WEBVTT 00:00:00.035 --> 00:00:04.157 A rainbow is not a coloured object hanging over a particular field. 00:00:04.470 --> 00:00:07.524 It is a viewing geometry that follows the observer. 00:00:08.576 --> 00:00:10.269 This is the destination. 00:00:10.352 --> 00:00:14.414 The visible bow is an arc centred on the point opposite the sun, 00:00:14.497 --> 00:00:18.454 and its outer red edge sits about forty two degrees from that direction. 00:00:18.537 --> 00:00:20.558 We now have to explain why. 00:00:21.262 --> 00:00:23.385 Here is the observation to explain. 00:00:23.468 --> 00:00:26.429 To see a rainbow, the sun must be behind you. 00:00:26.533 --> 00:00:29.877 That is a fact about geometry, and it is our first clue. 00:00:30.674 --> 00:00:34.936 Stand with the sun at your back and follow the shadow of your head away from the sun. 00:00:35.388 --> 00:00:40.404 That line is the anti-solar direction, the centre line for every rainbow you see. 00:00:41.108 --> 00:00:45.404 A drop out here catches the sunlight and sends its red light back to your eye. 00:00:46.018 --> 00:00:46.947 The angle between 00:00:46.947 --> 00:00:50.872 that returning ray and the centre line is about forty two degrees. 00:00:51.185 --> 00:00:55.051 Rotate that direction right around the centre line and the eligible drops form 00:00:55.051 --> 00:00:59.614 a thin conical shell, whose two edges are the red lines of this side view. 00:01:00.527 --> 00:01:02.362 Now step sideways. 00:01:02.675 --> 00:01:04.101 The cone moves with your eye, 00:01:04.184 --> 00:01:06.913 and a different drop joins the new line of sight. 00:01:07.365 --> 00:01:11.186 The rainbow is fixed by an angle, not by a place in the shower. 00:01:11.983 --> 00:01:15.571 Raise the sun and the anti-solar direction tilts down. 00:01:15.767 --> 00:01:18.775 The whole bow then follows it below the horizon. 00:01:19.088 --> 00:01:21.840 Lower the sun and the bow rises again. 00:01:22.544 --> 00:01:24.853 So the question has two parts. 00:01:24.936 --> 00:01:28.848 Why is the angle about forty two degrees, and why does the light bunch up there 00:01:28.848 --> 00:01:34.143 instead of spreading evenly across the sky? Both answers are inside one raindrop. 00:01:35.254 --> 00:01:38.854 Refraction is the bending of light as it crosses into water. 00:01:39.051 --> 00:01:40.606 An incoming ray and the centre 00:01:40.606 --> 00:01:45.007 of a spherical drop determine one flat slice where we can follow that bend. 00:01:45.204 --> 00:01:48.014 Seen head on, the slice is a circle. 00:01:48.211 --> 00:01:52.589 The same circle, centre, and ray will stay with us all the way to the answer. 00:01:53.386 --> 00:01:57.589 At the surface, the normal is the radius through the point of contact. 00:01:57.693 --> 00:02:01.641 The incoming ray makes the incidence angle i with that normal. 00:02:02.345 --> 00:02:06.023 At that surface the light turns toward the normal as it enters water, 00:02:06.106 --> 00:02:09.416 so the angle inside, r, is smaller than i. 00:02:10.328 --> 00:02:13.057 Snell's law gives the size of that bend. 00:02:13.254 --> 00:02:19.222 Sine i equals n times sine r, and water has n about one point three three. 00:02:19.926 --> 00:02:22.237 The ray reaches the back wall. 00:02:22.434 --> 00:02:26.672 Most light escapes and is lost, but a small fraction reflects. 00:02:26.869 --> 00:02:31.815 Symmetry makes the incoming and reflected angles there equal to the same r. 00:02:32.612 --> 00:02:36.340 At the front surface the reflected ray meets one more radius. 00:02:36.536 --> 00:02:42.470 It leaves water, opens from r back to i, and becomes the ray that can reach an eye. 00:02:43.173 --> 00:02:46.411 Each pair of angle marks appeared only for the event it named. 00:02:46.494 --> 00:02:51.289 The exit pair has now carried the same law to the last surface, so it can leave too. 00:02:52.086 --> 00:02:55.128 Now measure how far the ray turns at each event. 