1
00:00:00,000 --> 00:00:00,520


2
00:00:00,520 --> 00:00:02,370
We've been doing a
bunch of these videos

3
00:00:02,370 --> 00:00:05,380
with these convex lenses
where we drew parallel rays

4
00:00:05,380 --> 00:00:08,430
and rays that go
through the focal point

5
00:00:08,430 --> 00:00:11,552
to figure out what the
image of an object might be.

6
00:00:11,552 --> 00:00:13,010
But what I want to
do in this video

7
00:00:13,010 --> 00:00:15,570
is actually come up with
an algebraic relationship

8
00:00:15,570 --> 00:00:21,620
between the distance of the
object from the convex lens,

9
00:00:21,620 --> 00:00:24,900
the distance of the image
from the convex lens

10
00:00:24,900 --> 00:00:29,479
usually on the other side
and the focal length.

11
00:00:29,479 --> 00:00:30,770
So let's see if we can do this.

12
00:00:30,770 --> 00:00:33,500
And just to save you
the pain of having

13
00:00:33,500 --> 00:00:37,570
to watch me draw straight lines,
I drew this ahead of time.

14
00:00:37,570 --> 00:00:42,690
And so we can imagine this green
thing right here is the object.

15
00:00:42,690 --> 00:00:45,610
This is the object, and
these two little pink points

16
00:00:45,610 --> 00:00:48,340
right here are the focal points.

17
00:00:48,340 --> 00:00:50,640
They're a focal length away.

18
00:00:50,640 --> 00:00:51,900
And I did what we always drew.

19
00:00:51,900 --> 00:00:54,950
I drew one parallel ray
from the tip of that arrow

20
00:00:54,950 --> 00:00:58,960
to the actual convex lens,
and then it gets refracted,

21
00:00:58,960 --> 00:01:01,520
so it goes through the focal
point on the right-hand side

22
00:01:01,520 --> 00:01:03,420
and goes all the way over there.

23
00:01:03,420 --> 00:01:06,210
And then I drew a ray that
goes through the focal point

24
00:01:06,210 --> 00:01:07,570
on the left-hand side.

25
00:01:07,570 --> 00:01:09,970
And then when it gets
refracted, it becomes parallel.

26
00:01:09,970 --> 00:01:14,610
And then it actually intersects
with that previous ray

27
00:01:14,610 --> 00:01:15,890
right over here.

28
00:01:15,890 --> 00:01:21,690
And so this gives us a sense of
what the image will look like.

29
00:01:21,690 --> 00:01:22,440
It's inverted.

30
00:01:22,440 --> 00:01:23,150
It's real.

31
00:01:23,150 --> 00:01:26,281
In this case, it is larger
than the actual object.

32
00:01:26,281 --> 00:01:28,280
What I want to do is come
up with a relationship

33
00:01:28,280 --> 00:01:29,230
with these values.

34
00:01:29,230 --> 00:01:31,190
So let's see if we
can label them here.

35
00:01:31,190 --> 00:01:32,940
And then, just do a
little bit of geometry

36
00:01:32,940 --> 00:01:34,860
and a little bit of
algebra to figure out

37
00:01:34,860 --> 00:01:37,820
if there is an algebraic
relationship right here.

38
00:01:37,820 --> 00:01:40,890
So the first
number, the distance

39
00:01:40,890 --> 00:01:44,439
of the object-- that's this
distance from here to here,

40
00:01:44,439 --> 00:01:45,730
or we could just label it here.

41
00:01:45,730 --> 00:01:47,800
Since this is
already drawn for us,

42
00:01:47,800 --> 00:01:49,920
this is the distance
of the object.

43
00:01:49,920 --> 00:01:51,620
This is the way we drew it.

44
00:01:51,620 --> 00:01:53,330
This was the parallel light ray.

45
00:01:53,330 --> 00:01:57,420
But before it got refracted,
it traveled the distance

46
00:01:57,420 --> 00:02:00,480
from the object to
the actual lens.

47
00:02:00,480 --> 00:02:02,740
Now, the distance from the
image to the lens, that's

48
00:02:02,740 --> 00:02:03,720
this right over here.

