1 00:00:00,375 --> 00:00:01,860 - [Voiceover] So imagine you've got an object 2 00:00:01,860 --> 00:00:04,247 sitting in front of this concave mirror. 3 00:00:04,247 --> 00:00:07,035 If you wanted to figure out where the image is formed, 4 00:00:07,035 --> 00:00:08,583 you can draw ray tracings. 5 00:00:08,583 --> 00:00:10,227 And so one ray you can draw 6 00:00:10,227 --> 00:00:12,747 is a parallel ray that goes through the focal point. 7 00:00:12,747 --> 00:00:14,234 But these rays are reversible. 8 00:00:14,234 --> 00:00:15,811 I don't have to draw that one. 9 00:00:15,811 --> 00:00:19,149 Turns out, if you send rays back along the way they came, 10 00:00:19,149 --> 00:00:22,036 they'll just retrace the path they came along the other way. 11 00:00:22,036 --> 00:00:23,489 So these rays are reversible. 12 00:00:23,489 --> 00:00:26,306 I send a ray parallel, it gets sent through the focal point. 13 00:00:26,306 --> 00:00:28,223 But if I send a ray through the focal point, 14 00:00:28,223 --> 00:00:30,070 it will get sent parallel. 15 00:00:30,070 --> 00:00:31,904 In other words, I don't have to draw this ray. 16 00:00:31,904 --> 00:00:33,603 I can draw this ray right here. 17 00:00:33,603 --> 00:00:34,561 The one that goes 18 00:00:34,561 --> 00:00:36,607 from the tip of the object through the focal point. 19 00:00:36,607 --> 00:00:38,281 That's gonna get sent parallel. 20 00:00:38,281 --> 00:00:40,190 So I'm gonna draw this one just for fun 21 00:00:40,190 --> 00:00:41,213 and then I'm gonna draw another one. 22 00:00:41,213 --> 00:00:44,088 You need two in order to find where the image is. 23 00:00:44,088 --> 00:00:45,248 I'll draw this one here. 24 00:00:45,248 --> 00:00:48,249 So this white line is called the principal axis. 25 00:00:48,249 --> 00:00:51,148 It's drawn through the center of this curved mirror. 26 00:00:51,148 --> 00:00:53,354 I'm gonna draw a ray that goes from the tip of the object 27 00:00:53,354 --> 00:00:54,820 to that center point 28 00:00:54,820 --> 00:00:56,859 because I know the law of reflection 29 00:00:56,859 --> 00:01:00,270 says that the angle in has to equal the angle out. 30 00:01:00,270 --> 00:01:02,319 And the angles are measured form the normal line 31 00:01:02,319 --> 00:01:04,593 and it just so happens that this principal axis 32 00:01:04,593 --> 00:01:06,445 that's usually drawn here anyways, 33 00:01:06,445 --> 00:01:08,755 is serving as a perfectly good normal line 34 00:01:08,755 --> 00:01:10,478 for this center of the mirror 35 00:01:10,478 --> 00:01:12,667 since it's passing through that center of the mirror 36 00:01:12,667 --> 00:01:14,179 in a perpendicular way. 37 00:01:14,179 --> 00:01:15,422 I can just use that to my advantage. 38 00:01:15,422 --> 00:01:17,604 I know that the angle in's gotta equal the angle out. 39 00:01:17,604 --> 00:01:19,599 I just have to make sure that this angle out 40 00:01:19,599 --> 00:01:21,421 is about, looks like that, 41 00:01:21,421 --> 00:01:24,072 about equal to the angle that it came in at. 42 00:01:24,072 --> 00:01:26,222 So these two have to be the same angle theta. 43 00:01:26,222 --> 00:01:28,221 And now I can find where my image is. 44 00:01:28,221 --> 00:01:30,041 The image of this object is gonna be 45 00:01:30,041 --> 00:01:31,870 at the point where they cross. 