Book 5 in the Light and Matter series of free introductory physics textbooks www.com The Light and Matter series of introductory physics textbooks: 1 Newtonian Physics 2 Conservation Laws 3 Vibrations and Waves 4 Electricity and Magnetism 5 Optics 6 The Modern Revolution in Physics www.com Benjamin Crowell www.com Fullerton, California www.com copyright 1999-2006 Benjamin Crowell edition 2. 11th October 2006 This book is licensed under the Creative Com- mons Attribution-ShareAlike license, version 1.org/licenses/by-sa/1.0/, except for those photographs and drawings of which I am not the author, as listed in the photo credits. If you agree to the license, it grants you certain privileges that you would not otherwise have, such as the right to copy the book, or download the digital version free of charge from www. At your option, you may also copy this book under the GNU Free Documentation License version 1.org/licenses/fdl.txt, with no invariant sections, no front-cover texts, and no back-cover texts.com Brief Contents 1 The Ray Model of Light 11 2 Images by Reflection 31 3 Images, Quantitatively 43 4 Refraction 59 5 Wave Optics 77 www.com Contents Problems.
40 1 The Ray Model of Light 1.1 The Nature of Light. 12 The cause and effect relationship in vision, 12.—Light is a thing, and it travels from 3 Images, Quantitatively one point to another.—Light can travel 3.1 A Real Image Formed by a Converg- through a vacuum.2 Interaction of Light with Matter. 15 Location of the image, 44.—Magnification, Absorption of light, 15.—How we see non- 47.2 Other Cases With Curved Mirrors. 47 surement of the brightness of light, 17.3 The Ray Model of Light.
54 Models of light, 18.4 Geometry of Specular Reflection. 22 Reversibility of light rays, 23.5 ? The Principle of Least Time for Reflection .—Refractive properties of media, 61.—The index of refraction is related to the speed of 2 Images by Reflection light.—A mechanical model of Snell’s law, 64.—A derivation of Snell’s law, 64. 32 Color and refraction, 65.—How much light 2. 33 is reflected, and how much is transmitted?, 2.4 Images of Images .3 ? The Lensmaker’s Equation .4 ? The Principle of Least Time for 5.5 Double-Slit Diffraction .7 RSingle-Slit Diffraction .8 ? The Principle of Least Time .2 Scaling of Diffraction .3 The Correspondence Principle.
80 Appendix 2: Photo Credits 105 5. 81 Appendix 3: Hints and Solutions 106 9 www.com Chapter 1 The Ray Model of Light Ads for one Macintosh computer bragged that it could do an arith- metic calculation in less time than it took for the light to get from the screen to your eye. We find this impressive because of the contrast between the speed of light and the speeds at which we interact with physical objects in our environment. Perhaps it shouldn’t surprise us, then, that Newton succeeded so well in explaining the motion of objects, but was far less successful with the study of light.
These books are billed as the Light and Matter series, but only now, in the fifth of the six volumes, are we ready to focus on light. If you are reading the series in order, then you know that the climax of our study of electricity and magnetism was discovery that light is an electromagnetic wave. Knowing this, however, is not the same as knowing everything about eyes and telescopes. In fact, the full description of light as a wave can be rather cumbersome.
We will instead spend most of this book making use of a simpler model of light, the ray model, which does a fine job in most practical situations. Not only that, but we will even backtrack a little and 11 www.com start with a discussion of basic ideas about light and vision that predated the discovery of electromagnetic waves.1 The Nature of Light The cause and effect relationship in vision Despite its title, this chapter is far from your first look at light. That familiarity might seem like an advantage, but most people have never thought carefully about light and vision. Even smart people who have thought hard about vision have come up with incorrect ideas.
The ancient Greeks, Arabs and Chinese had theories of light and vision, all of which were mostly wrong, and all of which were accepted for thousands of years. One thing the ancients did get right is that there is a distinction between objects that emit light and objects that don’t. When you see a leaf in the forest, it’s because three different objects are doing their jobs: the leaf, the eye, and the sun. But luminous objects like the sun, a flame, or the filament of a light bulb can be seen by the eye without the presence of a third object.
Emission of light is often, but not always, associated with heat. In modern times, we are familiar with a variety of objects that glow without being heated, including fluorescent lights and glow-in-the-dark toys. How do we see luminous objects? The Greek philosophers Pythago- ras (b. 560 BC) and Empedocles of Acragas (b.
