Concepts of Modern | Physics
Sixth Edition
Arthur Beiser
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CONCEPTS OF MODERN PHYSICS, SIXTH EDITION
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Preface
C ontents
xii
CHAPTER 1 Relativity 1
Li
‘
1.10
Special Relativity 2
All motion is relative; the speed of light in free space is the same for all observers
Time Dilation 5
A moving clock ticks more slowly than a clock at rest Doppler Effect 10
Why the universe is believed to be expanding Length Contraction 15
Faster means shorter
Twin Paradox 17
A longer life, but it will not seem longer
Electricity and Magnetism 19
Relativity is the bridge
Relativistic Momentum 22
Redefining an important quantity
Mass and Energy 26
Where Ey = mc? comes from
Energy and Momentum 30
How they fit together in relativity * General Relativity 33
Gravity is a warping of spacetime
APPENDIX |: The Lorentz Transformation 37 APPENDIX II: Spacetime 46
Contents
CHAPTER 2 Particle Properties of Waves 52
2.1 Electromagnetic Waves 53 Coupled electric and magnetic oscillations that move with the speed of light and exhibit typical wave behavior
2.2 Blackbody Radiation 57 Only the quantum theory of light can explain its origin
2.3 Photoelectric Effect 62 The energies of electrons liberated by light depend on the frequency of the light
2.4 What Is Light? 67 Both wave and particle
2.5 X-Rays 68 They consist of high-energy photons
2.6 X-Ray Diffraction 72
“How x-ray wavelengths can be determined
2.7 Compton Effect 75 Further confirmation of the photon model
2.8 Pair Production 79 Energy into matter
2.9 Photons and Gravity 85 Although they lack rest mass, photons behave as though they have gravitational mass
CHAPTER 3 Wave Properties of Particles 92 3.1 De Broglie Waves 93 ; A moving body behaves in certain ways as though it has a wave nature 3.2 Waves of What? 95 Waves of probability 3.3 Describing a Wave 96 A general formula for waves 3.4 ‘Phase and Group Velocities 99 A group of waves need not have the same velocity as the waves - themselves 3.5 Particle Diffraction 104 An experiment that confirms the existence of de Broglie waves
3.6
3.7
3.8
3.9
Contents
Particle in a Box 106
Why the energy of a trapped particle is quantized
Uncertainty Principle 1 108
We cannot know the future because we cannot know the present Uncertainty Principle II 113
A particle approach gives the same result
Applying the Uncertainty Principle 114
A useful tool, not just a negative statement
CHAPTER 4 Atomic Structure 119
4.1
4.2
4.3
44
4.5
4.6
4.7
4.8
4.9
The Nuclear Atom 120
An atom is largely empty space
Electron Orbits 124
The planetary model of the atom and why it fails Atomic Spectra 127
Each element has a characteristic line spectrum The Bohr Atom 130
* Electron waves in the atom
Energy Levels and Spectra ~ 133
A photon is emitted when an electron jumps from one energy level to a lower level
Correspondence Principle 138
The greater the quantum number, the closer quantum physics approaches classical physics
Nuclear Motion 140
The nuclear mass affects the wavelengths of spectral lines
Atomic Excitation 142
How atoms absorb and emit energy
The Laser 145
How to produce light waves all in step
APPENDIX: Rutherford Scattering ~~ 152
CHAPTER 5 Quantum Mechanics 160
5.1
Quantum Mechanics 161 Classical mechanics is an approximation of quantum mechanics
vi Contents 5.2 The Wave Equation 163 It can have a variety of solutions, including complex ones 5.3 Schrédinger’s Equation: Time-Dependent Form 166 A basic physical principle that cannot be derived from anything else 5.4 Linearity.and Superposition 169 Wave functions add, not probabilities 5.5 Expectation Values 170 How to extract information from a wave function 5.6 Operators 172 , Another way to find expectation values 5.7 Schrodinger’s Equation: Steady-State Form 174 Eigenvalues and eigenfunctions 5.8 Particle in a Box 177 How boundary conditions and normalization determine wave functions 5.9 Finite Potential Well 183 The wave function penetrates the walls, which lowers the energy levels 5.10 Tunnel Effect 184 A particle without the energy to pass over a potential barrier may still tunnel through it 5.11 Harmonic Oscillator 187
Its energy levels are evenly spaced
APPENDIX: The Tunnel Effect 193
CHAPTER 6 Quantum Theory of the Hydrogen Atom 200
6.1
6.2
63
6.4
6.5
6.6
Schrodinger’s Equation for the Hydrogen Atom 201 Symmetry suggests spherical polar coordinates Separation of Variables 203
A differential equation for each variable Quantum Numbers 205
Three dimensions, three quantum numbers Principal Quantum Number 207 Quantization of energy
Orbital Quantum Number 208 Quantization of angular-momentum magnitude Magnetic Quantum Number 210 Quantization of angular-momentum direction
Contents vii
Electron Probability Density 212
No definite orbits
Radiative Transitions 218
What happens when an electron goes from one state to another Selection Rules 220
Some transitions are more likely to occur than others
Zeeman Effect 223
How atoms interact with a magnetic field
CHAPTER 7 Many-Electron Atoms 228
71
7.2
7.3
7.4
7.5
7.6
17
7.8
79
Electron Spin 229
Round and round it goes forever
Exclusion Principle 231
A different set of quantum numbers for each electron in an atom Symmetric and Antisymmetric Wave Functions 233 Fermions and bosons ; Periodic Table 235
Organizing the elements
Atomic Structures 238
Shells and subshells of electrons
Explaining the Periodic Table 240
How an atom’s electron structure determines its chemical behavior Spin-Orbit Coupling 247
Angular momenta linked magnetically
Total Angular Momentum 249
Both magnitude and direction are quantized
X-Ray Spectra 254
They arise from transitions to inner shells
APPENDIX: Atomic Spectra 259
CHAPTER 8 Molecules 266
8.1
The Molecular Bond 267 ; Electric forces hold atoms together to form molecules
viil
Contents
8.2
8.3
8.4
8.5
8.6
8.7
8.8
Electron Sharing 269
The mechanism of the covalent bond
The H,* Molecular Ion f 270
Bonding requires a symmetric wave function The Hydrogen Molecule = 274
The spins of the electrons must be antiparallel Complex Molecules 276
Their geometry depends on the wave functions of the outer electrons of their atoms
Rotational Energy Levels 282
Molecular rotational spectra are in the microwave region Vibrational Energy Levels 285
A molecule may have many different modes of vibration Electronic Spectra of Molecules 291
How fluorescence and phsophorescence occur
CHAPTER 9 Statistical Mechanics 296
9.1
9.2
9.3
9.4
92
9.6
9.7
9.8
9.9
Statistical Distributions 297
Three different kinds
Maxwell-Boltzmann Statistics 298
Classical particles such as gas molecules obey them. Molecular Energies in an Ideal Gas 300 They vary about an average of 2kT
Quantum Statistics 305
Bosons and fermions have different distribution functions Rayleigh-Jeans Formula 311
The classical approach to blackbody radiation Planck Radiation Law 313
How a photon gas'behaves
Einstein's Approach 318
Introducing stimulated emission
Specific Heats of Solids 320
Classical physics fails again
Free Electrons in a Metal 323
No more than one electron per quantum state
Contents
9.10 Electron-Energy Distribution 325 Why the electrons in a metal do not contribute to its specific heat except at very high and very low temperatures
9.11 Dying Stars 327 What happens when a star runs out of fuel
CHAPTER 10
The Solid State 335 10.1 Crystalline and Amorphous Solids 336 Long-range and short-range order 10.2 Ionic Crystals 338 The attraction of opposites can produce a stable union 10.3 Covalent Crystals 342 Shared electrons lead to the strongest bonds 10.4 Van der Waals Bond = 345 Weak but everywhere 10.5 Metallic Bond 348 ; A gas of free electrons is responsible for the characteristic properties of a metal 10.6 Band Theory of Solids 354 The energy band structure of a solid determines whether it is a conductor, an insulator, or a semiconductor 10.7 Semiconductor Devices 361 The properties of the p-n junction are responsible for the microelectronics industry 10.8 Energy Bands: Alternative Analysis 369 How the periodicity of a crystal lattice leads to allowed and forbidden bands 10.9 Superconductivity 376 No resistance at all, but only at very low temperatures (Go far) 10.10 Bound Electron Pairs 381 The key to superconductivity
CHAPTER 11 Nuclear Structure 387 11.1 - Nuclear Composition 388. Atomic nuclei of the same element have the same numbers of protons but can have different numbers of neutrons
Contents
Ta
Some Nuclear Properties 392
Small in size, a nucleus may have angular momentum and a magnetic moment
Stable Nuclei 396
Why some combinations of neutrons and protons are more stable than others -
Binding Energy 399
The missing energy that keeps a nucleus together
Liquid-Drop Model 403
A simple explanation for the binding-energy curve
Shell Model 408
Magic numbers in the nucleus
Meson Theory of Nuclear Forces 412
Particle exchange can produce either attraction or repulsion
CHAPTER 12 Nuclear Transformations 418
12.1
12.2
12.3
12.4
Radioactive Decay 419
Five kinds
Half-Life 424
Less and less, but always some left
Radioactive Series 430
Four decay sequences that each end in a stable daughter Alpha Decay 432
Impossible in classical physics, it nevertheless occurs Beta Decay 436
Why the neutrino should exist and how it was discovered Gamma Decay 440
Like an excited atom, an excited nucleus can emit a photon Cross Section 441
A measure of the likelihood of a particular interaction Nuclear Reactions 446
In many cases, a compound nucleus is formed first Nuclear. Fission 450
Divide and conquer
Nuclear Reactors 454
Eq = me* + $$$
Contents
xi
12.11 Nuclear Fusion in Stars 460 How the sun and stars get their energy 12.12 Fusion Reactors 463 The energy source of the future? APPENDIX: Theory of Alpha Decay 468
CHAPTER 13 Elementary Particles 474 13.1 Interactions and Particles 475 Which affects which 13.2 Leptons 477 Three pairs of truly elementary particles 13.3. Hadrons 481 Particles subject to the strong interaction 13.4 Elementary Particle Quantum Numbers 485 Finding order in apparent chaos 13.5 Quarks 489 The ultimate constituents of hadrons 13.6 Field Bosons 494 - Carriers of the interactions 13.7 The Standard Model and Beyond 496 Putting it all together 13.8 History of the Universe 498 It began with a bang 13.9 The Future 501 “In my beginning is my end.” (T. S. Eliot, Four Quartets)
APPENDIX Atomic Masses 507
Answers to Odd-Numbered Exercises’ 516 For Further Study 525
Credits 529
Index 531
Preface
quantization in blackbody radiation, a revolutionary idea soon followed by.
Albert Einstein's equally revolutionary theory of relativity and quantum the- ory of light. Students today must wonder why the label “modern” remains attached to this branch of physics. Yet it is not really all that venerable: my father was born: in 1900, for instance, and when I was leaming modem physics most of its founders, in- cluding Einstein, were still alive; | even had the privilege of meeting a number of them, :. including Heisenberg, Pauli, and Dirac. Few aspects of contemporary science—indeed, |: of contemporary life—are unaffected by the insights into matter and energy provided by modern physics, which continues as an active discipline as it enters its second © century. :
This book is intended to be used with a one-semester course in modern physics for 3: students who have already had ‘basic physics and calculus courses. Relativity and.” quantum ideas are considered first to provide a framework for understanding the - physics of atoms and nuclei. The theory of the atom is then developed with emphasis on quantum-mechanical notions. Next comes a discussion of the properties of aggre- .” gates of atoms, which includes a look at statistical mechanics. Finally atomic nuclei and elementary particles are examined. . ;
The balance in this book leans more toward ideas than toward experimental meth- - ods and practical applications, because I believe that the beginning student is better served by a conceptual framework than by a mass of details. For a similar reason the : sequence of topics follows a logical rather than strictly historical order. The merits of this approach have led to the extensive worldwide use of the five previous editions of Concepts of Modern Physics, including translations into a number of other languages, . since the first edition appeared nearly forty years ago. ;
Wherever possible, important subjects are introduced on an elementary level, which s enables even relatively unprepared students to understand what is going on from the». start and also encourages the development of physical intuition in readers in whom - the mathematics (rather modest) inspires no terror. More material is included than can”: easily be covered in one semester. Both factors give scope to an instructor to fashion the type of course desired, whether a general survey, a deeper inquiry into selected subjects, or a combination of both.
Like the text, the exercises are on all levels, from the quite easy (for practice and *:: reassurance) to those for which real thought is needed (for the joy of discovery), The: exercises are grouped to correspond to sections of the text with answers to the odd- © numbered exercises given at the back of the book. In addition, a Student Solutions Manual has been prepared by Craig Watkins that contains solutions to the odd- numbered exercises. :
Because the ideas of modern physics represented totally new directions in thought : when first proposed, rather than extensions of previous knowledge, the story of their development is exceptionally interesting. Although there is no room here for a full ac-. count, bits and pieces are included where appropriate, and thirty-nine brief -biogra: phies of important contributors are sprinkled through the text to help provide a hu- man persepctive: Many books on the history of modem physics are available for those
M odern physics began in 1900 with Max Planck’ discovery of the role of energy Skee
Preface
xiii
who wish to go further into this subject; those by Abraham Pais and by Emilio Segré, themselves distinguished physicists, are especially recommended.
For this edition of Concepts of Modern Physics the treatments of special relativity, quantum mechanics, and elementary particles received major revisions. In addition, numerous smaller changes and updates were made throughout the book, and several new topics were added, for instance Einstein's derivation of the Planck radiation law. There is more material on aspects of astrophysics that nicely illustrate important ele- ments of modem physics, which for this reason are discussed where relevant in the text rather than being concentrated in a single chapter.
Many students, although able to follow the arguments in the book, nevertheless may have trouble putting their knowledge to use. To help them, each chapter has a selec- tion of worked examples. Together with those in the Solutions Manual, over 350 solu- tions are thus available to problems that span all levels of difficulty. Understanding these solutions should bring the unsolved even-numbered exercises within reach.
In revising Concepts of Modern Physics for the sixth edition I have had the benefit of constructive criticism from the following reviewers, whose generous assistance was of great value: Steven Adams, Widener University; Amitava Bhattacharjee, The Univer- sity of Iowa; William E, Dieterle, California University of Pennsylvania; Nevin D. Gibson, Denison University; Asif Khand Ker, Millsaps College; Teresa Larkin-Hein, American University; Jorge A. Lopez, University of Texas at El Paso; Carl A. Rotter, West Virginia University; and Daniel Susan, Texas AGM University—Kingsville, 1 am also grateful to the following reviewers of previous editions for their critical reviews and comments: Donald R. Beck, Michigan Technological University; Ronald J. Bieniek, University of Missouri-Rolla; Lynn R. Cominsky, Sonoma State University; Brent Comstubble, United States Military Académy; Richard Gass, University of Cincinnati; Nicole Herbot, Arizona State Univer- sity; Vladimir Privman, Clarkson University; Arnold Strassenberg, State University of New York-Stony Brook; the students at Clarkson and Arizona State Universities who evaluated an earlier edition from their point of view; and Paul Sokol of Pennsylvania State Uni- versity who supplied a number of excellent exercises. I am especially indebted to Craig Watkins of Massachusetts Institute of Technology who went over the manuscript with a meticulous and skeptical eye and who checked the answers to all the exercises. Finally, I want to thank my friends at McGraw-Hill for their skilled and enthusiastic help throughout the project.
Arthur Beiser
contents of Modern ee
Sixth Edition
VW
1.2
1.3
1.4
1.5
1.6
_ CHAPTER 1
According to the theory of relativity, nothing can travel faster than light. Although today’s spacecraft can exceed 10 km/s, they are far from this ultimate speed limit,
SPECIAL RELATIVITY All motion is relative; the speed of light in free space is the same for all observers TIME DILATION A moving clock ticks more slowly than a clock at rest DOPPLER EFFECT Why the universe is believed to be expanding LENGTH CONTRACTION Faster means shorter TWIN PARADOX A longer life, but it will not seem longer ELECTRICITY AND MAGNETISM Relativity is the bridge
1.7 RELATIVISTIC MOMENTUM
Redefining an important quantity 1.8 MASS AND ENERGY
Where Eo = mc? comes from 1.8. ENERGY AND MOMENTUM
How they fit together in relativity 1.10 GENERAL RELATIVITY
Gravity is a warping of spacetime APPENDIX I: THE LORENTZ
TRANSFORMATION
APPENDIX II: SPACETIME
Chapter One
urements of time and space are affected by motion between an observer and what
is being observed. To say that Einstein’s theory of relativity revolutionized science is no exaggeration. Relativity connects space and time, matter and energy, electricity and magnetism—tinks that are crucial to our understanding of the physical universe. From relativity have come a host of remarkable predictions, all of which have been confirmed by experiment. For all their profundity, many of the conclusions of relativity can be reached with only the simplest of mathematics.
I n 1905 a young physicist of twenty-six named Albert Einstein showed how meas-
4.1. SPECIAL RELATIVITY
All motion is relative; the speed of light in free space is the same for all observers
When such quantities as length, time interval, and mass are considered in elementary physics, no special point is made about how they are measured. Since a standard unit exists for each quantity, who makes a certain determination would not seem to matter— everybody ought to get the same result. For instance, there is no question of principle involved in finding the length of an airplane when we are on board. All we have to do is put one end of a tape measure at the airplane's nose and look at the number on the tape at the airplane’ tail.
But what if the airplane is in flight and we are on the ground? It is not hard to de- termine the length of a distant object with a tape measure to establish a baseline, a surveyors transit to measure angles, and a knowledge of trigonometry. When we meas- ure the moving airplane from the ground, though, we find it to be shorter than it is to somebody in the airplane itself. To understand how this unexpected difference arises we must analyze the process of measurement when motion is involved.
Frames of Reference
The first step is to clarify what we mean by motion. When we say that something is moving, what we mean is that its position relative to something else is changing. A passenger moves relative to an airplane; the airplane moves relative to the earth; the earth moves relative to the sun; the sun moves relative to the galaxy of stars (the Milky Way) of which it is a member; and so on. In each case a frame of reference is part of the description of the motion. To say that something is moving always implies a specific frame of reference.
‘An inertial frame of reference is one in which Newton’ first law of motion holds. In such a frame, an object at rest remains at rest and an object in motion continues to move at constant velocity (constant speed and direction) if no force acts on it. Any frame of reference that moves at constant velocity relative to an inertial frame is itself an inertial frame.