46 00:01:31,870 --> 00:01:32,970 So the tip of this object 47 00:01:32,970 --> 00:01:35,364 gets mapped to this point right here 48 00:01:35,364 --> 00:01:37,270 and we get an image that's upside down 49 00:01:37,270 --> 00:01:38,283 and it looks like that. 50 00:01:38,283 --> 00:01:41,446 So ray tracing is cool; it lets you find where the image is. 51 00:01:41,446 --> 00:01:44,216 But I mean I just kinda eyeballed this angle here. 52 00:01:44,216 --> 00:01:45,938 This might have been off by a little bit. 53 00:01:45,938 --> 00:01:47,971 It might have been off by a degree or two. 54 00:01:47,971 --> 00:01:50,342 If I wanted to get exactly where the image is, 55 00:01:50,342 --> 00:01:52,940 I'd want an equation that I could just plug into. 56 00:01:52,940 --> 00:01:55,062 In other words, that I could plug into 57 00:01:55,062 --> 00:01:57,410 how far was the object from the mirror, 58 00:01:57,410 --> 00:01:59,442 and how long is this focal length, 59 00:01:59,442 --> 00:02:02,311 and it would spit out exactly where the image is gonna be. 60 00:02:02,311 --> 00:02:04,068 So that's what we're gonna derive in this video. 61 00:02:04,068 --> 00:02:06,344 It's called the mirror equation 62 00:02:06,344 --> 00:02:09,107 and it'll tell us how to relate the object distance, 63 00:02:09,107 --> 00:02:11,189 the image distance, and the focal length. 64 00:02:11,189 --> 00:02:12,022 So let's do this. 65 00:02:12,022 --> 00:02:13,007 How do you derive it? 66 00:02:13,007 --> 00:02:15,330 If you look in the textbook, it looks complicated. 67 00:02:15,330 --> 00:02:17,188 It's not actually as bad as it looks. 68 00:02:17,188 --> 00:02:19,001 When I used to look at the first time, 69 00:02:19,001 --> 00:02:21,653 I was like, "This is some sort of mathematical witchcraft 70 00:02:21,653 --> 00:02:22,805 "I don't want to deal with." 71 00:02:22,805 --> 00:02:24,581 But it's not nearly as bad as it looks. 72 00:02:24,581 --> 00:02:26,096 We're gonna start with drawing triangles. 73 00:02:26,096 --> 00:02:28,915 So we'll notice that these two angles are the same. 74 00:02:28,915 --> 00:02:31,156 We're gonna use this to our advantage. 75 00:02:31,156 --> 00:02:32,712 So we're gonna make two triangles 76 00:02:32,712 --> 00:02:35,712 that have these as one of their angles. 77 00:02:35,712 --> 00:02:37,632 So the first one, let's consider this one. 78 00:02:37,632 --> 00:02:39,098 Let's say one of the triangles will be 79 00:02:39,098 --> 00:02:41,963 from the base of this image to the center of the mirror, 80 00:02:41,963 --> 00:02:45,089 and then from the mirror to the tip of the image, 81 00:02:45,089 --> 00:02:46,388 and then from the tip of the image 82 00:02:46,388 --> 00:02:48,622 back down to the base of the image. 83 00:02:48,622 --> 00:02:50,503 So imagine this pink triangle right here, 84 00:02:50,503 --> 00:02:51,770 it's a right triangle 85 00:02:51,770 --> 00:02:54,155 'cause that angle right here is a right angle, 86 00:02:54,155 --> 00:02:56,298 and it's got theta as one of its angles. 87 00:02:56,298 --> 00:02:57,714 But I could draw another triangle 88 00:02:57,714 --> 00:02:59,351 that has this angle theta. 89 00:02:59,351 --> 00:03:01,232 I can go from the tip of the object 90 00:03:01,232 --> 00:03:02,871 to the center of the mirror, 91 00:03:02,871 --> 00:03:03,956 and then from the center of the mirror 92 00:03:03,956 --> 00:03:05,385 to the base of the object, 93 00:03:05,385 --> 00:03:06,495 and then from the base of the object 94 00:03:06,495 --> 00:03:09,445 to the tip of the object and I get this blue triangle. 95 00:03:09,445 --> 00:03:11,138 That's also a right triangle 96 00:03:11,138 --> 00:03:13,249 since this angle here is a right angle. 97 00:03:13,249 --> 00:03:15,670 So in other words, these triangles are similar. 98 00:03:15,670 --> 00:03:17,264 They both have an angle theta, 99 00:03:17,264 --> 00:03:18,711 they both have a right angle. 100 00:03:18,711 --> 00:03:21,002 Or in other words, if you don't like similar triangles, 101 00:03:21,002 --> 00:03:22,220 just think about it this way. 102 00:03:22,220 --> 00:03:23,361 You could use a trig function. 103 00:03:23,361 --> 00:03:24,816 Pick your favorite trig function. 104 00:03:24,816 --> 00:03:25,880 I'm gonna pick tangent. 