lens and send a beam from

46:03.288 --> 46:04.368
infinity.

46:04.369 --> 46:07.619
That means it comes in parallel
rays.

46:07.619 --> 46:12.199
That means u = infinity
and 1/v then becomes

46:12.204 --> 46:13.254
1/f.

46:13.250 --> 46:15.170
That means v = f.

46:15.170 --> 46:21.150
v is the image point,
so that f will be here.

46:21.150 --> 46:32.800
So that's what it means.

46:32.800 --> 46:37.630
Okay, so I've gotten these
formulas, spent a lot of time

46:37.626 --> 46:39.026
deriving them.

46:39.030 --> 46:43.420
I'm assuming you have used them
a lot in your younger days.

46:43.420 --> 46:45.260
I just want to give you a few
examples,

46:45.260 --> 46:47.590
but I don't want to do all of
them,

46:47.590 --> 46:51.580
because you can do endless
calculations involving

46:51.577 --> 46:54.817
1/u 1/v is
1/f,

46:54.820 --> 46:56.310
just punching in numbers.

46:56.309 --> 46:59.339
I will just tell you a couple
of things to bear in mind.

46:59.340 --> 47:02.350
Then I want to sort of wrap it
up.

47:02.349 --> 47:05.339
So let's take some examples.

47:05.340 --> 47:14.410
Let's take this focusing mirror
now.

47:14.409 --> 47:17.709
This is f and this is
2f.

47:17.710 --> 47:23.170
So let's see what happens when
I start with an object here.

47:23.170 --> 47:25.200
You should be able to sort of
draw the ray diagrams,

47:25.197 --> 47:26.827
and I'm just drawing you a few
of them.

47:26.829 --> 47:29.849
So from now on,
we quit Fermat's Principle of

47:29.847 --> 47:30.737
Least Time.

47:30.739 --> 47:32.769
We just go back to the usual
diagrammatics,

47:32.771 --> 47:35.631
having convinced ourselves it
all follows from the Principle

47:35.628 --> 47:36.498
of Least Time.

47:36.500 --> 47:39.330
So you don't want to go back to
that every time.

47:39.329 --> 47:41.739
So what will be the situation
here?

47:41.739 --> 47:43.849
You need only two rays to draw.

47:43.849 --> 47:46.929
So one ray, you draw a
horizontal ray,

47:46.929 --> 47:51.259
and you know that will go
through the focal point.

47:51.260 --> 47:53.530
That's the definition.

47:53.530 --> 47:56.360
Then you draw a ray through the
focal point.

47:56.360 --> 48:01.470
That will come out horizontal,
and again, why is that true?

48:01.469 --> 48:04.929
That's true because if you sent
a parallel ray at this point,

48:04.934 --> 48:08.004
it will go to the focal point,
and if it ends up here,

48:07.996 --> 48:09.956
then the reverse is also true.

48:09.960 --> 48:13.600
Any ray that's possible,
then the time reversed version

48:13.599 --> 48:16.799
of that's also possible,
because if one takes the least

48:16.804 --> 48:19.754
time to go from A to B,
other takes the least time to

48:19.751 --> 48:20.741
go from B to A.

48:20.739 --> 48:25.719
That's how you draw this and
they intersect somewhere here.

48:25.719 --> 48:29.249
In fact, one can show that if
you're to the right of

48:29.250 --> 48:32.770
2f,
the image will be to the left

48:32.769 --> 48:37.739
of 2f.  If you're
exactly at 2f,

48:37.739 --> 48:40.769
you can show the image will
also be at 2f.

48:40.769 --> 48:42.319
Let's see why.

48:42.320 --> 48:47.930
Now 1/u 1/v is
1/f.

48:47.929 --> 48:51.979
I'm saying if u is at
2f and I believe v

48:51.981 --> 48:54.611
is at 2f,
I get them to add up to

48:54.614 --> 48:56.914
1/f, so that's correct.

48:56.909 --> 49:00.899
So 2f is the point where
the image and the object cross.

49:00.900 --> 49:04.550
See, objects at infinity focus
on f.

49:04.550 --> 49:08.960
As you move the object to find
a distance, the image moves away

49:08.963 --> 49:12.243
and they cross when you put it
at 2f.

49:12.239 --> 49:14.639
Then you can of course go
closer than 2f and you

49:14.641 --> 49:16.821
can ask, "Now what image
will I get?"

49:16.820 --> 49:20.230
So maybe I'll draw one more
picture for that one.

49:20.230 --> 49:22.820
So this is closer than
2f.

49:22.820 --> 49:27.220
Closer than 2f but more
distant than f,

49:27.221 --> 49:28.661
something here.

