Simple Nature - Light and Matter

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12.2 Images by ReflectionInfants are always fascinated by the antics of the Baby in the Mirror.Now if you want to know something about mirror images that mostpeople don’t understand, try this. First bring this page closer andcloser to your eyes, until you can no longer focus on it withoutstraining. Then go in the bathroom and see how close you canget your face to the surface of the mirror before you can no longereasily focus on the image of your own eyes. You will find thatthe shortest comfortable eye-mirror distance is much less than theshortest comfortable eye-paper distance. This demonstrates thatthe image of your face in the mirror acts as if it had depth andexisted in the space behind the mirror. If the image was like a flatpicture in a book, then you wouldn’t be able to focus on it fromsuch a short distance.In this chapter we will study the images formed by flat andcurved mirrors on a qualitative, conceptual basis. Although thistype of image is not as commonly encountered in everyday life asimages formed by lenses, images formed by reflection are simpler tounderstand, so we discuss them first. In section 12.3 we will turnto a more mathematical treatment of images made by reflection.Surprisingly, the same equations can also be applied to lenses, whichare the topic of section 12.4.12.2.1 A virtual imagea / An image formed by amirror.We can understand a mirror image using a ray diagram. Figurea shows several light rays, 1, that originated by diffuse reflection atthe person’s nose. They bounce off the mirror, producing new rays,2. To anyone whose eye is in the right position to get one of theserays, they appear to have come from a behind the mirror, 3, wherethey would have originated from a single point. This point is wherethe tip of the image-person’s nose appears to be. A similar analysisapplies to every other point on the person’s face, so it looks asthough there was an entire face behind the mirror. The customaryway of describing the situation requires some explanation:Customary description in physics: There is an image of the facebehind the mirror.Translation: The pattern of rays coming from the mirror is exactlythe same as it would be if there were a face behind the mirror.Nothing is really behind the mirror.This is referred to as a virtual image, because the rays do notactually cross at the point behind the mirror. They only appear tohave originated there.self-check B748 Chapter 12 Optics
Imagine that the person in figure a moves his face down quite a bit —a couple of feet in real life, or a few inches on this scale drawing. Themirror stays where it is. Draw a new ray diagram. Will there still be animage? If so, where is it visible from? ⊲ Answer, p. 928The geometry of specular reflection tells us that rays 1 and 2are at equal angles to the normal (the imaginary perpendicular linepiercing the mirror at the point of reflection). This means thatray 2’s imaginary continuation, 3, forms the same angle with themirror as ray 3. Since each ray of type 3 forms the same angles withthe mirror as its partner of type 1, we see that the distance of theimage from the mirror is the same as that of the actual face fromthe mirror, and it lies directly across from it. The image thereforeappears to be the same size as the actual face.b / Example 2.An eye exam example 2Figure b shows a typical setup in an optometrist’s examinationroom. The patient’s vision is supposed to be tested at a distanceof 6 meters (20 feet in the U.S.), but this distance is larger thanthe amount of space available in the room. Therefore a mirror isused to create an image of the eye chart behind the wall.The Praxinoscope example 3Figure c shows an old-fashioned device called a praxinoscope,which displays an animated picture when spun. The removablestrip of paper with the pictures printed on it has twice the radiusof the inner circle made of flat mirrors, so each picture’s virtualimage is at the center. As the wheel spins, each picture’s imageis replaced by the next.c / The praxinoscope.Section 12.2 Images by Reflection 749
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Fullerton, Californiawww.lightandma
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Contents0 Introduction and Review0.
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5.5 More About Heat Engines . . . .
