Chapter 2 The Ray Model of Light
Table of Contents
2.1 Rays Don’t Rust
If you look at the winter night sky on a clear, moonless night far from any city lights, something strange will soon catch your eye. Near the constellation of Andromeda is a little white smudge. What is it? You can easily convince yourself that it’s not a cloud, because it moves along with the stars as they rise and set. What you’re seeing is the Andromeda galaxy, a fantastically distant group of stars very similar to our own Milky Way.1 We can see individual stars within the Milky Way galaxy because we’re inside it, but the Andromeda galaxy looks like a fuzzy patch because we can’t make out its individual stars. The vast distance to the Andromeda galaxy is hard to fathom, and it won’t help you to imagine it if I tell you the number of kilometers is 2 followed by 19 zeroes. Think of it like this: if the stars in our own galaxy were as close together as the hairs on your skin, the Andromeda galaxy would be thousands of kilometers away.
The light had a long journey to get to your eyeball! A wellmaintained car might survive long enough to accumulate a million kilometers on its odometer, but by that time it would be a rickety old rust-bucket, and the distance it had covered would still only amount to a fraction of a billionth of a billionth of the distance we’re talking about. Light doesn’t rust. A car’s tracks can’t go on forever, but the trail of a light beam can. We call this trail a “ray.”
a / How to locate the Andromeda galaxy.
2.2 Time-Reversal Symmetry
The neverending motion of a light ray is surprising compared with the behavior of everyday objects, but in a way it makes sense. A car is a complex system with hundreds of moving parts. Those parts can break, or wear down due to friction. Each part is itself made of atoms, which can do chemical reactions such as rusting. Light, however, is fundamental: as far as we know, it isn’t made of anything else. My wife’s car has a dent in it that preserves the record of the time she got rear-ended last year. As time goes on, a car accumulates more and more history. Not so with a light ray. Since a light ray carries no history, there is no way to distinguish its past from its future. Similarly, some brain-injured people are unable to form long-term memories. To you and me, yesterday is different from tomorrow because we can’t remember tomorrow, but to them there is no such distinction.
Experiments — including some of the experiments you’re going to do in this course — show that the laws of physics governing light rays are perfectly symmetric with respect to past and future. If a light ray can go from A to B, then it’s also possible for a ray to go from B to A. I remember as a child thinking that if I covered my eyes, my mommy couldn’t see me. I was almost right: if I couldn’t see her eyes, she couldn’t see mine.
b / The mirror left on the moon by the Apollo 11 astronauts.
Why light rays don’t stop
example 1
Once the experimental evidence convinces us of time-reversal symmetry, it’s easy to prove that light rays never get tired and stop moving. Suppose some light was headed our way from the Andromeda galaxy, but it stopped somewhere along the way and never went any farther. Its trail, which we call the “ray,” would be a straight line ending at that point in empty space. Now suppose we send a film crew along in a space ship to document the voyage, and we ask them to play back the video for us, but backwards. Time is reversed. The narration is backwards. Clocks on the wall go counterclockwise. In the reversed documentary, how does the light ray behave? At the beginning (which is really the end), the light ray doesn’t exist. Then, at some random moment in time, the ray springs into existence, and starts heading back towards the Andromeda galaxy. In this backwards version of the documentary, the light ray is not behaving the way light rays are supposed to. Light doesn’t just appear out of nowhere in the middle of empty space for no reason. (If it did, it would violate rotational symmetry, because there would be no physical reason why this out-of-nowhere light ray would be moving in one direction rather than another.) Since the backwards video is impossible, and all our accumulated data have shown that light’s behavior has time-reversal symmetry, we conclude that the forward video is also impossible. Thus, it is not possible for a light ray to stop in the middle of empty space.
The Apollo lunar ranging experiment
example 2
In 1969, the Apollo 11 astronauts made the first crewed landing on the moon, and while they were there they placed a mirror on the lunar surface. Astronomers on earth then directed a laser beam at the landing site. The beam was reflected by the mirror, and retraced its own path back to the earth, allowing the distance to the moon to be measured extremely accurately (which turns out to provide important information about the earth-moon system). Based on time-reversal symmetry, we know that if the reflection is a 180-degree turn, the reflected ray will behave in the same way as the outgoing one, and retrace the same path. (Figure p on page 31 explains the clever trick used to make sure the reflection would be a 180-degree turn, without having to align the mirror perfectly.)