105 00:03:25,880 --> 00:03:27,261 So let's take tangent theta. 106 00:03:27,261 --> 00:03:28,871 I know tangent theta by definition 107 00:03:28,871 --> 00:03:31,540 is always the opposite over the adjacent. 108 00:03:31,540 --> 00:03:33,056 The opposite to this angle, 109 00:03:33,056 --> 00:03:35,331 we'll take this angle down here first, 110 00:03:35,331 --> 00:03:38,049 the opposite to that theta is this side. 111 00:03:38,049 --> 00:03:39,406 What does that side mean? 112 00:03:39,406 --> 00:03:40,803 That's the height of the image. 113 00:03:40,803 --> 00:03:43,600 So I'm gonna call that "hi" for image height. 114 00:03:43,600 --> 00:03:45,298 That's how tall the image is. 115 00:03:45,298 --> 00:03:47,623 So I know that this is gonna equal height of the image 116 00:03:47,623 --> 00:03:51,109 divided by the adjacent side to this angle here 117 00:03:51,109 --> 00:03:52,626 is this distance right here 118 00:03:52,626 --> 00:03:54,078 and we're gonna give that a name. 119 00:03:54,078 --> 00:03:56,694 That's just how far the image is from the mirror. 120 00:03:56,694 --> 00:03:57,527 So we give this a name, 121 00:03:57,527 --> 00:04:00,610 we call this distance from the mirror to the image, 122 00:04:00,610 --> 00:04:02,296 we call that the image distance. 123 00:04:02,296 --> 00:04:05,121 And it's measured from the center of the mirror. 124 00:04:05,121 --> 00:04:08,084 So not from this end right here, this little tip part, 125 00:04:08,084 --> 00:04:10,880 but from wherever the center of the mirror is. 126 00:04:10,880 --> 00:04:12,973 It's measured from this point right here. 127 00:04:12,973 --> 00:04:15,608 And that's this adjacent side since this is 128 00:04:15,608 --> 00:04:18,124 how far that image is from the center of the mirror, 129 00:04:18,124 --> 00:04:21,529 so I'll call that "di" since it's the image distance. 130 00:04:21,529 --> 00:04:23,301 But that was for this theta down here. 131 00:04:23,301 --> 00:04:26,458 I know that tangent of theta, this is also a theta up here. 132 00:04:26,458 --> 00:04:28,651 I can use the same relationship for this theta. 133 00:04:28,651 --> 00:04:30,338 And I know that tangent of this theta 134 00:04:30,338 --> 00:04:33,040 also has to be opposite over adjacent, 135 00:04:33,040 --> 00:04:36,224 but the opposite of this theta is this side right here. 136 00:04:36,224 --> 00:04:37,422 And what is that side? 137 00:04:37,422 --> 00:04:39,907 That's just the height of the object. 138 00:04:39,907 --> 00:04:41,335 So I'll call that "ho". 139 00:04:41,335 --> 00:04:42,590 That's the object height, 140 00:04:42,590 --> 00:04:45,984 so the opposite side for this theta is the object height. 141 00:04:45,984 --> 00:04:47,429 And then you divide by what? 142 00:04:47,429 --> 00:04:49,153 You divide by the adjacent side. 143 00:04:49,153 --> 00:04:50,397 That's just gonna be this distance 144 00:04:50,397 --> 00:04:51,933 from the mirror to the object. 145 00:04:51,933 --> 00:04:53,184 We'll give that a name, 146 00:04:53,184 --> 00:04:55,949 and if you guessed object distance, then you guessed right. 147 00:04:55,949 --> 00:04:58,674 This is gonna be the object distance. 148 00:04:58,674 --> 00:05:00,923 And again we measure it from the center of the mirror, 149 00:05:00,923 --> 00:05:03,713 not some curved portion that sticks out over here, 150 00:05:03,713 --> 00:05:06,298 but from the center part of the mirror. 151 00:05:06,298 --> 00:05:09,418 So that's the adjacent side to this theta down here, 152 00:05:09,418 --> 00:05:11,208 so I'm gonna write that as "do". 153 00:05:11,208 --> 00:05:13,007 And so this is an important relationship. 154 00:05:13,007 --> 00:05:15,920 This is actually given a name just by itself. 