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669.11.3 Magnetic Fields by Ampère
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The Mars Climate Orbiter is prepare
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temperatures and with many combinat
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idence that quarks have smaller par
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give clearer explanations.Finally,
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It can also be handy to have a rela
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The prefix centi-, meaning 10 −2
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goes like this:V = 1 3 Ah[1]A = πr
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the notation 10 0 to stand for one,
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calculation, 5.04 cm, was really no
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0.2 Scaling and Order-of-Magnitude
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to many times your own height. The
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g / 1. This plank is as long as it
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the front panels of the three violi
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Correct solution #4: The area of a
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here?” The scientific Mr. Spock w
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1. Don’t even attempt more than o
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Problem 10.mean, however, is define
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(d) Find the person’s acceleratio
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Albert Einstein, and his moustache,
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54 Chapter 0 Introduction and Revie
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a / Portrait of Monsieur Lavoisiera
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1.1.1 Problem-solving techniquesHow
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∆m = 0, where m is the total mass
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different substances will have diff
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c / Left: In a frame of referenceth
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f / Discussion question B.(discusse
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Self-check D.since the derivative o
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ProblemsThe symbols √ , , etc. ar
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difficult? ⊲ Solution, p. 932Key
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Heat energy can be convertedto ligh
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a new form of invisible “mystery
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f / A realistic drawing of Joule’
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numbers. With a purely numerical ap
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of gravity doesn’t change much if
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field. If the plane can start from
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m / Discussion question C.n / A hyd
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88 Chapter 2 Conservation of Energy
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A car drives over a cliff.new frame
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How long does it take to move 1 met
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a / Approximations to thebrachistoc
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a / An ellipse is circle that hasbe
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to deduce the general equation for
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to the mass of the object that inte
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The minimum velocity required for t
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and since sin θ dθ occurs in the
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that is deeper than r. Under the as
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2.3.6 ⋆ Evidence for repulsive gr
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a / A vivid demonstration thatheat
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ture. This is a very good question,
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Three functions with thesame curvat
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This looks like a cosine function,
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Problem 16.13 Anya and Ivan lean ov
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Problem 27.then we’d have U/m =
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37 Two springs with spring constant
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ExercisesExercise 2A: Reasoning wit
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128 Chapter 2 Conservation of Energ
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3.1 Momentum In One Dimensiona / Sy
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eing to the right. The initial mome
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3.1.3 Momentum compared to kinetic
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3.1.4 Collisions in one dimensiong
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could never do that again in a mill
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i / The highjumper’s body passeso
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where x 1 is the mass of the first
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3.1.6 The center of mass frame of r
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The airbag increases ∆tso as to r
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d / Two magnets exert forceson each
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x (m) t (s)10 1.8420 2.8630 3.8040
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3.2.4 Forces between solidsConserva
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Maximum acceleration of a car examp
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Discussion QuestionA Criticize the
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C A pool ball is rebounding from th
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forces, as if the hand were a pair
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This may not sound like an impressi
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stroke are both executed in straigh
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Accelerating a cart example 35If yo
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w / A wedge.x / Archimedes’ screw
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As in the preceding example, we hav
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that have the same frequency is a s
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around until I got this result, sin
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quality factor, Q, is defined as Q
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ight direction to add energy to the
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i / Example 45: a viola withouta mu
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Collapse of the Nimitz Freeway exam
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end up canceling out, however:Q =
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186 Chapter 3 Conservation of Momen
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c / Bullets are dropped and shot at
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the asteroid’s energy and boostin
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coordinate axes. Even though ∆x,
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of the three components,∆p x = 0
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A useless vector operation example
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Solving for the unknowns gives∆x
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o / Example 64.p / Adding vectors g
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How to generalize one-dimensional e
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needed to support the object in fig
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The easiest method is the one demon
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Incorrect solution #4:(same notatio
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The magnitude of the acceleration i
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dropped out like a trap door, showi
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af / Breaking trail, by WalterE. Bo
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know how to integrate with respect
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Problem 16.Problem 19.resistance.)1