All inertial frames are equally valid. Suppose we see something changing its posi- tion with respect to us at constant velocity. Is it moving or are we moving? Suppose we are in a closed laboratory in which Newton’ first law holds. Is the laboratory mov- ing or is it at rest? These questions are meaningless because all constant-velocity motion is relative. There is no universal frame of reference that can be used everywhere, no such thing as “absolute motion.”
The theory of relativity deals with the consequences of the lack ofa universal frame of reference. Special relativity, which is what Einstein published in 1905, treats
Relativity
problems that involve inertial frames of reference. General relativity, published by Einstein a decade later, describes the relationship between gravity and the geometrical structure of space and time. The special theory has had an enormous impact on much of physics, and we shall concentrate on it here.
Postulates of Special Relativity ‘Two postulates underlie special relativity. The first, the principle of relativity, states: The laws of physics are the same in all inertial frames of reference.
This postulate follows from the absence of a universal frame of reference. If the laws of physics were different for different observers in relative motion, the observers could find from these differences which of them were “stationary” in space and which were “moving.” But such a distinction does not exist, and the principle of relativity expresses this fact.
The second postulate is based on the results of many experiments:
The speed of light in free space has the same value in all inertial frames of reference.
This speed is 2.998 X 10° m/s to four significant figures,
‘To appreciate how remarkable these postulates are, let us look at a hypothetical experiinent basically no different from actual ones that have been carried out in a number of ways. Suppose I turn on a searchlight just as you fly past in a spacecraft at a speed of 2 X 10° m/s (Fig. 1.1). We both measure the speed of the light waves from the searchlight using identical instruments. From the ground 1 find their speed to be 3 X 10° m/s as usual. “Common sense” tells me that you ought to find a speed of (3 — 2) X 10° més, or only 1 X 108 més, for the same light waves. But you also find their speed to be 3 X 10° m/s, even though to me you seem to be moving parallel to the waves at 2 X 10° més.
FY c= 3 X108 mis
(a) (b) ©
Figure 1.1 The speed of light is the same to all observers.
4 Chapter One
eee eee
Albert A. Michelson (1852-1931) was born in Germany but came to the United States at the age of two with his parents, who settled in Nevada. He attended the U.S. Naval Academy at Annapolis where, after two years of sea duty, He became a science instructor. To improve his knowledge of optics, in which he wanted to specialize, Michelson went to Europe and stud- ied in Berlin and Paris. Then he left the Navy to work first at the Case School of Applied Science in Ohio, then at Clark University in Massachusetts, and finally at the University of Chicago, where he headed the physics de- partment from 1892 to 1929. Michelson’s speciality was high- precision measurement, and for many decades his successive figures for the speed of light were the best available. He rede- fined the meter in terms of wavelengths of a particular spectral line and devised an interferometer that could determine the diameter of a star (stars ‘appear as points of light in even the most powerful telescopes).
Michelson’s most significant achievement, carried out in 1887 in collaboration with Edward Morley, was an experiment ‘to measure the motion of the earth through the “ether,” a hy- pothetical medium pervading the universe in which light waves
were supposed to occur. The notion of the ether was a hang- :
over from the days before light waves were recognized as elec-
tromagnetic, but nobody at the time seemed willing to discard
the idea that light propagates relative to some sort of universal _ frame of reference.
Path A
Parallel light from F single source
Half-silvered mirror
To look for the earth’s motion through the ether, Michelson and Morley used a pair of light bearns formed by a half-silvered mirror, as in Fig. 1.2. One light beam is directed to a mirror along a path perpendicular to the ether current, and the other goes to a mirror along a path parallel to the ether current. Both beams end up at the same viewing screen. The clear glass plate ensures that both beams pass through the same thicknesses of air and glass. If the transit times of the two beams are the same, they will arrive at the screen in phase and will interfere con- structively. An ether current due to the earth’s motion parallel to one of the beams, however, would cause the beams to have different transit times and’ the result would be destructive in- terference at the screen. This is the essence of the experiment.
Although the experiment was sensitive enough to detect the expected ether drift, to everyone's surprise none was found. The negative result had two consequences. First, it showed that the ether does not exist and so there is no such thing as “ab- solute motion” relative to the ether: all motion is relative to a specified frame of reference, not to a universal one. Second, the result showed that the speed of light is the same for all ob- servers, which is not true of waves that need a material medium. in which to occur (such as sound and water waves).
The Michelson-Morley experiment set the stage for Einstein's 1905 special theory of relativity, a theory that Michelson him- self was reluctant to accept. Indeed, not long before the flow- ering of relativity and quantum theory revolutionized physics, Michelson announced that “physical discoveries in the future are a matter of the sixth decimal place.” This was a common opinion of the time. Michelson received a Nobel Prize in 1907, the first American to do so.
Mirror A
Glass plate
Path B = Mirror B
y, Hypothetical +, ether current
= - ee —— Viewing screen
Figure 1.2 The Michelson-Morley-experiment.
Relativity
There is only one way to account for these results without violating the principle of relativity. It must be true that measurements of space and time are not absolute but de- pend on the relative motion between an observer and what is being observed. If I were to measure from the ground the rate at which your clock ticks and the length of your meter stick, I would find that the clock ticks more slowly than it did at rest on the ground and that the meter stick is shorter in the direction of motion of the spacecraft. To you, your clock and meter stick are the same as they were on the ground before you took off, To me they are different because of the relative motion, different in such a way that the speed of light you measure is the same 3 X 10° m/s I measure. Time intervals and lengths are relative quantities, but the speed of light in free space is the same to all observers.
Before Einstein’ work, a conflict had existed between the principles of mechanics, which were then based on Newton's laws of motion, and those of electricity and magnetism, which had been developed into a unified theory by Maxwell. Newtonian mechanics had worked well for over two centuries. Maxwell's theory not only covered all that was then known about electric and magnetic phenomena but had also pre- dicted that electromagnetic waves exist and identified light as an example of them. However, the equations of Newtonian mechanics and those of electromagnetism differ in the way they relate measurements made in one inertial frame with those made ina different inertial frame.
Einstein showed that Maxwells theory is consistent with special relativity whereas Newtonian mechanics is not, and his modification of mechanics brought these branches of physics into accord. As we will find, relativistic and Newtonian mechanics agree for relative speeds much lower than the spéed of light, which is why Newtonian mechanics seemed correct for so long. At higher speeds Newtonian mechanics fails and must be replaced by the relativistic version.
1.2 TIME DILATION
A moving clock ticks more slowly than a clock at rest
Measurements of time intervals ate affected by relative motion between an observer and what is observed. As a result, a clock that moves with respect to an observer ticks more slowly than it does without such motion, and all processes (including those of life) occur more slowly to an observer whien they take place in a different inertial frame.
If someone in a moving spacecraft finds that the time interval between two events in the spacecraft is to, we on the ground would find that the same interval has the longer duration t, The quantity to, which is determined by events that occur at the same place in an observer's frame of reference, is called the proper time of the interval between the events. When witnessed from the ground, the events that mark the be- ginning and end of the time interval occur at different places, and in consequence the duration of the interval appears longer than the proper time. This effect is called time dilation (to dilate is to become larger).
To see how time dilation comes about, let us consider two clocks, both of the par- ticularly simple kind shown in Fig. 1.3. In each clock a puise of light is reflected back and forth between two mirrors Lo apart. Whenever the light strikes the lower mirror, an electric signal is produced that marks the recording tape. Each mark corresponds to the tick of an ordinary clock.
One clock is at rest in a laboratory on the ground and the other is in a spacecraft that moves at the speed v relative to the ground. An observer in the laboratory watches both clocks: does she find that they tick at the same rate?
6 Chapter One
nc! hc
Recording device
Meter stick
Photosensitive surface
Figure 1.3 A simple clock. Each “tick” corresponds to a round trip of the light pulse from the lower mirror to the upper one and back.
Figure 1.4 shows the laboratory clock in operation. The time interval between ticks is the proper time to and the time needed for the light pulse to travel between the mirrors at the speed of light ¢ is to/2. Hence to/2 = Lo/c and
= (LD
Figure 1.5 shows the moving clock with its mirrors perpendicular to the direction of motion relative to the ground. The time interval between ticks is t. Because the clock is moving, the light pulse, as seen from the ground, follows a zigzag path. On its way from the lower mirror to the upper one in the time ¢/2, the pulse travels a horizontal distance of u(t/2) and a total distance of c(t/2). Since Lo is the vertical distance between the mirrors,
Cc) Bes I a\2 vt \2 =o ()-a(5) Figure 1.4 A light-pulse clock at 2 2 rest on the ground as seen by an 2 observer on the ground, The dial f @-vw=B Tepresents a conventional clock on 4 the ground. a (2ly) Cow eave’) 2Lp/c es 27-2 (1.2) 1%
~ But 2Lo/c is the time interval tg between ticks on the clock on the ground, as in Eq. (1.1), and so
Relativity
t
=| Ss) ' v cy Ly > yy ee ‘
t ooo Figure 1.5 A light-pulse clock in a spacecraft as seen by an observer on. the ground, The mirrors are paraltel to the direction of motion of the spacecraft. The dial Tepresents a conventional clock on the ground. = fo
Time dilation t= Vi-ve- (1,3)
7
. Here is a reminder of what the symbols in Eq. (1.4) represent:
to = time interval on clock at rest relative to an observer = proper time t = time interval on clock in motion relative to an observer
uv = speed of relative motion
c = speed of light
Because the quantity ‘V1 ~ v’/c? is always smaller than 1 for a moving object, t is always greater than f. The moving clock in the spacecraft appears to tick at a slower rate than the stationary one on the ground, as seen by an observer on the ground,
Exactly the same analysis holds for measurements of the clock on the ground by the pilot of the spacecraft. To him, the light pulse of the ground clock follows a zigzag path that requires a total time ¢ per round trip. His own clock, at rest in the spacecraft, ticks at intervals of to. He too finds that
to V1 — uf?
so the effect is reciprocal: every observer finds that clocks in motion relative to him tick more slowly than clocks at rest relative to him.
Our discussion has been based on a somewhat unusual clock. Do the same conclusions apply to ordinary clocks that use machinery—spring-controlled escapements, tuning forks, vibrating quartz crystals, or whatever—to produce ticks at constant time intervals?
" The answer must be yes, since if a mirror clock and a conventional clock in the space- craft agree with each other on the ground but not when in flight, the disagreement between then could be used to find the speed of the spacecraft independently of any outside frame of reference—which contradicts the principle that all motion is relative,
t=
Chapter One
The Ultimate Speed Limit
T he earth and the other planets of the solar system seem to be natural products of the evolu- tion of the sun. Since the sun is a rather ordinary star in other ways, it is not surprising that other stars have been found to have planetary systems around them as well. Life developed here on earth, and there is no known reason why it should not also have done so on some of these planets. Can we expect ever to be able to visit them and meet our fellow citizens of the universe? The trouble is that nearly all stars are very far away—thousands or millions of light-years away. (A light-year, the distance light travels in a year, is 9.46 X 10° m.) But if we can build a spacecraft whose speed is thousands or millions of times greater than the speed of light c, such distances would not be an obstacle.
Alas, a simple argument based on Einstein’s postulates shows that nothing can move faster than c. Suppose you ate in a spacecraft traveling at a constant speed v relative to the earth that is greater than c. As I watch from the earth, the lamps in the spacecraft suddenly go out, You switch on a flashlight to find the fuse box at the front of the spacecraft and change the blown fuse (Fig. 1.6a). The lamps go on again.
From the ground, though, 1 would see something quite different. To me, since your speed v is greater than c, the light from your flashlight illuminates the back of the spacecraft (Fig. 1.6b). I can only conclude that the laws of physics are different in your inertial frame from what they are in my inertial frame—which contradicts the principle of relativity. The only way to avoid this contradiction is to assume that nothing can move faster than the speed of light. This as- sumption has been tested experimentally many times and has always been found to be correct.
The speed of light c in relativity is always its value in free space of 3.00 x 10° més. In all ma- terial media, such as air, water, or glass, light travels more slowly than this, and atomic particles are able to move faster in such media than does light. When an electrically charged particle moves through a transparent substance at a speed exceeding that of light in the substance, a cone of light waves is emitted that corresponds to the bow wave produced by a ship moving through the water faster than water waves do. These light waves are known as Cerenkov radiation and form the basis of a method of determining the speeds of such particles. The minimum speed a particle must have to emit Cerenkov radiation is c/n in a medium whose index of refraction is n. Cerenkov ra- diation is visible as a bluish glow when an intense beam of particles is involved. fk 4
@ ®)
Figure 1.6 A person switches on a flashlight in a spacecraft assumed to be moving relative to the earth faster than light. (a) In the spacecraft frame, the light goes to the front of the spacecraft. (b) In the earth frame, the light goes to the back of the spacecraft. Because observers in the spacecraft and on the earth would see different events, the principle of relativity would be violated. The conclusion is that the spacecraft cannot be moving faster than light relative to the earth (or relative to anything else).
Relativity 9
LT RAL CE LEA eS tenner rrr wenenresernteeemninereerenees
Albert Einstein (1879-1955), bitterly unhappy with the rigid discipline of the schools of his native Germany, went at sixteen to Switzerland to com- plete his education, and later got a job examining patent applications at the Swiss Patent Office. Then, in 1905, ideas that had been germinating in his mind for years when he snould have been paying attention to other matters (one of his math teachers called Einstein a “lazy dog”) blossomed into three short papers that were to change decisively the course not only of physics but of modern civilization as well.
The first paper, on the photoelectric effect, proposed that light has a dual character with both particle and wave properties. The subject of the second paper was Brownian motion, the irregular zigzag movement of tiny bits of suspended matter, such as pollen grains in water, Einstein showed that Brownian motion results from the bombardment of the particles by randomly moving mol- ecules in the fluid in which they are suspended. This provided the long-awaited definite link with experiment that convinced the remaining doubters of the molecular theory of matter. The third paper introduced the special theory of relativity.
Although much of the world of physics was originally either indifferent or skeptical, even the most unexpected of Einstein’ conclusions were soon confirmed and the development of what is now called modem physics began in earnest. After university posts in Switzerland and Czechoslovakia, in 1913 he took up an
{AIP Niels Bohr Library)
appointment at the Kaiser Wilhelm Institute in Berlin that left him able to do research free of financial worries and routine duties. Einstein’ interest was now mainly in gravitation, and he started where Newton had left off more than two centuries earlier.
Einstein’ general theory of relativity, published in 1916, re- lated gravity to the structure of space and time. In this theory the force of gravity can be thought of as arising from a warp- ing of spacetime around a body of matter so that a nearby mass tends to move tbward it, much as a marble rolls toward the bot- tom of a saucer-shaped hole. From general relativity came a number of temarkable predictions, such as that light should be subject to gravity, all of which were verified experimentally, The later discovery that the universe is expanding fit neatly into the theory. In 1917 Einstein introduced the idea of stimulated emis- sion of radiation, an idea that bore fruit forty years later in the invention of the laser.
The development of quantum mechanics in the 1920s dis- turbed Einstein, who never accepted its probabilistic rather than deterministic view of events on an atomic scale. “God does not play dice with the world,” he said, but for once his physical in- tuition seemed to be leading him in the wrong direction.
Einstein, by now a world celebrity, left Germany in 1933 af- ter Hitler came to power and spent the rest of his life at the In- stitute for Advanced Study in Princeton, New Jersey, thereby
“escaping the fate of millions of other European Jews at the hands
of the Germans. His last years were spent in an unsuccessful search for a theory that would bring gravitation and electro- magnetism together into a single picture, a problem worthy of his gifts but one that remains unsolved to this day.
eter niente sereetetrnmnarenunaviserriery erent eeeeneynershenreneerewn emer
Example 1.1
A spacecraft is moving relative to the earth. An observer on the earth finds that, between 1 p.m. and 2 p.m. according to her clock, 3601 s elapse on the spacecraft’s clock. What is the space-
crafts speed relative to the earth?
Solution
Here to = 3600 s is the proper time interval on the earth and t = 3601 s is the time interval in the moving frame as measured from the earth. We proceed as follows:
to .
v ty \? ree)
t=
Poe ee cae _ 736005 2 vee fi (=) (2.998 x 108 mis) _/1 (cars
=7.1X 10° ms Todays spacecraft are much slower than this. For instance, the highest speed of the Apollo 11 Space- craft that went to the moon was only 10,840 m/s, and its clocks differed from those on the earth by less than one part in 10°. Most of the experiments that have confirmed time dilation made use of unstable nuclei and elementary particles which readily attain speeds not far from that of light.
eon,
Apollo 11 lifts off its pad to begin the first human visit to the moon. At its highest speed of 10.8 km/s relative to the earth, its clocks differed from those on the earth by less than one part in a billion.
Although time is a relative quantity, not all the notions of time formed by every- day experience are incorrect. Time does not run backward to any observer, for in- stance. A sequence of events that occur at some particular point at ti, t2, t3,.-- will appear in the same order to all observers everywhere, though not necessarily with the same time intervals t) — t), fs — tz, . . - between each pair of events. Similarly, no distant observer, regardless of his or her state of motion, can see an event before it happens—more precisely, before a nearby observer sees it—since the speed of light is finite and signals require the minimum period of time L/c to travel a distance L. There is no way to peer into the future, although past events may appear different to different observers.
1.3. DOPPLER EFFECT
Why the universe is believed to be expanding
We are all familiar with the increase in pitch of a sound when its source approaches us (or we approach the source) and the decrease in pitch when the source recedes from us (or we recede from the source). These changes in frequency constitute the doppler effect, whose origin is straightforward. For instance, successive waves emitted by a source moving toward an observer are closer together than normal because of the advance of the source; because the separation of the waves is the wavelength of the sound, the corresponding frequency is higher. The relationship between the source frequency Vp and the observed frequency is
Relativity 11 Doppler effect in ‘s l+ufc sound ame oes Vic i
where c = speed of sound v = speed of observer (+ for motion toward the source, — for motion away from it) V = speed of the source (+ for motion toward the observer, — for motion away from him)
If the observer is stationary, v = 0, and if the source is stationary, V = 0.
The doppler effect in sound varies depending on whether the source, or the observer, or both are moving. This appears to violate the principle of relativity: all that should count is the relative motion of source and observer. But sound waves occur only ina material medium such as air or water, and this medium is itself a frame of reference with respect to which motions of source and observer are measurable. Hence there is no contradiction. In the case of light, however, no medium is involved and only rela- tive motion of source and observer is meaningful. The doppler effect in light must therefore differ from that in sound.