Looking the wrong way through your glasses
example 3
If you take off your glasses, turn them around, and look through them the other way, they still work. This is essentially a demonstration of time reversal symmetry, although an imperfect one. It’s imperfect because you’re not time-reversing the entire path of the rays. Instead of passing first through the front surface of the lenses, then through the back surface, and then through the surface of your eye, the rays are now going through the three surfaces in a different order. For this reason, you’ll notice that things look a little distorted with your glasses reversed. To make a perfect example of time-reversal, you’d have to have a little lamp inside your eyeball!
If light never gets tired, why is it that I usually can’t see the mountains from my home in Southern California? They’re far away, but if light never stops, why should that matter? It’s not that light just naturally stops after traveling a certain distance, because I can easily see the sun, moon, and stars from my house, and they’re much farther away than the mountains. The difference is that my line of sight to the mountains cuts through many miles of pollution and natural haze. The time-reversal argument in example 1 depended on the assumption that the light ray was traveling through empty space. If a light ray starts toward me from the mountains, but hits a particle of soot in the air, then the time-reversed story is perfectly reasonable: a particle of soot emitted a ray of light, which hit the mountains.
Discussion Questions
C If you watch a time-reversed soccer game, are the players still obeying the rules?
2.3 Applications
The inverse-square law
Yet another objection is that a distant candle appears dim. Why is this, if not because the light is getting tired on the way to us? Likewise, our sun is just a star like any other star, but it appears much brighter because it’s so much closer to us. Why are the other stars so dim if not because their light wears out? It’s not that the light rays are stopping, it’s that they’re getting spread out more thinly. The light comes out of the source in all directions, and if you’re very far away, only a tiny percentage of the light will go into your eye. (If all the light from a star went into your eye, you’d be in trouble.)
c / The light is four times dimmer at twice the distance.
Figure c shows what happens if you double your distance from the source. The light from the flame spreads out in all directions. We pick four representative rays from among those that happen to pass through the nearer square. Of these four, only one passes through the square of equal area at twice the distance. If the two equal-area squares were people’s eyes, then only one fourth of the light would go into the more distant person’s eye.
Another way of thinking about it is that the light that passed through the first square spreads out and makes a bigger square; at double the distance, the square is twice as wide and twice as tall, so its area is 2 × 2 = 4 times greater. The same light has been spread out over four times the area.
In general, the rule works like this:
To get the 4, we multiplied 2 by itself, 9 came from multiplying 3 by itself, and so on. Multiplying a number by itself is called squaring it, and dividing one by a number is called inverting it, so a relationship like this is known as an inverse square law. Inverse square laws are very common in physics: they occur whenever something is spreading out in all directions from a point.
Self-check A
Alice is one meter from the candle, while Bob is at a distance of five meters. How many times dimmer is the light at Bob’s location? ‣ Answer
An example with sound
example 4
‣ Four castaways are adrift in an open boat, and are yelling to try to attract the attention of passing ships. If all four of them yell at once, how much is their range increased compared to the range they would have if they took turns yelling one at a time?
‣ This is an example involving sound. Although sound isn’t the same as light, it does spread out in all directions from a source, so it obeys the inverse-square law. In the previous examples, we knew the distance and wanted to find the intensity (brightness). Here, we know about the intensity (loudness), and we want to find out about the distance. Rather than taking a number and multiplying it by itself to find the answer, we need to reverse the process, and find the number that, when multiplied by itself, gives four. In other words, we’re computing the square root of four, which is two. They will double their range, not quadruple it.
Astronomical distance scales
example 5
The nearest star, Alpha Centauri,2 is about 10,000,000,000,000,000 times dimmer than our sun when viewed from our planet. If we assume that Alpha Centauri’s true brightness is roughly the same as that of our own sun, then we can find the distance to Alpha Centauri by taking the square root of this number. Alpha Centauri’s distance from us is equal to about 100,000,000 times our distance from the sun.