49:28.659 --> 49:34.009
So draw that ray,
draw it parallel,

49:34.005 --> 49:40.605
draw that ray,
draw that and they will meet

49:40.608 --> 49:42.808
like this.

49:42.809 --> 49:45.779
And finally,
if the object is at f,

49:45.780 --> 49:50.200
the image will be at infinity,
because that's just the reverse

49:50.197 --> 49:52.007
of the other picture.

49:52.010 --> 49:54.230
If all the rays from infinity
focus here,

49:54.230 --> 49:56.780
if you put an object at the
focal point and run the lines

49:56.777 --> 49:58.717
backwards,
they'll go in parallel,

49:58.715 --> 50:00.055
so they'll never meet.

50:00.059 --> 50:02.739
They're trying to meet at
infinity.

50:02.739 --> 50:05.259
But of course,
you can also take an object,

50:05.260 --> 50:08.560
even closer to the mirror than
the distance f.

50:08.559 --> 50:10.469
That's what I want to talk
about now.

50:10.469 --> 50:14.469
What if you go even closer than
f, so let's try that.

50:14.469 --> 50:27.789
So here it is, same guy.

50:27.789 --> 50:29.389
This is f.

50:29.389 --> 50:36.049
I want to take an object here.

50:36.050 --> 50:42.220
So the first ray I can draw is
the parallel ray going through

50:42.219 --> 50:42.939
that.

50:42.940 --> 50:44.710
First of all,
when you do the algebra,

50:44.710 --> 50:46.050
you can see some problems.

50:46.050 --> 50:52.870
1/u 1/v is
1/f, therefore 1/v

50:52.873 --> 50:57.143
is 1/f −
1/u.

50:57.139 --> 51:01.769
But if u is smaller than
f, 1/u is larger

51:01.768 --> 51:04.498
than f,
and 1/f −

51:04.501 --> 51:06.931
1/u will be negative.

51:06.929 --> 51:11.439
That means v itself will
be negative.

51:11.440 --> 51:14.160
So v is the location of
the image,

51:14.159 --> 51:16.059
and it's negative,
which means on the other side

51:16.061 --> 51:17.251
of the mirror,
you can ask,

51:17.246 --> 51:18.736
how can light go to the other
side?

51:18.739 --> 51:21.959
It doesn't go to the other
side, but you will see that it

51:21.961 --> 51:25.531
looks like something is behind
the mirror on the other side.

51:25.530 --> 51:28.220
So let's see why that happens.

51:28.219 --> 51:30.539
You will have some difficulty
drawing the second ray.

51:30.539 --> 51:33.819
Remember, the second ray you
draw through the focal point,

51:33.824 --> 51:35.904
and you say it comes out
parallel.

51:35.900 --> 51:39.160
If I draw a ray through the
focal point, now I've got to go

51:39.155 --> 51:39.825
backwards.

51:39.829 --> 51:43.689
Previously, you drew an object
and you drew a line through the

51:43.686 --> 51:47.666
focal point heading towards the
mirror, where the focal point is

51:47.670 --> 51:48.620
behind you.

51:48.619 --> 51:49.859
So what's the fate of this?

51:49.860 --> 51:50.940
Here is what you do.

51:50.940 --> 51:55.760
Continue it backwards and I
claim it will come out like

51:55.764 --> 51:56.394
this.

51:56.389 --> 51:59.149
Why do I know that is correct?

51:59.150 --> 52:01.710
Because I know the reverse of
that is correct.

52:01.710 --> 52:03.410
In other words,
a parallel ray,

52:03.413 --> 52:06.033
going like this will end up at
the focal point,

52:06.025 --> 52:08.235
therefore this is to guide the
eye.

52:08.239 --> 52:11.849
If you want to know what
happens with one particular ray,

52:11.849 --> 52:15.329
join it to the focal point,
hit the mirror and come out

52:15.331 --> 52:16.301
horizontal.

52:16.300 --> 52:17.950
That's how you do the ray
diagram.

52:17.949 --> 52:20.469
So this is a little delicate
thing, where if you really went

52:20.467 --> 52:22.557
to the focal point,
you'd never hit the mirror.

52:22.559 --> 52:25.449
You just do that to get the
direction, but you continue the

52:25.452 --> 52:27.352
other way and you come out
parallel.

52:27.349 --> 52:29.899
And the logic is,
if I draw it backwards,

52:29.900 --> 52:31.430
I know that's correct.

52:31.429 --> 52:33.309
But now look what's happening.

52:33.309 --> 52:36.579
These rays are not focusing
anywhere.

52:36.579 --> 52:40.549
In fact, this ray seems to come
from somewhere in that

52:40.550 --> 52:41.450
direction.

52:41.449 --> 52:47.299
That ray seems to come out in
that direction and the two of

52:47.295 --> 52:49.005
them meet here.