‘We can analyze the doppler effect in light by considering a light source as a clock that ticks % times per second and emits a wave of light with each tick. We will examine the three situations shown in Fig. 1.7.
1 Observer moving perpendicular to a line between him and the light source. The proper time between _ti is fo = 1/¥p, so between one tick and the next the time t = to/V'1 ~ v7/c? elapses in the reference frame of the observer. The frequency he
finds is accordingly o. l_ Vi-v?
v(transverse) = — = t to
Transverse doppler effect. v= V1 —v? fc? (1.5) in light
The observed frequency » is always lower than the source frequency vo.
2 Observer receding from the light source. Now the observer travels the distance vt away from the source between ticks, which means that the light wave from a given tick takes
Observer
v ( fo: (fees @ Q) G)
Figure 1.7 The frequency of the light seen by an observer depends on the direction and speed of the observer's motion relative to its source. r
12
Chapter One
ut/c longer to reach him than the previous one. Hence the total time between the arrival of successive waves is
mre ony l+vfc = pete tole 2, [i tere nae Vive °VIF U/eV1 ~ vc °V 1 — ufc
and the observed frequency is
v(receding) = ; = l aE ule os Vo fs = vie (1.6)
to Vl t+ u/c 1+ u/c
The observed frequency v is lower than the source frequency vo. Unlike the case of sound waves, which propagate relative to a material medium it makes no difference whether the observer is moving away from the source or the source is moving away from the observer.
3 Observer approaching the light source? The observer here travels the distance vt toward the source between ticks, so each light wave takes ut/c le%& time to arrive than the previous one. In this case T = t ~ ut/c and the result is
+ v(approaching) = vo, | ties (1.7)
Spectra of the double star Mizar, which consists of two stars that circle their center of mass, taken 2 days apart. In a the stars are in line with no motion toward or away from the earth, so their spectral lines are superimposed. In b one star is moving toward the earth and the other is mov~ ing away from the earth, so the spectral lines of the former are doppler-shifted toward the blue end of the spectrum and those of the latter are shifted toward the ted end.
The observed frequency is higher than the source frequency. Again, the same formula holds for motion of the source toward the observer. Equations (1.6) and (1.7) can be combined in the single formula
Longitudinal
doppler effect v=u.f fies (1.8) in light 1- vc
by adopting the.converition that vis + for source and observer approaching each other and — for source and observer receding from each other.
Relativity
13
en EESEEEERRR EERE ee Example 1.2
A driver is caught going through a red light. The driver claims to the judge that the color she actually saw was green (v = 5.60 X 104 Hz) and not red (vp = 4.80 X 10"* Hz) because of the doppler effect. The judge accepts this explanation and instead fines her for speeding at the rate of $1 for each km/h she exceeded the speed limit of 80 km/h. What was the fine?
Solution
Solving Eq. (1.8) for v gives
- {er %)\_ 8 (ex = oor] v ct a oa (3.00 X 10° m/s) 6.60? + 4.80)"
= 4.59 X 10’ m/s = 1.65 X 10° km/h
since 1 m/s = 3.6 km/h. The fine is therefore $(1.65 X 10° — 80) = $164,999,920. armrest nr ener enna rasnnsnerneerreeret
Visible light consists of electromagnetic waves in a frequency band to which the eye is sensitive. Other electromagnetic waves, such as those used in radar and in radio communications, also exhibit the doppler effect in accord with Eq. (1.8). Doppler shifts in radar waves are used by police to measure vehicle speeds, and doppler shifts in the radio waves emitted by a set of earth satellites formed the basis of the highly accurate ‘Transit system of marine navigation.
The Expanding Universe
The doppler effect in light is an important‘tool in astronomy, Stars emit light of cer- tain characteristic frequencies called spectral lines, and motion of a star toward or away from the earth shows up as a doppler shift in these frequencies. The spectral lines of distant galaxies of stars are all shifted toward the low-frequency (red) end of the spectrum and hence are called “red shifts.” Such shifts indicate that the galaxies are re- ceding from us and from one another. The speeds of recession are observed to be
Edwin Hubble (1889- At Mt. Wilson Observatory in California, Hubble made
1953) was bom in Missouri and, although always inter- ested in astronomy, pursued a variety of other subjects as well at the University of Chicago. He then went as a Rhodes Scholar to Oxford University in England where he concentrated on law, Spanish, and heavyweight boxing. After two years of teaching at an Indiana high school, Hubble realized what his true vocation was and returned to the University of Chicago to study astronomy.
the first accurate measurements of the distances of spiral galaxies which showed that they are far away in space from our own Milky Way galaxy. It had been known for some time that such galaxies have red shifts in their spectra that indi- cate motion away from the Milky Way, and Hubble joined his distance figures with the observed red shifts to conclude that the recession speeds were proportional to distance. This im- plies that the universe is expanding, a remarkable discovery that has led to the modern picture of the universe. Hubble was the first to use the 200-inch telescope, for many years the world’s largest, at Mt. Palomar in California, in 1949. In his later work Hubble tried to determine the structure of the universe by finding how the concentration of remote galax- ies varies with distance, a very difficult task that only today is being accomplished.
14 Chapter One
8X 10°
Recession speed, km/s
0 1 2 3 4X 10° Approximate distance, light-years
(a)
(b)
Figure 1.8 (a) Graph of recession speed versus distance for distant galaxies. The speed of recession averages about 21 km/s per million light-years. (b) Two-dimensional analogy of the expanding uni- verse. As the balloon is inflated, the spots on it become farther apart. A bug on the balloon would find that the farther away a spot is from its location, the faster the spot seems to be moving away; this is true no matter where the bug is. In the case of the universe, the more distant a galaxy is from us, the faster it is moving away, which means that the universe is expanding uniformly. :
proportional to distance, which suggests that the entire universe is expanding (Fig. 1.8). This proportionality is called Hubble’s law.
The expansion apparently began about 13 billion years ago when a very small, in- tensely hot mass of primeval matter exploded, an event usually called the Big Bang. As described in Chap. 13, the matter soon turned into the electrons, protons, and neu- trons of which the present universe is composed. Individual aggregates that formed during the expansion became the galaxies of today. Present data suggest that the current expansion will continue forever.
gS Example 1.3
A distant galaxy in the constellation Hydra is receding from the earth at 6.12 10’ ms. By how much is a green spectral line of wavelength 500 nm (1 nm = 107° m) emitted by this galaxy shifted toward the red end of the spectrum?
Relativit
Solution
Since A = ¢/v and Ap = c/ rp, from Eq. (1.6) we have
l+ufe A=Ao l-uf Here v = 0,204c and Ag = 500 nm, so 1 + 0.204 A = 500 nm 10204 = 615 nm
which is in the orange part of the spectrum. The shift isA — Ag = 115 nm, This galaxy is believed to be 2.9 billion light-years away,
1.4 LENGTH CONTRACTION
Faster means shorter
Measurements of lengths as well as of time intervals are affected by relative motion, The length L of an object in motion with respect to an observer always appears to the observer to be shorter than its length Lo when it is at test with respect to him. This contraction occurs only in the direction of the relative motion. The length Lo of an object in its rest frame is called its proper length. (We note that in Fig. 1.5 the clock is moving perpendicular to v, hence L = Lp there.)
The length contraction can be derived in a number of ways. Perhaps the simplest is based on time dilation and the principle 6f relativity. Let us consider what happens to unstable particles called muons that are created at high altitudes by fast cosmic-ray particles (largely protons) from space when they collide with atomic nuclei in the earth’s atmosphere. A muon has a mass 207 times that of the electron and has a charge of either +e or —e; it decays into an electron or a positron after an average lifetime of 2.2 ys (2.2 X 1076s).
Cosmic-ray muons have speeds of about 2.994 X 108 m/s (0.998c) and reach sea level in profusion—one of them passes through each square centimeter of the earth’s surface on the average slightly more often than once a minute. But in to = 2.2 ps, their average lifetime, muons can travel a distance of only
Ulo = (2.994 X 10° m/s)(2.2 X 1076s) = 6.6 X 10? m = 0.66 kin
before decaying, whereas they are actually created at altitudes of 6 km or more.
To resolve the paradox, we note that the muon lifetime of to = 2.2 yas is what an observer at rest with respect to a muon would find. Because the muons are hurtling toward us at the considerable speed of 0.998c, their lifetimes are extended in our frame of reference by time dilation to
to _ 2.2 X 1078s Vi-v¥2 Vi = 0.998042
34.8 X 107° s = 34.8 ys
t=
The moving muons have lifetimes almost 16 times longer than those at rest. In a time interval of 34.8 js, a muon whose speed is 0,998c can cover the distance
ut = (2.994 x 108 m/s)(34.8 X 107° s) = 1.04 X 10*m = 10.4 km
Chapter One
As found by observer As found by an observer
on the ground, the moving with the muon, the
muon altitude isLp. ground is L below it, which is a shorter distance than Lo.
Figure 1.9 Muon decay as seen by diflerent observers. The muon size is greatly exaggerated here; in fact, the muon seems likely to be a point particle with no extension in space.
Although its lifetime is only to = 2.2 ps in its own frame of reference, a muon can reach the ground from altitudes of as much as 10.4 km because in the frame in which these altitudes are measured, the muon lifetime is t = 34.8 ps.
What if somebody were to accompany a muon in its descent at v = 0.998c, so that to him or her the muon is at rest? The observer and the muon are now in the same frame of reference, and in this frame the muon’ lifetime is only 2.2 ys. To the observer, the muon can travel only 0.66 km before decaying. The only way to account for the arrival of the muon at ground level is if the distance it travels, from the point of view of an observer in the moving frame, is shortened by virtue of its motion (Fig. 1.9). The principle of relativity tells us the extent of the shortening—it must be by the same factor of Wt — v?/c2 that the muon lifetime is extended from the point of view ofa . stationary observer.
We therefore conclude that an altitude we on the ground find to be hp must appear in the muon’ frame of reference as the lower altitude
h= ho V1 — ve
In our frame of reference the muon can travel ho = 10.4 km because of time dilation. In the muon’s frame of reference, where there is no time dilation, this distance is abbreviated to
Relativit 17 a rr 7 2
0 z Sea E 0.001 0.01 0.1 10 vie
Figure 1.10 Relativistic length contraction. Only lengths in the direction of motion are affected. The horizontal scale is logarithmic.
h = (10.4 km) V1 — (0.9980)? = 0.66 km
As we know, a muon traveling at 0.998c goes this far in 2.2 BS. The relativistic shortening of distances is an example of the general contraction of lengths in the direction of motion:
Length L=LhVi-we (1.9)
contraction
Figure 1.10 is a graph of L/Lo versus u/c. Clearly the length contraction is most
significant at speeds near that of light. A speed of 1000 km/s seems fast to us, but it
only results in a shortening in the direction of motion to 99.9994 percent of the proper
length of an objéct moving at this speed. On the other hand, something traveling at
nine-tenths the speed of light is shortened to 44 percent of its proper length, a " significant change. ;
Like time dilation, the length contraction is a teciprocal effect. To a person in a spacecraft, objects on the earth appear shorter than they did when he or she was on the ground by the same factor of V1 ~ v?/c that the spacecraft appears shorter to somebody at rest. The proper length Lp found in the rest frame is the maximum length any observer will measure. As mentioned earlier, only lengths in the direction of motion undergo contraction. Thus to an outside observer a spacecraft is shorter in flight than on the ground, but it is not narrower. .
1.5 TWIN PARADOX A longer life, but it will not seem longer
We are now in a position to understand the famous telativistic effect known as the twin paradox. This patadox involves two identical clocks, one of which remains on the earth while the other is taken on a voyage into space 2 at the speed v and eventu- ally is brought back. It is customary to replace the clocks with the pair of twins Dick and
18 Chapter One
Jane, a substitution that is perfectly acceptable because the processes of life—heartbeats, respiration, and so on—constitute biological clocks of reasonable regularity.
Dick is 20 y old when he takes off on a space voyage at a speed of 0.80c to a star 20 light-years away. To Jane, who stays behind, the pace of Dick’ life is slower than hers by a factor of
V1 w/e = V1 — 0800/2 = 0.60 = 60%
To Jane, Dick’ heart beats only 3 times for every 5 beats of her heart; Dick takes only 3 breaths for every 5 of hers; Dick thinks only 3 thoughts for every 5 of hers. Finally Dick returns after 50 years have gone by according to Janes calendar, but to Dick the trip has taken only 30 y. Dick is therefore 50 y old whereas Jane, the twin whe stayed home, is 70 y old (Fig. 1.11).
Where is the paradox? If we consider the situation from the point of view ‘of Dick in the spacecraft, Jane on the earth is in motion relative to him at a speed of 0.80c. Should not Jane then be 50 y old when the spacecraft returns, while Dick is then 70—the precise opposite of what was concluded above?
But the two situations are not equivalent. Dick changed from one inertial frame to a different one when he started out, when he reversed direction to head home, and when he landed on the earth. Jane, however, remained in the same inertial frame dur- ing Dick’s whole voyage. The time dilation formula applies to Jane’s observations of Dick, but not to Dick’s observations of her.
To look at Dick's voyage from his perspective, we must take into account that the distance L he covers is shortened to
L= Ip V1 — v¥/e = (20 light-years) ‘V1 — (0.80¢)"/c? = 12 light-years
To Dick, time goes by at the usual rate, but his voyage to the star has taken L/v = 15 y and his return voyage another 15 y, for a total of 30 y. Of course, Dick’ life span has
een, 6
Q
SZ yas
DX KX
Figure 1.11 An astronaut who returns from a space voyage will be younger than his or her twin who remains on earth. Speeds close to the speed of light (here v = 0.8¢) are needed for this effect to be conspicuous.
Relativity 19 Sn a
not been extended to him, because regardless of Jane’s 50-y wait, he has spent only 30 y on the roundtrip. :
The nonsyrametric aging of the twins has been verified by experiments in which accurate clocks were taken on an airplane trip around the world and then compared with identical clocks that had been left behind. An observer who departs from an in- ertial system and then returns after moving relative to that system will always find his or her clocks slow compared with clocks that stayed in the system.
Example 1.4
Dick and Jane each send out a radio signal once a year while Dick is away. How many signals does Dick receive? How many does Jane receive?
Solution
On the outward trip, Dick and Jane are being separated at a rate of 0.80c. With the help of the teasoning used to analyze the doppler effect in Sec. 1.3, we find that each twin receives signals
[T+ uk [14080 T= toV Tae =O Vy imogo 73
apart. On the return trip, Dick and Jane are getting closer together at the same rate, and each receives signals more frequently, namely
_ flavke _ fr=080 1 Tov toe “OD ViF080 737 apart.
To Dick, the trip to the star takes 15 y, and he receives 15/3 = 5 signals from Jane, During the 15 y of the return trip, Dick receives 15/(1/3) = 45 signals from Jane, for a total of 50 sig- nals. Dick therefore concludes that Jane has aged by 50 y in his absence. Both Dick and Jane agree that Jane is 70 y old at the end of the voyage. :
To Jane, Dick needs Lo/v = 25 y for the outward trip. Because the star is 20 light-years away. Jane on the earth continues to receive Dick’s signals at the original rate of one every 3 y for 20 y after Dick has arrived at the star. Hence Jane receives signals every 3 y for 25 y + 20y= 45 y to give a total of 45/3 = 15 signals. (These are the 15 signals Dick sent out on the outward trip.) Then, for the remaining 5 y of what is to Jane a 50-y voyage, signals arrive from Dick at the shorter intervals of 1/3 y for an additional 5/(1/3) = 15 signals. Jane thus receives 30 sig- nals in all and concludes that Dick has aged by 30 y during the time he was away—which agrees with Dick’ own figure. Dick is indeed 20 y younger than his twin Jane on his return.
aes nee eeeereeenneneeeetemememmmnceeenneenememmeomeerene armen ene
1.6 ELECTRICITY AND MAGNETISM Relativity is the bridge
One of the puzzles that set Einstein on the trail of special relativity was the connec- tion between electricity and magnetism, and the ability of his theory to clarify the na- ture of this connection is one of its triumphs.
Because the moving charges (usually electrons) whose interactions give rise to many of the magnetic forces familiar to us have speeds far smaller than c, it is not obvious that the operation of an electric motor, say, is based on a relativistic effect. The idea becomes less implausible, however, when we reflect on the strength of electric forces, The electric attraction between the electron and proton in a hydrogen atom, for instance,
20
Chapter One
is 10°° times greater than the gravitational attraction between them. Thus even a small change in the character of these forces due to relative motion, which is what magnetic forces represent, may have large consequences. Furthermore, although the effective speed of an individual electron in a current-carrying wire (<1 mm/s) is less than that of a tired caterpillar, there may be 10° or more moving electrons per centimeter in such a wire, so the total effect may be considerable.
Although the full story of how relativity links electricity and magnetism is mathe- matically complex, some aspects of it are easy to appreciate. An example is the origin of the magnetic force between two parallel currents. An important point is that, like the speed of light, :
Electric charge is relativistically invariant.
A charge whose magnitude is found te be Q in one frame of reference is also Q in all other frames.
Let us look at the two idealized conductors shown in Fig. 1.12a. They contain equal numbers of positive and negative charges at rest that are equally spaced. Because the conductors are electrically neutral, there is no force between them.
Figure 1.12b shows the same conductors when they carry currents i, and iy in the same direction. The positive charges move to the right and the negative charges move to the left, both at the same speed v as seen from the laboratory frame of reference, (Actual currents in metals consist of flows of negative electrons only, of course, but the electri- cally equivalent model here is easier to analyze and the results are the same.) Because the charges are moving, their spacing is smaller than before by the factor V1 — vie. Since v is the same for both sets of charges, their spacings shrink by the same amounts, and both conductors remain neutral to an observer in the laboratory. However, the con- ductors now attract each other. Why?
Let us look at conductor Il from the frame of reference of one of the negative charges in conductor I. Because the negative charges in II appear at rest in this frame, their spacing is not contracted, as in Fig. 1.12c. On the other hand, the positive charges in II now have the velocity 2v, and their spacing is accordingly contracted to a greater extent than they are in the laboratory frame. Conductor Il therefore appears to have a net positive charge, and an attractive force acts on the negative charge in I.
