Object image and focal distance relationship (proof of formula)

WEBVTT
Kind: captions
Language: en

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We've been doing a
bunch of these videos

00:00:02.370 --> 00:00:05.380
with these convex lenses
where we drew parallel rays

00:00:05.380 --> 00:00:08.430
and rays that go
through the focal point

00:00:08.430 --> 00:00:11.552
to figure out what the
image of an object might be.

00:00:11.552 --> 00:00:13.010
But what I want to
do in this video

00:00:13.010 --> 00:00:15.570
is actually come up with
an algebraic relationship

00:00:15.570 --> 00:00:21.620
between the distance of the
object from the convex lens,

00:00:21.620 --> 00:00:24.900
the distance of the image
from the convex lens

00:00:24.900 --> 00:00:29.479
usually on the other side
and the focal length.

00:00:29.479 --> 00:00:30.770
So let's see if we can do this.

00:00:30.770 --> 00:00:33.500
And just to save you
the pain of having

00:00:33.500 --> 00:00:37.570
to watch me draw straight lines,
I drew this ahead of time.

00:00:37.570 --> 00:00:42.690
And so we can imagine this green
thing right here is the object.

00:00:42.690 --> 00:00:45.610
This is the object, and
these two little pink points

00:00:45.610 --> 00:00:48.340
right here are the focal points.

00:00:48.340 --> 00:00:50.640
They're a focal length away.

00:00:50.640 --> 00:00:51.900
And I did what we always drew.

00:00:51.900 --> 00:00:54.950
I drew one parallel ray
from the tip of that arrow

00:00:54.950 --> 00:00:58.960
to the actual convex lens,
and then it gets refracted,

00:00:58.960 --> 00:01:01.520
so it goes through the focal
point on the right-hand side

00:01:01.520 --> 00:01:03.420
and goes all the way over there.

00:01:03.420 --> 00:01:06.210
And then I drew a ray that
goes through the focal point

00:01:06.210 --> 00:01:07.570
on the left-hand side.

00:01:07.570 --> 00:01:09.970
And then when it gets
refracted, it becomes parallel.

00:01:09.970 --> 00:01:14.610
And then it actually intersects
with that previous ray

00:01:14.610 --> 00:01:15.890
right over here.

00:01:15.890 --> 00:01:21.690
And so this gives us a sense of
what the image will look like.

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It's inverted.

00:01:22.440 --> 00:01:23.150
It's real.

00:01:23.150 --> 00:01:26.281
In this case, it is larger
than the actual object.

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What I want to do is come
up with a relationship

00:01:28.280 --> 00:01:29.230
with these values.

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So let's see if we
can label them here.

00:01:31.190 --> 00:01:32.940
And then, just do a
little bit of geometry

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and a little bit of
algebra to figure out

00:01:34.860 --> 00:01:37.820
if there is an algebraic
relationship right here.

00:01:37.820 --> 00:01:40.890
So the first
number, the distance

00:01:40.890 --> 00:01:44.439
of the object-- that's this
distance from here to here,

00:01:44.439 --> 00:01:45.730
or we could just label it here.

00:01:45.730 --> 00:01:47.800
Since this is
already drawn for us,

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this is the distance
of the object.

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This is the way we drew it.

00:01:51.620 --> 00:01:53.330
This was the parallel light ray.

00:01:53.330 --> 00:01:57.420
But before it got refracted,
it traveled the distance

00:01:57.420 --> 00:02:00.480
from the object to
the actual lens.

00:02:00.480 --> 00:02:02.740
Now, the distance from the
image to the lens, that's

00:02:02.740 --> 00:02:03.720
this right over here.

00:02:03.720 --> 00:02:09.860
This is how far this parallel
light ray had to travel.

00:02:09.860 --> 00:02:13.980
So this is the distance
from the image to the lens.

00:02:13.980 --> 00:02:17.860
And then we have the focal
distance, the focal length.

00:02:17.860 --> 00:02:20.270
And that's just this
distance right here.

00:02:20.270 --> 00:02:22.390
This right here is
our focal length.

00:02:22.390 --> 00:02:24.240
Or, we could view it
on this side as well.

00:02:24.240 --> 00:02:29.760
This right here is
also our focal length.

00:02:29.760 --> 00:02:32.200
So I want to come up
with some relationship.

00:02:32.200 --> 00:02:35.220
And to do that, I'm going
to draw some triangles here.

00:02:35.220 --> 00:02:38.035
So what we can do is--
and the whole strategy--

00:02:38.035 --> 00:02:40.090
I'm going to keep looking
for similar triangles,

00:02:40.090 --> 00:02:43.870
and then try to see if
I can find relationship,

00:02:43.870 --> 00:02:47.230
or ratios, that relate these
three things to each other.

00:02:47.230 --> 00:02:49.000
So let me find some
similar triangles.

00:02:49.000 --> 00:02:51.120
So the best thing I
could think of to do

00:02:51.120 --> 00:02:54.020
is let me redraw this
triangle over here.

00:02:54.020 --> 00:02:55.830
Let me just flip it over.

00:02:55.830 --> 00:02:57.290
Let me just draw
the same triangle

00:02:57.290 --> 00:02:59.960
on the right-hand
side of this diagram.

00:02:59.960 --> 00:03:01.330
So this.

00:03:01.330 --> 00:03:03.140
So if I were to draw
the same triangle,

00:03:03.140 --> 00:03:04.355
it would look like this.

00:03:08.180 --> 00:03:10.510
And let me just be clear,
this is this triangle right

00:03:10.510 --> 00:03:11.200
over here.

00:03:11.200 --> 00:03:12.440
I just flipped it over.

00:03:15.770 --> 00:03:17.270
And so if we want
to make sure we're

00:03:17.270 --> 00:03:20.970
keeping track of the same
sides, if this length right here

00:03:20.970 --> 00:03:23.870
is d sub 0, or d
naught sometimes we

00:03:23.870 --> 00:03:26.290
could call it, or d0,
whatever you want to call it,

00:03:26.290 --> 00:03:37.040
then this length up here
is also going to be d0.

00:03:37.040 --> 00:03:38.602
And the reason why
I want to do that

00:03:38.602 --> 00:03:40.560
is because now we can do
something interesting.

00:03:40.560 --> 00:03:43.010
We can relate this triangle
up here to this triangle

00:03:43.010 --> 00:03:43.830
down here.

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And actually, we can see that
they're going to be similar.

00:03:47.240 --> 00:03:49.170
And then we can get
some ratios of sides.

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And then what we're
going to do is

00:03:50.000 --> 00:03:51.666
try to show that this
triangle over here

00:03:51.666 --> 00:03:53.720
is similar to this
triangle over here,

00:03:53.720 --> 00:03:55.307
get a couple of more ratios.

00:03:55.307 --> 00:03:57.640
And then we might be able to
relate all of these things.

00:03:57.640 --> 00:03:59.640
So the first thing we
have to prove to ourselves

00:03:59.640 --> 00:04:02.140
is that those triangles
really are similar.

00:04:02.140 --> 00:04:05.000
So the first thing to
realize, this angle right

00:04:05.000 --> 00:04:06.780
here is definitely
the same thing

00:04:06.780 --> 00:04:08.080
as that angle right over there.

00:04:08.080 --> 00:04:09.746
They're sometimes
called opposite angles

00:04:09.746 --> 00:04:10.590
or vertical angles.

00:04:10.590 --> 00:04:13.310
They're on the opposite side
of lines that are intersecting.

00:04:13.310 --> 00:04:14.870
So they're going to be equal.

00:04:14.870 --> 00:04:17.329
Now, the next thing-- and
this comes out of the fact

00:04:17.329 --> 00:04:20.920
that both of these lines--
this line is parallel

00:04:20.920 --> 00:04:23.040
to that line right over there.

00:04:23.040 --> 00:04:27.502
And I guess you could call
it alternate interior angles,

00:04:27.502 --> 00:04:29.710
if you look at the angles
game, or the parallel lines

00:04:29.710 --> 00:04:32.317
or the transversal of
parallel lines from geometry.

00:04:32.317 --> 00:04:33.900
We know that this
angle, since they're

00:04:33.900 --> 00:04:36.590
alternate interior
angles, this angle

00:04:36.590 --> 00:04:41.572
is going to be the same
value as this angle.

00:04:41.572 --> 00:04:43.030
You could view this
line right here

00:04:43.030 --> 00:04:46.340
as a transversal of
two parallel lines.

00:04:46.340 --> 00:04:48.480
These are alternate
interior angles,

00:04:48.480 --> 00:04:50.470
so they will be the same.

00:04:50.470 --> 00:04:54.090
Now, we can make that
exact same argument

00:04:54.090 --> 00:04:57.720
for this angle and this angle.

00:04:57.720 --> 00:04:59.710
And so what we see is
this triangle up here

00:04:59.710 --> 00:05:04.510
has the same three angles
as this triangle down here.

00:05:04.510 --> 00:05:07.320
So these two
triangles are similar.

00:05:07.320 --> 00:05:10.770
These are both-- Is really
more of a review of geometry

00:05:10.770 --> 00:05:11.480
than optics.

00:05:11.480 --> 00:05:14.100
These are similar triangles.

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Similar-- I don't have
to write triangles.

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They're similar.

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And because they're similar, the
ratios of corresponding sides

00:05:21.330 --> 00:05:23.150
are going to be the same.

00:05:23.150 --> 00:05:25.470
So d0 corresponds to this.

00:05:25.470 --> 00:05:29.420
They're both opposite
this pink angle.

00:05:29.420 --> 00:05:31.530
They're both opposite
that pink angle.

00:05:31.530 --> 00:05:35.250
So the ratio of d0 to d1--
let me write this over here.

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So the ratio of d0.

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Let me write this a
little bit neater.

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The ratio of d0 to d1.

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So this is the ratio of
corresponding sides--

00:05:43.640 --> 00:05:45.250
is going to be the same thing.

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And let me make
some labels here.

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That's going to be the same
thing as the ratio of this side

00:05:50.370 --> 00:05:51.960
right over here.

00:05:51.960 --> 00:05:54.050
This side right
over here, I'll call

00:05:54.050 --> 00:06:00.942
that A. It's opposite this
magenta angle right over here.

00:06:00.942 --> 00:06:03.400
That's going to be the same
thing as the ratio of that side

00:06:03.400 --> 00:06:06.095
to this side over
here, to side B.

00:06:06.095 --> 00:06:07.720
And once again, we
can keep track of it

00:06:07.720 --> 00:06:09.770
because side B is
opposite the magenta

00:06:09.770 --> 00:06:11.900
angle on this bottom triangle.

00:06:11.900 --> 00:06:15.250
So that's how we know that this
side, it's corresponding side

00:06:15.250 --> 00:06:17.000
in the other similar
triangle is that one.

00:06:17.000 --> 00:06:20.050
They're both opposite
the magenta angles.

00:06:20.050 --> 00:06:27.584
So we know d0 is to d1 as
A is to B. As A is to B.

00:06:27.584 --> 00:06:28.500
So that's interesting.

00:06:28.500 --> 00:06:30.180
We've been able to
relate these two

00:06:30.180 --> 00:06:33.550
things to these kind of
two arbitrarily lengths.