52:49.010 --> 52:52.570
That means to a person looking
from here,

52:52.570 --> 52:56.400
from this side with the eye,
the two rays,

52:56.400 --> 52:59.830
when you extend them backwards,
seem to emanate from this

52:59.827 --> 53:01.997
point,
so it will look like there is a

53:02.003 --> 53:05.143
candle behind the mirror,
emitting.

53:05.139 --> 53:07.789
So that's a virtual image.

53:07.789 --> 53:13.469
So virtual image is when there
is no real image,

53:13.465 --> 53:20.225
but everything seems to come
from a point behind that.

53:20.230 --> 53:25.730
Here's another variation they
probably exposed you to in high

53:25.733 --> 53:26.563
school.

53:26.559 --> 53:40.279
That is, what if I have a
mirror like this and I sent some

53:40.284 --> 53:47.514
parallel rays into that guy.

53:47.510 --> 53:49.980
You agreed this was not a
focusing lens.

53:49.980 --> 53:53.720
It's not a focusing mirror,
because you can already see,

53:53.722 --> 53:57.262
this light ray will come here,
take off like that.

53:57.260 --> 53:59.290
That ray will go right back.

53:59.289 --> 54:03.809
This will come here,
take off like this.

54:03.809 --> 54:09.459
So it is not a converging lens.

54:09.460 --> 54:13.720
But what is true is that if you
extrapolated all these lines

54:13.719 --> 54:17.329
backwards, they seem to emanate
from this point.

54:17.329 --> 54:18.039
You understand?

54:18.039 --> 54:20.689
If you look from the other
side, if you have a sun,

54:20.690 --> 54:23.070
and the sun is forming--sending
parallel rays,

54:23.074 --> 54:25.094
the image of the sun will be
here.

54:25.090 --> 54:27.110
It will be behind the mirror.

54:27.110 --> 54:29.990
It will all seem to be coming
from here.

54:29.989 --> 54:34.849
So instead of the lines
actually meeting at one point,

54:34.853 --> 54:38.343
they'll seem to come from one
point.

54:38.340 --> 54:39.820
So that's what happens.

54:39.820 --> 54:42.140
But how do you do the
calculations in this problem?

54:42.139 --> 54:45.619
Does anybody know?

54:45.619 --> 54:51.089
What's the mirror equation with
this problem?

54:51.090 --> 54:51.840
Pardon me?

54:51.840 --> 54:53.030
Student:
>

54:53.030 --> 54:55.370
Prof: Yes,
you say f is negative.

54:55.369 --> 54:58.189
It will turn out that
everything will work out,

54:58.190 --> 55:01.850
1/u 1/v is still
1/f,

55:01.849 --> 55:03.799
but if you want,
you can write this 1 over mod

55:03.800 --> 55:06.050
f,
because this will be a problem

55:06.048 --> 55:09.228
where the focal length,
if that is 5 inches,

55:09.228 --> 55:12.268
f should be treated as -5
inches,

55:12.269 --> 55:18.949
then everything will work out.

55:18.949 --> 55:26.949
Similarly, a lens which looks
like this is a diverging lens.

55:26.949 --> 55:33.389
If you send a parallel beam,
it will spread out like that.

55:33.389 --> 55:37.459
But to a person on the other
side, everything will seem to

55:37.463 --> 55:39.253
come from a point here.

55:39.250 --> 55:43.330
You can do that lens again by
using focal length to be

55:43.331 --> 55:44.181
negative.

55:44.179 --> 55:48.109
So u and v and
f are positive for a

55:48.105 --> 55:51.885
certain standard Mickey Mouse
problem where you have a

55:51.887 --> 55:55.667
focusing lens and u
bigger than f,

55:55.670 --> 55:58.510
then v is positive and
everything is positive.

55:58.510 --> 56:01.570
But you can then move around in
this world of parameters where

56:01.565 --> 56:02.965
v becomes negative.

56:02.969 --> 56:05.239
That means it's a virtual image.

56:05.239 --> 56:07.489
u is never negative,
because you put the object

56:07.494 --> 56:09.804
where you like,
and you can also have the focal

56:09.800 --> 56:11.590
length being negative if the
lens,

56:11.590 --> 56:16.070
instead of being a focusing
one, is a convex one that causes

56:16.067 --> 56:20.517
divergence,
or a lens which is causing

56:20.519 --> 56:27.169
divergence also is described by
f negative.

56:27.170 --> 56:30.630
Okay, there's one interesting
point.

56:30.630 --> 56:32.590
You can take it or leave it,
so relax.

56:32.590 --> 56:35.970
Don't kill yourself over this,
but I thought I should mention

56:35.965 --> 56:37.985
it to you, which is the
following.