Next we look at conductor II from the frame of reference of one of the positive charges in conductor I. The positive charges in If are now at rest, and the negative charges there move to the left at the speed 2v. Hence the negative charges are closer together than the positive ones, as in Fig. 1.12d, and the entire conductur appears neg- atively charged. An attractive force therefore acts on the positive charges in I.
Identical arguments show that the negative and positive charges in Il are attracted to I, Thus all the charges in each conductor experience forces directed toward the other conductor. To each charge, the force on it is an “ordinary” electric force that arises be- cause the charges of opposite sign in the other conductor are closer together than the charges of the same sign, so the other conductor appears to have a net charge. From the laboratory frame the situation is less straightforward. Both conductors are electrically neutral in this frame, and it is natural to explain their mutual attraction by attributing it to a special “magnetic” interaction between the currents.
A similar analysis explains the repulsive force between parallel conductors that carry currents in opposite directions. Although it is convenient to think of magnetic forces as being different from electric ones, they both result from a single electromagnetic in- teraction that occurs between charged particles. :
Clearly a current-carrying conductor that is electrically neutral in one frame of reference might not be neutral in another frame. How can this observation be reconciled
Relativity 21 1 oO ° ° re) , © ® 6 @ (a) It r@) oO re) O° e e e 6 © Positive charge O Negative charge I <—O) oO Or Or fo) [o) aan) © @ © © o> ng | Force on I i i 5 7 Force on II 134 <—O; oO O Oo oO oO ® wv" «@ © @ © © oe id i I 9 Force on negative charge (©) H
a ee (eo) 8 ie} (e)
@e@ @ © C@ © © © © © G—>
ee ss Nee Se aid a eo
. Force on positive charge
—— Te (d@) u ees ( WP Plo oO oO oO oO oOo oOo o o@ 2u @ e e e a
Figure 1.12 How the magnetic attraction between parallel currents arises. (a) Idealized parallel con- ductors that contain equal numbers of positive and negative charges. (b) When the conductors carry currents, the spacing of their moving charges undergoes a relativistic contraction as seen from the lab- oratory, The conductors attract each other when i, and i, are in the same direction. (c) As seen bya negative charge in I, the negative charges in II are at rest whereas the positive charges are in motion. The contracted spacing of the latter leads to a net positive charge in II that attracts the negative charge in 1. (d) As seen by a positive charges in 1, the positive charges in Il are at rest whereas the negative charges are in motion. The contracted spacing of the latter leads to a net negative charge on II that attrats the positive charge in 1. The contracted spacings in b, c, and d are greatly exaggerated.
with charge invariance? The answer is that we must consider the entire circuit of which the conductor is a part. Because the circuit must be closed for a current to occur in it, for every current element in one direction that a moving observer finds to have, say, a positive charge, there must be another current element in the opposite direction which the same observer finds to have a negative charge. Hence magnetic forces always act between different parts of the same circuit, even though the circuit as a whole appears electrically neutral to all observers.
The preceding discussion considered only a particular magnetic effect. All other magnetic phenomena can also be interpreted on the basis of Coulomb's law, charge in- variance, and special relativity, although the analysis is usually more complicated.
22
Chapter One
1.7 RELATIVISTIC MOMENTUM
Redefining an important quantity
In classical mechanics linear momentum p = mv is a useful quantity because it is con- served in a system of particles not acted upon by outside forces. When an event such as a collision or an explosion occurs inside an isolated system, the vector sum of the momenta of its particles before the event is equal to their vector sum afterward. We now have to ask whether p = mv is valid as the definition of momentum in inertial frames in relative motion, and if not, what a relativistically correct definition is.
To start with, we require that p be conserved in a collision for all observers in rel- ative motion at constant velocity. Also, we know that p = mv holds in classical mechanics, that is, for vy << c. Whatever the relativistically correct p is, then, it must reduce to mv for such velocities.
Let us consider an elastic collision (that is, a collision in which kinetic energy is conserved) between two particles A and B, as witnessed by observers in the reference © frames S and S’ which are in uniform relative motion. The properties of A and B are identical when determined in reference frames in which they are at rest. The frames S and S are oriented as in Fig. 1.13, with S’ moving in the +x direction with respect to S at the velocity v. :
Before the collision, particle A had been at rest in frame S and particle B in frame 5! Then, at the same instant, A was thrown in the +y direction at the speed V4 while B was thrown in the —y’ direction at the speed Vs, where
Va= Ve — (1.10)
Hence the behavior of A as seen from S is exactly the same as the behavior of B as seen from S’.
When the two particles collide, A rebounds in the —y direction at the speed Va, - while B rebounds in the +y’ direction at the speed Vg. If the particles are thrown from positions Y apart, an observer in S finds that the collision occurs at y = 3Y and one in S! finds that it occurs at y’ = y = +Y. The round-trip time To for A as measured in frame S is therefore
To = 7 ; (1.11) and it is the same for B in S’: nee Vp In S the speed Vz is found from Y Vz = T (1.12)
where T is the time required for B to make its round trip as measured in S, In S’, however, Bs trip requires the time To, where
(1.13)
Relativity
23
Collision as seen from frame S:
Collision as seen from frame S’:
Figure 1.13 An elastic collision as observed in two different frames of reference. The balls are initially Y apart, which is the same distance in both frames since S’ moves only in the x direction.
according to our previous results. Although observers in both frames see the same event, they disagree about the length of time the particle thrown from the other frame Tequires to make the collision and return.
Replacing T in Eq. (1.12) with its equivalent in terms of To, we have
Y V1 -v7¥e
Vz = B To
Chapter One
From Eq. (1.11), Va=
If we use the classical definition of momentum, p = my, then in frame S
= maVa = m, Ee > PA AVA A To
Pp = mgVz = mg V1 — ve |
i)
This means that, in this frame, momentum will not be conserved if m4 = mp, where my, and mg are the masses as measured in S. However, if
ma nF SS (1.14) Lv
then momentum will be conserved.
In the collision of Fig. 1.13 both A and B are moving in both frames. Suppose now that V4 and Vj are very small compared with », the relative velocity of the two frames. In this case an observer in S will see B approach A with the velocity v, make a glanc-
‘ing collision (since Vg < v), and then continue on. In the limit of V4 = 0, if m is the mass in S of A when A is at rest, then mg = m. In the limit of Vg = 0, if m(v) is the mass in S of B, which is moving at the velocity v, then mp = m(v). Hence Eq. (1.14) becomes
m
1 ur?
m(v) = (1.15)
We can see that if linear momentum is defined as
Relativistic nv ‘
momentum Pe [i wye ve (1.16)
then conservation of momentum is valid in special relativity. When v < c, Eq. (1.16) becomes just p = mv, the classical momentum, as required. Equation (1.16) is often written as
Relativistic p= ymv (1.17) momentum where 1 Y= (1.18)
£. vc
In this definition, m is the proper mass (or rest mass) of an object, its mass when measured at test relative to an observer. (The symbol + is the Greek letter gamma.)
Relativity
25
“Relativistic Mass”
\ j 7 ¢ could alternatively regard the increase in an object’s momentum over the classical value
as being due to an increase in the object's mass. Then we would call mg = m the rest mass of the object and m = m(v) from Eq, (1.17) its relativistic mass, its mass when moving rel- ative to an observer, so that p = mv. This is the view often taken in the past, at one time even by Einstein. However, as Einstein later wrote, the idea of relativistic mass is “not good” because “no clear definition can be given. It is better to introduce no other mass concept than the ‘rest mass’ m.” In this book the term mass and the symbol m will always refer to proper (or rest) mass, which will be considered relativistically invariant.
Figure 1.14 shows how p varies with u/c for both ymu and mv. When v/c is small, mv and ‘ymv are very nearly the same. (For v = 0.01c, the difference is only 0.005 percent; for v = 0.1, it is 0.5 percent, still small). As v approaches c, however, the curve for ymv rises more and more steeply (for v = 0.9c, the difference is 229 percent). If v= ¢, p = ymv = &, which is impossible. We conclude that no material object can travel as fast as light.
But what if a spacecraft moving at v, = 0.5c relative to the earth fires a projectile at v2 = 0.5¢ in the same direction? We on earth might expect to observe the projec- tile’ speed as v, + v2 = c, Actually, as discussed in Appendix I to this chapter, velocity addition in relativity is not so simple a process, and we would find the projectile’s speed to be only 0.8¢ in such a case.
Relativistic Second Law
In relativity Newton's second law of motion is given by
Relativistic _ op _ d second law Fs dt dt (ymy) (1.19)
This is more complicated than the classical formula F = ma because y is a function of v. When vc, ¥ is very nearly equal to 1, and F is very nearly equal to mv, as it should be.
Relativistic momentum ymu
Linear momentum p v 3 a
f lassigal momentum my 0 0.2 0.4 0.6 08 10 Velocity ratio u/c
Figure 1.14 The momentum of an object moving at the velocity v relative to an observer. The mass m of the object is its value when it is at rest relative to the observer. The object's velocity can never reach ¢ because its momentum would then be infinite, which is impossible. The relativistic momen- tum ‘ymvu is always correct; the classical momentum mv is valid for velocities much smaller than c.
26
Chapter One
sec are SRR AE NS AACN AR EORASSOTORSET,
Example 1.5
Find the acceleration of a particle of mass m and velocity v when it is acted upon by the con- stant force F, where F is parallel to v.
Solution
From Eq. (1.19), sinte a = du/dt,
pa Omen ( See i ay ma ( Vi- =73) a nf es + Re | Vi-vje a -v¥ey/ i] at ma
~ Gv?
We note that F is equal to -y*ma, not to yma. Merely replacing m by ym in classical formulas does not always give a relativistically correct result. The acceleration of the particle is therefore
a= 2G = vey? m
Even though the force is constant, the acceleration of the particle decreases as its velocity in- creases. As u->c, a—> 0, so the particle can never reach the speed of light, a conclusion we expect.
acco AAR A MENA AE EE II CL COTE TLART EELS AIEEE LO AEC
1.8 MASS AND ENERGY
Where Eo = mc comes from
The most famous relationship Einstein obtained from the postulates of special relativity—how powerful they turn out to be!—concerns mass and energy. Let us see how this relationship can be derived from what we already know.
As we recall from elementary physics, the work W done on an object by a con- stant force of magnitude F that acts through the distance s, where F is in the same direction as s, is given by W = Fs. If no other forces act on the object and the ob- ject starts from rest, all the work done on it becomes kinetic energy KE, so KE = Fs. In the general case where F need not be constant, the formula for kinetic energy is the integral
‘= KE = | F ds 0 In nonrelativistic physics, the kinetic energy of an object of mass m and speed v is
KE = tmv*. To find the correct relativistic formula for KE we start from the relativistic form of the second law of motion, Eq. (1.19), which gives
=f ee a xe= [a= v dtm) = ['v a
Relativity 27 Integrating by parts (f x dy = xy — f y dx), mv v udu KE = m [ 1 v3? 0 1l-v 2 ew ae + [me Vi- vie] . Vi -— vf? ° Bee |8 V1 ~ v2 Kinetic energy KE = yme2 —me = Yy- 1m? (1.20)
This result states that the kinetic energy of an object is equal to the difference between -ymc? and mc”. Equation (1.20) may be written
Total energy E= ymc* = mc? + KE (1.21)
If we interpret -ymc? as the total energy E of the object, we see that when it is at rest and KE = 0, it nevertheless possesses the energy mc’. Accordingly mc? is called the rest energy Eo of something whose mass is m. We therefore have
E=E)+KE where : Rest energy Ey = me? ; (1.22) If the object is moving, its total energy is _
mc?
Total energy E= yc? = —————— (1.23) V1 - vf?
Example 1.6
A stationary body explodes into two fragments each of mass 1.0 kg that move apart at speeds of 0.6¢ relative to the original body. Find the mass of the original body.
Solution
The rest energy of the original body must equal the sum of the total energies of the fragments. Hence
myc? myc?
Eg = me? = ymyc? + ym? = sh V1 - vi/c? V1 — v3/c?
«
and Ey _ _(2)(1.0 kg) m= 2 = 25 Wi = (0.60) .
“ Since mass and energy are not independent entities, their separate conservation prin- ciples are properly a single one—the principle of conservation of mass energy. Mass can be created or destroyed, but when this happens, an equivalent amount of energy simultaneously vanishes or comes into being, and vice versa. Mass and energy are dif- ferent aspects of the same thing.
28 Chapter One
It is worth emphasizing the difference between a conserved quantity, such as total , energy, and an invariant quantity, such as proper mass. Conservation of E means that, © in a given reference frame, the total energy of some isolated systern remains the same regardless of what events occur in the system. However, the total energy may be dif- ferent as measured from another frame. On the other hand, the invariance of m means that m has the same value in all inertial frames.
The conversion-factor between the unit of mass (the kilogram, kg) and the unit of energy (the joule, J) is. 2, so 1 kg of matter—the mass of this book is about that-—has an energy content of mc? = (1 kg)(3 X 10° m/s)? = 9 X 10'° J. This is enough to send a payload of a million tons to the moon. How is it possible for so much energy to be bottled up in even a modest amount of matter without anybody having been aware of it until Einstein’s work?
In fact, processes in which rest energy is liberated are very familiar. It is simply that we do not usually think of them in such terms. In every chemical reaction that evolves energy, a certain amount of matter disappears, but the lost mass is so small a fraction of the total mass of the reacting substances that it is imperceptible. Hence the “law” of conservation of mass in chemistry. For instance, only about 6 X 107! kg of matter vanishes when 1 kg of dynamite explodes, which is impossible to measure directly, but the more than 5 million joules of energy that is released is hard to avoid noticing. _
Example 1.7 +
Solar energy reaches the earth at the rate of about 1.4 kW per square meter of surface perpen- dicular to the direction of the sun (Fig, 1.15). By how much does the mass of the sun decrease per second owing to this energy loss? The mean radius of the earth’ orbit is 1.5 X 10)! m.
Solar
Figure 1.15
Sohution
The surface area of a sphere of radius ris A = 4217. The total power radiated by the sun, which is equal to the power received by a sphere whose raditis is that of the earth's orbit, is therefore P. B.
Pa AT (4m?) = (1.4 X 10? Wim2)(4ar)(L.5 x 10"! m)? = 4.0 X 107° W Thus the sun loses Ep = 4.0 X 107° J of rest energy per second, which means that the sun's rest mass decreases by
Ep 4.0 x 1076 J 5 = 2 = I = 44x 10 m= "2 BOX 10° m/s Ks per second. Since the sun's mass is 2.0 X 10°° kg, it is in no immediate danger of running out of matter. The chief energy-producing process in the sun and most other stars is the conversion of hydrogen to helium in its interior. The formation of each helium nucleus is accompanied by the release of 4.0 X 107" J of energy, so 10°” helium nuclei are produced in the sun per second.
LL
Relativity 29
Kinetic Energy at Low Speeds
When the relative speed v is small compared with c, the formula for kinetic energy must reduce to the familiar }mv?, which has been verified by experiment at such speeds. Let us see if this is true. The relativistic formula for kinetic energy is
: 2 Kinetic me 2
KE = yme? — me = me energy ” Vi-vye Gat)
Sihce v7/c? < 1, we can use the binomial approximation (1 + x)"~ 1 + nx, valid for xt < 1, to obtain
vuZe
*
w= (145-5 )ne = me Lo? v<e 2¢ 2
At low speeds the relativistic expression for the kinetic energy of a moving object
does indeed reduce to the classical one. So far as is known, the correct formulation of
mechanics has its basis in relativity, with classical mechanics representing art approxi-
mation that is valid only when v < c. Figure 1.16 shows how the kinetic energy of
0 02 04 06 08 10 12 14 16 vie
Figure 1.16 A comparison between the classical and relativistic formulas for the ratio between kinetic energy KE of a moving body and its rest energy mc. At low speeds the two formulas give the same tesult, but they diverge at speeds approaching that of light. According to relativistic mechanics, a body would need an infinite kinetic energy to travel with the speed of light, whereas in classical mechan- ics it would need only a kinetic energy of half its rest energy to have this speed. :
30
Chapter One
a moving object varies with its speed according to both classical and relativistic mechanics.
The degree of accuracy required is what determines whether it is more appropri- ate to use the classical or to use the relativistic formulas for kinetic energy. For in- stance, when v = 107 m/s (0.033), the formula }mv? understates the true kinetic energy by only 0.08 percent; when v = 3 X 10’ m/s (0.10), it understates the true kinetic energy by 0.8 percent; but when u = 1.5 X 10° m/s (0.5c), the understate- ment is a significant 19 percent; and when v = 0.999c, the understatement is a whop- ping 4300 percent. Since 10” m/s is about 6310 mi/s, the nonrelativistic formula my? is entirely satisfactory for finding the kinetic energies of ordinary objects, and it fails only at the extremely high speeds reached by elementary particles under cer- tain circumstances.
1.9 ENERGY AND MOMENTUM
How they fit together in relativity Total energy and momentum are conserved in an isolated system, and the rest energy ~ of a particle is invariant. Hence these quantities are in some sense more fundamental than velocity or kinetic energy, which are neither. Let us look into how the total en-
ergy, rest energy, and momentum of a particle are related. We begin with Eq. (1.23) for total energy,
me (1.23)
Total energy E= Vine
and square it to give
From Eq. (1.17) for momentum,
(1.17)
Momentum p= Miewe l-vfc?
we find that
Now we subtract pc? from E”: mc — mv? _ mect( — v7/c*) 1-3/2 1- ve
= (ne?P
Ska pe Py ‘
Relativity 31 re a
Hence
Energy and a 2y2 2 , Scientia EF? = (me)? + p*c? (1.24)
which is the formula we want. We note that, because mc? is invariant, so is E27 — pe: this quantity for a particle has the same value in all frames of reference.
For a system of particles rather than a single particle, Eq. (1.24) holds provided that the rest energy mc?—and hence mass m-—is that of the entire system. If the particles in the system are moving with respect to one another, the sum of their ° individual rest energies may not equal the rest energy of the system. We saw this in Example 1.7 when a stationary body of mass 2.5 kg exploded into two smaller bodies, each of mass 1.0 kg, that then moved apart. If we were inside the system, we would interpret the difference of 0.5 kg of mass as representing its conversion into kinetic energy of the smaller bodies. But seen as a whole, the system is at rest both before and after the explosion, so the system did not gain kinetic energy. Therefore the rest energy of the system includes the kinetic energies of its internal motions and it cor- tesponds to a mass of 2.5 kg both before and after the explosion.