Pupils and camera diaphragms
example 6
In bright sunlight, your pupils contract to admit less light. At night they dilate, becoming bigger “light buckets.” Your perception of brightness depends not only on the true brightness of the source and your distance from it, but also on how much area your pupils present to the light. Cameras have a similar mechanism, which is easy to see if you detach the lens and its housing from the body of the camera, as shown in the figure. Here, the diameter of the largest aperture is about ten times greater than that of the smallest aperture. Making a circle ten times greater in radius increases its area by a factor of 100, so the light-gathering power of the camera becomes 100 times greater. (Many people expect that the area would only be ten times greater, but if you start drawing copies of the small circle inside the large circle, you’ll see that ten are not nearly enough to fill in the entire area of the larger circle. Both the width and the height of the bigger circle are ten times greater, so its area is 100 times greater.)
d / The same lens is shown with its diaphragm set to three different apertures.
Parallax
Example 5 on page 25 showed how we can use brightness to determine distance, but your eye-brain system has a different method. Right now, you can tell how far away this page is from your eyes. This sense of depth perception comes from the fact that your two eyes show you the same scene from two different perspectives. If you wink one eye and then the other, the page will appear to shift back and forth a little.
e / At double the distance, the parallax angle is approximately halved.
If you were looking at a fly on the bridge of your nose, there would be an angle of nearly 180 between the ray that went into your left eye and the one that went into your right. Your brain would know that this large angle implied a very small distance. This is called the parallax angle. Objects at greater distances have smaller parallax angles, and when the angles are small, it’s a good approximation to say that the angle is inversely proportional to the distance. In figure e, the parallax angle is almost exactly cut in half when the person moves twice as far away.
Parallax can be observed in other ways than with a pair of eyeballs. As a child, you noticed that when you walked around on a moonlit evening, the moon seemed to follow you. The moon wasn’t really following you, and this isn’t even a special property of the moon. It’s just that as you walk, you expect to observe a parallax angle between the same scene viewed from different positions of your whole head. Very distant objects, including those on the Earth’s surface, have parallax angles too small to notice by walking back and forth. In general, rays coming from a very distant object are nearly parallel.
If your baseline is long enough, however, the small parallaxes of even very distant objects may be detectable. In the nineteenth century, nobody knew how tall the Himalayas were, or exactly where their peaks were on a map, and the Andes were generally believed to be the tallest mountains in the world. The Himalayas had never been climbed, and could only be viewed from a distance. From down on the plains of India, there was no way to tell whether they were very tall mountains very far away, or relatively low ones that were much closer. British surveyor George Everest finally established their true distance, and astounding height, by observing the same peaks through a telescope from different locations far apart.
An even more spectacular feat of measurement was carried out by Hipparchus over twenty-one centuries ago. By measuring the parallax of the moon as observed from Alexandria and the Hellespont, he determined its distance to be about 90 times the radius of the earth.3
The earth circles the sun, f, and we can therefore determine the distances to a few hundred of the nearest stars by making observations six months apart, so that the baseline for the parallax measurement is the diameter of the earth’s orbit. For these stars, the distances derived from parallax can be checked against the ones found by the method of example 5 on page 25. They do check out, which verifies the assumption that the stars are objects analogous to our sun.
f / The nearer star has a larger parallax angle. By measuring the parallax angles, we can determine the distances to both stars. (The scale on this drawing is not realistic. If the earth’s orbit was really this size, the nearest stars would be several kilometers away.)
g / The soccer ball will never slow down.
h / Galileo Galilei (1564-1642)
2.4 The Speed of Light
How fast does light travel? Does it even take any time to go from one place to another? If so, is the speed different for light with different colors, or for light with different brightnesses? Can a particular ray of light speed up or slow down?