56:37.989 --> 56:46.929
Suppose I have a lens like
this, and people tell you it's

56:46.932 --> 56:53.962
going to all seem to come from
this point.

56:53.960 --> 56:56.730
How do we know that's true?

56:56.730 --> 57:00.710
You cannot use the Principle of
Least Time to argue.

57:00.710 --> 57:04.370
Principle of Least Time says if
you have same starting point and

57:04.369 --> 57:07.389
the same ending point,
then there can be many ways to

57:07.389 --> 57:07.969
reach.

57:07.969 --> 57:09.929
Then you have a focusing effect.

57:09.929 --> 57:12.969
But here, there is no end point
where they all come.

57:12.969 --> 57:14.419
They don't ever come to the
same point.

57:14.420 --> 57:17.240
They seem to be coming from
here, but the rays never meet.

57:17.239 --> 57:20.609
So starting from infinity,
you cannot na�vely use the

57:20.608 --> 57:22.228
Principle of Least Time.

57:22.230 --> 57:26.150
But I'll tell you why this is
still going to be the correct

57:26.150 --> 57:26.760
answer.

57:26.760 --> 57:30.910
Take any mirror surface you
like, some part of a mirror.

57:30.909 --> 57:32.919
That's the normal to that
mirror.

57:32.920 --> 57:36.390
Imagine that a light ray does
that.

57:36.389 --> 57:41.209
This is the silver side of the
mirror.

57:41.210 --> 57:44.700
And I ask you,
what if I silver this side and

57:44.697 --> 57:48.817
shine light from the other side,
what will happen?

57:48.820 --> 57:53.460
I claim the answer is that if
you continue this ray here and

57:53.463 --> 57:57.483
continue that ray there,
then that's what light rays

57:57.478 --> 58:00.548
coming from the other side will
do.

58:00.550 --> 58:02.490
Why is that?

58:02.489 --> 58:07.939
Because this light ray had i
= r.

58:07.940 --> 58:09.820
You know that.

58:09.820 --> 58:13.810
But the normal to this surface
is the same normal on the other

58:13.811 --> 58:17.541
side, and by opposite angles
being equal, this is i

58:17.543 --> 58:19.183
and this is r.

58:19.179 --> 58:23.699
Therefore in any mirror coated
one way, if you continue the

58:23.701 --> 58:27.521
lines to the other side,
they'll still obey i =

58:27.519 --> 58:28.299
r.

58:28.300 --> 58:32.450
Consequently,
take the following problem:

58:32.447 --> 58:37.007
take the focusing mirror coated
like this.

58:37.010 --> 58:40.750
We know these lines will come
and join here.

58:40.750 --> 58:42.750
Then use the result I showed
you.

58:42.750 --> 58:47.080
That will then show you that
the mirror image of that on the

58:47.083 --> 58:50.173
other side will also obey i =
r,

58:50.170 --> 58:53.090
because if you draw the normal
here,

58:53.090 --> 58:56.320
if these two angles are equal,
those two angles will be equal.

58:56.320 --> 59:00.800
Therefore I know this will lead
to that and that will match this

59:00.802 --> 59:01.232
one.

59:01.230 --> 59:03.800
Therefore rays coming this way
will go there,

59:03.795 --> 59:06.765
but this one continued,
we know goes to f.

59:06.768 --> 59:09.838
That's the reason why,
if you know the fate of one

59:09.840 --> 59:12.720
kind of mirror,
you can find the fate if you do

59:12.722 --> 59:14.292
it on the other face.

59:14.289 --> 59:16.649
And this is the construction
that tells you why,

59:16.646 --> 59:19.406
if you shine light from the
right, they will all seem to

59:19.405 --> 59:20.655
come from this point.

59:20.659 --> 59:22.079
That's something optional.

59:22.079 --> 59:25.009
You can take it or leave it,
but I wanted to tell you where

59:25.007 --> 59:26.267
that result comes from.

59:26.268 --> 59:29.568
If you just do the algebra,
everything will work out if

59:29.574 --> 59:32.764
f is negative,
but you need to know why that's

59:32.755 --> 59:33.975
a valid picture.

59:33.980 --> 59:39.960
Okay, so there's one final
gadget I want to talk about,

59:39.960 --> 59:42.950
and that's the human eye.

59:42.949 --> 59:47.999
So the human eye is pretty
impressive when you consider the

59:47.996 --> 59:49.906
following situation.

59:49.909 --> 59:55.969
So here you've got--look,
I don't want to even try this.

59:55.969 --> 1:00:01.119
I'm just going to focus on the
lens in the human eye.

1:00:01.119 --> 1:00:09.249
Here are the eyebrows and here
is my cornea over there.