In a given situation, the rest energy of an isolated system may be greater than, the same as, or less than the sum of the rest energies of its members. An important case in which the system rest energy is less than the rest energies of its members is that of a system of particles held together by attractive forces, such as the neutrons and pro- tons in an atomic nucleus. The rest energy of a nucleus (except that of ordinary hydrogen, which is a single proton) is less than the total of the rest energies of its constituent particles. The difference is called the binding energy of the nucleus. To break a nucleus up completely calls for an amount of energy at least equal to its binding energy. This topic will be explored in detail in Sec. 11.4. For the moment it is inter- esting to note how large nuclear binding energies are—nearly 10" kj per kg of nuclear matter is typical. By comparison, the binding energy of water molecules in liq- uid water is only 2260 kJ/kg; this is the energy needed to tum 1 kg of water at 100°C to steam at the same temperature.
.
Massless Particles
Can a massless particle exist? To be more precise, can a particle exist which has no rest mass but which nevertheless exhibits such particlelike properties as energy and mo- mentum? In classical mechanics, a particle must have rest mass in order to have en- ergy and momentum, but in relativistic mechanics this requirement does not hold.
From Eqs. (1.17) and (1.23), when m = 0 and v<c, it is clear that E = p=0. A massless particle with a speed less than that of light can have neither energy nor mo- mentum. However, when m = 0 and v = c, E = 0/0 and p = 0/0, which are inde- terminate: E and p can have any values. Thus Eqs. (1.17) and (1.23) are consistent with the existence of massless particles that possess energy and momentum provided that they travel with the speed of light.
Equation (1.24) gives us the relationship between E and p for a particle with m = 0:
Massless particle E= pe (2.25)
The conclusion is not that massless particles necessarily occur, only that the laws of physics do not exclude the possibility as long as v = c and E = pe for them. In fact,
32 Chapter One
6 a I ce
a massless particle—the photon—indeed exists and its behavior is as expected, as we shall find in Chap. 2.
Electronvolts
In atomic physics the usual unit of energy is the electronvolt (eV), where 1 eV is the energy gained by an'electron accelerated through a potential difference of 1 volt, Since W = QV,
1 eV = (1.602 X 107! C)(1.000 V) = 1.602 x 1077°J
‘Two quantities normally expressed in electronvolts are the ionization energy of an atom (the work needed to remove one of its electrons) and the binding energy of a mole- cule (the energy needed to break it apart into separate atoms). Thus the ionization energy of nitrogen is 14.5 eV and the binding energy of the hydrogen molecule Hy is 4.5 eV. Higher energies in the atomic realm are expressed in kiloelectronvolts (keV), where 1 keV = 10° eV
In nuclear and elementary-particle physics even the keV is too small a unit in most cases, and the megaelectronvolt (MeV) and gigaelectronvolt (GeV) are more appro- priate, where .
1MeV=10%eV 1 GeV = 10° eV
An example of a quantity expressed in MeV is the energy liberated when the nucleus of a certain type of uranium atom splits into two parts. Each such fission event releases about 200 MeV; this is the process that powers nuclear reactors and weapons.
The rest energies of elementary particles are often expressed in MeV and GeV and the corresponding rest masses in MeWc? and GeWc?. The advantage of the latter units is that the rest energy equivalent to a rest mass of, say, 0.938 GeWc? (the rest mass of the proton) is just Eg = mc = 0.938 GeV. If the proton’s kinetic energy is 5.000 GeV, finding its total energy is simple: .
E = Ey + KE = (0.938 + 5.000) GeV = 5.938 GeV In a similar way the MeWc and GeWc are sometimes convenient units of linear mo-
mentum, Suppose we want to know the momentum of a proton whose speed is.0.800c. From Eq. (1.17) we have
mu (0.938 GeV/c?)(0.800c) p= Vi = ve Vi = (0.8000)/? 0.750 GeW/c = = V, 0.600 25 GeV/c
LY Example 1.8
An electron (m, = 0.511 MeV?) and a photon (m = 0) both have momenta of 2.000 MeV/c. Find the total energy of each. ;
Relativity 33
SS rrr ee
Solution
(@ From Eq. (1.24) the electron’s total energy is
B= Vit + pe? = VOS1 MeWe)*c* + (2.000 Meer = V(0.511 MeV)* + (2.000 MeV)? = 2.064 MeV
(b) From Eq. (1.25) the photon’ total energy is E = pe = (2.000 MeWc)c = 2.000 MeV
.
1.10 GENERAL RELATIVITY
Gravity is a warping of spacetime
Special relativity is concerned only with inertial frames of reference, that is, frames that
are not accelerated. Einstein's 1916 general theory of relativity goes further by in-
cluding the effects of accelerations on what we observe. Its essential conclusion is that
the force of gravity arises from a warping. of spacetime around a body of matter
(Fig. 1.17). As a result, an object moving through such a region of space in general
follows a curved path rather than a straight one, and may even be trapped there. The principle of equivalence is central to general relativity:
An observer in a closed laboratory cannot distinguish between the effects pro- duced by a gravitational field and those produced by an acceleration of the laboratory.
This principle follows from the experimental observation (to better than 1 part in 10!) that the inertial mass of an object, which governs the object's acceleration when a force acts on it, is always equal to its gravitational mass, which governs the gravitational force another object exerts on it. (The two masses are actually proportional; the con- stant' of proportionality is set equal to 1 by an appropriate choice of the constant of gravitation G.)
Figure 1.17 General relativity pictures gravity as a warping of spacetime due to the presence of a body of matter. An object nearby experiences an attractive force as a result of this distortion, much as a marble tolls toward the bottom of a depression in a rubber sheet. To paraphrase J. A. Wheeler, space- time tells mass how to move, and mass tells spacetime how to curve.
Chapter One
Laboratory in Accelerated laboratory gravitational field
Figure 1.18 According to the principle of equivalence, events that take place in an accelerated laboratory cannot be distinguished from those which take place in a gravitational field. Hence the deflection of a light beam relative to an observer in an accelerated laboratory means that light must be similarly deflected in a gravitational field.
Gravity and Light
It follows from the principle of equivalence that light should be subject to gravity. Ifa light beam is directed across an accelerated laboratory, as in Fig, 1,18, its path relative to the laboratory will be curved. This means that, if the light beam is subject to the — gravitational field to which the laboratory’ acceleration is equivalent, the beam would - follow the same curved path.
According to general relativity, light rays that graze the sun should have their paths bent toward it by 0.005°—the diameter of a dime seen from a mile away. This pre- diction was first confirmed in 1919 by photographs of stars that appeared in the sky near the sun during an eclipse, when they could be seen because the sun’ disk was covered by the moon. The photographs were then compared with other photographs of the same part of the sky taken when the sun was in.a distant part of the sky (Fig, 1.19). Einstein became a world celebrity as a result.
Because light is deflected in a gravitational field, a dense concentration of mass— such as a galaxy of stars—can act as a Jens to produce multiple images of a distant light source located behind it (Fig. 1.20). A quasar, the nucleus of a young galaxy, is brighter than 100 billion stars but is no larger than the solar system. The first observation of gravitational lensing was the discovery in 1979 of what seemed to be a pair of nearby quasars but was actually a single one whose light was deviated by an intervening massive object. Since then a number of other gravitational lenses have been found; the effect occurs in radio waves from distant sources as well as in light waves.
The interaction between gravity and light also gives rise to the gravitational red shift and to black holes, topics that are considered in Chap. 2. :
; Relativity 35 rr
Apparent #% position 7 of star
Star ¥
Starlight
Figure 1.19 Starlight passing near the sun is deflected by its strong gravitational field. The deflection can be measured during a solar eclipse when the sun’ disk is obscured by the moon.
Apparent ~O position of source
§ Source
Light and radio waves from source
ot eS See e. Apparent “>> © position of source
Figure 1.20 A gravitational lens, Light and radio waves from a source such as a quasar are deviated by a massive object such as a galaxy so that they seem to come from two or more identical sources, A number of such gravitational lenses have been identified.
Other Findings of General Relativity
A further success of general relativity was the clearing up of a long-standing puzzle in astronomy. The perihelion of a planetary orbit is the point in the orbit nearest the sun. Mercury orbit has the peculiarity that its perihelion shifts (precesses) about 1.6° per century (Fig. 1.21). All but 43” (L” = 1 are second = jms Of a degree) of this shift is due to the attractions of other planets, and for a while the discrepancy was used as evidence for an undiscovered planet called Vulcan whose orbit was supposed to lie
36 Chapter One
inside that of Mercury. When gravity is weak, general relativity gives very nearly the
same results as Newton’s formula F = Gm,nt,/7*. But Mercury is close to the sun and
so moves in a strong gravitational field, and Einstein was able to show from general
relativity that a precession of 43” per century was to be expected for its orbit. Mercury The existence of gravitational waves that travel with the speed of light was the prediction of general relativity that had to wait the longest to be verified. To visualize gravitational waves, we can think in terms of the model of Fig. 1.17 in which two- dimensional space is represented by a rubber sheet distorted by masses embedded in it. If one of the masses vibrates, waves will be sent out in the sheet that set other masses in vibration. A vibrating electric charge similarly sends out electromagnetic waves that excite vibrations in other charges.
A big difference between the two kinds of waves is that gravitational waves are ex- tremely weak, so that despite much effort none have as yet been directly detected. Figure 1.21 The precession of the However, in 1974 strong evidence for gravitational waves was found in the behavior perihelion of Mercury's orbit. of a system of two nearby stars, one a pulsar, that revolve around each cther. A pulsar is a very small, dense star, composed mainly of neutrons, that spins rapidly and sends out flashes of light and radio waves at a regular rate, much as the rotating beam of a lighthouse does (see Sec. 9.11). The pulsar in this particular binary system emits pulses every 59 milliseconds (ms), and it and its companion (probably another neutron star) have an orbital period of about 8 h. According to general relativity, such a system should give off gravitational waves and lose energy as a result, which would reduce the orbital period as the stars spiral in toward each other. A change in orbital period means a change in the arrival times of the pulsar’ flashes, and in the case of the ob- served binary system the orbital period was found to be decreasing at 75 ms per year. This is so close to the figure that general relativity predicts for the system that there seems to be no doubt that gravitational radiation is responsible. The 1993 Nobel Prize in physics was awarded to Joseph Taylor and Russell Hulse for this work.
Much more powerful sources of gravitational waves ought to be such events as two black holes colliding and supernova explosions in which the remnant star cores col- lapse into neutron stars (again, see Sec. 9.11). A gravitational wave that passes through a body of matter will cause distortions to ripple through it due to fluctuations in the gravitational field. Because gravitational forces are feeble—the electric attraction be- tween a proton and an electron is over 10°? times greater than the gravitational at- traction between them—such distortions at the earth induced by gravitational waves from a supernova in our galaxy (which occurs an average of once every 30 years or so) would amount to only about 1 part in 10'8, even less for a more distant super- nova. This corresponds to a change in, say, the height of a person by well under the diameter of an atomic nucleus, yet it seems to be detectable-—just—with current technology.
In one method, a large metal bar cooled to a low temperature to minimize the ran- dom thermal motions of its atoms is monitored by sensors for vibrations due to grav- jtational waves. In another method, an interferometer similar to the one shown in Fig, 1.2 with a laser as the light source is used to look for changes in the lengths of the arms to which the mirrors are attached. Instruments of both kinds are operating, thus far with no success.
A really ambitious scheme has been proposed that would use six spacecraft in or- bit around the sun placed in pairs at the corners of a triangle whose sides are 5 million kilometers (km) long. Lasers, mirrors, and sensors in the spacecraft would detect changes in their spacings resulting from the passing of a gravitational wave. It may only be a matter of time before gravitational waves will be providing information about a variety of cosmic disturbances on the largest scale.
Perihelion of orbit
The Lorentz Transformation
_ Appendix | to Chapter 1
The Lorentz Transformation
uppose we are in an inertial frame of reference S and find the coordinates of
some event that occurs at the time ¢ are x, y, z. An observer located in a dif-
ferent inertial frame S’ which is moving with respect to S at the constant ve- locity v will find that the same event occurs at the time ¢’ and has the coordinates a5 y', 2’. Un order to simplify our work, we shall assume that v is in the +x direction, as in Fig. 1.22.) How are the measurements x, y, Z, t related to x’, y’, 2’, t'?
Galilean Transformation
Before special relativity, transforming measurements from one inertial system to an- other seemed obvious. If clocks in both systems are started when the origins of S and S' coincide, measurements in the x direction made is S will be greater than those made in S’ by the amount vt, which is the distance S’ has moved in the x direction, That is,
x'=x-vt (1.26) There is no relative motion in the y and @ directions, and so
yay (1.27)
z
Figure 1.22 Frame S‘ moves in the +x direction with the speed v relative to frame S. The Lorentz transformation must be used to convert measurements made in one of these frames to their equivalents in the other.
38 Appendix to Chapter 1
g=Zz (1.28)
In the absence of any indication to the contrary in our everyday experience, we fur- ther assume that
vst (1.29)
The set of Eqs. (1.26) to (1.29) is known as the Galilean transformation.
To convert velocity components measured in the S frame to their equivalents in the 5! frame according to the Galilean transformation, we simply differentiate x’, y’, and z’ with respect to time:
vi, = x FU v (1.30) v= a =y a3) v= = =u, (1.32) °
Although the Galilean transformation and the corresponding velocity transfor- mation seem straightforward enough, they violate both of the postulates of special relativity. The first postulate calls for the same equations of physics in both the S and S’ inertial frames, but the equations of electricity and magnetism become very different when the Galilean transformation is used to convert quantities measured in one frame into their equivalents in the other. The second postulate calls for the same value of the speed of light c whether determined in S or S'. If we measure the speed of light in the x direction in the S system to be c, however, in the S’ system it will be
"=c-v according to Eq. (1.30). Clearly a different transformation is required if the postulates
of special relativity are to be satisfied. We would expect both time dilation and length contraction to follow naturally from this new transformation.
Lorentz Transformation
A reasonable guess about the nature of the correct relationship between x and x’ is x’ = h& — vt) (1.33)
Here k is a factor that does not depend upon either x or t but may be a function of v. The choice of Eq. (1.33) follows from several considerations:
L Itis linear in x and x’, so that a single event in frame S corresponds to a single event in frame S’, as it must.
2. It is simple, and a simple solution to a problem should always be explored first.
3 It has the possibility of reducing to Eq. (1.26), which we know to be correct in ordinary mechanics.
The Lorentz Transformation 39
Because the equations of physics must have the same form in both S and S’, we need only change the sign of v (in order to take into account the difference in the direction of relative motion) to write the corresponding equation for x in terms of x’ and ¢’:
x= kh’ + vt’) (1.34) The factor k must be the same in both frames of reference since there is no difference between S and S’ other than in the sign of v,
As in the case of the Galilean transformation, there is nothing to indicate that there might be differences between the corresponding coordinates y,y' and z, z' which are perpendicular to the direction of v. Hence we again take
yy (1.35) Zaz (1.36)
The time coordinates t and t’, however, are not equal. We can see this by substi- tuting the value of x’ given by Eq. (1.33) into Eq. (1.34). This gives
x= h(x — vt) + kot’
from which we find that
ER tame (2a )s "a3 Equations (1.33) and (1.35) to (1.37) constitute a coordinate transformation that satisfies the first postulate of special relativity.
The second postulate of relativity gives us a way to evaluate k. At the instant t = 0, the origins of the two frames of reference S and S’ are in the same place, according to our initial conditions, and t’ = 0 then also. Suppose that a flare is set off at the com- mon origin of S and S’ at t = ¢’ = 0, and the observers in each system measure the speed with which the flare’s light spreads out, Both observers must find the same speedc (Fig. 1.23), which means that in the S frame
x=ct (1.38) and in the S’ frame x =ct" (1.39)
Substituting for x’ and t’ in Eq. (1.39) with the help of Eqs. (1.33) and (1.37) gives
_ 42 We — v= ae + (2 Jee
and solving for x,
cht + vkt
OE LGR LG s
40 Appendix to Chapter 1
AU Appenn x to $< $<
s
(a)
(b)
Each observer detects light waves spreading out from own boat
He ’ sy gE>—> gED—> HED Ss
Pattern of ripples ° from stone dropped Each observer sees pattern in water spreading from boat S
Figure 1.23 (a) Inertial frame S’ is a boat moving at speed v in the +x direction relative to another boat, which is the inertial frame S. When t = & = 0, S' is next to S, and x = x9 = 0. At this moment a flare is fired from one of the boats. An observer on boat $ detects light waves spreading out at speed c from his boat. An observer on boat $’ also detects light waves spreading out at speed ¢ from her boat, even though S’ is moving to the right relative to S. (b) If instead a stone were dropped in the water at t = to = 0, the observers would find a pattern of ripples spreading out around S at different speeds relative to their boats. The difference between (a) and (b) is that water, in which the ripples move, is itself a frame of reference whereas space, in which light moves, is not.
This expression for x will be the same as that given by Eq. (1,38), namely, x = ct, provided that the quantity in the brackets equals 1. Therefore
The Lorentz Transformation 41
(1.42) (1.43)
(1.44)
These equations comprise the Lorentz transformation. They were first obtained by the Dutch physicist H.A. Lorentz, who showed that the basic formulas of electromagnetism are the same in all inertial frames only when Eggs. (1.41) to (1.44) are used. It was not until several years later that Einstein discovered their full significance. It is obvious that the Lorentz transformation reduces to the Galilean transformation when the relative velocity v is small compared with the velocity of
light c.
eee pSSrET SET tUPIUOAASS S
Example 1.9
Derive the relativistic length contraction using the Lorentz transformation.