The principle of inertia
We can answer the last question based on fundamental principles. All the experimental evidence supports time-reversal symmetry for light rays. Suppose that a beam of light traveling through a vacuum slowed down. After all, a rolling soccer ball starts to slow down immediately after you kick it. Even a rifle bullet slows down between the muzzle and the target. Why shouldn’t light slow down gradually? It can’t slow down, because of time-reversal symmetry. If the laws of physics said that a ray of light slowed down while traveling through a vacuum, then the time-reversed motion of the ray would violate the laws of physics. In the time-reversed version, the ray is moving the opposite direction and speeding up. Since all the experimental evidence shows that time-reversal symmetry is valid for light rays, we conclude that a ray will never speed up or slow down while traveling through a vacuum.
Why, then, do the ball and the bullet slow down? They wouldn’t slow down at all if they were traveling through interstellar space. It’s only due to friction that they lose speed. The ball slows down because of friction with the grass, and air friction is what decelerates the bullet. The laws of physics are not complicated, and in many ways they’re not even different for light rays than for material objects. The laws of physics are simple and consistent. We can now state the following important principle, first proposed by Florentine physicist Galileo Galilei:
The principle of inertia
A ray of light or a material object continues moving in the same direction and at the same speed if it is not interacting with anything else.
Measuring the speed of light
Observations also show that in a vacuum, all light moves at the same speed, regardless of its color, its brightness, or the manner in which it was emitted. The best evidence comes from supernovae, which are exploding stars. Supernovae are so bright that we can see them even when they occur in distant galaxies whose normal stars are too dim to resolve individually. When we observe a supernova, all the light gets to us at the same time, so it must all have traveled at the same speed.
Galileo made the first serious attempt to measure the speed of light. In his experiment, two people with lanterns stood a mile apart. The first person opened the shutter of his lantern, and the second person opened the shutter on his as soon as he saw the light from the first person’s. A third observer stood at an equal distance from both of them, and tried to measure the time lag between the two. No such time lag was observed, so you could say that the experiment failed, but in science a failure can still be important. This is known as a negative experiment. Galileo’s results showed that the speed of light must be at least ten times the speed of sound. It was important that he published his negative result, both because it convinced people that the problem was scientifically interesting and because it told later workers that the speed of light must be very fast, which would help them to design experiments that might actually work.
The first person to prove that light’s speed was finite, and to determine it numerically, was Ole Roemer, in a series of measurements around the year 1675. Roemer observed Io, one of Jupiter’s moons, over a long period. Since Io presumably took the same amount of time to complete each orbit of Jupiter, it could be thought of as a very distant, very accurate clock. A practical and accurate pendulum clock had recently been invented, so Roemer could check whether the ratio of the two clocks’ cycles, about 42.5 hours to one orbit, stayed exactly constant or changed a little. If the process of seeing the distant moon was instantaneous, there would be no reason for the two to get out of step. Even if the speed of light was finite, you might expect that the result would be only to offset one cycle relative to the other. The earth does not, however, stay at a constant distance from Jupiter and its moons. Since the distance is changing gradually due to the two planets’ orbital motions, a finite speed of light would make the “Io clock” appear to run faster as the planets drew near each other, and more slowly as their separation increased. Roemer did find a variation in the apparent speed of Io’s orbits, which caused Io’s eclipses by Jupiter (the moments when Io passed in front of or behind Jupiter) to occur about 7 minutes early when the earth was closest to Jupiter, and 7 minutes late when it was farthest. Based on these measurements, Roemer estimated the speed of light to be approximately 200,000 kilometers per second, which is in the right ballpark compared to modern measurements of 300,000 km/s.
i / A modern image of Jupiter and its moon Io (right) from the Voyager 1 probe.
j / The earth is moving towards Jupiter and Io. Since the distance is shrinking, it’s taking less and less time for light to get to us from Io. Io appears to circle Jupiter more quickly than normal. Six months later, the earth will be on the opposite side of the sun, and receding from Jupiter and Io, so Io will appear to go around more slowly.
Discussion Questions
A When phenomena like X-rays and cosmic rays were first discovered, nobody knew what they were. Suggest one way of testing the hypothesis that they were forms of light.
k / Two self-portraits of the author, one taken in a mirror and one with a piece of aluminum foil.
l / The incident and reflected rays are both perpendicular to the surface.
m / This doesn’t happen.