492 BC), who unfortunately were very influential, claimed that when you looked at a candle flame, the flame and your eye were both sending out some kind of mysterious stuff, and when your eye’s stuff collided with the candle’s stuff, the candle would become evident to your sense of sight. Bizarre as the Greek “collision of stuff theory” might seem, it had a couple of good features. It explained why both the candle and your eye had to be present for your sense of sight to function. The theory could also easily be expanded to explain how we see nonluminous objects.
If a leaf, for instance, happened to be present at the site of the collision between your eye’s stuff and the candle’s stuff, then the leaf would be stimulated to express its green nature, allowing you to perceive it as green. Modern people might feel uneasy about this theory, since it sug- gests that greenness exists only for our seeing convenience, implying a human precedence over natural phenomena. Nowadays, people would expect the cause and effect relationship in vision to be the other way around, with the leaf doing something to our eye rather than our eye doing something to the leaf. But how can you tell? The most common way of distinguishing cause from effect is to de- termine which happened first, but the process of seeing seems to occur too quickly to determine the order in which things happened.
12 Chapter 1 The Ray Model of Light www.com Certainly there is no obvious time lag between the moment when you move your head and the moment when your reflection in the mirror moves. Today, photography provides the simplest experimental evidence that nothing has to be emitted from your eye and hit the leaf in order to make it “greenify.” A camera can take a picture of a leaf even if there are no eyes anywhere nearby. Since the leaf appears green regardless of whether it is being sensed by a camera, your eye, or an insect’s eye, it seems to make more sense to say that the leaf’s greenness is the cause, and something happening in the camera or eye is the effect. Light is a thing, and it travels from one point to another.
Another issue that few people have considered is whether a can- dle’s flame simply affects your eye directly, or whether it sends out light which then gets into your eye. Again, the rapidity of the effect makes it difficult to tell what’s happening. If someone throws a rock at you, you can see the rock on its way to your body, and you can tell that the person affected you by sending a material substance your way, rather than just harming you directly with an arm mo- tion, which would be known as “action at a distance.” It is not easy to do a similar observation to see whether there is some “stuff” that travels from the candle to your eye, or whether it is a case of action at a distance. Newtonian physics includes both action at a distance (e.
the earth’s gravitational force on a falling object) and contact forces a / Light from a candle is bumped such as the normal force, which only allow distant objects to exert off course by a piece of glass. forces on each other by shooting some substance across the space Inserting the glass causes the between them (e., a garden hose spraying out water that exerts a apparent location of the candle force on a bush). The same effect can be produced by taking off your One piece of evidence that the candle sends out stuff that travels eyeglasses and looking at which to your eye is that as in figure a, intervening transparent substances you see near the edge of the can make the candle appear to be in the wrong location, suggesting lens, but a flat piece of glass that light is a thing that can be bumped off course. Many peo- works just as well as a lens for ple would dismiss this kind of observation as an optical illusion, this purpose.
(Some optical illusions are purely neurological or psycho- logical effects, although some others, including this one, turn out to be caused by the behavior of light itself.) A more convincing way to decide in which category light belongs is to find out if it takes time to get from the candle to your eye; in Newtonian physics, action at a distance is supposed to be instan- taneous. The fact that we speak casually today of “the speed of light” implies that at some point in history, somebody succeeded in showing that light did not travel infinitely fast. Galileo tried, and failed, to detect a finite speed for light, by arranging with a person in a distant tower to signal back and forth with lanterns.1 The Nature of Light 13 www.com uncovered his lantern, and when the other person saw the light, he uncovered his lantern. Galileo was unable to measure any time lag that was significant compared to the limitations of human reflexes.
The first person to prove that light’s speed was finite, and to determine it numerically, was Ole Roemer, in a series of measure- ments around the year 1675. Roemer observed Io, one of Jupiter’s moons, over a period of several years. 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 ac- curate pendulum clock had recently been invented, so Roemer could check whether the ratio of the two clocks’ cycles, about 42.5 hours to 1 orbit, stayed exactly constant or changed a little.
If the process b / An image of Jupiter and of seeing the distant moon was instantaneous, there would be no its moon Io (left) from the Cassini probe. 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.