00:06:33.550 --> 00:06:36.720
But we need to somehow connect
those to the focal length.

00:06:36.720 --> 00:06:39.480
And to connect them to
a focal length, what

00:06:39.480 --> 00:06:42.386
we might want to do
is relate A and B.

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A sits on the same triangle
as the focal length

00:06:45.050 --> 00:06:45.750
right over here.

00:06:45.750 --> 00:06:48.490
So let's look at this
triangle right over here.

00:06:48.490 --> 00:06:49.910
Let me put in a better color.

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So let's look at this
triangle right over here

00:06:51.980 --> 00:06:54.980
that I'm highlighting in green.

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This triangle in green.

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And let's look at that in
comparison to this triangle

00:06:59.610 --> 00:07:01.440
that I'm also highlighting.

00:07:01.440 --> 00:07:04.695
This triangle that I'm
also highlighting in green.

00:07:04.695 --> 00:07:06.320
Now, the first thing
I want to show you

00:07:06.320 --> 00:07:09.790
is that these are also
similar triangles.

00:07:09.790 --> 00:07:12.520
This angle right over
here and this angle

00:07:12.520 --> 00:07:14.470
are going to be the same.

00:07:14.470 --> 00:07:17.900
They are opposite angles
of intersecting lines.

00:07:17.900 --> 00:07:20.290
And then, we can make
a similar argument--

00:07:20.290 --> 00:07:21.600
alternate interior angles.

00:07:21.600 --> 00:07:22.930
Well, there's a couple
arguments we could make.

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One, you can see that this is
a right angle right over here.

00:07:25.480 --> 00:07:26.910
This is a right angle.

00:07:26.910 --> 00:07:29.670
If two angles of two
triangles are the same,

00:07:29.670 --> 00:07:31.390
the third angle also
has to be the same.

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So we could also say
that this thing-- let

00:07:33.520 --> 00:07:35.690
me do this in another
color because I don't want

00:07:35.690 --> 00:07:37.630
to be repetitive too
much with the colors.

00:07:37.630 --> 00:07:39.350
We can say that
this thing is going

00:07:39.350 --> 00:07:41.467
to be the same
thing as this thing.

00:07:41.467 --> 00:07:43.050
Or another way you
could have said it,

00:07:43.050 --> 00:07:46.610
is you could have said, well,
this line over here, which

00:07:46.610 --> 00:07:48.550
is kind of represented
by the lens,

00:07:48.550 --> 00:07:50.452
or the lens-- the
line that is parallel

00:07:50.452 --> 00:07:51.910
to the lens or
right along the lens

00:07:51.910 --> 00:07:55.170
is parallel to kind of the
object right over there.

00:07:55.170 --> 00:07:57.670
And then you could make the
same alternate interior argument

00:07:57.670 --> 00:07:58.110
there.

00:07:58.110 --> 00:07:59.560
But the other thing
is just, look.

00:07:59.560 --> 00:08:00.540
I have two triangles.

00:08:00.540 --> 00:08:02.790
Two of the angles in those
two triangles are the same,

00:08:02.790 --> 00:08:05.020
so the third angle
has to be the same.

00:08:05.020 --> 00:08:07.500
Now, since all three
angles are the same,

00:08:07.500 --> 00:08:10.360
these are also both
similar triangles.

00:08:10.360 --> 00:08:12.180
So we can do a similar thing.

00:08:12.180 --> 00:08:15.260
We can say A is to B.
Remember, both A and B

00:08:15.260 --> 00:08:16.740
are opposite the 90-degree side.

00:08:16.740 --> 00:08:19.700
They're both the hypotenuse
of the similar triangle.

00:08:19.700 --> 00:08:25.255
So A is to B as-- we could say
this base length right here.

00:08:25.255 --> 00:08:26.850
And it got overwritten
a little bit.

00:08:26.850 --> 00:08:29.310
But this base length
right over here is f.

00:08:29.310 --> 00:08:32.730
That's our focal length.

00:08:32.730 --> 00:08:35.630
As f in this triangle is
related to this length

00:08:35.630 --> 00:08:37.250
on this triangle.

00:08:37.250 --> 00:08:41.470
They are both opposite
that white angle.

00:08:41.470 --> 00:08:44.000
So as f is to this
length right over here.

00:08:44.000 --> 00:08:45.350
Now, what is this length?

00:08:45.350 --> 00:08:47.877
So this whole distance is
di, all the way over here.

00:08:47.877 --> 00:08:49.460
But this length is
that whole distance

00:08:49.460 --> 00:08:51.220
minus the focal length.

00:08:51.220 --> 00:08:54.680
So this is di minus
the focal length.

00:08:54.680 --> 00:09:01.106
So A is to B as f is to
di minus the focal length.

00:09:01.106 --> 00:09:02.980
And there you have it,
we have a relationship

00:09:02.980 --> 00:09:05.220
between the distance
of the object,

00:09:05.220 --> 00:09:07.280
the distance of the image,
and the focal length.

00:09:07.280 --> 00:09:08.670
And now we just have to do
a little bit of algebra.

00:09:08.670 --> 00:09:10.795
If this is equal to this
and this is equal to that,

00:09:10.795 --> 00:09:13.774
then this blue thing has to be
equal to this magenta thing.

00:09:13.774 --> 00:09:15.440
And now we just have
to do some algebra.

00:09:15.440 --> 00:09:16.540
So let's do that.

00:09:16.540 --> 00:09:22.290
So we got d naught, or d0
I guess we'd call it, to di

00:09:22.290 --> 00:09:24.990
is equal to the ratio
of the focal length

00:09:24.990 --> 00:09:28.440
to the difference of
the image distance

00:09:28.440 --> 00:09:32.290
to the-- the focal
length to the image

00:09:32.290 --> 00:09:34.220
distance minus the focal length.

00:09:34.220 --> 00:09:36.180
And now here, we just
have to do some algebra.

00:09:36.180 --> 00:09:39.140
So let's-- just to simplify
this, let's cross multiply it.

00:09:39.140 --> 00:09:46.460
So if we multiply d0 times this
thing over here, we get d0 di.

00:09:46.460 --> 00:09:52.110
I'm really just
distributing it, minus d0 f.

00:09:52.110 --> 00:09:55.570
I'm just distributing this
d0, just cross multiplying,

00:09:55.570 --> 00:09:57.710
which is really just the
same thing as multiplying

00:09:57.710 --> 00:10:00.862
both sides by both
denominators, or multiplying

00:10:00.862 --> 00:10:01.820
the denominators twice.

00:10:01.820 --> 00:10:04.120
Either way, that's all
cross multiplying is.

00:10:04.120 --> 00:10:10.990
That is going to be
equal to di times f.

00:10:14.440 --> 00:10:16.400
And now, we can add this
term right over here

00:10:16.400 --> 00:10:17.860
to both sides of this equation.

00:10:17.860 --> 00:10:19.820
I'm just going to switch
to a neutral color.

00:10:19.820 --> 00:10:23.830
So we get d0 di is
equal to-- I'm just

00:10:23.830 --> 00:10:24.960
adding this to both sides.

00:10:24.960 --> 00:10:27.418
When you add it to the left
side, it obviously cancels out.

00:10:27.418 --> 00:10:33.477
Is equal to di f, this
thing over here, plus d0 f.

00:10:33.477 --> 00:10:34.310
And then, let's see.

00:10:34.310 --> 00:10:38.480
We could factor out
an f, a focal length.

00:10:38.480 --> 00:10:46.510
So we get d0 di is equal
to f times di plus d0.

00:10:46.510 --> 00:10:47.970
And then, what can we do?

00:10:47.970 --> 00:10:51.630
We could divide both sides by f.

00:10:51.630 --> 00:10:53.740
So this will become over f.

00:10:53.740 --> 00:10:57.580
This becomes over f.

00:10:57.580 --> 00:11:00.940
Essentially, canceling it out.

00:11:00.940 --> 00:11:03.315
And then, I just don't want
to skip too many steps,

00:11:03.315 --> 00:11:05.910
so let me just rewrite
what we have here.

00:11:05.910 --> 00:11:07.090
So these cancel out.

00:11:07.090 --> 00:11:16.140
So we have d0 di over f
is equal to di plus d0.

00:11:16.140 --> 00:11:19.320
And now, let's divide
both sides by d0 di.

00:11:19.320 --> 00:11:22.190
So 1 over d0 di.

00:11:22.190 --> 00:11:25.020
Divide this side by d0 di.

00:11:25.020 --> 00:11:26.860
It cancels out over here.

00:11:26.860 --> 00:11:30.060
And so we are left with,
on the left-hand side,

00:11:30.060 --> 00:11:33.742
1 over the focal length is
equal to this thing over here.

00:11:33.742 --> 00:11:35.200
And we can separate
this thing out.

00:11:35.200 --> 00:11:37.682
This thing over here,
this is the same thing--

00:11:37.682 --> 00:11:39.140
we just separate
out the numerator.

00:11:39.140 --> 00:11:47.235
Is the same thing as di over
d0 di plus d0 over d0 di.

00:11:47.235 --> 00:11:49.820
But di over d0 di,
the di's cancel out.

00:11:49.820 --> 00:11:50.970
We just have a 1.

00:11:50.970 --> 00:11:52.300
Here, the d0's cancel out.

00:11:52.300 --> 00:11:53.460
You just have a 1.

00:11:53.460 --> 00:11:57.550
So this is equal to 1 over
the distance of the object.

00:11:57.550 --> 00:12:02.417
And this is plus 1 over
the distance the image.

00:12:02.417 --> 00:12:05.000
So right from the get-go, this
was a completely valid formula.

00:12:05.000 --> 00:12:06.850
We actually had
achieved what we wanted.

00:12:06.850 --> 00:12:08.920
But this is a neater formula.

00:12:08.920 --> 00:12:11.740
You don't have the di's
repeat and the f repeated.

00:12:11.740 --> 00:12:14.000
Right here, we have an
algebraic relationship

00:12:14.000 --> 00:12:16.840
for a convex mirror that
relates the focal length

00:12:16.840 --> 00:12:22.410
to the distance of the object
and the distance of the image.

00:12:22.410 --> 00:12:25.730
Anyway, I think that's
pretty neat how it came out

00:12:25.730 --> 00:12:28.810
to be at least a
pretty clean formula.


Concave lenses

WEBVTT
Kind: captions
Language: en

00:00:00.030 --> 00:00:02.700
We've talked a lot about convex lenses,

00:00:02.700 --> 00:00:05.680
so I thought I would do a quick video on concave lenses,

00:00:05.680 --> 00:00:08.440
although there aren't as many combinations

00:00:08.440 --> 00:00:10.000
of what a concave lens can do.