49
00:02:03,720 --> 00:02:09,860
This is how far this parallel
light ray had to travel.

50
00:02:09,860 --> 00:02:13,980
So this is the distance
from the image to the lens.

51
00:02:13,980 --> 00:02:17,860
And then we have the focal
distance, the focal length.

52
00:02:17,860 --> 00:02:20,270
And that's just this
distance right here.

53
00:02:20,270 --> 00:02:22,390
This right here is
our focal length.

54
00:02:22,390 --> 00:02:24,240
Or, we could view it
on this side as well.

55
00:02:24,240 --> 00:02:29,760
This right here is
also our focal length.

56
00:02:29,760 --> 00:02:32,200
So I want to come up
with some relationship.

57
00:02:32,200 --> 00:02:35,220
And to do that, I'm going
to draw some triangles here.

58
00:02:35,220 --> 00:02:38,035
So what we can do is--
and the whole strategy--

59
00:02:38,035 --> 00:02:40,090
I'm going to keep looking
for similar triangles,

60
00:02:40,090 --> 00:02:43,870
and then try to see if
I can find relationship,

61
00:02:43,870 --> 00:02:47,230
or ratios, that relate these
three things to each other.

62
00:02:47,230 --> 00:02:49,000
So let me find some
similar triangles.

63
00:02:49,000 --> 00:02:51,120
So the best thing I
could think of to do

64
00:02:51,120 --> 00:02:54,020
is let me redraw this
triangle over here.

65
00:02:54,020 --> 00:02:55,830
Let me just flip it over.

66
00:02:55,830 --> 00:02:57,290
Let me just draw
the same triangle

67
00:02:57,290 --> 00:02:59,960
on the right-hand
side of this diagram.

68
00:02:59,960 --> 00:03:01,330
So this.

69
00:03:01,330 --> 00:03:03,140
So if I were to draw
the same triangle,

70
00:03:03,140 --> 00:03:04,355
it would look like this.

71
00:03:04,355 --> 00:03:08,180


72
00:03:08,180 --> 00:03:10,510
And let me just be clear,
this is this triangle right

73
00:03:10,510 --> 00:03:11,200
over here.

74
00:03:11,200 --> 00:03:12,440
I just flipped it over.

75
00:03:12,440 --> 00:03:15,770


76
00:03:15,770 --> 00:03:17,270
And so if we want
to make sure we're

77
00:03:17,270 --> 00:03:20,970
keeping track of the same
sides, if this length right here

78
00:03:20,970 --> 00:03:23,870
is d sub 0, or d
naught sometimes we

79
00:03:23,870 --> 00:03:26,290
could call it, or d0,
whatever you want to call it,

80
00:03:26,290 --> 00:03:37,040
then this length up here
is also going to be d0.

81
00:03:37,040 --> 00:03:38,602
And the reason why
I want to do that

82
00:03:38,602 --> 00:03:40,560
is because now we can do
something interesting.

83
00:03:40,560 --> 00:03:43,010
We can relate this triangle
up here to this triangle

84
00:03:43,010 --> 00:03:43,830
down here.

85
00:03:43,830 --> 00:03:47,240
And actually, we can see that
they're going to be similar.

86
00:03:47,240 --> 00:03:49,170
And then we can get
some ratios of sides.

87
00:03:49,170 --> 00:03:50,000
And then what we're
going to do is

88
00:03:50,000 --> 00:03:51,666
try to show that this
triangle over here

89
00:03:51,666 --> 00:03:53,720
is similar to this
triangle over here,

90
00:03:53,720 --> 00:03:55,307
get a couple of more ratios.

91
00:03:55,307 --> 00:03:57,640
And then we might be able to
relate all of these things.

92
00:03:57,640 --> 00:03:59,640
So the first thing we
have to prove to ourselves

93
00:03:59,640 --> 00:04:02,140
is that those triangles
really are similar.

94
00:04:02,140 --> 00:04:05,000
So the first thing to
realize, this angle right

95
00:04:05,000 --> 00:04:06,780
here is definitely
the same thing

96
00:04:06,780 --> 00:04:08,080
as that angle right over there.