1:00:09.250 --> 1:00:11.620
And you put some object here.

1:00:11.619 --> 1:00:17.389
It goes and forms an inverted
image somewhere here.

1:00:17.389 --> 1:00:20.569
You must know that the image of
everything you see is actually

1:00:20.568 --> 1:00:22.598
physically upside down in your
brain.

1:00:22.599 --> 1:00:24.719
Did you know that?

1:00:24.719 --> 1:00:26.169
Or you did not know that.

1:00:26.170 --> 1:00:28.870
That's a very operationally
well defined thing.

1:00:28.869 --> 1:00:33.759
If I can look into your cornea,
I look at the candles upright

1:00:33.755 --> 1:00:38.795
here, the candle flame is here,
the flame of the candle will be

1:00:38.802 --> 1:00:40.352
here like that.

1:00:40.349 --> 1:00:44.639
But it doesn't seem to bother
us, because where do you really

1:00:44.643 --> 1:00:45.793
see something?

1:00:45.789 --> 1:00:46.629
It's not very clear.

1:00:46.630 --> 1:00:49.620
Do we see it in the cornea or
do you see it in the brain?

1:00:49.619 --> 1:00:52.349
There's a 1:1 correlation
between what's registered in the

1:00:52.351 --> 1:00:54.511
cornea and what I run into in my
real life.

1:00:54.510 --> 1:00:57.260
I'm walking around,
I bump into the upside down

1:00:57.260 --> 1:00:57.740
table.

1:00:57.739 --> 1:00:59.359
After the while,
the fact that it's upside down

1:00:59.360 --> 1:01:00.490
in the cornea is not relevant.

1:01:00.489 --> 1:01:03.219
I know if I take two steps,
I'm going to hit this object.

1:01:03.219 --> 1:01:06.959
The brain has learned how to
translate that into what you

1:01:06.963 --> 1:01:08.103
will encounter.

1:01:08.099 --> 1:01:10.599
In fact, I'm told in some
bizarre experiment,

1:01:10.599 --> 1:01:14.059
they put glasses on people so
that everything got inverted one

1:01:14.063 --> 1:01:14.863
more time.

1:01:14.860 --> 1:01:18.950
And after a few days,
those guys were just fine.

1:01:18.949 --> 1:01:21.289
So the eye has got a lot of
software.

1:01:21.289 --> 1:01:25.369
It's not purely hardware,
but software.

1:01:25.369 --> 1:01:28.189
I learned it the hard way when
I had an eye operation and the

1:01:28.186 --> 1:01:30.906
doctor pulled out the stuff and
said, "Ta-da!"

1:01:30.909 --> 1:01:32.899
and I couldn't see anything.

1:01:32.900 --> 1:01:34.490
So I was in a panic.

1:01:34.489 --> 1:01:36.699
My doctor didn't seem
particularly worried,

1:01:36.699 --> 1:01:38.909
and I said, "Are you
worried?"

1:01:38.909 --> 1:01:40.959
He said, "No,
no, you'll be fine in a few

1:01:40.958 --> 1:01:41.548
days."

1:01:41.550 --> 1:01:45.360
Now I've heard that before so I
didn't believe that lie.

1:01:45.360 --> 1:01:48.470
But slowly, I got better and
better.

1:01:48.469 --> 1:01:51.529
So it's not that my eye was
operated any more.

1:01:51.530 --> 1:01:54.260
He already fixed it,
but the brain began to

1:01:54.264 --> 1:01:57.524
reprogram it with respect to the
new parameters.

1:01:57.518 --> 1:02:00.818
You change the parameters of
the eye and he reprogrammed it

1:02:00.815 --> 1:02:03.995
so the brain had to correlate
everything that it saw with

1:02:03.996 --> 1:02:06.606
everything that's registered in
the brain.

1:02:06.610 --> 1:02:08.130
So you just need to 1:1 map.

1:02:08.130 --> 1:02:10.300
You can flip it any way you
like.

1:02:10.300 --> 1:02:12.320
You can flip it once,
or twice or any number of

1:02:12.318 --> 1:02:12.668
times.

1:02:12.670 --> 1:02:14.740
After a while,
you will learn how to use it.

1:02:14.739 --> 1:02:17.009
But that's not the only
impressive thing.

1:02:17.010 --> 1:02:23.170
If you have an object here,
you go to the Lens equation.

1:02:23.170 --> 1:02:29.190
You want the image to be at
that distance,

1:02:29.188 --> 1:02:33.298
and that's not negotiable.

1:02:33.300 --> 1:02:37.660
The distance from your lens to
your retina is fixed.