00:00:10.000 --> 00:00:10.930
So just remember,

00:00:10.930 --> 00:00:13.050
concave has the word 'cave' in it

00:00:13.050 --> 00:00:16.410
so I always imagine that it's kind of caving inwards

00:00:16.410 --> 00:00:17.940
caving inwards

00:00:17.940 --> 00:00:19.770
It's a little bit of an exaggerated drawing

00:00:19.770 --> 00:00:21.590
but you get the general idea

00:00:21.590 --> 00:00:25.990
this is a concave lens right over here

00:00:25.990 --> 00:00:29.440
and let me draw its principal axis

00:00:29.440 --> 00:00:33.480
so this is its principal axis

00:00:33.480 --> 00:00:35.060
shouldn't curve at the beginning

00:00:35.060 --> 00:00:39.240
just redraw it, so that's its principal axis

00:00:39.240 --> 00:00:41.710
and let me draw the two focal points

00:00:41.710 --> 00:00:45.400
let me draw one over here on the left side

00:00:45.400 --> 00:00:49.810
and then an equal distance focal distance on the right side

00:00:49.810 --> 00:00:53.370
and I'm assuming that the two surfaces are both concave

00:00:53.370 --> 00:00:54.850
and they are symmetric over here

00:00:54.850 --> 00:00:56.630
Just an assumption I'm making

00:00:56.630 --> 00:00:58.440
Now lets think about what's gonna happen

00:00:58.440 --> 00:01:04.000
if I put an object some place

00:01:04.000 --> 00:01:09.060
on the left side of this concave lens

00:01:09.060 --> 00:01:10.960
So I can put it, let me just stick it anywhere

00:01:10.960 --> 00:01:13.400
So let me just stick it right over there

00:01:13.400 --> 00:01:15.980
So, if I do and like always,

00:01:15.980 --> 00:01:18.500
we will do our two rays but with a concave lens,

00:01:18.500 --> 00:01:20.030
I'm only gonna do two rays

00:01:20.030 --> 00:01:22.000
one that is parallel to the principal axis

00:01:22.000 --> 00:01:25.020
and then I'm gonna do one that does not get diffracted

00:01:25.020 --> 00:01:27.020
so it goes right through the center of the lens

00:01:27.020 --> 00:01:28.040
and let's think about what happens.

00:01:28.040 --> 00:01:32.460
So, if we go parallel, to the principal axis,

00:01:32.460 --> 00:01:35.380
this will get diffracted away, since it's coming in parallel

00:01:35.380 --> 00:01:38.370
and it will get diffracted outwards

00:01:38.370 --> 00:01:40.960
so that it looks like it's coming from this focus.

00:01:40.960 --> 00:01:43.370
So it will get diffracted in a way

00:01:43.370 --> 00:01:46.070
so it looks like its coming from that focus

00:01:46.070 --> 00:01:49.000
So its getting diverged outwards

00:01:49.000 --> 00:01:54.000
Let me draw another point or another ray, I should say

00:01:54.000 --> 00:01:56.030
I will do that in yellow

00:01:56.030 --> 00:01:57.260
and this will just gonna straight

00:01:57.260 --> 00:01:59.690
through the centre of the lens and not get diffracted

00:01:59.690 --> 00:02:03.000
So its just gonna go straight through the lens

00:02:03.000 --> 00:02:04.780
and not get diffracted

00:02:04.780 --> 00:02:07.170
So what will the image be here?

00:02:07.170 --> 00:02:10.880
Well clearly, these two rays don't converge

00:02:10.880 --> 00:02:13.490
so we won't have a real image

00:02:13.490 --> 00:02:16.740
but they do both look like they are diverging from someplace

00:02:16.740 --> 00:02:21.960
and they look like they are diverging from right over there

00:02:21.960 --> 00:02:25.700
so look to an observer out here

00:02:25.700 --> 00:02:27.790
if their eyeball is right over there

00:02:27.790 --> 00:02:35.390
I could draw their nose, just make it clear where the eye is

00:02:35.390 --> 00:02:37.670
If they look at from this side,

00:02:37.670 --> 00:02:44.100
they are going to see a virtual image of the image right over here

00:02:44.100 --> 00:02:45.050
Just like that

00:02:45.050 --> 00:02:49.010
So you're actually gonna see is a virtual image of the object

00:02:49.010 --> 00:02:52.130
that is not inverted and it's gonna be smaller

00:02:52.130 --> 00:02:56.250
its gonna look like it's closer to the lens than it really is

00:02:56.250 --> 00:02:59.160
Anyway, that's all it really is to know,

00:02:59.160 --> 00:03:01.400
obviously there's a lot you can know about anything.

00:03:01.400 --> 00:03:04.580
but that's the general gist about the concave lenses


Convex lens examples

WEBVTT
Kind: captions
Language: en

00:00:00.010 --> 00:00:04.170
Like we did with parabolic mirrors, what I really want to do in this video

00:00:04.210 --> 00:00:09.800
is just for put objects at different distances relative to this convex lens

00:00:10.040 --> 00:00:12.920
and just think about what its image will look like

00:00:12.920 --> 00:00:16.950
And the whole point of doing that is going through all the different situations but

00:00:16.960 --> 00:00:19.990
more is getting practice of how to think about it

00:00:19.990 --> 00:00:23.340
So let's first put an object out here

00:00:23.510 --> 00:00:28.540
That's more than two focal lengths away from the lens

00:00:28.540 --> 00:00:30.720
I'll put the object right here

00:00:30.760 --> 00:00:32.670
We'll do our classic arrow

00:00:32.770 --> 00:00:37.280
Actually I'll make a point here. When we dealt with parabolic mirrors

00:00:37.290 --> 00:00:42.100
we talked about the distance of two focal lengths being our center of curvature

00:00:42.180 --> 00:00:45.480
Over here we're just gonna call it 2 focal length, because it's really not

00:00:45.480 --> 00:00:47.540
the center of curvature--

00:00:47.550 --> 00:00:51.020
or this distance really isn't the radius of curvature of

00:00:51.040 --> 00:00:53.640
each of these curves. So we're just gonna call it 2 focal lengths

00:00:53.950 --> 00:00:59.150
So with that said, let's actually try to figure out what the image of this thing would look like

00:00:59.150 --> 00:01:02.710
as the light from it gets refracted through this lens

00:01:02.710 --> 00:01:06.390
So like always, it's useful to draw one ray

00:01:06.490 --> 00:01:10.690
Every point of this object is emitting rays in every direction

00:01:10.690 --> 00:01:12.830
because it's diffusely reflecting light

00:01:12.850 --> 00:01:15.320
So we can just pick rays that are convenient

00:01:15.320 --> 00:01:17.280
So we can go from the tip of the arrow

00:01:17.290 --> 00:01:22.360
and go parallel to our principal axis just like that

00:01:22.440 --> 00:01:26.470
I'm not gonna show all the internal refraction within this lens right here

00:01:26.600 --> 00:01:31.270
But we know that if we enter the lens, parallel to the principal axis

00:01:31.270 --> 00:01:41.630
when we get refracted, we will go through the focal point on the other side of the lens

00:01:41.990 --> 00:01:45.100
And on the left side of the lens

00:01:45.100 --> 00:01:48.340
we'll do another ray that goes through the focal point on the left side

00:01:48.340 --> 00:01:59.770
and then comes out parallel

00:02:00.190 --> 00:02:04.530
The incident ray goes through the left focal point

00:02:04.700 --> 00:02:10.710
and when it gets refracted, it will now be parallel

00:02:10.790 --> 00:02:18.790
And so the light that came from that point of the object will reconverge right over here

00:02:18.840 --> 00:02:21.530
So if you did this for every point of this object

00:02:21.530 --> 00:02:25.310
if you did a point in the middle right over here, it would reconverge right over here

00:02:25.350 --> 00:02:30.100
If you did this point right over here, it would reconverge right over there

00:02:30.100 --> 00:02:38.170
So the image of this object is going to look like this

00:02:38.190 --> 00:02:46.220
So it's going to be a real image. The rays actually converge here. So it's a real inverted image

00:02:46.230 --> 00:02:48.980
It was pointing up before. Now it's pointing that down

00:02:48.980 --> 00:02:53.370
And in this situation, it's actually going to be smaller than the original

00:02:53.490 --> 00:02:57.700
So, real, smaller, inverted image

00:02:57.700 --> 00:03:00.270
Let's do a couple of other scenarios

00:03:00.390 --> 00:03:05.580
Let me copy and paste this before starting this video to save on time

00:03:05.980 --> 00:03:12.910
Let's do a situation now where the object is at the 2 focal distances. I guess we can call that

00:03:13.480 --> 00:03:20.430
Let's put the object right over here

00:03:20.890 --> 00:03:22.850
We can do the exact same thing

00:03:22.850 --> 00:03:26.470
You might want to do it on paper on your own to get some practice doing it

00:03:26.480 --> 00:03:31.960
We'll do one ray that's parallel

00:03:31.980 --> 00:03:36.560
When it gets refracted, it will go through the focal point on the other side

00:03:36.580 --> 00:03:40.750
And then we'll do another ray that goes through the focal point on the left side

00:03:40.750 --> 00:03:45.110
and then it will become parallel

00:03:45.130 --> 00:03:48.080
And they can reconverge or they converge right over there

00:03:48.090 --> 00:03:53.340
So you can see that this is actually a parallelogram

00:03:53.350 --> 00:03:58.490
This distance right there is going to be the same thing as that distance over here

00:03:58.490 --> 00:04:01.690
It is actually--I don't want to say symmetry. You can't flip it over

00:04:01.690 --> 00:04:05.190
But this is a--and this distance right over here is going to be the same

00:04:05.190 --> 00:04:08.080
thing as this distance over here. I won't go into all the geometry

00:04:08.090 --> 00:04:17.210
Anyway, this would be an inverted image of exactly the same size at the exact same distance

00:04:17.260 --> 00:04:19.780
Actually I didn't talk about distance in this one over here

00:04:19.800 --> 00:04:24.340
Over here, it's a real image; it's smaller and it's also going to be closer in

00:04:24.340 --> 00:04:27.150
than this one was. It's gonna be closer to lens

00:04:27.160 --> 00:04:33.090
But here the image is going to be the same size as the original object

00:04:33.220 --> 00:04:37.620
It's going to be inverted, but it's going to be at the same distance from the lens

00:04:37.620 --> 00:04:41.470
just on the other side. Once again this is a real image

00:04:41.830 --> 00:04:43.920
Let's do another one

00:04:45.610 --> 00:04:47.810
Copy and paste

00:04:48.080 --> 00:04:52.480
So let's stick something between one focal length and two focal lengths

00:04:53.230 --> 00:05:03.020
So what's put my object right over there

00:05:03.040 --> 00:05:06.600
So once again, let's go parallel

00:05:06.620 --> 00:05:16.460
then we will refract through the focus on the other side

00:05:16.480 --> 00:05:19.250
And then let's go through the focus on the left side

00:05:19.870 --> 00:05:25.330
And then we will refract and go parallel

00:05:25.370 --> 00:05:30.830
So here, we will have--remember, I could use any point. Here I just used the tip

00:05:30.850 --> 00:05:34.250
because I know the base will converge over there

00:05:34.250 --> 00:05:38.410
If I took 2 points, it would go converge right back there

00:05:38.560 --> 00:05:40.760
And then if I did with a point right here

00:05:40.770 --> 00:05:43.360
and I did that same exercise, it would go right over there

00:05:43.360 --> 00:05:47.150
And so the whole image would show up over here

00:05:47.150 --> 00:05:50.580
So here, it's real; it's inverted; and it's larger

00:05:50.770 --> 00:05:57.220
and it is now further away from the lens than the object on the other side