97
00:04:08,080 --> 00:04:09,746
They're sometimes
called opposite angles

98
00:04:09,746 --> 00:04:10,590
or vertical angles.

99
00:04:10,590 --> 00:04:13,310
They're on the opposite side
of lines that are intersecting.

100
00:04:13,310 --> 00:04:14,870
So they're going to be equal.

101
00:04:14,870 --> 00:04:17,329
Now, the next thing-- and
this comes out of the fact

102
00:04:17,329 --> 00:04:20,920
that both of these lines--
this line is parallel

103
00:04:20,920 --> 00:04:23,040
to that line right over there.

104
00:04:23,040 --> 00:04:27,502
And I guess you could call
it alternate interior angles,

105
00:04:27,502 --> 00:04:29,710
if you look at the angles
game, or the parallel lines

106
00:04:29,710 --> 00:04:32,317
or the transversal of
parallel lines from geometry.

107
00:04:32,317 --> 00:04:33,900
We know that this
angle, since they're

108
00:04:33,900 --> 00:04:36,590
alternate interior
angles, this angle

109
00:04:36,590 --> 00:04:41,572
is going to be the same
value as this angle.

110
00:04:41,572 --> 00:04:43,030
You could view this
line right here

111
00:04:43,030 --> 00:04:46,340
as a transversal of
two parallel lines.

112
00:04:46,340 --> 00:04:48,480
These are alternate
interior angles,

113
00:04:48,480 --> 00:04:50,470
so they will be the same.

114
00:04:50,470 --> 00:04:54,090
Now, we can make that
exact same argument

115
00:04:54,090 --> 00:04:57,720
for this angle and this angle.

116
00:04:57,720 --> 00:04:59,710
And so what we see is
this triangle up here

117
00:04:59,710 --> 00:05:04,510
has the same three angles
as this triangle down here.

118
00:05:04,510 --> 00:05:07,320
So these two
triangles are similar.

119
00:05:07,320 --> 00:05:10,770
These are both-- Is really
more of a review of geometry

120
00:05:10,770 --> 00:05:11,480
than optics.

121
00:05:11,480 --> 00:05:14,100
These are similar triangles.

122
00:05:14,100 --> 00:05:16,740
Similar-- I don't have
to write triangles.

123
00:05:16,740 --> 00:05:18,110
They're similar.

124
00:05:18,110 --> 00:05:21,330
And because they're similar, the
ratios of corresponding sides

125
00:05:21,330 --> 00:05:23,150
are going to be the same.

126
00:05:23,150 --> 00:05:25,470
So d0 corresponds to this.

127
00:05:25,470 --> 00:05:29,420
They're both opposite
this pink angle.

128
00:05:29,420 --> 00:05:31,530
They're both opposite
that pink angle.

129
00:05:31,530 --> 00:05:35,250
So the ratio of d0 to d1--
let me write this over here.

130
00:05:35,250 --> 00:05:37,050
So the ratio of d0.

131
00:05:37,050 --> 00:05:38,840
Let me write this a
little bit neater.

132
00:05:38,840 --> 00:05:41,765
The ratio of d0 to d1.

133
00:05:41,765 --> 00:05:43,640
So this is the ratio of
corresponding sides--

134
00:05:43,640 --> 00:05:45,250
is going to be the same thing.

135
00:05:45,250 --> 00:05:47,230
And let me make
some labels here.

136
00:05:47,230 --> 00:05:50,370
That's going to be the same
thing as the ratio of this side

137
00:05:50,370 --> 00:05:51,960
right over here.

138
00:05:51,960 --> 00:05:54,050
This side right
over here, I'll call

139
00:05:54,050 --> 00:06:00,942
that A. It's opposite this
magenta angle right over here.

140
00:06:00,942 --> 00:06:03,400
That's going to be the same
thing as the ratio of that side

141
00:06:03,400 --> 00:06:06,095
to this side over
here, to side B.