1:02:37.659 --> 1:02:42.419
So no matter what you have,
as u varies,

1:02:42.422 --> 1:02:47.702
you want v to be fixed
to obey this equation,

1:02:47.702 --> 1:02:52.572
so how do you think that's
going to happen?

1:02:52.570 --> 1:02:54.160
Pardon me?

1:02:54.159 --> 1:02:55.999
You've got to change f.

1:02:56.000 --> 1:02:58.670
You've got to change the focal
length of the lens,

1:02:58.668 --> 1:03:01.338
and that's the amazing thing
about the human lens,

1:03:01.335 --> 1:03:03.455
is that it's not made out of
glass.

1:03:03.460 --> 1:03:08.070
It's made out of some jelly
like stuff, and there are some

1:03:08.070 --> 1:03:09.770
muscles pulling it.

1:03:09.768 --> 1:03:11.988
And if the muscles pull it,
it will become longer and

1:03:11.989 --> 1:03:13.739
thinner, it will have one focal
length.

1:03:13.739 --> 1:03:16.069
If they relax,
it will have another focal

1:03:16.065 --> 1:03:16.585
length.

1:03:16.590 --> 1:03:20.660
So as I look at an object which
is very far, that's the normal

1:03:20.664 --> 1:03:22.474
relaxed state of the eye.

1:03:22.469 --> 1:03:25.699
As the object comes closer,
it takes a certain effort to

1:03:25.697 --> 1:03:28.457
focus on the object,
to keep reading whatever is

1:03:28.456 --> 1:03:29.626
coming near you.

1:03:29.630 --> 1:03:33.270
And that's the process in which
your eye is trying to change its

1:03:33.271 --> 1:03:34.141
focal length.

1:03:34.139 --> 1:03:37.429
But it cannot do more than some
amount.

1:03:37.429 --> 1:03:41.429
The eye cannot focus anything
that is closer than a certain

1:03:41.431 --> 1:03:43.711
distance called the near point.

1:03:43.710 --> 1:03:47.540
And the near point is 25
centimeters, well,

1:03:47.539 --> 1:03:48.999
approximately.

1:03:49.000 --> 1:03:52.890
This is not a law of nature,
but this number has crept into

1:03:52.887 --> 1:03:54.227
all the textbooks.

1:03:54.230 --> 1:03:57.800
So 25 centimeters is all you
can do.

1:03:57.800 --> 1:04:00.800
See, if you didn't have the
limitation, you don't need

1:04:00.797 --> 1:04:01.587
microscopes.

1:04:01.590 --> 1:04:04.820
You want to see a piece of
bacteria, just pull the sucker

1:04:04.820 --> 1:04:06.380
really close to your eye.

1:04:06.380 --> 1:04:08.420
Why doesn't it work?

1:04:08.420 --> 1:04:11.100
Because what we really see with
the eye is the following:

1:04:11.099 --> 1:04:14.349
if this is your eye,
what the eye cares about is the

1:04:14.347 --> 1:04:18.227
angle subtended by the object
with respect to your eye.

1:04:18.230 --> 1:04:19.980
We all know that.

1:04:19.980 --> 1:04:22.930
So if you have the little
bacteria guy here,

1:04:22.929 --> 1:04:27.249
you can see it just as well if
you can bring it that closely.

1:04:27.250 --> 1:04:30.220
That will look as big as a
pencil 1 meter away,

1:04:30.215 --> 1:04:32.855
except the eye cannot
accommodate that.

1:04:32.860 --> 1:04:35.590
That's why you need a
magnifying lens,

1:04:35.594 --> 1:04:39.074
and that's the last thing I'm
going to tell you,

1:04:39.068 --> 1:04:42.318
is the principle of the
magnifying lens.

1:04:42.320 --> 1:04:46.700
So here is the deal:
the eye really cares about the

1:04:46.701 --> 1:04:51.781
angle subtended by some object,
but there is a limit to how

1:04:51.784 --> 1:04:54.154
close you can bring it.

1:04:54.150 --> 1:05:01.050
So if you have an object of
height h--you don't have

1:05:01.050 --> 1:05:02.360
any lens.

1:05:02.360 --> 1:05:05.530
You just have your eye and
you're looking at it--

1:05:05.530 --> 1:05:10.970
then if that's the height
h and that's the angle

1:05:10.974 --> 1:05:14.244
θ,
the best you can do,

1:05:14.235 --> 1:05:16.885
put θ_0,
is

1:05:16.885 --> 1:05:23.795
tanθ_0 =
h over this 25

1:05:23.804 --> 1:05:25.804
centimeters.

1:05:25.800 --> 1:05:27.410
If you can come to 15
centimeters,

1:05:27.411 --> 1:05:29.661
θ_0 will
be even bigger,

1:05:29.661 --> 1:05:30.591
but you cannot.