00:02:55.325 --> 00:02:58.321 At the entrance it turns through i minus r. 00:02:58.804 --> 00:03:01.243 The reflection is the large turn. 00:03:01.347 --> 00:03:05.735 Equal angles r leave a straight angle minus two r between the old direction 00:03:05.735 --> 00:03:07.013 and the new one. 00:03:08.065 --> 00:03:11.189 Leaving the drop adds another i minus r. 00:03:11.385 --> 00:03:14.497 All three local turns now stand beside the path: 00:03:14.694 --> 00:03:19.525 i minus r, one hundred eighty minus two r, and i minus r. 00:03:19.834 --> 00:03:21.750 Add those three turns. 00:03:22.063 --> 00:03:26.429 The deviation is one hundred eighty degrees plus two i minus four r. 00:03:26.625 --> 00:03:30.224 The angle D compares the original forward direction with the direction 00:03:30.224 --> 00:03:32.245 that actually leaves the drop. 00:03:33.297 --> 00:03:37.045 A ray through the centre has i and r both zero, 00:03:37.128 --> 00:03:41.135 so the expression gives a turn of one hundred eighty degrees, straight back. 00:03:41.331 --> 00:03:45.927 Snell's law also makes r the inverse sine of sine i over n, 00:03:46.010 --> 00:03:48.437 leaving i as the only free angle. 00:03:49.350 --> 00:03:52.810 Set the entry low on the drop, at twenty degrees. 00:03:53.007 --> 00:03:55.281 The ray returns almost the way it arrived, 00:03:55.364 --> 00:03:58.662 with a deviation of about one hundred sixty degrees. 00:03:59.458 --> 00:04:01.444 Slide the entry point upward. 00:04:01.641 --> 00:04:05.331 The deviation falls through one hundred fifty, one hundred forty five, 00:04:05.414 --> 00:04:07.133 and one hundred forty. 00:04:07.237 --> 00:04:08.654 Then it slows. 00:04:08.850 --> 00:04:11.254 There, it has stopped coming down. 00:04:11.865 --> 00:04:13.700 Keep sliding toward the rim. 00:04:14.013 --> 00:04:16.473 Only after the minimum does the curve climb again, 00:04:16.556 --> 00:04:19.923 and it keeps climbing as the entry point approaches the edge. 00:04:20.836 --> 00:04:23.554 The flat bottom makes the minimum important. 00:04:23.657 --> 00:04:26.558 These two rays enter at forty eight and seventy degrees, 00:04:26.641 --> 00:04:28.801 far apart on the face of the drop. 00:04:29.505 --> 00:04:33.311 Yet their outgoing directions differ by less than a quarter of a degree. 00:04:33.394 --> 00:04:37.490 A broad band of entry points therefore sends light back in nearly one direction, 00:04:37.573 --> 00:04:40.337 so the light piles up instead of spreading thin. 00:04:41.134 --> 00:04:43.503 That concentration is the rainbow. 00:04:43.700 --> 00:04:47.323 To locate it exactly, use the flat curve condition: 00:04:47.519 --> 00:04:51.886 at the minimum, the derivative of D with respect to i is zero. 00:04:52.682 --> 00:04:57.455 Solving that condition says d r by d i must equal one half. 00:04:57.767 --> 00:05:02.296 The angle inside must change half as fast as the incidence angle outside. 00:05:03.092 --> 00:05:05.229 Differentiate Snell's law. 00:05:05.391 --> 00:05:09.894 Cosine i equals n cosine r times d r by d i. 00:05:09.977 --> 00:05:14.389 Substituting one half gives cosine i equal to n cosine r over two. 00:05:15.151 --> 00:05:17.195 Now square both sides. 00:05:17.357 --> 00:05:21.747 Four cosine squared i equals n squared cosine squared r. 00:05:22.509 --> 00:05:26.721 Replace cosine squared r with one minus sine squared r. 00:05:26.804 --> 00:05:31.995 Snell's law then replaces n squared sine squared r with sine squared i. 00:05:32.191 --> 00:05:36.674 Finally, sine squared i is one minus cosine squared i. 00:05:37.470 --> 00:05:39.445 Collect the cosine terms. 00:05:39.641 --> 00:05:44.936 Cosine i is the square root of n squared minus one, divided by three. 00:05:45.849 --> 00:05:51.608 For water, i is fifty nine point four degrees and r is forty point two. 00:05:51.921 --> 00:05:55.625 Putting both into the turn formula gives a minimum deviation of about 00:05:55.625 --> 00:05:57.680 one hundred thirty eight degrees. 00:05:58.477 --> 00:06:01.032 Return to the special ray at the minimum. 00:06:01.228 --> 00:06:04.187 Its path is still the same path through the same drop; 00:06:04.270 --> 00:06:06.720 only the chosen entry point has moved. 00:06:07.517 --> 00:06:10.478 Carry the incoming direction through the exit point. 