142
00:06:06,095 --> 00:06:07,720
And once again, we
can keep track of it

143
00:06:07,720 --> 00:06:09,770
because side B is
opposite the magenta

144
00:06:09,770 --> 00:06:11,900
angle on this bottom triangle.

145
00:06:11,900 --> 00:06:15,250
So that's how we know that this
side, it's corresponding side

146
00:06:15,250 --> 00:06:17,000
in the other similar
triangle is that one.

147
00:06:17,000 --> 00:06:20,050
They're both opposite
the magenta angles.

148
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.

149
00:06:27,584 --> 00:06:28,500
So that's interesting.

150
00:06:28,500 --> 00:06:30,180
We've been able to
relate these two

151
00:06:30,180 --> 00:06:33,550
things to these kind of
two arbitrarily lengths.

152
00:06:33,550 --> 00:06:36,720
But we need to somehow connect
those to the focal length.

153
00:06:36,720 --> 00:06:39,480
And to connect them to
a focal length, what

154
00:06:39,480 --> 00:06:42,386
we might want to do
is relate A and B.

155
00:06:42,386 --> 00:06:45,050
A sits on the same triangle
as the focal length

156
00:06:45,050 --> 00:06:45,750
right over here.

157
00:06:45,750 --> 00:06:48,490
So let's look at this
triangle right over here.

158
00:06:48,490 --> 00:06:49,910
Let me put in a better color.

159
00:06:49,910 --> 00:06:51,980
So let's look at this
triangle right over here

160
00:06:51,980 --> 00:06:54,980
that I'm highlighting in green.

161
00:06:54,980 --> 00:06:56,850
This triangle in green.

162
00:06:56,850 --> 00:06:59,610
And let's look at that in
comparison to this triangle

163
00:06:59,610 --> 00:07:01,440
that I'm also highlighting.

164
00:07:01,440 --> 00:07:04,695
This triangle that I'm
also highlighting in green.

165
00:07:04,695 --> 00:07:06,320
Now, the first thing
I want to show you

166
00:07:06,320 --> 00:07:09,790
is that these are also
similar triangles.

167
00:07:09,790 --> 00:07:12,520
This angle right over
here and this angle

168
00:07:12,520 --> 00:07:14,470
are going to be the same.

169
00:07:14,470 --> 00:07:17,900
They are opposite angles
of intersecting lines.

170
00:07:17,900 --> 00:07:20,290
And then, we can make
a similar argument--

171
00:07:20,290 --> 00:07:21,600
alternate interior angles.

172
00:07:21,600 --> 00:07:22,930
Well, there's a couple
arguments we could make.

173
00:07:22,930 --> 00:07:25,480
One, you can see that this is
a right angle right over here.

174
00:07:25,480 --> 00:07:26,910
This is a right angle.

175
00:07:26,910 --> 00:07:29,670
If two angles of two
triangles are the same,

176
00:07:29,670 --> 00:07:31,390
the third angle also
has to be the same.

177
00:07:31,390 --> 00:07:33,520
So we could also say
that this thing-- let

178
00:07:33,520 --> 00:07:35,690
me do this in another
color because I don't want

179
00:07:35,690 --> 00:07:37,630
to be repetitive too
much with the colors.

180
00:07:37,630 --> 00:07:39,350
We can say that
this thing is going

181
00:07:39,350 --> 00:07:41,467
to be the same
thing as this thing.

182
00:07:41,467 --> 00:07:43,050
Or another way you
could have said it,

183
00:07:43,050 --> 00:07:46,610
is you could have said, well,
this line over here, which

184
00:07:46,610 --> 00:07:48,550
is kind of represented
by the lens,

185
00:07:48,550 --> 00:07:50,452
or the lens-- the
line that is parallel

186
00:07:50,452 --> 00:07:51,910
to the lens or
right along the lens

187
00:07:51,910 --> 00:07:55,170
is parallel to kind of the
object right over there.

188
00:07:55,170 --> 00:07:57,670
And then you could make the
same alternate interior argument

189
00:07:57,670 --> 00:07:58,110
there.

190
00:07:58,110 --> 00:07:59,560
But the other thing
is just, look.