2.5 Reflection
Seeing by reflection
So far we’ve only talked about how you see things that emit light: stars, candles, and so on. If you’re reading this book on a computer screen, that’s how you’re seeing it right now. But what if you’re reading this book on paper? The paper doesn’t emit light, and it would be invisible if you turned out the lights in the room. The light from the lamp is hitting the paper and being reflected to your eyes.
Most people only think of reflection as something that happens with mirrors or other shiny, smooth surfaces, but it happens with all surfaces. Consider figure k. The aluminum foil isn’t as smooth as the mirror, so my reflection is blurry and jumbled. If I hadn’t told you, you probably wouldn’t have known that it was a reflection of a person at all. If the paper you’re reading from was as smooth as a mirror, you would see a reflection of the room in it, and the brightest object in the reflection would probably be the lamp that’s lighting the room. Paper, however, is not that smooth. It’s made of wood pulp. The reflection of the room is so blurry and jumbled that it all looks like one big, washed-out, white blur. That white blur is what you see when you see the paper. This is called diffuse reflection. In diffuse reflection, the reflected rays come back out at random angles.
Specular reflection
Reflection from a smooth surface is called specular reflection, from the Latin word for mirror. (The root, a verb meaning “to look at,” is the same as the root of “spectacular” and “spectacle.”) When a light ray is reflected, we get a new ray at some new angle, which depends on the angle at which the incident (original) ray came in. What’s the rule that determines the direction of the reflected ray? We can determine the answer by symmetry.
First, if the incident ray comes in perpendicular to the surface, l, then there is perfect left-right reflection symmetry. (It’s just a coincidence that we have reflection symmetry occurring in our analysis of reflection.) If the reflected ray came back at some angle to the left or right, it would violate this symmetry. Therefore the reflected ray must be right on top of the incident ray, straight back up. Because this is the simplest possible specular reflection, we define these angles as zero: all rays have their angles measured with respect to perpendicular, not with respect to the surface itself. Typically the rays themselves will not be perpendicular to the surface, but we still measure their angles with respect to an imaginary line perpendicular to the surface, which we call the normal. (“Normal” is simply another word for perpendicular.)
Now what if the incident angle isn’t zero? Figure m shows what doesn’t happen. It’s not possible for the reflected angle r to be unequal to the incident angle i, because of symmetry. Suppose we lived in a goofy universe, where the laws of physics gave the result shown in the figure: r is always less than i. What would happen if we did a time-reversal on the diagram? Oops — then we’d have r greater than i ! Since experiments support time-reversal symmetry for light rays, we conclude that this is impossible.4 The actual laws of physics give equal angles of incidence and reflection,
r = i .
n / This does happen.
o / example 7
p / A corner reflector
q / example 8
Reflecting a pool ball
example 7
The proof of r = i for light rays works equally well for pool balls, provided that the effects that violate symmetry are small. For instance, we assume that the ball doesn’t have lots of spin put on it, because that would break the left-right reflection symmetry.
Self-check B
Continue the ray in figure p through its second reflection. In what direction is the returning ray? How does this relate to example 2 on page 22? ‣ Answer
An image
example 8
Figure q shows some representative rays spreading out from one point on the flame. These rays strike the mirror and are reflected. To the observer on the left, the reflected rays are indistinguishable from the ones that would have originated from an actual flame on the far side of the mirror. Rays don’t carry any history, so there is no way for the eye to know that the rays underwent reflection along the way. (The rays shown in the diagram form an image of one point on the flame, but every other point on the flame sends out a similar bundle of rays, and has its own image formed.)
Self-check C
What happens in figure q if you replace the flame with an object that doesn’t emit light, and can only be seen by diffuse reflection? ‣ Answer
Discussion Questions
A Laser beams are made of light. In science fiction movies, laser beams are often shown as bright lines shooting out of a laser gun on a spaceship. Why is this scientifically incorrect?
Problem 3.
Problem 4a.
Problem 4b.
Problems
1 The natives of planet Wumpus play pool using light rays on an eleven-sided table with mirrors for bumpers. Trace this shot accurately with a ruler and protractor to reveal the hidden message.