155 00:05:15,920 --> 00:05:17,923 This isn't the equation we're hunting for, 156 00:05:17,923 --> 00:05:19,886 but it's so important it gets its own name. 157 00:05:19,886 --> 00:05:21,967 Just part of this derivation gets its own name. 158 00:05:21,967 --> 00:05:24,007 It's called the magnification equation 159 00:05:24,007 --> 00:05:25,504 but it's usually not written this way. 160 00:05:25,504 --> 00:05:29,099 People usually write it as hi and then they divide by ho. 161 00:05:29,099 --> 00:05:32,371 So you get hi divided by ho equals, 162 00:05:32,371 --> 00:05:34,470 and then you multiply both sides by di, 163 00:05:34,470 --> 00:05:36,586 you get di over do. 164 00:05:36,586 --> 00:05:39,280 So a lot of times this is called the magnification equation. 165 00:05:39,280 --> 00:05:41,798 It gives you a way to find what the height of the image is. 166 00:05:41,798 --> 00:05:43,341 So we were lookin' for a way to find 167 00:05:43,341 --> 00:05:45,356 how far the image is from the mirror, 168 00:05:45,356 --> 00:05:46,838 but this lets you figure out, okay, 169 00:05:46,838 --> 00:05:48,869 you also need to know how big is it gonna be? 170 00:05:48,869 --> 00:05:50,847 So if we solve this height of the image, 171 00:05:50,847 --> 00:05:52,130 we get the height of the image is 172 00:05:52,130 --> 00:05:54,772 the height of the object times some factor, 173 00:05:54,772 --> 00:05:56,275 and that factor's just gonna be 174 00:05:56,275 --> 00:05:59,640 the image distance divided by the object distance. 175 00:05:59,640 --> 00:06:00,473 Here's the thing, though. 176 00:06:00,473 --> 00:06:01,306 Look at. 177 00:06:01,306 --> 00:06:02,575 This image got flipped over. 178 00:06:02,575 --> 00:06:05,289 So if we define this image distance as positive 179 00:06:05,289 --> 00:06:07,413 if it's on the same side as the object, 180 00:06:07,413 --> 00:06:09,341 we get a flipped over image, 181 00:06:09,341 --> 00:06:10,568 that'd be like a negative height. 182 00:06:10,568 --> 00:06:11,925 So since we want to represent 183 00:06:11,925 --> 00:06:14,209 flipped over images with a negative value, 184 00:06:14,209 --> 00:06:16,157 we actually write this equation down 185 00:06:16,157 --> 00:06:17,615 with a negative inside of here. 186 00:06:17,615 --> 00:06:18,817 We say that the height of the image 187 00:06:18,817 --> 00:06:22,215 is equal to negative ho, di over do, 188 00:06:22,215 --> 00:06:23,723 or you could put the negative up here, too. 189 00:06:23,723 --> 00:06:25,418 So we stick a negative over here. 190 00:06:25,418 --> 00:06:26,886 That way we know that if you get 191 00:06:26,886 --> 00:06:28,318 a negative value for the image height, 192 00:06:28,318 --> 00:06:29,870 you know it's flipped over. 193 00:06:29,870 --> 00:06:30,703 So in other words, 194 00:06:30,703 --> 00:06:34,815 if the value I got for hi was negative three centimeters, 195 00:06:34,815 --> 00:06:36,098 that means I'd get an image 196 00:06:36,098 --> 00:06:38,008 that had a height of three centimeters, 197 00:06:38,008 --> 00:06:39,185 but it'd be flipped over 198 00:06:39,185 --> 00:06:40,802 and that's what the negative represents. 199 00:06:40,802 --> 00:06:42,257 Okay, so that's kind of a side note. 200 00:06:42,257 --> 00:06:44,111 This is not what we're trying to derive. 201 00:06:44,111 --> 00:06:45,110 We're trying to derive a formula 202 00:06:45,110 --> 00:06:47,086 that would give us the image distance 203 00:06:47,086 --> 00:06:49,619 based on object distance and focal length. 204 00:06:49,619 --> 00:06:51,602 So we need to do another set of triangles. 