191
00:07:59,560 --> 00:08:00,540
I have two triangles.

192
00:08:00,540 --> 00:08:02,790
Two of the angles in those
two triangles are the same,

193
00:08:02,790 --> 00:08:05,020
so the third angle
has to be the same.

194
00:08:05,020 --> 00:08:07,500
Now, since all three
angles are the same,

195
00:08:07,500 --> 00:08:10,360
these are also both
similar triangles.

196
00:08:10,360 --> 00:08:12,180
So we can do a similar thing.

197
00:08:12,180 --> 00:08:15,260
We can say A is to B.
Remember, both A and B

198
00:08:15,260 --> 00:08:16,740
are opposite the 90-degree side.

199
00:08:16,740 --> 00:08:19,700
They're both the hypotenuse
of the similar triangle.

200
00:08:19,700 --> 00:08:25,255
So A is to B as-- we could say
this base length right here.

201
00:08:25,255 --> 00:08:26,850
And it got overwritten
a little bit.

202
00:08:26,850 --> 00:08:29,310
But this base length
right over here is f.

203
00:08:29,310 --> 00:08:32,730
That's our focal length.

204
00:08:32,730 --> 00:08:35,630
As f in this triangle is
related to this length

205
00:08:35,630 --> 00:08:37,250
on this triangle.

206
00:08:37,250 --> 00:08:41,470
They are both opposite
that white angle.

207
00:08:41,470 --> 00:08:44,000
So as f is to this
length right over here.

208
00:08:44,000 --> 00:08:45,350
Now, what is this length?

209
00:08:45,350 --> 00:08:47,877
So this whole distance is
di, all the way over here.

210
00:08:47,877 --> 00:08:49,460
But this length is
that whole distance

211
00:08:49,460 --> 00:08:51,220
minus the focal length.

212
00:08:51,220 --> 00:08:54,680
So this is di minus
the focal length.

213
00:08:54,680 --> 00:09:01,106
So A is to B as f is to
di minus the focal length.

214
00:09:01,106 --> 00:09:02,980
And there you have it,
we have a relationship

215
00:09:02,980 --> 00:09:05,220
between the distance
of the object,

216
00:09:05,220 --> 00:09:07,280
the distance of the image,
and the focal length.

217
00:09:07,280 --> 00:09:08,670
And now we just have to do
a little bit of algebra.

218
00:09:08,670 --> 00:09:10,795
If this is equal to this
and this is equal to that,

219
00:09:10,795 --> 00:09:13,774
then this blue thing has to be
equal to this magenta thing.

220
00:09:13,774 --> 00:09:15,440
And now we just have
to do some algebra.

221
00:09:15,440 --> 00:09:16,540
So let's do that.

222
00:09:16,540 --> 00:09:22,290
So we got d naught, or d0
I guess we'd call it, to di

223
00:09:22,290 --> 00:09:24,990
is equal to the ratio
of the focal length

224
00:09:24,990 --> 00:09:28,440
to the difference of
the image distance

225
00:09:28,440 --> 00:09:32,290
to the-- the focal
length to the image

226
00:09:32,290 --> 00:09:34,220
distance minus the focal length.

227
00:09:34,220 --> 00:09:36,180
And now here, we just
have to do some algebra.

228
00:09:36,180 --> 00:09:39,140
So let's-- just to simplify
this, let's cross multiply it.

229
00:09:39,140 --> 00:09:46,460
So if we multiply d0 times this
thing over here, we get d0 di.

230
00:09:46,460 --> 00:09:52,110
I'm really just
distributing it, minus d0 f.

231
00:09:52,110 --> 00:09:55,570
I'm just distributing this
d0, just cross multiplying,

232
00:09:55,570 --> 00:09:57,710
which is really just the
same thing as multiplying

233
00:09:57,710 --> 00:10:00,862
both sides by both
denominators, or multiplying

234
00:10:00,862 --> 00:10:01,820
the denominators twice.

235
00:10:01,820 --> 00:10:04,120
Either way, that's all
cross multiplying is.