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of gravity on Mars.(a) Find the tim
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partner force. (a) A swimmer speeds
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Problem 45the 0.4-gram masses would
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53 If you walk 35 km at an angle 25
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(b) Interpret this equation in the
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Problem 71.232 Chapter 3 Conservati
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77 A car accelerates from rest. At
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Problem 81.81 Complete example 71 o
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ExercisesExercise 3A: Force and Mot
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Exercise 3C: Worksheet on Resonance
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Exercise 3D: Vectors and MotionEach
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244 Chapter 3 Conservation of Momen
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An overhead view of apiece of putty
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inward toward the hinge will have n
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special case, we can choose to visu
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j / Two asteroids collide.k / Every
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Note that although the factors of 2
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Relationship between force and torq
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is straight down, which is perpendi
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⊲ All three objects in the figure
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aa / Example 10.to either side of e
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momentum tells us that L = mrv sin
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In the absence of any torque, a rig
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Radial acceleration at the surface
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The parallel axis theorem example 1
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differential. The result isarea ===
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of inertia as if the object was smo
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h / Moments of inertia of somegeome
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4.3 Angular Momentum In Three Dimen
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14 deg, and it points along an axis
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manner like this, then the definiti
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k / Example 27.torque to the left w
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We can also generalize the plane-ro
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Problem 1.Problem 6.Problem 8.Probl
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14 (a) The bar of mass m is attache
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Problem 22.22 The sun turns on its
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35 The nucleus 168 Er (erbium-168)
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ExercisesExercise 4A: TorqueEquipme
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pesky set of constraints on heat en
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mit more air into the cavity behind
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Pressure of lava underneath a volca
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h / A simplified version of anideal
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for us to construct a simple connec
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For the first time we have an inter
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a / 1. The temperature differencebe
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Carnot engines operating between a
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from A to B, he lets it by, but whe
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B When we run the Carnot engine in
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f / A phase space for a singleatom
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time, the energy sharing is very un
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will the the one that maximizes the
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If the gas is monoatomic, then we k
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5.4.4 The arrow of time, or “this
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5.4.6 Summary of the laws of thermo
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may be in the form of microscopic d
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significant amount of heat to flow
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while the smaller area under the bo
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7 (a) Determine the ratio between t
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338 Chapter 5 Thermodynamics
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end of this book, we’ll even see
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c / As the wave pattern passes the
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the wave move ahead faster and get
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k / Hitting a key on a pianocauses
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Our final result for the speed of t
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care, because the delay is the same
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An easy way to visualize this is in
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can be constructed as a superpositi
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⊲ Looking up the speed of light i
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scientists, to speak of the Big Ban
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6.2 Bounded WavesSpeech is what sep
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d / An uninverted reflection. There
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our intuitive expectation of strong
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valid solutions. In the following s
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j / In the mirror image, theareas o
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m / A pulse bounces backand forth.i
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q / Graphs of loudness versusfreque
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s / Standing waves on a rope. (PSSC
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“cavity” and “neck” parts o
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Problem 5.Problem 8.5 The figure sh
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Problem 16.16 A Fabry-Perot interfe
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c / Newton’s laws do not distingu
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f / The correspondence principlereq
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d / A Galilean version of therelati
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g / Three types of transformations
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system, velocities are always unitl
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p / Apparatus used for the testof r
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sumption, the assumption that it ma
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at rest relative to the water. But
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7.2.4 No action at a distanceThe Ne
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so that Alice can complete her moti
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time into different regions accordi
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position relative to the sun at exa
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7.2.7 ⋆ Four-vectors and the inne
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would depend on the relative veloci
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7.3 DynamicsSo far we have said not
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In this frame, as expected, the sma
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But this whole argument was based o
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meters per second, so converting to
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Expressing γ as ( 1 − v 2 /c 2)
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7.3.4 ⋆ ProofsThis optional secti
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these lines. For example, a car sit
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An Einstein’s ring. Thedistant ob