Solution
Let us consider a rod lying along the x’ axis in the moving frame S’, An observer in this frame determines the coordinates of its ends to be x{ and x}, and so the proper length of the rod is
Lp = xb ~ x}
Tt TE hth pWSnssthpSWGSPURSSSsAS
Hendrik A. Lorentz (1853-1928) was born in Armhem, Holland, and studied at the University of Leyden. At nineteen he returned to Arnhem and taught at the high school there while preparing a doctoral thesis that extended Maxwell's theory of elec- tromagnetism to cover the details of the refraction and reflection of light. In 1878 he became professor of the- oretical physics at Leyden, the first such post in Holland, where he remained for thirty-four years until he moved to Haarlem. Lorentz went on to reformulate and simplify Maxwell's theory and to introduce the idea that electromagnetic fields are created by electric charges on the atomic level. He proposed that the emission of light by atoms and various optical phenomena could be traced to the mo- tions and interactions of atomic electrons. The discovery in
1896 by Pieter Zeeman, a student of his, that the spectral lines of atoms that radiate in a magnetic field are split into components of slightly different frequency confirmed Lorentz’s work and led to a Nobel Prize for both of them in 1902,
The set of equations that enables electromagnetic quantities in one frame of reference to be transformed into their values in another frame of reference moving relative to the first were found by Lorentz in 1895, although their full significance was not realized until Einstein’s theory of special relativity ten years afterward. Lorentz (and, independently, the Irish physicist G. E Fitzgerald) suggested that the negative result of the Michelson- Morley experiment could be understood if lengths in the direction of motion relative to an observer were contracted. Sub- sequent experiments showed that although such contractions do occur, they are not the real reason for the Michelson- Morley result, which is that there is no “ether” to serve as a universal frame of reference.
An Sherer SSSGpeSSPUNERCSS
42 Appendix to Chapter 1
In order to find L = x2 — x, the length of the rod as measured in the stationary frame S at the” time £, we make use of Eq. (1.41) to give
= et = i~ : ed
Vi - vee V1 - vf? Hence L=xy -x = 6h —-x) V1 - v2 = V1 - vf?
This is the same as*Eq. (1.9)
a TT TS EA
Inverse Lorentz Transformation
In Example 1.9 the coordinates of the ends of the moving rod were measured in the stationary frame S at the same time t, and it was easy to use Eq. (1.41) to find L in terms of Lo and v. If we want to examine time dilation, though, Eq. (1.44) is not con- venient, because t and ty, the start and finish of the chosen time interval, must be measured when the moving clock is at the respective different positions x, and x . In situations of this kind it is easier to use the inverse Lorentz transformation, which converts measurements made in the moving frame S’ to their equivalents in S.
To obtain the inverse transformation, primed and unprimed quantities in Eqs. (1.41) to (1.44) are exchanged, and v is replaced by —v:
Inverse Lorentz
transformation = (1.45) yay, (1.46)
gaz (1.47)
Jom (1.48)
Example 1.10 Derive the formula for time dilation using the inverse Lorentz transformation. Solution
Let us consider a clock at the point x’ in the moving frame S’. When an observer in S' finds that the time is t), an observer in S will find it to be 4), where, from Eq. (1.48),
' i+ _ ¢
po : Vi — vf
After a time interval of fg (to him), the observer in the moving system finds that the time is now th according to his clock. That is,
f=
The Lorentz Transformation 43
OOO nk _ _c eee
The observer in S, however, measures the end of the same time interval to be
’
ux! tht 2t°3 $$$ V1 =v? so to her the duration of the interval t is a—-th to
t=bh-h= =
Vi-vfe Vi = wfc
This is what we found earlier with the help of a light-pulse clock.
Pe sh he AC
Velocity Addition
Special relativity postulates that the speed of light c in free space has the same value for all observers, regardless of their relative motion.“Common sense” (which means here the Galilean transformation) tells us that if we throw a ball forward at 10 m/s from a car moving at 30 m/s, the ball’s speed relative to the road will be 40 m/s, the sum of the two speeds. What if we switch on the car's headlights when its speed is v? The same reasoning suggests that their light, which is emitted from the reference frame S’ (the car) in the direction of its motion relative to another frame S (the road), ought to have 4 speed of c + vas measured in S. But this violates the above postulate, which has had ample experimental verification. Common sense is no more reliable as a guide in science than it is elsewhere, and we must turn to the Lorentz transformation equa- tions for the correct scheme of velocity addition.
Suppose something is moving relative to both $ and S’. An observer in S measures its three velocity components to be
dx dy dz aE Vy = =~ V,= >
Y= fe dt
while to an observer in S’ they are
ad! dy’ dz’ Vs Vos Vis x dt’ - dat’ z dt’ By differentiating the inverse Lorentz transformation equations for x, y, z, and t, we obtain dz’ a’ +2 dx' + vdt' 2 dk = dy=dy’ dzg=dz’ dt= 1— vf? V1 - vc tu dx' +udt' ‘ and so V.= ee ke Ls
44 Appendix to Chapter 1
APE OR
Relativistic velocity 1 transformation
Similarly,
ve asp
If Vi, = ¢, that is, if light is emitted in the moving frame S' in its direction of motion relative to S, an observer in frame S$ will measure the speed
ctu cc + v)
14% ‘ctv
2
Thus observers in the cat and on the road both find the same value for the speed of
light, as they must.
AAT
Example 1.11
Spacecraft Alpha is moving at 0.90c with respect to the earth. If spacecraft Beta is to pass Alpha at a relative speed of 0.50c in the same direction, what speed must Beta have with respect to
the earth?
Solution
According to the Galilean transformation, Beta would need a speed relative to the earth of, 0.90c + 0.50c = 1.40c, which we know is impossible. According to Eq. (1.49), however, with Vi, = 0.50 and v = 0,90c, the required speed is only
0,50c + 0.90¢
~ T, ©.9009(0.500) 2
= 0.97¢
which is less than c. It is necessary to go less than 10 percent faster than a spacecraft traveling at 0.90c in order to pass it at a relative speed of 0.50c.
I
Simultaneity
The relative character of time as well as space has many implications. Notably, events that seem to take place simultaneously to one observer may not be simultaneous to another observer in relative motion, and vice versa.
Let us examine two events—the setting off of a pair of flares, say—that occur at the same time to to somebody on the earth but at the different locations x, and x2. What does the pilot of a spacecraft in flight see? To her, the flare at x, and to appears at the
The Lorentz Transformation
according to Eq. (1.44), while the flare at x. and to appears at the time
to — vx_fc?
V1 ~ u/c?
Hence two events that occur simultaneously to one observer are separated by a time interval of
b=
pa Von — x) /? p= UOT IC
) vc
to an observer moving at the speed v relative to the other observer. Who is right? The question is, of course, meaningless: both observers are “right” since each simply meas- ures what he or she sees.
Because simultaneity is a relative concept and not an absolute one, physical theo- ties that require simultaneity in events at different locations cannot be valid. For in- stance, saying that total energy is conserved in an isolated system does not rule out a process in which an amount of energy AE vanishes at one place while an equal amount of energy AE comes into being somewhere else with no actual transport of energy from one place to the other. Because simultaneity is relative, some observers of the process will find energy not being conserved. To rescue conservation of energy in the light of special relativity, then, we have to say that, when energy disappears somewhere and appears elsewhere, it has actually flowed from the first location to the second. Thus energy is conserved locally everywhere, not merely when an isolated system is considered—a much stronger statement of this principle.
45
Appendix to Chapter 1
Appendix II to C
Spacetime
nature, A length that one observer can measure with only a meter stick may have to be measured with both a meter stick and a clock by another observer. A convenient and elegant way to express the results of special relativity is to regard events as occurring in a four-dimensional spacetime in which the usual three coordi- nates x, y, z refer to space and a fourth coordinate ict refers to time, where i = VA1. Although we cannot visualize spacetime, it is no harder to deal with mathematically than three-dimensional space. The reason that ict is chosen as the time coordinate instead of just ¢ is that the quantity
A s we have seen, the concepts of space and time are inextricably mixed in
vaxrty+7— (ctr (1.52)
is invariant under a Lorentz transformation. That is, if an event occurs at x, y, 2, t in an inertial frame S and at x’, y’, 2’, t’ in another inertial frame S’, then
Part ~Pt2— (ax? ty% +27 -(eP
Because s* is invariant, we can think of a Lorentz transformation merely as a rotation in spacetime of the coordinate axes x, y, z, ict (Fig, 1.24).
The four coordinates x, y, z, ict define a vector in spacetime, and this four-vector remains fixed in spacetime regardless of any rotation of the coordinate system—that is, regardless of any shift in point of view from one inertial frame S to another S’.
Another four-vector whose magnitude remains constant under Lorentz transforma- tions has the components py, Py, Pz» iE/c. Here px, Py, pz are the usual components of the linear momentum of a body whose total energy is E. Hence the value of
2 Be + By + Be - E
9 eras
Figure 1.24 Rotating a two-dimensional coordinate system does not change the quantity s? = 27 + y? =x’? + y’, where s is the length of the vector s. This result can be generalized to the four- dimensional spacetime coordinate system x, y, Z, ict.
Spacetime 47 —_-—-—___ tin
is the same in all inertial frames even though p,, Py: Pz and E separately may be dif- ferent. This invariance was noted earlier in connection with Eq. (1.24); we note that P= pet py + pe.
A more mathematically elaborate formulation brings together the electric and mag- netic fields E and B into an invariant quantity called a tensor. This approach to incorporating special relativity into physics has led both to a deeper understanding of natural laws and to the discovery of new phenomena and relationships.
Spacetime Intervals
The statements made at the end of Sec. 1.2 (P. 10) are easy to confirm using the idea of spacetime. Figure 1.25 shows two events plotted on the axes x and ct. Event 1 oc- curs atx = 0, t = 0 and event 2 occurs at x = Ax, t = At. The spacetime interval As between them is defined by
Spacetime interval 2 Bad 3 between events (As)* = AD? — (Ax) (1.53)
The virtue of this definition is that (As)*, like the s* of Eq. 1.52, is invariant under Lorentz transformations. If Ax and At are the differences in space and time between two events measured in the S frame and Ax’ and At’ are the same quantities meas- ured in the S’ frame,
(As)? = (cA? ~ (Ax)? = (cAt'y? — (Ax’?
Therefore whatever conclusions we arrive at in the S frame in which event 1 is at the origin hold equally well in any other frame in relative motion at constant velocity.
Figure 1.25 The past and future light cones in spacetime of event 1.
48 Appendix to Chapter 1 °
Now let us look into the possible relationships between events 1 and 2, Event 2 can be related causally in some way to event 1 provided that a signal traveling slower than the speed of light can connect these events, that is, provided that
cAt > |Ax| or . Timelike interval (AsP > 0 (1.53) An interval in which (As)? > 0 is said to be timelike. Every timelike interval that connects event 1 with another event lies within the light cones bounded by x = +ctin Fig. 1.25. All events that could have affected event 1 lie in the past light cone; all events
that event 1 is able to affect lie in the future light cone. (Events connected by timelike intervals need not necessarily be related, of course, but it is possible for them to be
related.) Conversely, the criterion for there being no causal relationship between events 1 and 2 is that cAt < |Ax| or Spacelike interval (As)? <0 (1.54)
An interval in which (As)* <0 is said to be spacelike. Every event that is connected with event 1 by a spacelike interval lies outside the light cones of event 1 and neither has interacted with event 1 in the past nor is capable of interacting with it in the future; the two events must be entirely unrelated.
When events 1 and 2 can be connected with a light signal only,
cAt = |Ax| or Lightlike interval As =0 (1.55)
‘An interval in which As = 0 is said to be lightlike. Events that can be connected with event 1 by lightlike intervals lie on the boundaries of the light cones.
These conclusions hold in terms of the light cones of event 2 because (As)? is invariant; for example, if event 2 is inside the past light cone of event 1, event | is inside the future light cone of event 2. In general, events that lie in the future of an event as seen in one frame of reference S lie in its future in every other frame S’, and events that lie in the past of an event in S lie in its past in every other frame S’. Thus “future” and “past” have invariant meanings. However, “simultaneity” is an ambiguous concept, because all events that lie outside the past and future light cones of event 1 (that is, all events connected by spacelike intervals with event 1) can appear to occur simultaneously with event 1 in some particular frame of reference.
‘The path of a particle in spacetime is called its world line (Fig. 1.26). The world line of a particle must lie within its light cones. :
Exercises 49
* ABSOLUTE FUT
ABSOLUTELY
Here and now
ABSOLUTELY
UNRELATED
Figure 1.26 The world line of a particle in spacetime.
UNRELATED
But be ye doers of the word, and not hearers only, deceiving your own selves. —James 1:22
1,1 Special Relativity
1, If the speed of light were smaller than it is, would relativistic phenomena be more or less conspicuous than they are now?
2. It is possible for the electron beam in a television picture tube to move across the screen at a speed faster than the speed of light. Why does this not contradict special relativity?
1.2 Time Dilation
3. An athlete has learned enough physics to know that if he meas- ures from the earth a time interval on a moving spacecraft, what he finds will be greater than what somebody on the spacecraft would measure. He therefore proposes to set a world record for the 100-m dash by having his time taken by an observer on a moving spacecraft. Is this a good idea?
4. An observer on a spacecraft moving at 0.700c relative to the earth finds that a car takes 40.0 min to make a trip. How long does the trip take to the driver of the car?
5. Two observers, A on earth and B in a spacecraft whose speed is 2.00 x 10° m/s, both set their watches to the same time when the ship is abreast of the earth. (2) How much time must elapse by A’s reckoning before the watches differ by 1,00 s? (b) To A, Bs watch seems to run slow. To B, does A’s watch seem to run fast, run slow, or keep the same time as his own watch?
6, An airplane is flying at 300 m/s (672 mi/h). How much time must elapse before a clock in the airplane and one on the ground differ by 1.00 s?
7. How fast must a spacecraft travel relative to the earth for each day on the spacecraft to correspond to 2 d on the earth?
8. The Apolto 11 spacecraft that landed on the moon in 1969 traveled there at a speed relative to the earth of 1.08 X 10* ms. ‘To an observer on the earth, how much longer than his own day was a day on the spacecraft?
9, A certain particle has a lifetime of 1.00 x 10-7 s when meas- ured at rest. How far does it go before decaying if its speed is 0.99¢ when it is created?
1.3 Doppler Effect
10. A spacecraft receding from the earth at 0.97c transmits data at the rate of 1.00 X 10° pulses/s. At what rate are they received?
11. A galaxy in the constellation Ursa Major is receding from the earth at 15,000 km/s. If one of the characteristic wavelengths of the light the galaxy emits is 550 nm, what is the corresponding wavelength measured by astronomers on the earth?
12. The frequencies of the spectral lines in light from a distant galaxy are found to be two-thirds as great as those of the same lines in light from nearby stars, Find the recession speed of the distant galaxy.
50 Appendix to Chapter 1
13.
14,
15,
16.
14 17.
18.
19.
20.
21.
15
22.
23.
A spacecraft receding from the earth emits radio waves at a constant frequency of 10° Hz. If the receiver on earth can measure frequencies to the nearest hertz, at what spacecraft speed can the difference between the relativistic and classical doppler effects be detected? For the classical effect, assume the earth is stationary. .
A car moving at 150 km/h (93 mi/h) is approaching a station- ary police car whose radar speed detector operates at a fre- quency of 15 GHz. What frequency change is found by the speed detector?
If the angle between the direction of motion of a light source of frequency ¥o and the direction from it to an observer is 8, the frequency v the observer finds is given by
os Mi - vie
= 1 — (ufo) cos 8
where v is the relative speed of the source. Show that this for- mula includes Eqs. (1.5) to (1.7) as special cases.
(a) Show that when v < c¢, the formulas for the doppler effect both in light and in sound for an observer approaching a source, and vice versa, all reduce to v * v9(1 + v/c), so that Av/y = u/c. [Hint: For x 1, 1/(1 + x) * 1 — x.) (b) What do the formulas for an observer receding from a source, and vice versa, reduce to when v << c?
Length Contraction
An astronaut, whose height on the earth is exactly 6 ft is lying parallel to the axis of a spacecraft moving at 0.90c relative to the earth. What is his height as measured by an observer in the same spacecraft? By an observer on the earth?
An astronaut is standing in a spacecraft parallel to its direction of motion. An observer on the earth finds that the spacecraft speed is 0,60¢ and the astronaut is 1.3 m tall. What is the as- tronaut’s height as measured in the spacecraft?
How much time does a meter stick moving at 0,100c relative to an observer take to pass the observer? The meter stick is paral- lel to its direction of motion.
A meter stick moving with respect to an observer appears only
500 mm long to her, What is its relative speed? How long does it take to pass her? The meter stick is parallel to its direction of motion.
A spacecraft antenna is at an angle of 10° relative to the axis of the spacecralt. If the spacecraft moves away from the earth at a speed of 0.70c, what is the angle of the antenna as seen from the earth?
Twin Paradox
‘Twin A makes a round trip at 0.6¢ to a star 12 light-years away, while twin B stays on the earth. Each twin sends the other a signal once a yeat by his own reckoning. (a) How many signals does A send during the trip? How many does B send? (b) How many signals does A receive? How many does B receive?
A woman leaves the earth in a spacecraft that makes a round trip to the nearest star, 4 light-years distant, at a speed of 0,9¢.
17
24.
25,
26.
18
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
How much younger is she upon her return than her twin sister who remained behind? :
Relativistic Momentum
(a) An electron’s speed is doubled from 0.2c to 0.4c. By what ratio does its momentum increase? (b) What happens to the momentum ratio when the electron’s speed is doubled again from 0.4¢ to 0.8¢?
All definitions are arbitrary, but some are more useful than oth- ers. What is the objection to defining linear momentum as p = my instead of the more complicated p = ymv?
Verify that
Mass and Energy Dynamite liberates about 5.4 X 10° jfkg when it explodes. What fraction of its total energy content is this?
A certain quantity of ice at 0°C melts into water at 0°C and in so doing gains 1.00 kg of mass. What was its initial mass?
At what speed does the kinetic energy of a particle equal its rest energy?
How many joules of energy per kilogram of rest mass are needed to bring a spacecraft from rest to a speed of 0.90¢?
An electron has a kinetic energy of 0.100 MeV, Find its speed according to classical and relativistic mechanics,
Verify that, for E >> Fo, eee (2) ¢€ 2\E A particle has a kinetic energy 20 times its rest energy. Find the
speed of the particle in terms of c.
(a) The speed of a proton is increased from 0.20¢ to 0.40c. By what factor does its kinetic energy increase? (b) The proton speed is again doubled, this time to 0.80c. By what factor does its kinetic energy increase now?
How much work (in MeV) must be done to increase the speed of an electron from 1.2 X 10° m/s to 2.4 X 10° m/s?
(a) Derive a formula for the minimum kinetic energy needed by a particle of rest mass m to emit Cerenkov radiation in a medium of index of refraction n. [Hint: Start from Eqs. (1.21) and (1.23).} (6) Use this formula to find KEmia for an electron in a medium of n = 1.5.
Prove that Lym, does not equal the kinetic energy of a particle moving at relativistic speeds.