236
00:10:04,120 --> 00:10:10,990
That is going to be
equal to di times f.

237
00:10:10,990 --> 00:10:14,440


238
00:10:14,440 --> 00:10:16,400
And now, we can add this
term right over here

239
00:10:16,400 --> 00:10:17,860
to both sides of this equation.

240
00:10:17,860 --> 00:10:19,820
I'm just going to switch
to a neutral color.

241
00:10:19,820 --> 00:10:23,830
So we get d0 di is
equal to-- I'm just

242
00:10:23,830 --> 00:10:24,960
adding this to both sides.

243
00:10:24,960 --> 00:10:27,418
When you add it to the left
side, it obviously cancels out.

244
00:10:27,418 --> 00:10:33,477
Is equal to di f, this
thing over here, plus d0 f.

245
00:10:33,477 --> 00:10:34,310
And then, let's see.

246
00:10:34,310 --> 00:10:38,480
We could factor out
an f, a focal length.

247
00:10:38,480 --> 00:10:46,510
So we get d0 di is equal
to f times di plus d0.

248
00:10:46,510 --> 00:10:47,970
And then, what can we do?

249
00:10:47,970 --> 00:10:51,630
We could divide both sides by f.

250
00:10:51,630 --> 00:10:53,740
So this will become over f.

251
00:10:53,740 --> 00:10:57,580
This becomes over f.

252
00:10:57,580 --> 00:11:00,940
Essentially, canceling it out.

253
00:11:00,940 --> 00:11:03,315
And then, I just don't want
to skip too many steps,

254
00:11:03,315 --> 00:11:05,910
so let me just rewrite
what we have here.

255
00:11:05,910 --> 00:11:07,090
So these cancel out.

256
00:11:07,090 --> 00:11:16,140
So we have d0 di over f
is equal to di plus d0.

257
00:11:16,140 --> 00:11:19,320
And now, let's divide
both sides by d0 di.

258
00:11:19,320 --> 00:11:22,190
So 1 over d0 di.

259
00:11:22,190 --> 00:11:25,020
Divide this side by d0 di.

260
00:11:25,020 --> 00:11:26,860
It cancels out over here.

261
00:11:26,860 --> 00:11:30,060
And so we are left with,
on the left-hand side,

262
00:11:30,060 --> 00:11:33,742
1 over the focal length is
equal to this thing over here.

263
00:11:33,742 --> 00:11:35,200
And we can separate
this thing out.

264
00:11:35,200 --> 00:11:37,682
This thing over here,
this is the same thing--

265
00:11:37,682 --> 00:11:39,140
we just separate
out the numerator.

266
00:11:39,140 --> 00:11:47,235
Is the same thing as di over
d0 di plus d0 over d0 di.

267
00:11:47,235 --> 00:11:49,820
But di over d0 di,
the di's cancel out.

268
00:11:49,820 --> 00:11:50,970
We just have a 1.

269
00:11:50,970 --> 00:11:52,300
Here, the d0's cancel out.

270
00:11:52,300 --> 00:11:53,460
You just have a 1.

271
00:11:53,460 --> 00:11:57,550
So this is equal to 1 over
the distance of the object.

272
00:11:57,550 --> 00:12:02,417
And this is plus 1 over
the distance the image.

273
00:12:02,417 --> 00:12:05,000
So right from the get-go, this
was a completely valid formula.

274
00:12:05,000 --> 00:12:06,850
We actually had
achieved what we wanted.

275
00:12:06,850 --> 00:12:08,920
But this is a neater formula.

276
00:12:08,920 --> 00:12:11,740
You don't have the di's
repeat and the f repeated.

277
00:12:11,740 --> 00:12:14,000
Right here, we have an
algebraic relationship

278
00:12:14,000 --> 00:12:16,840
for a convex mirror that
relates the focal length

279
00:12:16,840 --> 00:12:22,410
to the distance of the object
and the distance of the image.

280
00:12:22,410 --> 00:12:25,730
Anyway, I think that's
pretty neat how it came out

281
00:12:25,730 --> 00:00:00,000
to be at least a
pretty clean formula.