00:05:57.220 --> 00:06:00.020
This is really kind of the reverse of the first example

00:06:00.030 --> 00:06:04.210
The first example, the object was larger and more than two focal lengths away

00:06:04.210 --> 00:06:06.250
and the image was in this range

00:06:06.250 --> 00:06:09.870
Now the object is here and now the image's on the other side. So they're kind of--

00:06:10.040 --> 00:06:13.790
that's really just the other side of that first examples

00:06:13.820 --> 00:06:18.850
Let's do a couple more

00:06:18.870 --> 00:06:27.070
So let's put the object at the focal point and see what would happen

00:06:27.090 --> 00:06:31.310
And sometimes people memorize this type of things for physics exams

00:06:31.330 --> 00:06:35.310
You don't need to. All you have to do is remember just think about 2 rays

00:06:35.310 --> 00:06:38.830
usually from the tip of the arrow that gives you a sense what the image will look like

00:06:38.950 --> 00:06:42.380
and you do one parallel and one that goes through the focal point

00:06:42.410 --> 00:06:43.800
Well, with that said

00:06:43.800 --> 00:06:47.340
we're gonna do something slightly different when something is sitting at the focal point

00:06:48.080 --> 00:06:50.910
When something is sitting at the focal point

00:06:50.930 --> 00:06:54.130
one is we can do the parallel

00:06:54.440 --> 00:06:58.660
It will be refracted through the focal point on the other side

00:06:58.990 --> 00:07:04.920
And then instead of doing a ray that goes through the focal point on this side

00:07:04.920 --> 00:07:08.130
because actually you can't. You're sitting on the focal point right there

00:07:08.230 --> 00:07:13.610
Let's do a ray that does not get refracted. We did a similar thing with the parabolic mirrors

00:07:13.620 --> 00:07:19.920
What I want do is a ray that'll go right to the center of the lens

00:07:19.920 --> 00:07:22.450
where it won't get refracted

00:07:22.860 --> 00:07:28.260
It's just go straight through the center of the lens

00:07:28.710 --> 00:07:35.440
So what you have here is, both of these rays that were diverging from this point from this tip

00:07:35.440 --> 00:07:37.100
they don't reconverge anywhere

00:07:37.120 --> 00:07:39.690
And they don't even look like they're diverging from another point

00:07:39.690 --> 00:07:43.050
if someone's eyeball is right over here

00:07:43.410 --> 00:07:47.010
They're just going to see these as 2 parallel rays of light

00:07:47.010 --> 00:07:50.820
So no image will form. Not a real image or virtual image

00:07:50.840 --> 00:07:56.920
So we can say no image is going to form

00:07:57.990 --> 00:08:08.290
Now the last case is, let's put the object less than one focal distance away

00:08:09.140 --> 00:08:16.880
Let's put the object right over here. And think about what happens

00:08:16.900 --> 00:08:19.520
Once again, the light ray from this tip

00:08:19.520 --> 00:08:24.390
where to go parallel. It will be refracted through the focal point on the other side

00:08:24.810 --> 00:08:26.720
And then let's do another light ray, going

00:08:26.720 --> 00:08:30.590
in a direction as if it were coming the direction of the focal point on this side

00:08:30.610 --> 00:08:32.270
So it comes from that direction

00:08:32.290 --> 00:08:34.230
The light ray would go like that

00:08:34.590 --> 00:08:39.790
And then it would come out on the other side parallel

00:08:40.000 --> 00:08:43.300
So clearly, these two light rays are not converging

00:08:43.310 --> 00:08:45.560
So no real image's going to form

00:08:45.690 --> 00:08:51.390
But they do look like they're diverging from some point

00:08:51.720 --> 00:08:59.350
Continue these lines. They both look like they're coming from a point right over there

00:08:59.350 --> 00:09:04.020
So what's going to happen? If someone's eyes are processing these light rays

00:09:04.100 --> 00:09:06.880
they're gonna see the tip of this arrow all the way up here

00:09:06.880 --> 00:09:11.200
They're gonna see the base of this arrow down here

00:09:11.580 --> 00:09:29.700
And essentially, they're going to see a magnified virtual image of the actual arrow

00:09:29.810 --> 00:09:33.720
Anyway, hopefully you found that interesting


Convex lenses

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:02.150
We've talked a
lot about mirrors,

00:00:02.150 --> 00:00:05.110
in particular parabolic
mirrors, that reflect light.

00:00:05.110 --> 00:00:07.740
What I want to do now
is talk about lenses,

00:00:07.740 --> 00:00:09.270
or talk about what a lens is.

00:00:09.270 --> 00:00:13.830
And think about how they
transmit or refract light.

00:00:13.830 --> 00:00:15.690
So a simple lens, and
we've all seen them.

00:00:15.690 --> 00:00:17.930
Maybe it's made of glass,
maybe something else.

00:00:20.700 --> 00:00:23.440
And I'm going to focus
on convex lenses first.

00:00:23.440 --> 00:00:27.140
So remember, concave means
it opens inward, like a cave.

00:00:27.140 --> 00:00:29.900
Convex means it kind
of opens outward.

00:00:29.900 --> 00:00:33.110
And in a convex lens,
it'll be symmetric.

00:00:33.110 --> 00:00:35.610
So let me see if I can draw it.

00:00:35.610 --> 00:00:37.711
One side of the lens
will look like that.

00:00:37.711 --> 00:00:39.460
And this one, you could
kind of view this.

00:00:39.460 --> 00:00:42.480
And oftentimes, most
lenses, the simpler lenses,

00:00:42.480 --> 00:00:43.430
are made this way.

00:00:43.430 --> 00:00:45.770
So this is kind of the
surface of a sphere,

00:00:45.770 --> 00:00:47.250
or part of the
surface of a sphere.

00:00:47.250 --> 00:00:49.660
Let me see if I can draw
that a little bit better.

00:00:49.660 --> 00:00:55.740
So part of the surface of a
sphere, and it's symmetric.

00:00:55.740 --> 00:00:59.010
So it has some center, right
over here, just like that.

00:00:59.010 --> 00:01:01.110
And then you have
another surface

00:01:01.110 --> 00:01:03.575
of a sphere that's
exactly the same.

00:01:03.575 --> 00:01:08.860
I'm doing my best to draw this
convex lens, just like that.

00:01:08.860 --> 00:01:12.254
That is a pretty good job here.

00:01:12.254 --> 00:01:14.170
And let me copy and paste
it so I can actually

00:01:14.170 --> 00:01:18.340
use this drawing in the
future, before I mark it up.

00:01:18.340 --> 00:01:18.860
All right.

00:01:18.860 --> 00:01:19.845
So I've copied it.

00:01:19.845 --> 00:01:21.220
So let's think
about what's going

00:01:21.220 --> 00:01:24.620
to happen as light
goes through this lens,

00:01:24.620 --> 00:01:27.150
as it's transmitted
through it and maybe gets

00:01:27.150 --> 00:01:28.800
diffracted by it.

00:01:28.800 --> 00:01:31.250
So we're assuming this is air
out here, and this is glass.

00:01:31.250 --> 00:01:34.680
Something that has a higher
index of refraction, something

00:01:34.680 --> 00:01:36.510
in which light travels slower.

00:01:36.510 --> 00:01:38.140
So you can imagine
that some light that

00:01:38.140 --> 00:01:42.140
is going parallel-- I
guess you could view it

00:01:42.140 --> 00:01:44.020
to the principal
axis of the lens.

00:01:44.020 --> 00:01:46.940
This would be the principal
axis of the lens right here,

00:01:46.940 --> 00:01:49.140
just like we talked
about the principal axis

00:01:49.140 --> 00:01:51.890
of our parabolic mirrors.

00:01:51.890 --> 00:01:54.690
But if you imagine light
that's going parallel to that,

00:01:54.690 --> 00:01:57.160
right when it hits this
surface over here-- Remember,

00:01:57.160 --> 00:01:59.390
the perpendicular
at this point is

00:01:59.390 --> 00:02:03.170
going to look like this because
the lens is actually curved.

00:02:03.170 --> 00:02:05.575
And remember, it's moving
faster on the outside.

00:02:05.575 --> 00:02:07.450
So the right side is
going to be able to stay

00:02:07.450 --> 00:02:09.660
outside a little bit longer.

00:02:09.660 --> 00:02:11.480
Or actually I should
say, the top side

00:02:11.480 --> 00:02:14.509
of the light-- if you
imagine the car analogy--

00:02:14.509 --> 00:02:16.300
is going to be able to
stay out of the lens

00:02:16.300 --> 00:02:19.200
a little bit longer
than the bottom side,

00:02:19.200 --> 00:02:20.110
or the bottom wheels.

00:02:20.110 --> 00:02:22.490
Or if we go in the
direction of the light,

00:02:22.490 --> 00:02:25.230
the left side of the car
is going to be able to--

00:02:25.230 --> 00:02:28.875
And just so we can visualize the
car, there's the left wheels.

00:02:28.875 --> 00:02:30.000
Those are the right wheels.

00:02:30.000 --> 00:02:31.999
The left wheels are going
to be able to stay out

00:02:31.999 --> 00:02:35.210
a little bit longer and travel
faster a little bit longer.

00:02:35.210 --> 00:02:37.290
So this is the
perpendicular again.

00:02:37.290 --> 00:02:42.630
So it will it be
refracted downwards

00:02:42.630 --> 00:02:44.254
like that, a little bit.

00:02:44.254 --> 00:02:45.920
And then once you get
to this interface,

00:02:45.920 --> 00:02:50.860
now you're going to move into
a faster medium, into the air

00:02:50.860 --> 00:02:51.750
again.

00:02:51.750 --> 00:02:54.230
And let me draw our
perpendicular over here.

00:02:58.160 --> 00:03:03.200
And you could imagine that
the right side of this ray

00:03:03.200 --> 00:03:09.010
is going to-- Actually,
the left side of this ray

00:03:09.010 --> 00:03:10.800
is going to come out first.

00:03:10.800 --> 00:03:13.880
And since the left side of
this ray, or the left side

00:03:13.880 --> 00:03:15.790
of these tires are
going to come out first,

00:03:15.790 --> 00:03:18.060
or maybe the top tires are
going to come out first,

00:03:18.060 --> 00:03:20.150
they're going to be
able to travel faster.

00:03:20.150 --> 00:03:22.610
And so you'll be deflected
even more downwards.

00:03:22.610 --> 00:03:27.190
So it will look
something like this.

00:03:27.190 --> 00:03:34.670
And the light ray would
do something like that.

00:03:34.670 --> 00:03:38.300
Now there is a point
out here someplace

00:03:38.300 --> 00:03:43.540
that whenever I take
any ray that is parallel

00:03:43.540 --> 00:03:50.760
to the principal
axis of the lens,

00:03:50.760 --> 00:03:55.430
it will be refracted through
the lens to that same point.

00:03:55.430 --> 00:03:56.930
So here, we're going
to be refracted

00:03:56.930 --> 00:03:57.930
a little bit like that.

00:03:57.930 --> 00:03:59.500
And there we'll
be refracted more.

00:03:59.500 --> 00:04:01.550
And then we're going to
go to that same point.

00:04:01.550 --> 00:04:03.240
So that's another ray.