1:05:30.590 --> 1:05:32.800
So for an object of given
height h,

1:05:32.795 --> 1:05:34.405
the smallest distance is 25.

1:05:34.409 --> 1:05:37.549
The biggest angle is
tanθ = h/25.

1:05:37.550 --> 1:05:40.700
And for small angles,
we forget the tan and we say

1:05:40.699 --> 1:05:43.529
θ_0 is
h/25.

1:05:43.530 --> 1:05:48.190
That's the best you can do
without a magnifier.

1:05:48.190 --> 1:05:50.220
So what does a magnifier do?

1:05:50.219 --> 1:05:54.139
A magnifier,
this is the focal length here.

1:05:54.139 --> 1:05:57.959
You put the object here,
somewhere there,

1:05:57.960 --> 1:06:01.320
and you try to form the image
for that one,

1:06:01.320 --> 1:06:05.980
so the parallel ray goes
through the focal point on the

1:06:05.978 --> 1:06:07.098
other side.

1:06:07.099 --> 1:06:10.459
Then, let's see,
a parallel ray going to the

1:06:10.463 --> 1:06:13.983
focal point here,
that's where you're going to

1:06:13.983 --> 1:06:16.803
have to get a virtual image now.

1:06:16.800 --> 1:06:17.310
Understand?

1:06:17.309 --> 1:06:20.239
Another ray you can draw is one
through the center that goes

1:06:20.244 --> 1:06:21.194
straight through.

1:06:21.190 --> 1:06:25.910
If you continue these guys
here, you will find there's an

1:06:25.909 --> 1:06:28.439
image here, a virtual image.

1:06:28.440 --> 1:06:30.770
In other words,
again what I'm telling you is

1:06:30.773 --> 1:06:34.113
the rays will not actually meet,
but they will seem to be coming

1:06:34.112 --> 1:06:35.282
from a point here.

1:06:35.280 --> 1:06:38.490
So if this is
h_2 and this is

1:06:38.490 --> 1:06:41.770
v, the θ you
are getting is

1:06:41.773 --> 1:06:45.203
h_2/v,
which is h/u.

1:06:45.199 --> 1:06:50.079
This is your actual height and
this is u.

1:06:50.079 --> 1:06:53.239
So basically what you've done
is the following:

1:06:53.244 --> 1:06:57.584
you managed to bring the object
closer to your eye than the near

1:06:57.577 --> 1:06:59.157
point, if you want.

1:06:59.159 --> 1:07:04.039
That's because the actual image
seems to be much further where

1:07:04.039 --> 1:07:05.399
you can see it.

1:07:05.400 --> 1:07:08.850
So the magnification,
the angle of opening you get

1:07:08.847 --> 1:07:14.527
with the magnifier is this,
so the magnification for angle

1:07:14.534 --> 1:07:20.734
is θ over
θ_0 and

1:07:20.728 --> 1:07:26.088
that equals h over u divided by
h over 25.

1:07:26.090 --> 1:07:31.990
That's 25 times 1/u.

1:07:31.989 --> 1:07:35.399
25 is in centimeters.

1:07:35.400 --> 1:07:45.210
That we're going to write as
1/f −

1:07:45.208 --> 1:07:47.978
1/v.

1:07:47.980 --> 1:07:49.380
Now you've got to be careful
here.

1:07:49.380 --> 1:07:52.760
v is a negative number
because you're forming a virtual

1:07:52.757 --> 1:07:53.197
image.

1:07:53.199 --> 1:08:01.789
So I'm going to rewrite this as
the absolute value of v.

1:08:01.789 --> 1:08:07.239
Therefore the gain you get from
using the microscope can be

1:08:07.240 --> 1:08:12.600
written as 25 over f  25
over the absolute value of

1:08:12.597 --> 1:08:13.817
v.

1:08:13.820 --> 1:08:16.580
v is where the image is.

1:08:16.578 --> 1:08:19.168
Now where do you want this
image to be?

1:08:19.170 --> 1:08:22.630
Of course, the smaller the
v, better your

1:08:22.630 --> 1:08:25.800
magnification is,
but v cannot be any

1:08:25.796 --> 1:08:27.266
smaller than 25.

1:08:27.270 --> 1:08:30.370
The best you can do with
v is 25,

1:08:30.368 --> 1:08:34.498
in which case you will get 1 25
over f as your

1:08:34.498 --> 1:08:35.928
magnification.

1:08:35.930 --> 1:08:38.980
But if you're a jeweler working
with jewels all the time and you

1:08:38.984 --> 1:08:40.784
want to see your piece of
diamond,

1:08:40.779 --> 1:08:44.149
you don't want the image to be
at the near point of your eye,

1:08:44.149 --> 1:08:46.129
because you're really
struggling to see it,

1:08:46.130 --> 1:08:48.130
because that's the best your
eye can accommodate.