A moving electron collides with a stationary electron and an electron-positron pair comes into being as a result (a positron is a positively charged electron). When all four particles have the same velocity alter the collision, the kinetic energy requited for this process is a minimum. Use a relativistic calculation to show that KEnin = 6mc?, where m is the rest mass of the electron.
Exercises 51 eS
39,
ra Initial center of mass
MAT} —» ° ‘ a Burst of radiation is emitted
| —-M/2
K L > ; | < New center of mass 1S kK [- Radiation is . |— absorbed and oN box stops Figure 1.27 The box has moved the distance $ to the left when it stops. An alternative derivation of the mass-energy formula Ey = me2,
also given by Einstein, is based on the principle that the location of the center of mass (CM) of an isolated system cannot be changed by any process that occurs inside the system, Figure 1.27 shows a rigid box of length L that rests on a frictionless surface; the mass M of the box is equally divided between its two ends. A burst of electromagnetic radiation of energy Eo is emitted by one end of the box. According to classical physics, the radiation has the momen- tum p = Eo/c, and when it is emitted, the box recoils with the speed v = Fy/Mc so that the total momeniuitir of the system remains zero. After a time ¢ ~ L/c the radiation reaches the other end of the box and is absorbed there, which brings the box to a stop after having moved the distance 5. If the CM of the box is to remain in its original place, the radiation must have transferred mass from one end to the other, Show that this amount of mass is m = Fy/c?.
1.9 Energy and Momentum
40.
41,
42.
43,
44,
Find the SI equivalents of the mass unit MeWc? and the momentum unit MeWe.
In its own frame of reference, a proton takes 5 min to cross the Milky Way galaxy, which is about 10° light-years in dianieter. | (a) What is the approximate energy of the proton in electronvalts? @®) About how long would the proton take to cross the galaxy as measured by an observer in the galaxy’ reference frame?
What is the energy of a photon whose momentum is the same as that of a proton whose kinetic energy is 10.0 MeV?
Find the momentum (in MeV/c) of an electron whose speed is 0.600c.
Find the total energy and kinetic energy (in GeV) and the momentum (in GeWc) of a proton whose speed is 0.900c. The mass of the proton is 0.938 GeWc?.
48.
49,
50.
Find the momentum of an electron whose kinetic energy equals its rest energy of 511 keV.
Verify that u/c = pc/E.
Find the speed and momentum (in GeV/c) of a proton whose total energy is 3.500 GeV.
Find the total energy of a neutron (m = 0.940 GeWc?) whose momentum is 1.200 GeWc.
A particle has a kinetic energy of 62 MeV and a momentum of 335 MeWec. Find its mass (in MeWe?) and speed (as a fraction ofc).
(a) Find the mass (in GeV?) of a particle whose total energy is 4.00 GeV and whose momentum js 1.45 GeWe. (b) Find the total energy of this particle in a reference frame in which its momentum is 2.00 GeWc.
Appendix I: The Lorentz Transformation
51.
52.
53.
34.
55.
An observer detects two explosions, one that occurs near her at a certain time and another that occurs 2.00 ms later 100 km away. Another observer finds that the two explosions occur at the same place. What time interval separates the explosions to the second observer?
An observer detects two explosions that occur at the same time, one near her and the other 100 km away. Another observer finds that the two explosions occur 160 km apart. What time interval separates the explosions to the second observer?
A spacecraft moving in the +x direction receives a light sig- nal from a source in the xy plane. In the reference frame of the fixed stars, the speed of the spacecraft is v and the signal arrives at an angle 6 to the axis of the spacecraft. (a) With the help of the Lorentz transformation find the angle 6’ at which the signal arrives in the reference frame of the space- craft. (b) What would you conclude from this result about the view of the stars from-a porthole on the side of the spacecraft?
A body moving at 0.500¢ with respect to an observer disinte- grates into two fragments that move in opposite directions rela- tive to their center of mass along the same line of motion as the original body. One fragment has a velocity of 0.600c in the backward direction relative to the center of mass and the other has a velocity of 0.500¢ in the forward direction. What veloci-
“ties will the observer find?
A man on the moon sees two spacecraft, A and B, coming to- ward him from opposite directions at the respective speeds of 0,800c and 0.900c. (a) What does a man on A measure for the speed with which he is approaching the moon? For the speed with which he is approaching B? (b) What does a man on
B measure for the speed with which he is approaching the moon? For the :peed with which he is approaching A?
An electron whose speed relative to an observer ina laboratory is 0.800c is also being studied by an observer moving in the same direction as the electron at a speed of 0.500c relative to the laboratory. What is the kinetic energy (in MeV) of the elec- tron to each observer? z
21
22
2.3
24
52
” Eiee
The penetrating ability of x-rays enabled them to reveal the frog which this snake had
swallowed. The snake's jaws are very loosely joined
ELECTROMAGNETIC WAVES Coupled electric and magnetic oscillations that move with the speed of light and exhibit typical wave behavior
BLACKBODY RADIATION Only the quantum theory of light can explain its origin
PHOTOELECTRIC EFFECT The energies of electrons liberated by light depend on the frequency of the light
WHAT IS LIGHT? Both wave and particle
2.5
2.6
27
2.8
2.9
and so can open widely.
X-RAYS
They consist of high-energy photons X-RAY DIFFRACTION
How x-ray wavelengths can be determined
COMPTON EFFECT Further confirmation of the photon model
PAIR PRODUCTION Energy into matter
PHOTONS AND GRAVITY Although they lack rest mass, photons behave though they have gravitational mass
: Particle Properties of Waves 53 TT nner ES
n our everyday experience there is nothing mysterious or ambiguous about the T concepts of particle and wave. A stone dropped into a lake and the tipples that
spread out from its point of impact apparently have in common only the ability to carry energy and momentum from one place to another. Classical physics, which mirrors the “physical reality” of our sense impressions, treats particles and waves as separate components of that reality. The mechanics of particles and the optics of waves are traditionally independent disciplines, each with its own chain of experiments and principles based on their results,
The physical reality we perceive has its roots in the microscopic world of atoms and molecules, electrons and nuclei, but in this world there are neither particles nor waves in our sense of these terms, We regard electrons as particles because they possess charge and mass and behave according to the laws of particle mechanics in such familiar de- vices as television picture tubes. We shall see, however, that it is just as correct to in- terpret a moving electron as a wave manifestation as it is to interpret it as a particle manifestation. We regard electromagnetic waves as waves because under suitable cir- cumstances they exhibit diffraction, interference, and polarization. Similarly, we shall see that under other circumstances electromagnetic waves behave as though they con- sist of streams of particles. Together with special relativity, the wave-particle duality is central to an understanding of modern physics, and in this book there are few argu- ments that do not draw upon either or both of these fundamental ideas.
2.1. ELECTROMAGNETIC WAVES
Coupled electric and magnetic oscillations that move with the speed of light and exhibit typical wave behavior~
In 1864 the British physicist James Clerk Maxwell made the remarkable suggestion that accelerated electric charges generate linked electric and magnetic disturbances that can travel indefinitely through space. If the charges oscillate periodically, the distur- bances are waves whose electric and magnetic components are perpendicular to each other and to the direction of propagation, as in Fig. 2.1,
From the earlier work of Faraday, Maxwell knew that a changing magnetic field can induce a current in a wire loop. Thus a changing magnetic field is equivalent in its effects to an electric field. Maxwell proposed the converse: a changing electric field has a magnetic field associated with it. The electric fields produced by electromagnetic induction ate easy to demonstrate because metals offer little resistance to the flow of charge. Even a weak field can lead to a measurable current in a metal. Weak magnetic fields are much harder to detect, however, and Maxwell's hypothesis was based on a symmetry argument rather than on experimental findings.
Electric field
» Direction of wave
Magnetic field
Figure 2.1 The electric and magnetic fields in an electromagnetic wave vary together. The fields are perpendicular to each other and to the direction of propagation of the wave.
54 Chapter Two
re
James Clerk Maxwell (1831- 1879) was born in Scotland shortly before Michael Faraday discovered electromagnetic induc- tion. At nineteen he entered Cam- bridge University to study physics and mathematics. While still a stu- dent, he investigated the physics of color vision and later used his ideas to make the first color pho- tograph. Maxwell became known to the scientific world at twenty-four when he showed that the rings of Saturn could not be solid or liquid but must consist of separate small bodies. At about this time Maxwell became in- terested in electricity and magnetism and grew convinced that the wealth of phenomena Faraday and others had discovered were not isolated effects but had an underlying unity of some kind. Maxwell’ initial step in establishing that unity came in 1856 with the paper “On Faraday’s Lines of Force,” in which he developed a mathematical description of electric and mag- netic fields.
Maxwell left Cambridge in 1856 to teach at a college in Scotland and later at King’s College in London. In this period he expanded his ideas on electricity and magnetism to create a single comprehensive theory of electromagnetism. The funda- mental equations he arrived at remain the foundations of the subject today, From these equations Maxwell predicted that electromagnetic waves should exist that travel with the speed
of light, described the properties the waves should have, and surmised that light consisted of electromagnetic waves. Sadly, he did not live to see his work confirmed in the experiments of the German physicist Heinrich Hertz.
Maxwells contributions to kinetic theory and statistical mechanics were on the same profound level as his contribu- tions to electromagnetic theory. His calculations showed that the viscosity of a gas ought to be independent of its pressure, a surprising result that Maxwell, with the help of his wife, con- firmed in the laboratory. They also found that the viscosity was proportional to the absolute temperature of the gas. Maxwell's explanation for this proportionality gave him a way to estimate the size and mass of molecules, which until then could only be guessed at. Maxwell shares with Boltzmann credit for the equa- tion that gives the distribution of molecular energies in a gas.
In 1865 Maxwell returned to his family’s home in Scotland. There he continued his research and also composed a treatise on electromagnetism that was to be the standard text on the subject for many decades. It was still in print a century later. In 1871 Maxwell went back to Cambridge to establish and direct the Cavendish Laboratory, named in honor of the pio- neering physicist Henry Cavendish. Maxwell died of cancer at the age of forty-eight in 1879, the year in which Albert Ein- stein was born. Maxwell had been the greatest theoretical physi- cist of the nineteenth century; Einstein was to be the greatest theoretical physicist of the twentieth century. By a similar coincidence, Newton was born in the year of Galileo's death.)
If Maxwell was right, electromagnetic (em) waves must occur in which constantly
varying electric and magnetic fields are coupled together by both electromagnetic in- duction and the converse mechanism he proposed. Maxwell was able to show that the speed c of electromagnetic waves in free space is given by
= 2.998 X 10° m/s
V €o#o0
where €p is the electric permittivity of free space and to is its magnetic permeability. This is the same as the speed of light waves. The correspondence was too great to be accidental, and Maxwell concluded that light consists of electromagnetic waves. -
During Maxwell’ lifetime the notion of em waves remained without direct experi- mental support. Finally, in 1888, the German physicist Heinrich Hertz showed that em waves indeed exist and behave exactly as Maxwell had predicted. Hertz generated the waves by applying an alternating current to an air gap between two metal balls. The width of the gap was such that a spark occurred each time the current reached a peak. A wire loop with a small gap was the detector; em waves set up oscillations in the loop that produced sparks in the gap. Hertz determined the wavelength and speed of the waves he generated, showed that they have both electric and magnetic components, and found that they could be reflected, refracted, and diffracted.
Light is not the only example of an em wave. Although all such waves have the same fundamental nature, many features of their interaction with matter depend upon
Particle Properties of Waves
Frequency, Photon Wavelength, Hz energy, eV Radiation m 22.1 ete 10 107 Boo 19 B+ 1074 g 10? (1 Mev)10° &e (1pm) 107+ 10! 10° ig N+ 018 10* 10714 | (1 key) 10° (nm) 10° 4 107 ioe 4 16. 1 10 107 4 10!34 | seo | ol 1 Visible (pm) 1076 | 107? 10° 1034 10? 107 10 4 10"4 103 (1mm) 10 4 10+ (lem) 107 4 10104 . 10° be 107 4 (1 GHz) 10° + ape ie 10° 107 i g io 4 -8 ag 2 4 (1 MHz) 108 i 2 oes | 109 19 : Standard km) 1 . 1910 * broadcast 10° 4 4 c ‘ 10 to! sisi 10° (kez) 103 1 |
Figure 2.2 The spectrum of electromagnetic radiation.
their frequencies, Light waves, which are em waves the eye responds to, span only a brief frequency interval, from about 4.3 X 10" Hz for red light to about 7.5 X 10" Hz for violet light. Figure 2.2 shows the em wave spectrum from the low frequencies used in radio communication to the high frequencies found in x-Tays and gamma rays.
A characteristic property of all waves is that they obey the principle of superposition:
When two or more waves of the same nature travel past a point at the same time, the instantaneous amplitude there is the sum of the instantaneous amplitudes of the individual waves.
Instantaneous amplitude refers to the value at a certain place and time of the quan- tity whose variations constitute the wave. (‘Amplitude” without qualification refers to the maximum value of the wave variable.) Thus the instantaneous amplitude of a wave in a stretched string is the displacement of the string from its normal position; that of a water wave is the height of the water surface relative to its normal level: that of a sound wave is the change in pressure relative to the normal pressure. Since the elec- tric and magnetic fields in a light wave are related by E = cB, its instantaneous amplitude can be taken as either E or B. Usually E is used, since it is the electric fields of light waves whose interactions with matter give rise to nearly all common optical effects,
5
5
56 Chapter Two :
The interference of water waves. Constructive interference occurs along the line AB and destructive interference occurs along the line CD.
When two or more trains of light waves meet in a region, they interfere to produce a new wave there whose instantaneous amplitude is the sum of those of the original waves. Constructive interference refers to the reinforcement of waves with the same phase to produce a greater amplitude, and destructive interference refers to the partial or complete cancellation of waves whose phases differ (Fig. 2.3). If the original waves have different frequencies, the result will be a mixture of constructive and destructive interference, as in Fig. 3.4.
The interference of light waves was first demonstrated in 1801 by Thomas Young, who used a pair of slits illuminated by monochromatic light from a single source (Fig. 2.4). From each slit secondary waves spread out as though originating at the slit; this is an ex- ample of diffraction, which, like interference, is a characteristic wave phenomenon. Ow- ing to interference, the screen is not evenly lit but shows a pattern of alternate bright and dark lines. At those places on the screen where the path lengths from the two slits differ by an odd number of half wavelengths Q. /2, 3A/2, 5/2, .. .), destructive inter- ference occurs and a dark line is the result. At those places where the path lengths are
(a) (b)
Figure 2.3 (@) In constructive interference, superposed waves in phase reinforce each other. (4) In destructive interference, waves out of phase partially or completely cancel each other.
Particle Properties of Waves 57
“”O Constructive interference
produces bright line Destructive Monochromatic © interference light source produces dark line
\ly ; —e ‘@) Constructive 7\\ interference
produces bright line
Appearance of screen
Figure 2.4 Origin of the interference pattern in Young’ experiment. Constructive interference occurs where the difference in path lengths from the slits to the screen is @,A, 2A, . . . . Destructive interference occurs where the path difference is A/2, 34/2, 5A/2,....
equal or differ by a whole number of wavelengths (A, 2A, 3A, . . .), constructive inter- ference occurs and a bright line is the result. At intermediate places the interference is only partial, so the light intensity on the screen varies gradually between the bright and dark lines.
Interference and diffraction are found only in waves—the particles we are familiar with do not behave in those ways. If light consisted of a stream of classical particles, the entire screen would be dark. Thus Young’ experiment is proof that light consists of waves. Maxwell's theory further tells us what kind of waves they are: electromag- netic. Until the end of the nineteenth century the nature of light seemed settled forever.
2.2 BLACKBODY RADIATION Only the quantum theory of light can explain its origin
Following Hertz’s experiments, the question of the fundamental nature of light seemed clear: light consisted of em waves that obeyed Maxwell's theory. This cer- tainty lasted only a dozen years, The first sign that something was seriously amiss came from attempts to understand the origin of the radiation emitted by bodies of matter.
We are all familiar with the glow of a hot piece of metal, which gives off visible light whose color varies with the temperature of the metal, going from red to yellow to white as it becomes hotter and hotter. In fact, other frequencies to which our eyes do not respond are present as well. An object need not be so hot that it is luminous for it to be radiating em energy; all objects radiate such energy continuously whatever their temperatures, though which frequencies predominate depends on the temperature. At Toom temperature most of the radiation is in the infrared part of the spectrum and hence is invisible.
The ability of a body to radiate is closely related to its ability to absorb radiation. This is to be expected, since a body at a constant temperature is in thermal equilib- tium with its surroundings and must absorb energy from them at the same rate as it emits energy. It is convenient to consider as an ideal body one that absorbs all radi- ation incident upon it, regardless of frequency. Such a body is called a blackbody.
The point of introducing the idealized blackbody in a discussion of thermal ra- diation is that we can now disregard the precise nature of whatever is radiating, since
58 Chapter Two
all blackbodies behave identically. In the laboratory a blackbody can be approximated by a hollow object with a very small hole leading to its interior (Fig. 2.5). Any ra- diation striking the hole enters the cavity, where it is trapped by reflection back and forth until it is absorbed. The cavity walls are constantly emitting and absorbing ra- diation, and it is in the properties of this radiation (blackbody radiation) that we are interested. Experimentally we can sample blackbody radiation simply by inspecting what ; ; emerges fromi the hole in the cavity. The results agree with everyday experience. A ae ee blackbody radiates more when it is hot than when it is cold, and the spectrum’ of a proximation of a blackbody P- hot blackbody has its peak at a higher frequency than the peak in the spectrum ofa cooler one. We recall the behavior of an iron bar as it is heated to progressively higher temperatures: at first it glows dull red, then bright orange-red, and eventually it be- comes “white hot.” The spectrum of blackbody radiation is shown in Fig. 2.6 for two temperatures.
The Ultraviolet Catastrophe
Why does the blackbody spectrum have the shape shown in Fig. 2.6? This prob- lem was examined at the end of the nineteenth century by Lord Rayleigh and James Jeans. The details of their calculation are given in Chap. 9. They started by con- sidering the radiation inside a cavity of absolute temperature T whose walls are perfect reflectors to be a series of standing em waves (Fig. 2.7). This is a three- dimensional generalization of standing waves in a stretched string. The condition
T= 1800K
Spectral energy density, u(u)dv
0 2x10 4x10! 6x10%Hz Sitio? The color and brightness of an Visible light object heated until it glows, such as the filament of this light bulb, depends upon its tempetature, which here is about 3000 K. An Figure 2.6 Blackbody spectra. The spectral distribution of energy in the radiation depends only or object that glows white is hotter the temperature of the body. The higher the temperature, the greatet the amount of radiation and the than it is when it glows red, and higher the frequency at which the maximum emission occurs. The dependence of the latter frequency it gives off more light as well. on temperature follows a formula called Wien’ displacement law, which is discussed in Sec. 9.6.