00:04:03.240 --> 00:04:05.400
And then this is
another parallel ray.

00:04:05.400 --> 00:04:07.510
It'll be refracted a
little bit over here,

00:04:07.510 --> 00:04:08.640
and then a little bit more.

00:04:08.640 --> 00:04:10.490
And it'll go to that same point.

00:04:10.490 --> 00:04:13.080
And I think you could guess what
I'm about to call this point.

00:04:13.080 --> 00:04:15.959
I wish I could draw my lines
a little bit straighter.

00:04:15.959 --> 00:04:17.500
It's refracted a
little bit, and then

00:04:17.500 --> 00:04:20.124
refracted a little bit more, and
goes straight into that point.

00:04:20.124 --> 00:04:22.720
This point, where all of the
parallel rays-- Sometimes

00:04:22.720 --> 00:04:26.850
you'll hear them talked
of as collimated rays.

00:04:26.850 --> 00:04:30.740
Those are rays of light
that are roughly parallel.

00:04:30.740 --> 00:04:33.680
They all converge at this point
on the other side of the lens.

00:04:33.680 --> 00:04:36.010
They're essentially all
being focused on that point.

00:04:36.010 --> 00:04:42.600
And this right here you can
view as the focus of the lens.

00:04:42.600 --> 00:04:45.080
Or you can view this
length from the lens

00:04:45.080 --> 00:04:48.290
to that point as
the focal length.

00:04:48.290 --> 00:04:50.660
Now this lens is
completely symmetric.

00:04:50.660 --> 00:04:52.130
Anything you can
do from one side,

00:04:52.130 --> 00:04:54.010
you end up getting
focused on the right side.

00:04:54.010 --> 00:04:56.450
If you had collimated
rays, or parallel rays,

00:04:56.450 --> 00:05:01.395
coming from the right side,
the same thing would happen.

00:05:01.395 --> 00:05:03.020
But it would just be
on the other side.

00:05:03.020 --> 00:05:04.740
So that ray would go like that.

00:05:04.740 --> 00:05:07.000
And then it would be
refracted some more.

00:05:07.000 --> 00:05:11.450
And maybe it would go to
this point, right over here.

00:05:11.450 --> 00:05:17.810
And so you actually have
two foci for a lens.

00:05:17.810 --> 00:05:21.400
Two actual points where,
if parallel rays are coming

00:05:21.400 --> 00:05:23.199
from one side,
they'll be focused

00:05:23.199 --> 00:05:24.490
on the point on the other side.

00:05:24.490 --> 00:05:27.010
And if parallel arrays are
coming from the left side,

00:05:27.010 --> 00:05:30.590
they'll be focused
at the focal length

00:05:30.590 --> 00:05:33.952
or at the focus point
on the right-hand side.

00:05:33.952 --> 00:05:35.410
And this goes the
other way around.

00:05:35.410 --> 00:05:38.034
Let me draw another lens.

00:05:38.034 --> 00:05:39.450
And actually, one
thing that we're

00:05:39.450 --> 00:05:41.730
going to assume while
we're dealing with lenses,

00:05:41.730 --> 00:05:43.700
and this is kind of a
simplifying assumption,

00:05:43.700 --> 00:05:45.600
is called a thin
lens assumption.

00:05:45.600 --> 00:05:47.030
There is a difference
in distance

00:05:47.030 --> 00:05:51.440
it travels, depending on where
the light travels in the lens.

00:05:51.440 --> 00:05:56.330
For example, here there's
less distance than over here.

00:05:56.330 --> 00:05:58.477
And in an introductory
physics-- and we're

00:05:58.477 --> 00:06:00.060
going to do that
here, as well-- we're

00:06:00.060 --> 00:06:02.101
just going to ignore that
difference in distance,

00:06:02.101 --> 00:06:06.200
because that would lead to some
differences in how the light is

00:06:06.200 --> 00:06:07.950
refracted and transmitted
and all of that.

00:06:07.950 --> 00:06:09.780
Because it has to
travel a smaller

00:06:09.780 --> 00:06:11.100
distance here than over here.

00:06:11.100 --> 00:06:13.300
So we're going to ignore
those differences,

00:06:13.300 --> 00:06:15.980
and we're just going to make
the thin lens assumption.

00:06:15.980 --> 00:06:17.800
But using a thin
lens assumption,

00:06:17.800 --> 00:06:19.550
let's think a little
bit about what's

00:06:19.550 --> 00:06:21.010
going to happen with the light.

00:06:21.010 --> 00:06:22.150
And in the next
few examples, I'm

00:06:22.150 --> 00:06:23.780
not going to worry about
this kind of two-step.

00:06:23.780 --> 00:06:25.738
I'm just going to say,
look, it just in general

00:06:25.738 --> 00:06:29.820
gets refracted in that direction
when it exits the lens.

00:06:29.820 --> 00:06:35.570
So let me just draw a
simple lens right over here.

00:06:35.570 --> 00:06:37.410
It is symmetric.

00:06:37.410 --> 00:06:40.620
And it has two focal
points, one on this side,

00:06:40.620 --> 00:06:42.390
so that is one focal point.

00:06:42.390 --> 00:06:44.970
And then it has another focal
point, the exact same distance,

00:06:44.970 --> 00:06:45.720
on the other side.

00:06:45.720 --> 00:06:47.670
This lens is symmetric.

00:06:47.670 --> 00:06:49.850
So let's think about
what this lens will

00:06:49.850 --> 00:06:53.940
do to the images of
different objects.

00:06:53.940 --> 00:06:58.580
So let me draw its
principal axis again.

00:06:58.580 --> 00:07:02.270
So both focal points lie
along that principal axis.

00:07:02.270 --> 00:07:06.757
Now let's stick an object out
here, beyond the focal length.

00:07:06.757 --> 00:07:08.590
So let's think about
what's going to happen.

00:07:08.590 --> 00:07:10.965
So first, remember, we can
pick any point on this object.

00:07:10.965 --> 00:07:13.811
Light is being diffusely
reflected off of every point.

00:07:13.811 --> 00:07:15.310
I like to pick
points that are going

00:07:15.310 --> 00:07:17.520
to do something that's
kind of predictable.

00:07:17.520 --> 00:07:19.699
So let's pick a point.

00:07:19.699 --> 00:07:20.740
Well, let's take the tip.

00:07:20.740 --> 00:07:23.390
And take a ray that does
something that's predictable.

00:07:23.390 --> 00:07:27.622
So let's take a ray that is
parallel to the principal axis.

00:07:27.622 --> 00:07:29.080
I mean, I could
draw this two steps

00:07:29.080 --> 00:07:29.990
so it gets refracted once.

00:07:29.990 --> 00:07:31.406
And then it'll get
refracted again

00:07:31.406 --> 00:07:35.120
through the focal point on
the other side of the lens.

00:07:35.120 --> 00:07:39.680
So then it gets refracted
through there, just like that.

00:07:39.680 --> 00:07:43.290
And then, I could
take another ray

00:07:43.290 --> 00:07:45.690
from the tip of
that arrow that goes

00:07:45.690 --> 00:07:47.820
through the focal
point on this side.

00:07:47.820 --> 00:07:51.330
So it goes through the
focal point on this side.

00:07:51.330 --> 00:07:54.710
And so that is going to
get refracted like this,

00:07:54.710 --> 00:07:56.434
and then get refracted again.

00:07:56.434 --> 00:07:58.350
So it comes out on the
other side of the lens,

00:07:58.350 --> 00:07:59.800
going parallel.

00:07:59.800 --> 00:08:02.150
And hopefully this
makes sense to you,

00:08:02.150 --> 00:08:05.754
because it's kind
of a symmetric deal

00:08:05.754 --> 00:08:07.170
that we're dealing
with over here.

00:08:07.170 --> 00:08:09.620
Something coming in
parallel on the right side

00:08:09.620 --> 00:08:10.970
will go through the focal point.

00:08:10.970 --> 00:08:12.803
Then something going
through the focal point

00:08:12.803 --> 00:08:14.750
will come out on the
other side parallel.

00:08:14.750 --> 00:08:18.430
So whatever light is coming out
radially outward onto this side

00:08:18.430 --> 00:08:22.400
and going through the lens
will converge at this point,

00:08:22.400 --> 00:08:25.319
right over here, on the
other side of the lens.

00:08:25.319 --> 00:08:27.360
And so you could do even
light that goes straight

00:08:27.360 --> 00:08:29.517
through the lens would
end up right over there.

00:08:29.517 --> 00:08:31.100
It actually won't
be refracted at all.

00:08:31.100 --> 00:08:33.850
It'll just be able to go
straight through the lens.

00:08:33.850 --> 00:08:35.780
And so the image
that gets formed

00:08:35.780 --> 00:08:40.690
on the other side of the
lens will look like that.

00:08:40.690 --> 00:08:44.059
So in this example,
it looks like we

00:08:44.059 --> 00:08:46.110
have an inverted real image.

00:08:50.255 --> 00:08:51.630
And once again,
it's a real image

00:08:51.630 --> 00:08:54.260
because the light is actually
converging at that point.

00:08:54.260 --> 00:08:57.210
You would actually be able
to put some type of a screen

00:08:57.210 --> 00:09:00.082
and project the image there.

00:09:00.082 --> 00:09:01.540
In the next video,
we're just going

00:09:01.540 --> 00:09:05.580
to practice this idea of
drawing these rays to figure out

00:09:05.580 --> 00:09:09.260
what type of images we'll get,
depending where the object is,

00:09:09.260 --> 00:09:11.660
whether it's at the focal
point, beyond the focal point,

00:09:11.660 --> 00:09:13.380
beyond two times
the focal point,

00:09:13.380 --> 00:09:14.777
or within the focal point.

00:09:14.777 --> 00:09:16.360
And the best thing
there is we'll just

00:09:16.360 --> 00:09:19.180
get a lot of practice doing
this, drawing these rays

00:09:19.180 --> 00:09:21.723
and thinking about how
they'll get refracted.


Convex parabolic mirrors

WEBVTT
Kind: captions
Language: en

00:00:00.560 --> 00:00:02.380
All the parabolic
mirror examples

00:00:02.380 --> 00:00:05.530
we've been doing so
far have been concave.

00:00:05.530 --> 00:00:08.740
And that just means,
you might know already

00:00:08.740 --> 00:00:13.390
what concave means, but
just to make it clear,

00:00:13.390 --> 00:00:16.590
they were opening out
towards the objects,

00:00:16.590 --> 00:00:21.150
and in most of the
cases, out towards where

00:00:21.150 --> 00:00:22.520
the images were formed.

00:00:22.520 --> 00:00:27.730
So what we cared about the
inside surface of the mirror

00:00:27.730 --> 00:00:28.589
right here.

00:00:28.589 --> 00:00:30.130
And the way I always
remember concave

00:00:30.130 --> 00:00:32.030
is the kind of forms that cave.

00:00:32.030 --> 00:00:34.860
You could view this part right
here as the inside of the cave.

00:00:34.860 --> 00:00:38.220
And we had a bunch of examples
where we had focal points.

00:00:38.220 --> 00:00:41.710
And then this one here
was our principal axis.

00:00:41.710 --> 00:00:42.730
And we had objects.