205 00:06:51,602 --> 00:06:52,435 What we'll do now 206 00:06:52,435 --> 00:06:54,595 is instead of considering these thetas here, 207 00:06:54,595 --> 00:06:56,332 we'll consider these angles here. 208 00:06:56,332 --> 00:06:57,692 Now I can't call them theta 209 00:06:57,692 --> 00:06:59,430 'cause we already called those theta, 210 00:06:59,430 --> 00:07:00,714 so I'll call these phi. 211 00:07:00,714 --> 00:07:02,850 So these angles also have to be the same 212 00:07:02,850 --> 00:07:04,519 because any time you have a line 213 00:07:04,519 --> 00:07:06,297 and then you cut another line through it, 214 00:07:06,297 --> 00:07:08,679 these angles here will always be the same. 215 00:07:08,679 --> 00:07:09,722 So we could do the same game. 216 00:07:09,722 --> 00:07:11,840 We could play the same game we played for theta 217 00:07:11,840 --> 00:07:13,250 with these phis. 218 00:07:13,250 --> 00:07:14,783 We'll form two triangles, 219 00:07:14,783 --> 00:07:17,765 each triangle's gonna have phi as one of the angles. 220 00:07:17,765 --> 00:07:18,723 So for the first one, 221 00:07:18,723 --> 00:07:21,361 we'll do this object height as one side 222 00:07:21,361 --> 00:07:23,099 over here to the base of phi 223 00:07:23,099 --> 00:07:24,727 and then back up over to here. 224 00:07:24,727 --> 00:07:25,560 And then for the other one 225 00:07:25,560 --> 00:07:27,744 we need to also have a triangle that has phi in it, 226 00:07:27,744 --> 00:07:29,866 so we'll do this from here to here, 227 00:07:29,866 --> 00:07:32,786 down to there, and then back up to phi. 228 00:07:32,786 --> 00:07:34,065 So we've got two triangles. 229 00:07:34,065 --> 00:07:36,510 This triangle and this triangle, 230 00:07:36,510 --> 00:07:39,009 and they both have phi and they both have a right angle. 231 00:07:39,009 --> 00:07:40,905 So these are also similar triangles. 232 00:07:40,905 --> 00:07:42,927 In other words, we'll play the same game. 233 00:07:42,927 --> 00:07:45,321 We'll say that tangent of this phi 234 00:07:45,321 --> 00:07:48,126 is gonna have to equal the opposite over the adjacent. 235 00:07:48,126 --> 00:07:51,964 The opposite to this phi is this side which is just ho. 236 00:07:51,964 --> 00:07:54,354 So we can write ho divided by 237 00:07:54,354 --> 00:07:56,710 the adjacent now is not do, 238 00:07:56,710 --> 00:07:58,595 'cause this side only goes to here. 239 00:07:58,595 --> 00:08:01,071 It doesn't go the whole way to the mirror. 240 00:08:01,071 --> 00:08:03,075 It only goes that far. 241 00:08:03,075 --> 00:08:05,223 So that's the entire object distance 242 00:08:05,223 --> 00:08:07,764 minus this piece right over here. 243 00:08:07,764 --> 00:08:11,577 So if I subtracted this much from the object distance, 244 00:08:11,577 --> 00:08:13,156 I'd get the remaining amount 245 00:08:13,156 --> 00:08:15,576 which is the adjacent part of this triangle. 246 00:08:15,576 --> 00:08:17,137 So this distance from the mirror 247 00:08:17,137 --> 00:08:19,351 to the focal point is given a name. 248 00:08:19,351 --> 00:08:22,305 It's called the focal length and we represent it with an f. 249 00:08:22,305 --> 00:08:23,138 So a little confusing 250 00:08:23,138 --> 00:08:25,245 'cause f represents both a point 251 00:08:25,245 --> 00:08:28,394 and it represents the length from the mirror to that point. 252 00:08:28,394 --> 00:08:30,842 So f is gonna represent that length as well. 253 00:08:30,842 --> 00:08:33,371 So this adjacent side we could write as 254 00:08:33,371 --> 00:08:36,659 the object distance minus the focal length 255 00:08:36,659 --> 00:08:39,106 since this remaining part right here 256 00:08:39,106 --> 00:08:40,323 is the adjacent side 257 00:08:40,323 --> 00:08:42,638 which is object distance minus the focal length. 258 00:08:42,638 --> 00:08:45,520 But we know that this phi is also equal to this phi. 259 00:08:45,520 --> 00:08:47,654 So I can do tangent of theta for this phi, 260 00:08:47,654 --> 00:08:49,967 the opposite side would now be this side. 261 00:08:49,967 --> 00:08:50,848 What is that? 262 00:08:50,848 --> 00:08:52,557 What is this side of the triangle? 