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dq-radar-corner-2d

p / Discussion question B.


dq-radar-corner-3d

q / Discussion question C.

The inverse-square law

Energy is conserved, so a ray of light should theoretically be able to cross an infinite distance without losing any of its intensity, provided that it's traveling through empty space, so that there's no matter that it can give its energy away to. In that case, why does a distant candle appear dim? 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 gets “tired,” or “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.)

inversesquare

k / The light is four times dimmer at twice the distance.

Figure k 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:

equation

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. Physicists already knew about this kind of inverse square law, for light, before Newton found out that the force of gravity varied as an inverse square, so his law of gravity made sense to them intuitively, and they were ready to accept it. However, Newton's law of gravity doesn't describe gravity as a substance that physically travels outward through space, so it's only a rough analogy. (One modern hypothesis about gravity is that the messages of gravitational attraction between two objects are actually carried by little particles, called gravitons, but nobody has ever detected a graviton directly.)

self-check:

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?

Example 1: An example with sound

◊ 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.

Example 2: Astronomical distance scales

The nearest star, Alpha Centauri,1 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.

Example 3: Pupils and camera diaphragms

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.

camera-diaphragm

l / The same lens is shown with its diaphragm set to three different apertures.

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.)

Parallax

Example 2 on page 144 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.

parallax

m / 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 m, 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.2

The earth circles the sun, n, 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 2 on page 144. They do check out, which verifies the assumption that the stars are objects analogous to our sun.

stellarparallax

n / 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.)

Reversibility of light rays

The fact that specular reflection displays equal angles of incidence and reflection means that there is a symmetry: if the ray had come in from the right instead of the left in the figure above, the angles would have looked exactly the same. This is not just a pointless detail about specular reflection. It's a manifestation of a very deep and important fact about nature, which is that the laws of physics do not distinguish between past and future. Cannonballs and planets have trajectories that are equally natural in reverse, and so do light rays. This type of symmetry is called time-reversal symmetry.

Typically, time-reversal symmetry is a characteristic of any process that does not involve heat. For instance, the planets do not experience any friction as they travel through empty space, so there is no frictional heating. We should thus expect the time-reversed versions of their orbits to obey the laws of physics, which they do. In contrast, a book sliding across a table does generate heat from friction as it slows down, and it is therefore not surprising that this type of motion does not appear to obey time-reversal symmetry. A book lying still on a flat table is never observed to spontaneously start sliding, sucking up heat energy and transforming it into kinetic energy.

Similarly, the only situation we've observed so far where light does not obey time-reversal symmetry is absorption, which involves heat. Your skin absorbs visible light from the sun and heats up, but we never observe people's skin to glow, converting heat energy into visible light. People's skin does glow in infrared light, but that doesn't mean the situation is symmetric. Even if you absorb infrared, you don't emit visible light, because your skin isn't hot enough to glow in the visible spectrum.

Example 4: Ray tracing on a computer

A number of techniques can be used for creating artificial visual scenes in computer graphics. Figure o shows such a scene, which was created by the brute-force technique of simply constructing a very detailed ray diagram on a computer. This technique requires a great deal of computation, and is therefore too slow to be used for video games and computer-animated movies. One trick for speeding up the computation is to exploit the reversibility of light rays. If one was to trace every ray emitted by every illuminated surface, only a tiny fraction of those would actually end up passing into the virtual “camera,” and therefore almost all of the computational effort would be wasted. One can instead start a ray at the camera, trace it backward in time, and see where it would have come from. With this technique, there is no wasted effort.

computer-ray-tracing

o / This photorealistic image of a nonexistent countertop was produced completely on a computer, by computing a complicated ray diagram.

Discussion Questions
  • If a light ray has a velocity vector with components cx and cy, what will happen when it is reflected from a surface that lies along the y axis? Make sure your answer does not imply a change in the ray's speed.
  • Generalizing your reasoning from discussion question A, what will happen to the velocity components of a light ray that hits a corner, as shown in the figure, and undergoes two reflections?
  • Three pieces of sheet metal arranged perpendicularly as shown in the figure form what is known as a radar corner. Let's assume that the radar corner is large compared to the wavelength of the radar waves, so that the ray model makes sense. If the radar corner is bathed in radar rays, at least some of them will undergo three reflections. Making a further generalization of your reasoning from the two preceding discussion questions, what will happen to the three velocity components of such a ray? What would the radar corner be useful for?

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