00:00:42.730 --> 00:00:47.030
And I showed you how the light
would reflect off of the mirror

00:00:47.030 --> 00:00:49.304
and go through the focal points.

00:00:49.304 --> 00:00:50.720
So thus we did a
bunch of examples

00:00:50.720 --> 00:00:52.490
like that in the last
couple of videos.

00:00:52.490 --> 00:00:54.590
What I want to do in this
video is a quick example

00:00:54.590 --> 00:00:58.570
of using a convex
parabolic mirror.

00:00:58.570 --> 00:01:03.480
So let's do a convex
parabolic mirror.

00:01:03.480 --> 00:01:06.880
And here we care about the
other side of the mirror.

00:01:06.880 --> 00:01:10.960
We care about the side that's
the outside of the bowl,

00:01:10.960 --> 00:01:13.570
if you want to view the
parabolic mirror as a ball.

00:01:13.570 --> 00:01:15.320
So let's think about
that as a little bit.

00:01:15.320 --> 00:01:17.170
So let's imagine
that once again we

00:01:17.170 --> 00:01:19.810
have something that's
the shape of a parabola.

00:01:19.810 --> 00:01:22.170
It has this-- let me draw
a better version of it--

00:01:22.170 --> 00:01:25.200
so it has the shape
of a parabola.

00:01:25.200 --> 00:01:30.160
This is the principal
axis right over here.

00:01:30.160 --> 00:01:32.960
You could almost view that
as the line of symmetry.

00:01:32.960 --> 00:01:35.400
This is still the focal point.

00:01:35.400 --> 00:01:38.050
That right there
is still the focus.

00:01:38.050 --> 00:01:41.310
But now we're going to assume
that the reflective surface is

00:01:41.310 --> 00:01:42.920
on the outside.

00:01:42.920 --> 00:01:44.206
It's on the outside.

00:01:44.206 --> 00:01:45.580
So the reflective
surface is kind

00:01:45.580 --> 00:01:49.807
of jutting out towards us
as opposed to caving in.

00:01:49.807 --> 00:01:51.640
That's another way to
remember it-- concave,

00:01:51.640 --> 00:01:53.010
looks like it's caved in.

00:01:53.010 --> 00:01:55.335
Here, it's jutting
out towards us.

00:01:55.335 --> 00:01:56.960
And let's think about
what would happen

00:01:56.960 --> 00:02:01.680
if I put an object over
here on the outside assuming

00:02:01.680 --> 00:02:04.370
that this is a
reflective surface.

00:02:04.370 --> 00:02:07.750
So if I put an object over
here, what is going to happen?

00:02:07.750 --> 00:02:09.810
So let's just do
the same exercise.

00:02:09.810 --> 00:02:12.160
But what we're
going to do is we're

00:02:12.160 --> 00:02:15.050
going to have one parallel ray.

00:02:15.050 --> 00:02:17.240
We could do rays that go
in any direction from any

00:02:17.240 --> 00:02:22.180
of these points because there's
some light source over here.

00:02:22.180 --> 00:02:23.680
They never draw
the light source,

00:02:23.680 --> 00:02:26.440
but it reflects diffusely
off of this object.

00:02:26.440 --> 00:02:29.610
So this object is emitting
light rays in every direction.

00:02:29.610 --> 00:02:33.130
But the useful
diffused light rays

00:02:33.130 --> 00:02:36.990
being emitted by this object
are the ones that are parallel

00:02:36.990 --> 00:02:39.030
to the principal
axis and the ones

00:02:39.030 --> 00:02:41.292
that would go through the focus.

00:02:41.292 --> 00:02:42.500
Let's do one that's parallel.

00:02:42.500 --> 00:02:45.040
So if something is parallel
to the principal axis--

00:02:45.040 --> 00:02:47.290
and I'm not doing
the map over here--

00:02:47.290 --> 00:02:48.980
but if it reflects
on the outside

00:02:48.980 --> 00:02:53.360
of this parabolic mirror,
it will reflect in a way

00:02:53.360 --> 00:02:56.079
so it looks like it's
coming from the focus.

00:02:56.079 --> 00:02:58.620
So I would see the focus as on
the other side of this mirror,

00:02:58.620 --> 00:03:00.161
but it would reflect
in a way that it

00:03:00.161 --> 00:03:03.300
looks like it's
coming from the focus.

00:03:03.300 --> 00:03:05.340
So that ray will
reflect like that.

00:03:05.340 --> 00:03:08.500
And then if we have another ray
from the head of this object,

00:03:08.500 --> 00:03:10.650
from the tip of that
arrow, and that ray

00:03:10.650 --> 00:03:13.610
is going in the
direction of the focus--

00:03:13.610 --> 00:03:16.920
so the focus is there-- so
let me draw the direction.

00:03:16.920 --> 00:03:18.790
So let's say I have
an incident ray going

00:03:18.790 --> 00:03:21.010
in the direction of the focus.

00:03:21.010 --> 00:03:23.950
When that gets reflected,
it will reflect parallel

00:03:23.950 --> 00:03:25.100
to the principal axis.

00:03:30.000 --> 00:03:32.230
And so what type of image
is going to be formed here?

00:03:32.230 --> 00:03:34.650
Clearly, these two rays
will never converge.

00:03:34.650 --> 00:03:36.190
So we can't form a real image.

00:03:36.190 --> 00:03:41.290
We cannot project that image
onto a screen or cloth and then

00:03:41.290 --> 00:03:41.950
see it.

00:03:41.950 --> 00:03:43.460
These two rays are converging.

00:03:43.460 --> 00:03:45.920
But if one were to
observe the rays,

00:03:45.920 --> 00:03:50.570
they look like they are
diverging from a single point.

00:03:50.570 --> 00:03:52.885
This ray looks like,
let me make it clear,

00:03:52.885 --> 00:03:55.010
this one that just got
reflected out parallel looks

00:03:55.010 --> 00:03:57.560
like it's coming from
if you go straight

00:03:57.560 --> 00:03:58.762
back behind the mirror.

00:03:58.762 --> 00:04:00.220
And then this one
that's coming out

00:04:00.220 --> 00:04:02.780
looks like it's coming
out of this point there.

00:04:02.780 --> 00:04:04.740
So it looks like
they're diverging

00:04:04.740 --> 00:04:08.365
from this point on the
other side of the mirror.

00:04:08.365 --> 00:04:10.740
And not only doesn't look like
they're coming from there,

00:04:10.740 --> 00:04:13.870
but the actual image
will look like that.

00:04:13.870 --> 00:04:16.689
We could do it with other
points on this arrow if we want.

00:04:16.689 --> 00:04:18.730
If you take the bottom of
the arrow, that's maybe

00:04:18.730 --> 00:04:26.910
the easiest, light that goes
straight to the actual mirror

00:04:26.910 --> 00:04:29.050
will then be reflected
straight back.

00:04:29.050 --> 00:04:32.310
So it would look
like it's coming

00:04:32.310 --> 00:04:35.250
from a point at the
mirror back over here.

00:04:35.250 --> 00:04:36.839
And we could do other things.

00:04:36.839 --> 00:04:38.380
We could draw stuff
so that you could

00:04:38.380 --> 00:04:40.935
see what the whole image-- we
could take points over here,

00:04:40.935 --> 00:04:42.310
and you would say
that that would

00:04:42.310 --> 00:04:44.070
correspond to a
point over there.

00:04:44.070 --> 00:04:46.120
But I guess the thing
that hopefully you'll

00:04:46.120 --> 00:04:48.370
realize from this
video is when you're

00:04:48.370 --> 00:04:51.750
dealing with a convex
parabolic mirror,

00:04:51.750 --> 00:04:53.777
the outside is a
reflective surface.

00:04:53.777 --> 00:04:55.360
You're not going to
form a real image,

00:04:55.360 --> 00:04:58.110
you're going to form
a virtual image.

00:04:58.110 --> 00:05:00.280
This is a virtual
image, just like you

00:05:00.280 --> 00:05:02.200
would see in your
bathroom mirror,

00:05:02.200 --> 00:05:04.810
although that's probably
not parabolic, I'm guessing.

00:05:04.810 --> 00:05:06.655
And it's also going
to be a smaller image.

00:05:12.060 --> 00:05:14.620
And you see these types
of mirrors all the time,

00:05:14.620 --> 00:05:16.040
especially around corners.

00:05:16.040 --> 00:05:17.610
If you see a
corner-- let's let me

00:05:17.610 --> 00:05:22.550
draw a hallway with
corners-- you'll

00:05:22.550 --> 00:05:26.110
sometimes see mirrors-- let me
do it the other way-- you'll

00:05:26.110 --> 00:05:28.670
sometimes see mirrors so that
people-- well, the mirror might

00:05:28.670 --> 00:05:30.720
be out here-- so that
you can see people

00:05:30.720 --> 00:05:32.220
as they're coming
around the corner.

00:05:32.220 --> 00:05:34.011
And the reason why
these mirrors are useful

00:05:34.011 --> 00:05:36.770
is that they reflect light
from a lot of directions.

00:05:36.770 --> 00:05:39.877
And so you can kind
of see around corners.

00:05:39.877 --> 00:05:41.460
This is not the best
of drawing, but I

00:05:41.460 --> 00:05:44.950
think you've seen these
convex mirrors, mirrors

00:05:44.950 --> 00:05:46.399
that look something like this.

00:05:46.399 --> 00:05:48.690
Sometimes you'll also see
them at the aisles of stores,

00:05:48.690 --> 00:05:50.150
at the head of aisles of stores.

00:05:50.150 --> 00:05:53.740
And this way, the store owners
can have a good field of view.

00:05:53.740 --> 00:05:56.710
And they can see if
anyone is shoplifting.

00:05:56.710 --> 00:05:59.860
Anyway, hopefully, you
found that interesting.


Parabolic mirrors 2

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:02.455
Let's draw a bunch
of parabolic mirrors.

00:00:02.455 --> 00:00:04.080
And what I want to
do in this video is,

00:00:04.080 --> 00:00:06.630
do a bunch of
examples of objects

00:00:06.630 --> 00:00:08.140
in front of parabolic mirrors.

00:00:08.140 --> 00:00:10.800
And think about what the
images of those objects

00:00:10.800 --> 00:00:14.530
will be based on how
far those objects are.

00:00:14.530 --> 00:00:17.010
And besides just giving
us a better understanding

00:00:17.010 --> 00:00:21.240
of parabolic mirrors, this will
also, it 'll hopefully also

00:00:21.240 --> 00:00:23.940
give us a sense of how do
we manipulate or how can we

00:00:23.940 --> 00:00:25.640
conceptualize these light rays?

00:00:25.640 --> 00:00:27.550
Which will be a
pretty useful tool

00:00:27.550 --> 00:00:30.570
when we tackle other types
of reflective or refractive

00:00:30.570 --> 00:00:32.509
devices like lenses.

00:00:32.509 --> 00:00:33.990
So that's a parabolic mirror.

00:00:33.990 --> 00:00:37.180
I've drawn its principal
axis right over here.

00:00:37.180 --> 00:00:39.763
And let me just copy and paste
this just so that I-- actually,

00:00:39.763 --> 00:00:41.180
let me draw the
focal point, too.