263 00:08:52,557 --> 00:08:54,710 That's just equal to the height of the image. 264 00:08:54,710 --> 00:08:57,068 This side is the same as the image height 265 00:08:57,068 --> 00:08:59,510 so I can say that this whole thing, tangent of phi, 266 00:08:59,510 --> 00:09:01,014 has to equal opposite over adjacent. 267 00:09:01,014 --> 00:09:03,881 This time the opposite of phi is the image height. 268 00:09:03,881 --> 00:09:06,507 And we divide by this distance right here, 269 00:09:06,507 --> 00:09:08,152 which is just the focal length. 270 00:09:08,152 --> 00:09:11,046 So this adjacent side for this triangle 271 00:09:11,046 --> 00:09:13,331 is simply the focal length. 272 00:09:13,331 --> 00:09:14,860 So I'll just divide by f 273 00:09:14,860 --> 00:09:17,024 and so what I've got are two equations 274 00:09:17,024 --> 00:09:18,161 that I'm gonna put together 275 00:09:18,161 --> 00:09:20,860 and we will get the mirror equation out of this. 276 00:09:20,860 --> 00:09:23,182 There's different ways to proceed at this point. 277 00:09:23,182 --> 00:09:24,205 What I'm gonna do is 278 00:09:24,205 --> 00:09:27,041 I don't want ho or hi in either of these, 279 00:09:27,041 --> 00:09:30,795 so I'm just gonna solve this one here for ho over hi. 280 00:09:30,795 --> 00:09:33,253 And I get ho over hi, 281 00:09:33,253 --> 00:09:35,811 so imagine dividing both sides by hi, 282 00:09:35,811 --> 00:09:38,623 and then multiplying both sides by do minus f, 283 00:09:38,623 --> 00:09:42,983 and I'd get ho over hi equals do minus f, 284 00:09:42,983 --> 00:09:43,816 the focal length, 285 00:09:43,816 --> 00:09:45,441 divided by the focal length. 286 00:09:45,441 --> 00:09:46,694 But I could do the same thing up here. 287 00:09:46,694 --> 00:09:48,679 I can get ho over hi. 288 00:09:48,679 --> 00:09:51,422 It's just gonna equal do over di. 289 00:09:51,422 --> 00:09:54,381 So that's all you have to do is divide both sides by hi 290 00:09:54,381 --> 00:09:56,894 and then multiply both sides by do. 291 00:09:56,894 --> 00:09:58,814 But this left-hand side right here 292 00:09:58,814 --> 00:10:01,322 is the same as this left-hand side right here. 293 00:10:01,322 --> 00:10:03,053 So we know that ho over hi 294 00:10:03,053 --> 00:10:05,786 already equals do minus f over f, 295 00:10:05,786 --> 00:10:09,806 and up here we know that ho over hi equals do over di, 296 00:10:09,806 --> 00:10:12,388 so that means that do minus f over f 297 00:10:12,388 --> 00:10:15,834 has to also equal do over di 298 00:10:15,834 --> 00:10:19,821 since both of these expressions equal ho over hi. 299 00:10:19,821 --> 00:10:21,121 So we can set them all equal 300 00:10:21,121 --> 00:10:22,889 'cause they're all equal to the same thing: 301 00:10:22,889 --> 00:10:24,440 ho over hi. 302 00:10:24,440 --> 00:10:26,135 And now we're gonna solve this. 303 00:10:26,135 --> 00:10:27,270 We're just gonna clean it up. 304 00:10:27,270 --> 00:10:29,008 The left-hand side I can write, 305 00:10:29,008 --> 00:10:30,481 I'll just stop using colors here, 306 00:10:30,481 --> 00:10:34,922 the left-hand side is gonna be do over f minus one, 307 00:10:34,922 --> 00:10:37,701 since f over f just equals one. 308 00:10:37,701 --> 00:10:40,320 And that's gotta equal do over di. 309 00:10:40,320 --> 00:10:43,277 And now we can divide both sides by do. 310 00:10:43,277 --> 00:10:46,251 If I divide both sides by do I get one over f, 311 00:10:46,251 --> 00:10:47,790 since the do cancels, 312 00:10:47,790 --> 00:10:50,755 minus one over do equals, 313 00:10:50,755 --> 00:10:53,576 and then the do will cancel with this do on top, 314 00:10:53,576 --> 00:10:55,427 and I just get one over di. 315 00:10:55,427 --> 00:10:57,981 And now we made it except it's usually written 316 00:10:57,981 --> 00:11:00,566 with this one over do added on the right. 