1:08:48.130 --> 1:08:50.590
It's okay for a few seconds,
but it hurts.

1:08:50.590 --> 1:08:55.020
Ideally you want to see the
object at infinity.

1:08:55.020 --> 1:08:57.630
Therefore typically,
you pick v to be

1:08:57.627 --> 1:09:01.447
infinity that means you put the
thing you're looking at right at

1:09:01.447 --> 1:09:02.657
the focal point.

1:09:02.658 --> 1:09:05.808
If you put it at the focal
point, image will go to

1:09:05.811 --> 1:09:06.521
infinity.

1:09:06.520 --> 1:09:08.680
Now you might say,
"Look, I've never seen an

1:09:08.676 --> 1:09:09.616
object at infinity.

1:09:09.618 --> 1:09:11.918
What are you talking
about?"

1:09:11.920 --> 1:09:14.670
It turns out that infinity
could be 100 meters,

1:09:14.673 --> 1:09:17.133
or 10 meters is good enough for
a lens.

1:09:17.130 --> 1:09:19.380
It's far enough so that it
looks to be infinite.

1:09:19.380 --> 1:09:21.280
How can you see something at
infinity?

1:09:21.279 --> 1:09:23.419
The answer is,
as it goes farther out,

1:09:23.417 --> 1:09:26.247
it also gets taller,
so you don't lose anything by

1:09:26.248 --> 1:09:28.618
moving it farther out,
as long as it grows

1:09:28.618 --> 1:09:29.888
proportionately.

1:09:29.890 --> 1:09:33.600
It is just that distant objects
are when the eye muscles have to

1:09:33.595 --> 1:09:34.355
do no work.

1:09:34.359 --> 1:09:37.239
In other words,
the human eye is designed to

1:09:37.235 --> 1:09:39.305
focus on objects at infinity.

1:09:39.310 --> 1:09:42.560
It can manage if you bring it
closer by those muscles working,

1:09:42.560 --> 1:09:44.960
but if you don't want the
muscles to be tired,

1:09:44.957 --> 1:09:47.247
you want to have the image at
infinity.

1:09:47.250 --> 1:09:49.710
So you take this little thing
you're looking at,

1:09:49.710 --> 1:09:52.840
and you use the lens to create
an image at infinity,

1:09:52.840 --> 1:09:57.120
and again, forgetting the extra
1, most people don't bother with

1:09:57.118 --> 1:09:58.068
the extra 1.

1:09:58.069 --> 1:10:03.049
So the point to remember is the
angle of magnification is really

1:10:03.046 --> 1:10:07.626
given by 25 centimeters divided
by the focal length of your

1:10:07.627 --> 1:10:08.337
lens.

1:10:08.340 --> 1:10:12.400
So if you have a lens,
a focal length 2.5 centimeters,

1:10:12.395 --> 1:10:15.605
you get an angle or
magnification of 10.

1:10:15.609 --> 1:10:18.209
Most of the time,
the magnification you get is 3

1:10:18.206 --> 1:10:18.866
or 4 or 5.

1:10:18.868 --> 1:10:23.198
Any of these things you buy in
a supermarket to read

1:10:23.197 --> 1:10:25.587
something--
you don't know what I'm talking

1:10:25.587 --> 1:10:27.747
about,
but you will in a few

1:10:27.747 --> 1:10:32.617
years--then that will give you
magnification of 3 or 4.

1:10:32.618 --> 1:10:35.318
All right, so this is the end
of geometrical optics,

1:10:35.318 --> 1:10:38.548
and next time I'm going to tell
you why you want to change the

1:10:38.545 --> 1:10:39.705
rules of the game.

1:10:39.710 --> 1:10:43.290
But before you go,
I won't let you go until you

1:10:43.288 --> 1:10:46.868
tell me why I would want to
change anything?

1:10:46.869 --> 1:10:50.489
Why would anybody change this?

1:10:50.488 --> 1:10:52.448
The minute you answer me,
you go to lunch.

1:10:52.448 --> 1:10:55.678
What's the reason I would
change all this?

1:10:55.680 --> 1:10:56.280
Yes?

1:10:56.279 --> 1:10:57.289
Student:
>

1:10:57.289 --> 1:10:59.839
Prof: Yes, he saved you.

1:10:59.840 --> 1:11:02.540
There are some experiments that
contradict this.

1:11:02.538 --> 1:11:04.308
That's the only reason to
change anything,

1:11:04.305 --> 1:11:06.195
and I'll tell you what they are
next time.

1:11:06.199 --> 1:11:10.999