00:00:41.180 --> 00:00:43.560
So this is the focus
right over here.

00:00:43.560 --> 00:00:46.260
So let me just draw the
focus right over there.

00:00:46.260 --> 00:00:47.950
And this is the
center of curvature.

00:00:47.950 --> 00:00:51.450
It's twice the distance from
this point as the focus.

00:00:51.450 --> 00:00:54.580
So that is the-- let me make
it as close as possible.

00:00:54.580 --> 00:00:57.899
So that is the center of
curvature right over there.

00:00:57.899 --> 00:00:59.190
And let me copy and paste this.

00:00:59.190 --> 00:01:01.050
So we can reuse this
later on in the video.

00:01:01.050 --> 00:01:02.760
I don't have to keep drawing it.

00:01:02.760 --> 00:01:04.739
So copy it.

00:01:04.739 --> 00:01:06.017
So I've copied that.

00:01:06.017 --> 00:01:07.600
Now, let's put an
object-- and I think

00:01:07.600 --> 00:01:08.920
we did this in the last video.

00:01:08.920 --> 00:01:12.660
Let's put an object beyond
the center of curvature.

00:01:12.660 --> 00:01:13.950
So let's put an object here.

00:01:13.950 --> 00:01:17.510
And the convention is to use
an upward pointing arrow.

00:01:17.510 --> 00:01:18.560
This isn't a light ray.

00:01:18.560 --> 00:01:19.910
It's used to show an object.

00:01:19.910 --> 00:01:22.290
And we use the tip of
the arrow to really show

00:01:22.290 --> 00:01:23.450
the top of the object.

00:01:23.450 --> 00:01:26.737
And that's usually where we
trace our light rays from.

00:01:26.737 --> 00:01:28.070
But it doesn't have to be there.

00:01:28.070 --> 00:01:28.710
We could do the middle.

00:01:28.710 --> 00:01:29.740
You could do the bottom.

00:01:29.740 --> 00:01:31.760
And you could figure out what
the image of the object's

00:01:31.760 --> 00:01:32.390
going to be.

00:01:32.390 --> 00:01:33.630
So let's do that.

00:01:33.630 --> 00:01:36.040
And when you're dealing
with parabolic mirrors,

00:01:36.040 --> 00:01:39.490
it's easiest to have to
show just two light rays.

00:01:39.490 --> 00:01:41.851
One that goes parallel
to the principal axis.

00:01:41.851 --> 00:01:43.350
And one that goes
through the focus.

00:01:43.350 --> 00:01:45.850
Because you know what's going
to happen to each of those

00:01:45.850 --> 00:01:46.600
when they reflect.

00:01:46.600 --> 00:01:48.550
And you don't have
to do any math there.

00:01:48.550 --> 00:01:50.960
So let's have the parallel ray.

00:01:50.960 --> 00:01:51.970
That's the parallel ray.

00:01:51.970 --> 00:01:56.280
When it reflects-- the parallel
incident ray, when it reflects,

00:01:56.280 --> 00:02:00.180
it will go through the focus.

00:02:00.180 --> 00:02:01.840
And then, let's
have an incident ray

00:02:01.840 --> 00:02:03.200
that goes through the focus.

00:02:05.990 --> 00:02:07.805
And when it reflects,
it will be parallel.

00:02:10.310 --> 00:02:13.060
And this is the example we
actually saw in the last video.

00:02:13.060 --> 00:02:14.670
And so whatever light
is being emitted

00:02:14.670 --> 00:02:17.020
from this point over
here in that direction,

00:02:17.020 --> 00:02:19.830
it will come back and
converge again at this point.

00:02:19.830 --> 00:02:22.720
And we could actually do it
at every point along the--

00:02:22.720 --> 00:02:24.019
we could do it halfway.

00:02:24.019 --> 00:02:26.310
This halfway point of the
arrow, just to make it clear.

00:02:26.310 --> 00:02:28.640
This halfway point of
the arrow, same thing.

00:02:28.640 --> 00:02:30.700
You have a parallel.

00:02:30.700 --> 00:02:34.280
Something parallel will go,
will reflect a parallel incident

00:02:34.280 --> 00:02:36.319
ray will reflect
through the focus.

00:02:36.319 --> 00:02:38.360
And I'll just do it for
this one right over here.

00:02:38.360 --> 00:02:40.191
It will reflect
through the focus.

00:02:40.191 --> 00:02:41.690
And then, if you
have something that

00:02:41.690 --> 00:02:44.360
goes through the focus,
an incident ray that

00:02:44.360 --> 00:02:48.840
goes through a focus, it
will reflect parallel.

00:02:48.840 --> 00:02:51.760
So this point will correspond
to this point over here.

00:02:51.760 --> 00:02:55.480
I think that makes it clearer,
that this image, the image

00:02:55.480 --> 00:02:57.000
of this object
when it's reflected

00:02:57.000 --> 00:03:02.660
by this parabolic mirror
will look just like that.

00:03:02.660 --> 00:03:05.830
So it will actually
form a real image.

00:03:05.830 --> 00:03:09.109
It'll form a real
image that is smaller

00:03:09.109 --> 00:03:10.150
than this original image.

00:03:10.150 --> 00:03:12.260
It's not so clear, the
way I did it over here.

00:03:12.260 --> 00:03:14.949
But you could push this
even further back out.

00:03:14.949 --> 00:03:16.490
And it'll be clear
that this is going

00:03:16.490 --> 00:03:19.580
to be a smaller real image
than that right over there.

00:03:19.580 --> 00:03:21.560
Now, let's do a
couple more examples.

00:03:21.560 --> 00:03:23.860
So let me just paste my drawing.

00:03:23.860 --> 00:03:25.170
So I don't have to redraw it.

00:03:25.170 --> 00:03:26.194
Let's see what happens.

00:03:26.194 --> 00:03:26.860
Let me write it.

00:03:26.860 --> 00:03:30.070
So here the image, just so
we can keep track of things.

00:03:30.070 --> 00:03:39.580
Here, the image is real and
smaller than the actual object

00:03:39.580 --> 00:03:44.110
when the actual object is
beyond the center of curvature.

00:03:44.110 --> 00:03:47.547
And actually, let me make it a
little bit clearer by drawing.

00:03:47.547 --> 00:03:49.880
Let me do another example
like that where I do something

00:03:49.880 --> 00:03:52.900
big way out here just
to make it clear.

00:03:52.900 --> 00:04:00.420
So once again, we go parallel,
reflect through the focus.

00:04:00.420 --> 00:04:03.090
And then, we can go
through the focus.

00:04:03.090 --> 00:04:06.882
We go through the focus and
then reflect out like that.

00:04:06.882 --> 00:04:08.590
And there you see,
now, it's much clearer

00:04:08.590 --> 00:04:12.010
that the image is going
to be much smaller.

00:04:12.010 --> 00:04:16.190
And, of course, inverted
relative to the actual object.

00:04:16.190 --> 00:04:18.490
Now, let's do this again.

00:04:18.490 --> 00:04:20.470
But this time, let's
place the object

00:04:20.470 --> 00:04:22.890
at the center of curvature.

00:04:22.890 --> 00:04:29.090
So let's place the
object right over here.

00:04:29.090 --> 00:04:32.010
So right at that distance
that twice the distance

00:04:32.010 --> 00:04:35.820
from the vertex of the
parabola to the focus.

00:04:35.820 --> 00:04:38.150
So we do a parallel line.

00:04:38.150 --> 00:04:39.990
These lines are the
hardest thing to do.

00:04:39.990 --> 00:04:43.957
So parallel incident ray,
parallel to the principal axis.

00:04:43.957 --> 00:04:45.540
This is the principal
axis right here.

00:04:45.540 --> 00:04:49.820
Principal axis-- that's what
this line is right over here.

00:04:49.820 --> 00:04:52.900
It's kind of the line of
symmetry of the parabola.

00:04:52.900 --> 00:04:55.300
When it reflects, it will
reflect through the focus.

00:04:58.050 --> 00:05:00.330
And then, let's take
another ray that goes,

00:05:00.330 --> 00:05:02.847
the incident ray goes
through the focus.

00:05:02.847 --> 00:05:04.805
And when it reflects, it
will reflect parallel.

00:05:07.440 --> 00:05:11.140
And my drawing isn't the
neatest drawing on the planet.

00:05:11.140 --> 00:05:14.540
And actually, let me draw it
a little bit better than that.

00:05:14.540 --> 00:05:15.850
Well, that's pretty good.

00:05:15.850 --> 00:05:16.800
Let me just.

00:05:16.800 --> 00:05:19.620
So that's the incident
ray that was parallel.

00:05:19.620 --> 00:05:23.860
And then, an incident
ray that goes through--

00:05:23.860 --> 00:05:26.375
let me-- I'm having
trouble drawing these.

00:05:26.375 --> 00:05:28.570
An incident ray that
goes through the focus

00:05:28.570 --> 00:05:32.120
will then come out and
reflect right over there.

00:05:32.120 --> 00:05:33.340
And they'll converge.

00:05:33.340 --> 00:05:36.200
And the way I've drawn it,
my drawing isn't ideal.

00:05:36.200 --> 00:05:39.219
But the way, but
the reality is, is

00:05:39.219 --> 00:05:41.260
that they'll converge so
that the image will just

00:05:41.260 --> 00:05:44.420
be in an inverted same size
version as this thing up here.

00:05:44.420 --> 00:05:45.700
Because it's symmetric.

00:05:45.700 --> 00:05:47.900
Let me see if I can
redraw this whole thing,

00:05:47.900 --> 00:05:50.020
so it comes out neater.

00:05:50.020 --> 00:05:51.770
So far, that looks good.

00:05:51.770 --> 00:05:57.450
So then, you want to
reflect like that.

00:05:57.450 --> 00:05:59.360
You got-- you're coming
through the focus.

00:05:59.360 --> 00:06:01.910
And then, you have another ray
that goes through the focus.

00:06:01.910 --> 00:06:03.451
This whole thing
should be symmetric.

00:06:03.451 --> 00:06:06.190
And then, when it reflects,
it comes back out like that.

00:06:06.190 --> 00:06:07.800
So that makes a
little bit clearer.

00:06:07.800 --> 00:06:09.270
So this is the object.

00:06:09.270 --> 00:06:11.690
And now, its image is
just an inverted version

00:06:11.690 --> 00:06:12.430
of this object.

00:06:12.430 --> 00:06:17.370
The image comes into
focus, or the rays

00:06:17.370 --> 00:06:21.670
converge at the same distance
from the actual mirror

00:06:21.670 --> 00:06:23.470
as the actual object.

00:06:23.470 --> 00:06:26.110
And it's going to be the
same size, just inverted.

00:06:26.110 --> 00:06:35.800
So that's-- so here the image
is real and the same size

00:06:35.800 --> 00:06:37.010
as the object.

00:06:37.010 --> 00:06:39.300
Let's do a couple more of these.

00:06:39.300 --> 00:06:41.000
I think you get the hang of it.

00:06:41.000 --> 00:06:42.000
And you might want
to try them out.

00:06:42.000 --> 00:06:43.374
You might want to
pause the video

00:06:43.374 --> 00:06:46.380
and try it on paper because
really nothing beats practice.