317 00:11:00,566 --> 00:11:02,746 If we add one over do to both sides, 318 00:11:02,746 --> 00:11:04,656 we finally get the expression we wanted, 319 00:11:04,656 --> 00:11:07,636 which is one over the object distance, 320 00:11:07,636 --> 00:11:10,828 plus one over the image distance, 321 00:11:10,828 --> 00:11:13,786 equals one over the focal length. 322 00:11:13,786 --> 00:11:15,570 It's a pretty simple formula. 323 00:11:15,570 --> 00:11:17,925 It took a little bit of effort to get to this, 324 00:11:17,925 --> 00:11:21,130 but this is called the mirror equation 325 00:11:21,130 --> 00:11:23,666 and it relates the focal length of the mirror 326 00:11:23,666 --> 00:11:26,126 to the image distance and the object distance. 327 00:11:26,126 --> 00:11:26,959 In other words, 328 00:11:26,959 --> 00:11:29,554 if you now how far you put the object from the mirror, 329 00:11:29,554 --> 00:11:31,321 and you know the focal length of the mirror, 330 00:11:31,321 --> 00:11:35,126 that lets you figure out exactly where the image distance is 331 00:11:35,126 --> 00:11:38,583 instead of just kinda eyeballing it and using a protractor. 332 00:11:38,583 --> 00:11:41,461 This'll let you solve for where that image is exactly. 333 00:11:41,461 --> 00:11:42,438 And if you couple it 334 00:11:42,438 --> 00:11:44,800 with this magnification equation up here, 335 00:11:44,800 --> 00:11:47,620 you can also figure out the exact height of the image. 336 00:11:47,620 --> 00:11:48,963 Now you might be feeling a little sketchy 337 00:11:48,963 --> 00:11:50,125 about this negative sign. 338 00:11:50,125 --> 00:11:51,811 I mean I kinda threw it in quick over here. 339 00:11:51,811 --> 00:11:53,438 What's going on with this negative? 340 00:11:53,438 --> 00:11:54,952 In fact how do we know if any of this stuff 341 00:11:54,952 --> 00:11:56,290 is negative or positive? 342 00:11:56,290 --> 00:11:58,188 Well the convention that I use, 343 00:11:58,188 --> 00:11:59,856 that a lot of textbooks use now, 344 00:11:59,856 --> 00:12:01,236 is the one where 345 00:12:01,236 --> 00:12:04,664 focal lengths will be positive for concave mirrors, 346 00:12:04,664 --> 00:12:07,375 just like this mirror right here, this is a concave mirror. 347 00:12:07,375 --> 00:12:08,878 But if it was bent the other way, 348 00:12:08,878 --> 00:12:12,377 if the mirror looked like this, it'd be a convex mirror, 349 00:12:12,377 --> 00:12:14,232 and that would be a negative focal length. 350 00:12:14,232 --> 00:12:16,736 And it kinda makes sense if the mirror was bent this way, 351 00:12:16,736 --> 00:12:18,727 the focal point is kinda back this way 352 00:12:18,727 --> 00:12:19,843 which is kinda behind it, 353 00:12:19,843 --> 00:12:21,573 so it makes sense that it's sort of negative. 354 00:12:21,573 --> 00:12:23,337 And the object distance over here, 355 00:12:23,337 --> 00:12:24,991 if it's on the same side as your eye, 356 00:12:24,991 --> 00:12:26,276 if you're using this correctly, 357 00:12:26,276 --> 00:12:28,198 your eye should be over on this side, 358 00:12:28,198 --> 00:12:30,090 you would see an object right here 359 00:12:30,090 --> 00:12:32,642 but you'd also see an image of that object right here. 360 00:12:32,642 --> 00:12:34,702 And if this object is on the same side 361 00:12:34,702 --> 00:12:36,502 as the mirror as your eye, which it should be, 362 00:12:36,502 --> 00:12:38,496 then this object distance is positive. 363 00:12:38,496 --> 00:12:40,077 It's basically always positive 364 00:12:40,077 --> 00:12:42,237 unless you've got some double set of mirrors 365 00:12:42,237 --> 00:12:43,509 or something weird happening. 