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two-dimensional universe as if it w
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h / 1. A ray of light is emittedupw
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object that that isn’t influenced
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it impossible for any observer to b
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a variant on the Penrose singularit
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in it. Instead, it is currently spe
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2 Astronauts in three different spa
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(d) Simplify your answer to part c
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Problem 25b. Redrawn fromVan Baak,
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ExercisesExercise 7A: The Michelson
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Exercise 7B: Sports in Slowlightlan
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448 Chapter 7 Relativity
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Exercise 7D: Misconceptions about R
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452 Chapter 7 Relativity
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sun, moon, stars, and planets were
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Two types of chargeWe can easily co
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the other acquires an equal amount
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to get the beam up to speed in the
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telling us that we know about matte
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the muddle. The row-and-column sche
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the force of air friction canceled
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ectly evaluate the implications of
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that they were indeed electrically
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C Thomson found that the m/q of an
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the object with a net positive char
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thickness of material the radioacti
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It was already known that although
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particle. It turned out that it was
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For example they could easily strip
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cording to Mosely, the atomic numbe
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that fly off to see what was inside
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solution was found by measuring the
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o / A nuclear power plant at Catten
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neutrons. In a nuclear fission bomb
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We can now list all four of the kno
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1. Our sun’s source of energy is
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a / The known nuclei, represented o
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killed, because the DNA becomes una
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e / Wild Przewalski’s horsesprosp
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Problem 1. Top: A realisticpicture
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12 The subatomic particles called m
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ExercisesExercise 8A: Nuclear decay
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saxophone, every technological tool
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Example 1Ions moving across a cell
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ut the original meaning was to trav
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Force depends only on position. Sin
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Here are a few questions and answer
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flow through it.For many substances
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superconductivity in metals that wo
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j / Example 9. In 1 and 2,charges t
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look like the usual resistor. The f
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9.1.5 Current-conducting properties
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attery acid becomes depleted of hyd
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9.2 Parallel and Series CircuitsIn
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to each resistance, resulting inI t
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where “...” means that the sum
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the two resistors in figure h/3.We
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Choice of high voltage for power li
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A complicated circuit example 20⊲
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Problem 2.ProblemsThe symbols √ ,
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Problem 16.only count that as one u
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knob turned all the way clockwise,
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A printed circuit board, likethe ki
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ExercisesExercise 9A: Voltage and C
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Exercise 9C: Reasoning About Circui
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556 Chapter 9 Circuits
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a / A bar magnet’s atoms are(part
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defined. The ship’s captain can m
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Reduction in gravity on Io due to J
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j / Example 3.self-check AFind an e
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a dipole moment — they are define
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10.2.1 One dimensionVoltage is elec
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The interpretation is that if you b
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Figure c shows some examples of way
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For large values of d, this express
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where Q is the total charge of the
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g / Example 12: variation of the fi
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corners to the disk and transform i
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10.4 Energy In Fieldsa / Two opposi
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self-check DWe can think of the qua
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of the inward field contributed by
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space, 2 while charge doesn’t, we
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a / The symbol for a capacitor.b /
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ounding each capacitor will be half
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in time:x ↔ qv ↔ Iself-check GH
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due to its own momentum. It perform
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or|V | =∣ LdIdt ∣ ,which in man
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the inductor resists such a sudden
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Example 26.Death by solenoid; spark
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ordering.( 1 √2+√ i ) 2 = √ 1
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x / The complex number e iφlies on
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Figure aa shows a useful way to vis
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Inductors tend to be big, heavy, ex
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The reason for using the trig ident
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a purely inductive or capacitive lo
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Resonance with damping example 33
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orE outward, on side 1 A + E outwar
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|E| = kq totalr 2 ,where r is the r
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y considering its point charges ind
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Discussion Questionsg / Discussion
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with charge, change the Coulomb con
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The symmetry between the two sides
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where dv is the volume of the cube.