Problem 1.
2 Sketch a copy of figure q on page 31. There are some places from which the image is visible, and some from which it isn’t. Show these regions on your sketch by outlining their borders and filling them with two different kinds of shading.
3 (a) Draw a ray diagram showing why a small light source (a candle, say) produces sharper shadows than a large one (e.g. a long fluorescent bulb). Draw a cross-section — don’t try to draw in three dimensions. Your diagram needs to show rays spreading in many directions from each point on the light source, and you need to track the rays until they hit the surface on which the shadow is being cast.
(b) Astronaut Mary goes to Mercury, while Gary visits Jupiter’s moon Ganymede. Unfortunately it’s hard to tell whose vacation pictures are whose, because everybody looks the same in a space suit. Which picture is which? (Note that the brightness of the light is irrelevant. As you can see, the pictures look equally bright, because they took longer or shorter exposures to compensate for the amount of sunlight.)
4 (a) The first figure shows a surface that is mostly smooth, but has a few irregularities in it. Use a ray diagram to show how reflection from this surface would work. (b) The second figure shows an onion on an old chair. What evidence do you see in this picture that there are surfaces like the one in part a?
5 Many astronomers made attempts to detect the parallax of the stars before anyone finally measured their very small parallax angles. The early results were used as an argument against models of the universe in which the earth orbited the sun. Were all these efforts a waste? Should we criticize the astronomers who made them for producing incorrect results? How does this resemble the story of Galileo’s attempt to measure the speed of light? Galileo’s result could be stated as a lower limit on the speed of light, i.e., a mathematical inequality rather than an equality; could you do something similar with the early parallax measurements?
6 If a mirror on a wall is only big enough for you to see yourself from your head down to your waist, can you see your entire body by backing up? Test this experimentally and come up with an explanation for your observations using ray diagrams. Note that it’s easy to confuse yourself if the mirror is even a tiny bit off of vertical; check whether you are able to see more of yourself both above and below. (To make this test work, you may need to lower the mirror so that you can’t see the top of your head, or put a piece of tape on the mirror, and pretend that’s the top of it.)
7 The diagram shows the moon orbiting the earth (not to scale) with sunlight coming in from the right. (a) Why are the sun’s rays shown coming in parallel? Explain. (b) Figure out the phase of the moon when the moon is at each point in its orbit. In other words, when is it a new moon, when is it a crescent, when is it a half moon, when is it gibbous, and when is it full?
8 (a) You’re photographing some people around a campfire. If you step back three times farther from the fire to frame the shot differently, how many times longer will the exposure have to be? Explain. (b) You’re worried that with the longer exposure, the dancing flames will look blurry. Rather than compensating for the greater distance with a longer exposure, you decide to open the diaphragm of the camera wider. How many times greater will the diameter of the aperture have to be? Explain.
9 Why did Roemer only need to know the radius of the earth’s orbit, not Jupiter’s?
10 Suggest a simple experiment or observation, without any special equipment, to show that light isn’t a form of sound. (Note that there are invisible forms of light such as ultraviolet and infrared, so the invisibility of sound doesn’t prove anything. Likewise, you can’t conclude anything from the inaudibility of light.)
In problems 11 and 12, you need to know that radio waves are fundamentally the same phenomenon as light, and travel at the same speed.
11 The Stealth bomber is designed with flat, smooth surfaces. Why would this make it difficult to detect via radar? Explain using a ray diagram.
12 A Global Positioning System (GPS) receiver is a device that lets you figure out where you are by receiving radio signals from satellites. It’s accurate to within a few meters. The details are a little complicated, but for our present purposes, let’s imagine a simplified version of the system in which the satellite sends a signal at a known time, and your handheld unit receives it at a time that is also very accurately measured. The time delay indicates how far you are from the satellite. As a further simplification, let’s assume that everything is one-dimensional: the satellite is low on the eastern horizon, and we’re only interested in determining your east-west position (longitude).5 How accurate does the measurement of the time delay have to be, to determine your position to this accuracy of a few meters?
Problem 6
Problem 7.