366 00:12:43,509 --> 00:12:46,141 If you got a single mirror, there's no craziness goin' on, 367 00:12:46,141 --> 00:12:49,605 this object distance is just always defined to be positive 368 00:12:49,605 --> 00:12:51,651 using the convention that we're using up here. 369 00:12:51,651 --> 00:12:54,047 And again, there are other conventions you can use. 370 00:12:54,047 --> 00:12:56,864 But this is the one used in a lot of textbooks these days. 371 00:12:56,864 --> 00:12:58,707 The di is a little trickier. 372 00:12:58,707 --> 00:13:01,384 The di is also positive if 373 00:13:01,384 --> 00:13:03,774 that image is on the same side as your eye, 374 00:13:03,774 --> 00:13:04,791 like it is right here. 375 00:13:04,791 --> 00:13:05,932 So this image 376 00:13:05,932 --> 00:13:08,226 would be considered a positive image distance 377 00:13:08,226 --> 00:13:09,708 since it's on this side. 378 00:13:09,708 --> 00:13:12,693 If the image got formed on this side of the mirror, 379 00:13:12,693 --> 00:13:14,255 say you formed an image right here, 380 00:13:14,255 --> 00:13:15,970 that would be a negative image distance. 381 00:13:15,970 --> 00:13:18,374 If this was five centimeters behind the mirror, 382 00:13:18,374 --> 00:13:19,237 we'd consider that 383 00:13:19,237 --> 00:13:22,035 a negative five centimeter image distance. 384 00:13:22,035 --> 00:13:25,498 And if you use that convention with this mirror equation, 385 00:13:25,498 --> 00:13:27,122 you'll get a correct relationship 386 00:13:27,122 --> 00:13:28,416 between the object distance, 387 00:13:28,416 --> 00:13:30,119 image distance, and the focal length. 388 00:13:30,119 --> 00:13:32,187 And if you use that same sign convention 389 00:13:32,187 --> 00:13:34,379 with this magnification equation, 390 00:13:34,379 --> 00:13:36,613 you'll also get the exact height of the image 391 00:13:36,613 --> 00:13:38,517 and if that image height comes out negative, 392 00:13:38,517 --> 00:13:40,248 you'll know that it got flipped upside down. 393 00:13:40,248 --> 00:13:41,980 And if the image height comes out positive, 394 00:13:41,980 --> 00:13:44,277 you know that the image will stay right-side up. 395 00:13:44,277 --> 00:13:47,285 So recapping, using a bunch of similar triangles, 396 00:13:47,285 --> 00:13:49,452 we were able to derive a mirror equation 397 00:13:49,452 --> 00:13:51,218 that relates the object distance, 398 00:13:51,218 --> 00:13:53,015 image distance, and focal length, 399 00:13:53,015 --> 00:13:56,084 and along the way we derived a magnification equation 400 00:13:56,084 --> 00:13:58,710 that relates the heights of the image and object 401 00:13:58,710 --> 00:14:00,877 to the distances of the image and object. 402 00:14:00,877 --> 00:14:02,923 And you have to be very careful with signs. 403 00:14:02,923 --> 00:14:04,415 Even though the object distance 404 00:14:04,415 --> 00:14:06,205 is basically always positive, 405 00:14:06,205 --> 00:14:08,710 focal length can be positive or negative. 406 00:14:08,710 --> 00:14:11,618 Focal length will be positive for concave mirrors, 407 00:14:11,618 --> 00:14:14,094 and it'll be negative for convex mirrors. 408 00:14:14,094 --> 00:14:15,864 Image distances will be positive 409 00:14:15,864 --> 00:14:18,835 if they're on the same side of the mirror as your eye, 410 00:14:18,835 --> 00:14:20,192 if they're in front of the mirror. 411 00:14:20,192 --> 00:14:22,579 But they'll be negative if they're behind the mirror. 412 00:14:22,579 --> 00:14:23,469 And again this is not 413 00:14:23,469 --> 00:14:25,636 the only sign convention you could use, 414 00:14:25,636 --> 00:14:27,995 but it's as good as any other sign convention 415 00:14:27,995 --> 00:00:00,000 and it's the one used in a lot of textbooks these days.