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The three terms in the divergence a
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Problem 19.Problem 20.proton, for e
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the lightning strike.(b) Based on y
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units.(b) Verify that RC has units
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lating freely (without any driving
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ExercisesExercise 10A: Field Vector
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5. Now hook up the two solenoids in
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A large current is createdby shorti
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time I used it implicitly was in fi
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f / A standard dipole madefrom a sq
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and substituting q = λh and v = m/
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o / Magnetic forces cause abeam of
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electric field, a magnetic one, can
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u / In this scene from SwanLake, th
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so the total field in the z directi
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For the y component, we havee / A s
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We can pin down the result even mor
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i / The field of a dipole.where r i
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circulates around the y axis, so at
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to be, not where it is now. Coulomb
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We have found one specific example
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to a flagpole, we can cancel out a
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11.4 Ampère’s Law In Differentia
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y evaluating the field at the midpo
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i.e.,F = axˆx + byˆx + cˆx + dx
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k / A summary of the derivative, gr
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c / Detail from Ascending andDescen
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the magnet. Are these atomic curren
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field in her region of space has be
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A changing magnetic flux makes a cu
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nearly zero. By Faraday’s law, th
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c / An Ampèrian surface superimpos
Page 698 and 699: and Maxwell’s equations becomeΦ
Page 700 and 701: k / Red and blue light travelat the
Page 702 and 703: second, the zero-point is located a
Page 704 and 705: is the magnitude of the momentum ve
Page 706 and 707: Discussion question A.Discussion qu
Page 708 and 709: only penetrates to a very small dep
Page 710 and 711: elationship D = ɛE would actually
Page 712 and 713: esulting in cancellation.The opposi
Page 714 and 715: a low permeability, while the other
Page 716 and 717: n / A fluxgate compass.is externall
Page 718 and 719: 6 Two parallel wires of length L ca
Page 720 and 721: an instant at the upper right, but
Page 722 and 723: Problem 25.20 Four long wires are a
Page 724 and 725: Problem 35.Problem 37.32 Verify Amp
Page 726 and 727: Problem 42.beam of light usually co
Page 728 and 729: (c) Discuss the relationship betwee
Page 730 and 731: (f) Use conservation of energy to r
Page 732 and 733: 5. Now position yourself with your
Page 734 and 735: 12.1.1 The nature of lightThe cause
Page 736 and 737: An image of Jupiter andits moon Io
Page 738 and 739: d / Two self-portraits of theauthor
Page 740 and 741: ultimate truth about light, but the
Page 742 and 743: • There is a tendency to conceptu
Page 744 and 745: self-check AEach of these diagrams
Page 746 and 747: m / Discussion question B.Discussio
Page 750 and 751: Discussion QuestionA The figure sho
Page 752 and 753: down magnification is AB/DE. A repe
Page 754 and 755: i / A Newtonian telescopebeing used
Page 756 and 757: B Locate the images formed by two p
Page 758 and 759: 12.3 Images, QuantitativelyIt sound
Page 760 and 761: ⊲ The object and image angles are
Page 762 and 763: 12.3.2 Other cases with curved mirr
Page 764 and 765: signs also have to be memorized for
Page 766 and 767: h / A diverging mirror in the shape
Page 768 and 769: j / Spherical mirrors are thecheape
Page 770 and 771: general type of eye that we share w
Page 772 and 773: shown on this graph and then attemp
Page 774 and 775: the mechanical model would predict
Page 776 and 777: that your calculator will flash an
Page 778 and 779: D Classify the examples shown in fi
Page 780 and 781: 12.4.6 ⋆ Microscopic description
Page 782 and 783: 12.5 Wave OpticsElectron microscope
Page 784 and 785: of a crystal? Sound waves are used
Page 786 and 787: j / Thomas Youngk / Double-slit dif
Page 788 and 789: object that diffracts it, so the tr
Page 790 and 791: Although the equation λ/d = sin θ
Page 792 and 793: things we’ve learned about diffra
Page 794 and 795: apidly changing distances; on reuni
Page 796 and 797: 6 The figure on the next page shows
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14 Here’s a game my kids like to
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27 Suppose a converging lens is con
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same depth, but not quite. [Check:
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Problem 42.42 Panel 1 of the figure