00:06:10.930 --> 00:06:15.064 Forward to backward is a straight angle of one hundred eighty degrees. 00:06:15.261 --> 00:06:19.639 The ray has already turned through D, about one hundred thirty eight degrees. 00:06:20.691 --> 00:06:23.548 Subtract that deviation from the straight angle. 00:06:23.744 --> 00:06:25.741 The remaining angle between the outgoing ray 00:06:25.741 --> 00:06:28.482 and straight back is forty two degrees. 00:06:29.093 --> 00:06:30.859 That is the promised number. 00:06:31.055 --> 00:06:35.560 Snell's law fixes the path inside one circle, the three turns create a minimum 00:06:35.560 --> 00:06:37.346 at one hundred thirty eight degrees, 00:06:37.429 --> 00:06:40.715 and the view back toward the sun leaves forty two degrees. 00:06:41.779 --> 00:06:43.925 White sunlight contains many colours, 00:06:44.009 --> 00:06:47.272 and water bends each one by a slightly different amount. 00:06:47.468 --> 00:06:49.025 Red bends least; 00:06:49.221 --> 00:06:50.836 violet bends most. 00:06:51.633 --> 00:06:56.486 Follow the least-deviated red path through the drop, then the violet path. 00:06:56.683 --> 00:07:00.221 They enter close together, separate through the two refractions, 00:07:00.305 --> 00:07:02.477 and leave in different directions. 00:07:03.273 --> 00:07:06.173 For red, n is one point three three one 00:07:06.257 --> 00:07:09.474 and the viewing angle is forty two point three degrees. 00:07:09.671 --> 00:07:12.594 For violet, n is one point three four four 00:07:12.678 --> 00:07:15.883 and the viewing angle is forty point six degrees. 00:07:16.680 --> 00:07:20.709 A single drop therefore does not send an entire rainbow to one eye. 00:07:20.813 --> 00:07:23.480 Its red and violet leave along different lines, 00:07:23.564 --> 00:07:27.593 so one viewpoint receives at most one narrow part of that colour fan. 00:07:28.390 --> 00:07:30.608 Put two drops back in the shower. 00:07:30.805 --> 00:07:32.871 One is higher above the anti-solar line 00:07:32.871 --> 00:07:37.319 and one is lower, while the sunlight reaches both from behind the observer. 00:07:38.115 --> 00:07:41.088 The higher drop can send its red ray to the eye. 00:07:41.285 --> 00:07:44.862 Its violet ray passes over the observer's head and is missed. 00:07:45.566 --> 00:07:48.500 The lower drop can send violet to the same eye. 00:07:48.584 --> 00:07:51.604 Its red ray passes below the observer's feet. 00:07:52.818 --> 00:07:56.603 Red therefore arrives from the larger viewing angle and sits on the outside 00:07:56.603 --> 00:07:58.241 of the primary bow. 00:07:58.437 --> 00:08:01.700 Violet arrives from the smaller angle and sits inside. 00:08:02.613 --> 00:08:05.760 The anti-solar point is the centre of the geometry. 00:08:06.073 --> 00:08:09.661 The horizon lies above it whenever the sun is above the horizon. 00:08:10.458 --> 00:08:13.698 Rotate the primary viewing directions around that centre. 00:08:13.894 --> 00:08:19.119 Red traces the outer arc and violet traces the inner arc, with every point supplied 00:08:19.119 --> 00:08:21.267 by a different drop in the shower. 00:08:22.064 --> 00:08:24.499 The complete geometry is circular, 00:08:24.583 --> 00:08:27.475 but the ground hides the portion below the horizon. 00:08:27.788 --> 00:08:31.423 What remains visible is an arc, not a full ring. 00:08:31.848 --> 00:08:36.399 Look just inside the primary bow, then just outside it. The inside is brighter 00:08:36.399 --> 00:08:40.103 because a broad band of rays returns on that side of the minimum. 00:08:40.300 --> 00:08:44.063 The region just outside receives far fewer primary rays. 00:08:44.766 --> 00:08:48.993 A second internal reflection creates a secondary bow farther out. 00:08:49.306 --> 00:08:51.373 Its colour order reverses: 00:08:51.570 --> 00:08:54.159 violet is outside and red is inside. 00:08:54.956 --> 00:08:59.052 One drop supplies the path, the minimum deviation supplies the brightness, 00:08:59.136 --> 00:09:00.968 dispersion supplies the colours, 00:09:01.052 --> 00:09:04.361 and rotation around the anti-solar point supplies the arc. 00:09:04.558 --> 00:09:07.403 Together they make the rainbow in the sky.