00:06:46.380 --> 00:06:51.720
So let's stick our object
between the center of curvature

00:06:51.720 --> 00:06:53.966
and the focus and
the focal point.

00:06:53.966 --> 00:06:55.340
So if we put our
object there, we

00:06:55.340 --> 00:06:59.000
could have a light ray that goes
parallel to the principal axis.

00:06:59.000 --> 00:07:02.020
And then, it will reflect
out through the focus.

00:07:02.020 --> 00:07:04.810
And then, you could
have another point.

00:07:04.810 --> 00:07:06.675
It goes through another ray.

00:07:06.675 --> 00:07:08.010
It goes through the focus.

00:07:08.010 --> 00:07:09.540
And then, it reflects.

00:07:09.540 --> 00:07:11.140
And then, it will reflect out.

00:07:16.340 --> 00:07:17.980
And let me draw it
better than that.

00:07:17.980 --> 00:07:20.580
Actually, this is--
I should probably

00:07:20.580 --> 00:07:23.470
have used a more precise
tool when I did all of this.

00:07:23.470 --> 00:07:27.060
So let me draw it
right over here.

00:07:27.060 --> 00:07:29.070
So I have the parallel one.

00:07:29.070 --> 00:07:31.360
And then, it goes
through the focus

00:07:31.360 --> 00:07:34.936
just like that when
it gets reflected.

00:07:34.936 --> 00:07:36.810
I really should have
had a line tool for this

00:07:36.810 --> 00:07:38.690
to have neater drawings.

00:07:38.690 --> 00:07:42.840
And then, a ray that
goes through the focus

00:07:42.840 --> 00:07:45.300
will be reflected out parallel.

00:07:45.300 --> 00:07:47.150
It would be reflected
out parallel.

00:07:47.150 --> 00:07:50.820
And at least for the light
that comes from that tip,

00:07:50.820 --> 00:07:53.120
they will re-converge
at that tip.

00:07:53.120 --> 00:07:56.070
And if you did it for
every point on this arrow,

00:07:56.070 --> 00:07:57.720
the image would be
an inverted arrow

00:07:57.720 --> 00:07:59.510
that is bigger
than the original.

00:07:59.510 --> 00:08:00.620
And it is beyond.

00:08:00.620 --> 00:08:03.530
It's almost the opposite of the
first example that we showed.

00:08:03.530 --> 00:08:06.560
And so now the image is
bigger than the original.

00:08:06.560 --> 00:08:10.140
So the image is real.

00:08:10.140 --> 00:08:11.820
And it is bigger.

00:08:11.820 --> 00:08:17.260
And the image will,
where it converges

00:08:17.260 --> 00:08:20.494
is going to be beyond
the center of curvature.

00:08:20.494 --> 00:08:21.410
And you could imagine.

00:08:21.410 --> 00:08:23.126
If this was the
object right here,

00:08:23.126 --> 00:08:24.750
then this would be
the image over here.

00:08:24.750 --> 00:08:26.500
If you just trace
the lines backwards.

00:08:26.500 --> 00:08:29.470
So there's a symmetry
here between this example

00:08:29.470 --> 00:08:31.600
and the first one
we did up here.

00:08:31.600 --> 00:08:33.070
Now, let's just
do a couple more.

00:08:33.070 --> 00:08:36.580
Let's imagine if the object is
actually at the focal point.

00:08:36.580 --> 00:08:38.679
It's actually at the focus.

00:08:38.679 --> 00:08:40.190
So let's draw an object there.

00:08:40.190 --> 00:08:42.049
Let's think about
what would happen.

00:08:42.049 --> 00:08:47.640
So if we're at the focus, a
ray that comes out parallel

00:08:47.640 --> 00:08:53.370
will go through the focal point
and come out just like that.

00:08:53.370 --> 00:08:56.940
And then, here we're going to--
you can't-- we can't have a ray

00:08:56.940 --> 00:08:57.820
that just goes.

00:08:57.820 --> 00:09:00.760
Well, actually we could have
a ray that goes through.

00:09:00.760 --> 00:09:04.141
Well, that-- you can't
go into the object.

00:09:04.141 --> 00:09:05.890
So here, I'll do a
slightly different ray.

00:09:05.890 --> 00:09:10.100
I'll do a ray that intersects
the parabolic mirror

00:09:10.100 --> 00:09:10.859
right over there.

00:09:10.859 --> 00:09:12.400
And the reason why
I want to do there

00:09:12.400 --> 00:09:14.290
is because there
the parabolic mirror

00:09:14.290 --> 00:09:17.070
is essentially flat and
essentially vertical.

00:09:17.070 --> 00:09:18.770
So you can imagine
that the incident

00:09:18.770 --> 00:09:21.411
ray is going to be the same
thing as the reflected ray.

00:09:21.411 --> 00:09:23.410
So you could draw a ray
that comes in like that.

00:09:23.410 --> 00:09:25.520
So this is a departure
from what we did before.

00:09:25.520 --> 00:09:28.120
And then the reflected ray
will come out like that.

00:09:28.120 --> 00:09:29.930
So what happens
when an object is

00:09:29.930 --> 00:09:35.200
at the focal point is that all
of the light that's coming off

00:09:35.200 --> 00:09:41.120
of this object in any
direction will all be made,

00:09:41.120 --> 00:09:43.330
it will all be made parallel.

00:09:43.330 --> 00:09:44.990
And so it won't converge.

00:09:44.990 --> 00:09:46.070
So it won't converge.

00:09:46.070 --> 00:09:47.920
So it won't be able
to form a real image.

00:09:47.920 --> 00:09:49.790
And it's not, it
doesn't look like it's

00:09:49.790 --> 00:09:53.300
diverging from some
point in the mirror.

00:09:53.300 --> 00:09:55.600
So it won't even
form a virtual image.

00:09:55.600 --> 00:10:00.060
So here, there will
actually be no image

00:10:00.060 --> 00:10:02.780
when the object is actually
at the focal point.

00:10:02.780 --> 00:10:05.510
And then, the last case,
as you can imagine,

00:10:05.510 --> 00:10:08.410
is if an object is closer
than the focal point.

00:10:08.410 --> 00:10:09.980
So let's draw that.

00:10:09.980 --> 00:10:13.770
So let's put an object
at the focal point.

00:10:13.770 --> 00:10:15.840
So right over here.

00:10:15.840 --> 00:10:20.060
And here, just for the sake
of argument, one I can draw.

00:10:20.060 --> 00:10:22.040
I can always draw a parallel.

00:10:22.040 --> 00:10:24.200
And anything that, any
light that goes parallel

00:10:24.200 --> 00:10:26.930
will then look to come
out in a direction that

00:10:26.930 --> 00:10:28.750
would go through
the focal point.

00:10:28.750 --> 00:10:30.470
So it would go through
the focal point.

00:10:30.470 --> 00:10:32.320
It would come out
in that direction.

00:10:32.320 --> 00:10:34.070
Although the object
itself is blocking.

00:10:34.070 --> 00:10:36.100
But it would look to
go in that direction.

00:10:36.100 --> 00:10:38.130
It would be reflected
in that direction.

00:10:38.130 --> 00:10:40.120
And then, you could
imagine a ray of light

00:10:40.120 --> 00:10:42.240
that would have been coming
from the focal point,

00:10:42.240 --> 00:10:44.890
or would have been coming
from the same direction.

00:10:44.890 --> 00:10:47.510
So you could imagine coming
from the same direction

00:10:47.510 --> 00:10:50.000
as the focal point
would be reflected

00:10:50.000 --> 00:10:51.730
parallel, in a
parallel direction

00:10:51.730 --> 00:10:52.815
to the principal axis.

00:10:56.060 --> 00:10:58.710
Now, these two light
rays are not converging.

00:10:58.710 --> 00:11:00.400
But they look like
they're diverging

00:11:00.400 --> 00:11:03.932
from some point
behind the mirror.

00:11:03.932 --> 00:11:05.390
They'll look like
they're diverging

00:11:05.390 --> 00:11:07.070
from some point
behind the mirror.

00:11:07.070 --> 00:11:09.930
So in this case, we are
forming a virtual image.

00:11:15.352 --> 00:11:16.810
And the virtual
image will actually

00:11:16.810 --> 00:11:18.510
look something like this.

00:11:18.510 --> 00:11:24.162
And so it'll be larger than
the original virtual image.

00:11:24.162 --> 00:11:25.370
So it's kind of a magnifying.

00:11:25.370 --> 00:11:26.995
If you were to go
to-- what are called?

00:11:26.995 --> 00:11:30.717
--a fun house at the circus or
the amusement park, whatever.

00:11:30.717 --> 00:11:33.050
And if you were to get close
enough to parabolas mirror,

00:11:33.050 --> 00:11:37.960
it would show a
magnified version of you.

00:11:37.960 --> 00:11:39.067
A virtual version of you.

00:11:39.067 --> 00:11:41.150
And actually, let me draw
that a little bit bigger

00:11:41.150 --> 00:11:42.750
just because it
might not be clear.

00:11:42.750 --> 00:11:44.120
So let me draw.

00:11:44.120 --> 00:11:48.570
So if this is the mirror,
that is the focal point.

00:11:48.570 --> 00:11:51.450
This right here is
the principal axis.

00:11:51.450 --> 00:11:55.670
This is maybe you,
maybe whatever object.

00:11:55.670 --> 00:11:57.600
You could draw a ray
that goes parallel.

00:11:57.600 --> 00:12:00.130
It will reflect in the
direction of the focal point.

00:12:00.130 --> 00:12:02.180
So it'll reflect out like that.

00:12:02.180 --> 00:12:04.397
It'll be blocked by
the object, though.

00:12:04.397 --> 00:12:05.980
And then, something
that looks like it

00:12:05.980 --> 00:12:07.521
would have come from
the focal point,

00:12:07.521 --> 00:12:09.450
from the same direction
as the focal point

00:12:09.450 --> 00:12:12.920
would then be reflected
parallel to the principal axis.

00:12:12.920 --> 00:12:16.080
So these two rays, once
again, there are diverging.

00:12:16.080 --> 00:12:19.390
But they look like, to the
human brain, to the human eye,

00:12:19.390 --> 00:12:22.380
they look like they came
from that point over there.

00:12:22.380 --> 00:12:24.690
And so this would
correspond to that point

00:12:24.690 --> 00:12:26.070
on the virtual image.

00:12:28.499 --> 00:12:30.290
So hopefully, that
gives you some practice.

00:12:30.290 --> 00:12:32.040
But the most important
thing, it gives you

00:12:32.040 --> 00:12:34.220
some practice dealing
with these arbitrary rays

00:12:34.220 --> 00:12:36.986
that we're showing emanating
from the tip of this arrow.

00:12:36.986 --> 00:12:38.610
And we could do it
for the whole arrow.

00:12:38.610 --> 00:12:41.235
But the reason why we're picking
these rays in these directions

00:12:41.235 --> 00:12:42.760
is that they're
easy to work with.

00:12:42.760 --> 00:12:45.027
They go through the focus,
they'll come out parallel.

00:12:45.027 --> 00:12:46.610
If the incident ray
is parallel, it'll

00:12:46.610 --> 00:12:48.860
come out through the focus.