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46 The figure below shows two diffr
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53 The beam of a laser passes throu
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Problem 59.the ancient problem of i
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3. Now imagine the following new si
Page 814 and 815:
Exercise 12B: Object and Image Dist
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Exercise 12D: Double-Source Interfe
Page 818 and 819:
Exercise 12E: Single-slit diffracti
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Exercise 12F: Diffraction of LightE
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energy, instead of being spread out
Page 824 and 825:
correlated. If they have been playi
Page 826 and 827:
small.The statement that the rule f
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But the y axis can no longer be a u
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for a long time once it gets there,
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j / Calibration of the 14 C dating
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given by(probability of a ≤ x ≤
Page 836 and 837:
k / In recent decades, a huge hole
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A wave is partially absorbed.c / A
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f / The hamster in her hamsterball
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How would you extract h from the gr
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F Does E = hf imply that a photon c
Page 846 and 847:
of approaching this issue.)j / Bull
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We assume v is small enough so that
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the probability distribution will b
Page 852 and 853:
trons that we have already used for
Page 854 and 855:
equations of general validity are t
Page 856 and 857:
wave carries high probability and w
Page 858 and 859:
An infinite sine wave can only tell
Page 860 and 861:
momentum implies a definite kinetic
Page 862 and 863:
of snapshots would amount to a desc
Page 864 and 865:
close are the electrons to the limi
Page 866 and 867:
dimensional particle in a box, and
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Applying this to conservation of en
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the probability of making it throug
Page 872 and 873:
Three dimensionsFor simplicity, we
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1. Oscillations can go backand fort
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C The figure shows a skateboarder t
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a / Eight wavelengths fit aroundthi
Page 880 and 881:
As shown by these examples, the unc
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e / The three states of the hydroge
Page 884 and 885:
f / The energy levels of a particle
Page 886 and 887:
∂r/∂x = x/r comes in handy. Com
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50% of its time in each atom. It’
Page 890 and 891:
tus to the z axis) and more memorab
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ut why does that have anything to d
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the data are only inaccurate due to
Page 896 and 897:
14 The photoelectric effect can occ
Page 898 and 899:
Problem 25.(b) Sketch a graph showi
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conservation of energy and momentum
Page 902 and 903:
Problem 43.43 On pp. 884-885 of sub
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ExercisesExercise 13A: Quantum Vers
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⊲ First we convert the equation i
Page 908 and 909:
Programming With PythonThe purpose
Page 910 and 911:
Appendix 2: MiscellanyUnphysical
Page 912 and 913:
18 bestt = t19 c1 = bestc120 c2 = b
Page 914 and 915:
p i + ∑ iThe spin theoremTheorem:
Page 916 and 917:
Appendix 3: Photo CreditsExcept as
Page 918 and 919:
Appendix 4: Hints and SolutionsHint
Page 920 and 921:
Hints for Chapter 6Page 377, proble
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Answers to Self-Checks for Chapter
Page 924 and 925:
Page 301: (1) Not valid. The equati
Page 926 and 927:
Page 584:N −1 m −2 C 2 V 2 m
Page 928 and 929:
the dashed ray.Page 748: You should
Page 930 and 931:
direction it was initially going (i
Page 932 and 933:
Page 52, problem 38: The cone of mi
Page 934 and 935:
Page 224, problem 37: (a) Spring co
Page 936 and 937:
object, we have static friction, wh
Page 938 and 939:
Page 549, problem 29:(a) Conservati
Page 940 and 941:
Page 798, problem 19: (a) The objec
Page 942 and 943:
.0.8 Notation and unitsquantity uni
Page 944 and 945:
surfaces. For comparison, a typical
Page 946 and 947:
elations:a = ∆v∆tx = 1 2 at2 +
Page 948 and 949:
we are moving along with the projec
Page 950 and 951:
the number of oscillations required
Page 952 and 953:
In general, the cross product of ve
Page 954 and 955:
Sound waves consist of increases an
Page 956 and 957:
signs for charge is that with this
Page 958 and 959:
know that there is a delay in time
Page 960 and 961:
is related byΦ = 4πkq into the ch
Page 962 and 963:
to use sophisticated models such as
Page 964 and 965:
Chapter 13, Quantum Physics, page 8
Page 966 and 967:
oth position and momentum, the Heis
Page 968 and 969:
Schrödinger’s, 864cathode rays,
Page 970 and 971:
unstable, 87equipartition theorem,
Page 972 and 973:
Joule, James, 73paddlewheel experim
Page 974 and 975:
Einstein’s early theory, 838energ
Page 976 and 977:
Nikola, 178tesla (unit), 652thermal
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Simple Nature - Light and Matter

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