Frequency, v
Particle Properties of Waves 59 F
for standing waves in such a cavity is that the path length from wall to wall, whatever the direction, must be a whole number of half-wavelengths, so that a node occurs at each reflecting surface. The number of independent standing waves G(v)dy in the frequency interval between vy and dy per unit volume in the cavity turned out to be s
Density of standing 8av7dv
waves in cavity GO)dv = C (2.1)
This formula is independent of the shape of the cavity. As we would expect, the higher the frequency », the shorter the wavelength and the greater the number of possible standing waves.
The next step is to find the average energy per standing wave. According to the theorem of equipartition of energy, a mainstay of classical physics, the average energy ES) per degree of freedom of an eritity (such as a molecule of an ideal gas) that is a mem- , ber of a system of such entities in thermal equilibrium at the temperature T is kT. kL ——> Here k is Boltzmann's constant:
Figure 2.7 Em radiation in a cav-
ity whose walls are perfect reflec-
tors consists of standing waves
that have nodes at the walls,
A degree of freedom is a mode of energy possession. Thus a monatomic ideal gas which restricts their possible
molecule has three degrees of freedom, corresponding to kinetic energy of motion in Wavelengths. Shown are three 4 aie 3 possible wavelengths when the
three independent directions, for an average total energy of 2kT. distance between opposite walls
A one-dimensional harmonic oscillator has two degrees of freedom, one that corre- wi
sponds to its kinetic energy and one that corresponds to its potential energy. Because
each standing wave in a cavity originates.in an oscillating electric charge in the cavity
wall, two degrees of freedom are associated with the wave and it should have an average
energy of 2()kT:
Boltzmann's constant k= 1381 X10 4K
Classical average energy et per standing wave e=kT (2.2)
The total energy u(v) dv per unit volume in the cavity in the frequency interval from v to v + dv is therefore
Rayleigh-Jeans = _ 8akT formula u(v) dv = €G(v) dv = 3 Pdvi (2.3)
This radiation rate is proportional to this energy density for frequencies between v and vy + dv. Equation (2.3), the Rayleigh-Jeans formula, contains everything that classi- cal physics can say about the spectrum of blackbody radiation.
Even a glance at Eq. (2.3) shows that it cannot possibly be correct. As the fre- quency v increases toward the ultraviolet end of the spectrum, this formula predicts that the energy density should increase as v7. In the limit of infinitely high fre- quencies, u(v) dy therefore should also go to infinity. In reality, of course, the energy density (and radiation rate) falls to 0 as » —> « (Fig. 2.8). This discrepancy became known as the ultraviolet catastrophe of classical physics. Where did Rayleigh and Jeans go wrong?
60 Chapter Two
a Lh
i i Rayleigh-Jeans
Spectral energy density, u(v)dv
2x104 3x10 = 4x10"4
Frequency, v (Hz)
0 1x10%
Figure 2.8 Comparison of the Rayleigh-Jeans formula for the spectrum of the radiation from a black- body at 1500 K with the observed spectrum, The discrepancy is known as the ultraviolet catastrophe because it increases with increasing frequency. This failure of classical physics led Planck to the dis- covery that radiation is emitted in quanta whose energy is hv.
Planck Radiation Formula
In 1900 the German physicist Max Planck used “lucky guesswork” (as he later called it)
to come up with a formula for the spectral energy density of blackbody radiation:
Planck radiation formula
8rh_ v dv u(y) dv = 2 PM] (2.4)
Here h is a constant whose value is
Planck’s constant
h = 6.626 X 10*J+s
Max Planck (1858-1947) was bom in Kiel and educated in Mu- nich and Berlin. At the University of Berlin he studied under Kirch- hoff and Helmholtz, as Hertz had done earlier. Planck realized that blackbody radiation was important because it was a fundamental effect independent of atomic structure, which was still a mystery in the late nineteenth century, and worked at understanding it for six years be-
* fore finding the formula the radiation obeyed. He “strived from
the day of its discovery to give it a real physical interpretation.” The result was the discovery that radiation is emitted in energy steps of hy. Although this discovery, for which he received the Nobel Prize in 1918, is now considered to mark the start of
modem physics, Planck himself remained skeptical for a long time of the physical reality of quanta. As he later wrote, “My vain attempts to somehow reconcile the elementary quantum with classical theory continued for many years and cost me great effort. . . . Now I know for certain that the quantum of action has a much more fundamental significance than I orig- inally suspected.”
Like many physicists, Planck was a competent musician (he sometimes played with Einstein) and in addition enjoyed moun- tain climbing, Although Planck remained in Germany during the Hitler era, he protested the Nazi treatment of Jewish scien- tists and lost his presidency of the Kaiser Wilhelm Institute as a result. In 1945 one of his sons was implicated in a plot to kill Hitler and was executed. After World War II the Institute was renamed after Planck and he was again its head until his death.
Particle Properties of Waves 61
At high frequencies, hy >> kT and e*”/*T +00, which means that u(y) dv—>0 as observed. No more ultraviolet catastrophe. At low frequencies, where the Rayleigh- Jeans formula is a good approximation to the data (see Fig. 2.8), hv < kT and hy/kT <1. In general,
rx Fah epi Ses a SA TE a
If x is small, e* + 1 + x, and so for hv/kT <1 we have
1 1 kT Pry ~ be fad hp hy < kT
Thus at low frequencies Planck’ formula becomes
dp
uly) dv = oat (5) dp = Sapl.
hv oo”
which is the Rayleigh-Jeans formula. Planck's formula is clearly at least on the right track; in fact, it has turned out to be completely correct.
Next Planck had the problem of justifying Eq. (2.4) in terms of physical principles. A new principle seemed needed to explain his formula, but what was it? After several weeks of “the most strenuous work of my life,” Planck found the answer: The oscilla- tors in thé cavity walls could not have a continuous distribution of possible energies € but must have only the specific energies _.
Oscillator energies €, = nhv n=0,1,2,... (2.5)
- An oscillator emits radiation of frequency » when it drops from one energy state to the next lower one, and it jumps to the next higher state when it absorbs radiation of frequency v. Each discrete bundle of energy hy is called a quantum (plural quanta) from the Latin for “how much.”
With oscillator energies limited to nhy, the average energy per oscillator in the cavity walls—and so per standing wave—turned out to be not € = kT as for a continuous distribution of oscillator energies, but instead
Actual average energy = hy per standing wave cm goat (2.6)
This average energy leads to Eq. (2.4). Blackbody radiation is further discussed in Chap. 9.
eee Example 2.1 ;
Assume that a certain 660-Hz tuning fork can be considered as a harmonic oscillator whose vi- brational energy is 0.04 J. Compare the energy quanta of this tuning fork with those of an atomic oscillator that emits and absorbs orange light whose frequency is 5.00 X 10" Hz.
62 Chapter Two
Solution
(a) For the tuning fork, hy, = 6.63 X 10" J +s) (660 5-4) = 4.38 X 10°77 J
The total energy of the vibrating tines of the fork is therefore about 10”? times the quantum energy hv. The quantization of energy in the tuning fork is obviously far too small to be observed, and we are justified in regarding the fork as obeying classical physics. }
(b) For the atomic oscillator, hv, = (6.63 X 10724 J + s) (5.00 X 10" s“) = 3,32 X 109 J
In electronvolts, the usual energy unit in atomic physics,
3.32 X 1079 J
hv, = 1.60 X 107? Jev
= 2.08 eV
This is a significant amount of energy on an atomic scale, and it is not surprising that classical physics fails to account for phenomena on this scale. TS
The concept that the oscillators in the cavity walls can interchange energy with standing waves in the cavity only in quanta of hy is, from the point of view of classi- cal physics, impossible to understand. Planck regarded his quantum hypothesis as an “act of desperation” and, along with other physicists of his time, was unsure of how seriously to regard it as an element of physical reality. For many years he held that, although the energy transfers between electric oscillators and em waves apparently are quantized, em waves themselves behave in an entirely classical way with a continuous range of possible energies.
2.3 PHOTOELECTRIC EFFECT
The energies of electrons liberated by light depend on the frequency of the light
During his experiments on em waves, Hertz noticed that sparks occurred more readily in the air gap of his transmitter when ultraviolet light was directed at one of the metal balls. He did not follow up this observation, but others did. They soon discovered that the cause was electrons emitted when the frequency of the light was sufficiently high. This phe- nomenon is known as the photoelectric effect and the emitted electrons are called pho- toelectrons, It is one of the ironies of history that the same work to demonstrate that light consists of em waves also gave the first hint that this was not the whole story.
Figure 2.9 shows how the photoelectric effect was studied. An evacuated tube con- tains two electrodes connected to a source of variable voltage, with the metal plate whose surface is irradiated as the anode. Some of the photoelectrons that emerge from this sur- face have enough energy to reach the cathode despite its negative polarity, and they con- stitute the measured current. The slower photoelectrons are repelled before they get to the cathode. When the voltage i$ increased to a certain value Vo, of the order of several volts, no more photoelectrons arrive, as indicated by the current dropping to zero. This extinction voltage corresponds to the maximum photoelectron kinetic energy.
Particle Properties of Waves 63
Electrons ee
Evacuated quartz tube
: ‘Frequency ap
—— |
Figure 2.9 Experimental observation of the photoelectric effect.
Photoelectron current
The existence of the photoelectric effect is not surprising. After all, light waves carry energy, and some of the energy absorbed by the metal may somehow concentrate on individual electrons and reappear as their kinetic energy. The situation should be like water waves dislodging pebbles from a beach. But three experimental findings show that no such simple explanation is possible. Figure 2.10 Photoelectron cur-
= tent is proportional to light in- 1 Within the limits of experimental accuracy (about 107? s), there is no time interval tensity J for all retarding voltages. between the arrival of light at a metal surface and the emission of photoelectrons. How- The stopping potential Vo, which ever, because the energy in an em wave is supposed to be spread across the wavefronts, C"esponds to the maximum . i ge ape photoelectron energy, is the same a period of time should elapse before an individual electron accumulates enough ENETBY for all intensities of light of the (several eV) to leave the metal. A detectable photoelectron current results when 107° same frequency v. W/m? of em energy is absorbed by a sodium surface. A layer of sodium 1 atom thick and 1 m? in area contains about 10" atoms, so if the incident light is absorbed in the uppermost atomic layer, each atom receives energy at an average rate of 10-29 W At this rate over a month would be needed for an atom to accumulate energy of the mag- nitude that photoelectrons from a sodium surface are observed to have. 2 A bright light yields more photoelectrons than a dim one of the same frequency, but the electron energies remain the same. (Fig. 2.10). The em theory of light, on the con- trary, predicts that the more intense the light, the greater the energies of the electrons. 3 The higher the frequency of the light, the more energy the photoelectrons have (Fig, 2.11), Blue light results in faster electrons than red light. At frequencies below a certain critical frequency v9, which is characteristic of each particular metal, no elec-
Retarding potential
“Light intensity... : “oo co= constant SU > Uy > U3 |
Photoelectron current
0 Vp@G) VQ) Wav
trons are emitted. Above vp the photoelectrons range in energy from 0 to a maximum Retarding potential value that increases linearly with increasing frequency (Fig. 2.12). This observation, also, cannot be explained by the em theory of light. Figure 2.11 The stopping poten-
tial Vo, and hence the maximum photoelectron energy, depends on the frequency of the light. When the retarding potential is V = 0, the photoelectron current is the When Planck’s derivation of his formula appeared, Einstein was one of the first— same for light of a given intensity
perhaps the first—to understand just how radical the postulate of energy quantization regardless of its frequency.
.Quantum Theory of Light
Chapter Two
ee
Calcium
Maximum photoelectron Energy, eV
0 2 4 6 8 Ba 12X10" Frequency, Hz
Figure 2.12 Maximum photoelectron kinetic energy KEmax versus frequency of incident light for three metal surfaces.
of oscillators was: “It was as if the ground was pulled from under one.” A few years later, in 1905, Einstein realized that the photoelectric effect could be understood if the energy in light is not spread out over wavefronts but is concentrated in small packets, or photons. (The term photon was coined by the chemist Gilbert Lewis in 1926.) Each photon of light of frequency v has the energy hv, the same as Planck's quantum energy. Planck had thought that, although energy from an electric oscillator apparently had to be given to em waves in separate quanta of hy each, the waves themselves behaved exactly as in conventional wave theory. Einstein's break with classical physics was more drastic: Energy was not only given to em waves in separate quanta but was also car- ried by the waves in separate quanta.
The three experimental observations listed above follow directly from Einstein’s hy- pothesis. (1) Because em wave energy is concentrated in photons and not spread out, there should be no delay in the emission of photoelectrons. (2) All photons of fre- quency v have the same energy, so changing the intensity of a monochromatic light beam will change the number of photoelectrons but not their energies. (3) The higher the frequency », the greater the photon energy hy and so the more energy the photo- electrons have.
What is the meaning of the critical frequency % below which no photoelectrons are emitted? There must be a minimum energy ¢ for an electron to escape from a partic- ular metal surface or else electrons would pour out all the time. This energy is called the work function of the metal, and is related to v9 by the formula
Work function d= hyo (2.7)
The greater the work function of a metal, the more energy is needed for an electron to leave its surface, and the higher the critical frequency for photoelectric emission to occur.
Some examples of photoelectric work functions are given in Table 2.1. To pull an electron from a metal surface generally takes about half as much energy as that needed
Particle Properties of Waves
Table 2.1 Photoelectric Work Functions ete ic pea ee arena ae SE a
Metal Symbol Work Function, eV Cesiura Cs 19 Potassium K 22 Sodium Na 23 Lithium li 25 Calcium Ca 3.2 Copper Cu 47 Silver Ag 47 Platinum Pt 6.4
a a re
to pull an electron from a free atom of that metal (see Fig. 7.10); for instance, the ionization energy of cesium is 3.9 eV compared with its work function of 1.9 eV. Since the visible spectrum extends from about 4.3 to about 7.5 X 10!* Hz, which corre- sponds to quantum energies of 1.7 to 3.3 eV, it is clear from Table 2.1 that the pho- toelectric effect is a phenomenon of the visible and ultraviolet tegions,
According to Einstein, the photoelectric effect in a given metal should obey the equation
Photoelectric effect hv = KEmax + & (2.8)
where hy is the photon energy, KEmax is the maximum photoelectron energy (which is proportional to the stopping potential), and ¢ is the minimum energy needed for an
All light-sensitive detectors, including the eye and the one used in this video camera, are based on the absorption of energy from photons of light by electrons in the atoms the light falls on.
Chapter Two
TS
E=hug E=hv
KE pax = he — hug
KE=0 ; I ost. Metal
Figure 2.13 If the energy hv (the work function of the surface) is needed to remove an electron from a metal surface, the maximum electron kinetic energy will be hy — hvo when light of frequency v is directed at the surface. .
electron to leave the metal. Because @ = hyo, Eq. (2.8) can be rewritten (Fig. 2.13)
hy = KEmax + h¥o KE ax = hv ~ hv = h(v — Yo) (2.9)
This formula accounts for the relationships between KEmax and v plotted in Fig. 2.12 from experimental data. If Einstein was right, the slopes of the lines should all be equal to Planck's constant h, and this is indeed the case. In terms of electronvolts, the formula E = hy for photon energy becomes Photon 6.626 X 10 **J +s ) i . {2a “n 2? |, = 4, x 107! : 2. energy ( 1.602 X 10" ev v = (4.136 X 10°? )veV:s (2.10)
If we are given instead the wavelength A of the light, then since v = c/A we have
Photon 4.136 X 107} eV « s)(2.998 X 10% mvs) _ 1.240 X 107° eV m energy Ss ze a (2.11)
TL Sn: Example 2.2
Ultraviolet light of wavelength 350 nm and intensity 1.00 W/m? is directed at a potassium sur- face. (a) Find the maximum KE of the photoelectrons. (b) If 0.50 percent of the incident pho- tons produce photoelectrons, how many are emitted per second if the potassium surface has an area of 1,00 cm?? ;
Solution (a) From Eq. (2.11) the energy of the photons is, since 1 nm = 1 nanometer = 10° m,
1.24 X 10°%eV-m
= oO 3 ? (350 nm)(10~° m/nm)
: Particle Properties of Waves 67 ee ES OE
Table 2.1 gives the work function of potassium as 2.2 eV, so KEnax = hv ~ 6 = 3.5 eV —-2.2eV= 13 eV
{b) The photon energy in joules is 5.68 X 10719 J. Hence the number of photons that reach the surface per second is
— Elt _ @/AA) _ (1.00 Wim?) (1.00 x 107* m2)
PE E, 5.68 X 107” J/photon
= 1.76 X 10™ photons/s
The rate at which photoelectrons are emitted is therefore
n, = (0.0050)n, = 8.8 X 10" photoelectrons/s
saat epeeteeenenmeneeemeneener en
Thermionic Emission
E instein’s interpretation of the photoelectric effect is supported by studies of thermionic emis- sion. Long ago it was discovered that the presence of a very hot object increases the elec- tric conductivity of the surrounding air. Eventually the reason for this effect was found to be the emission of electrons from such an object. Thermionic emission makes possible the operation of such devices as television picture tubes, in which metal filaments or specially coated cathodes at high temperature supply dense streams of electrons.
The emitted electrons evidently obtain their energy from the thermal agitation of the parti- cles of the metal, and we would expect the electrons to need a certain minimum energy to escape. This minimum energy can be determined for many surfaces, and it is always close to the photoelectric work function for the same surfaces. In photoelectric emission, photons of light provide the energy required by an electroti to escape, while in thermionic emission heat does so.
7 ) ; Bees x} | 2.4 WHAT IS LIGHT? ; i cies
Both wave and particle
The concept that light travels as a series of little packets is directly opposed to the wave theory of light (Fig. 2.14). Both views have strong experimental support, as we have
seen. According to the wave theory, light waves leave a source with their energy spread
out continuously through the wave pattern. According to the quantum theory, light ff consists of individual photons, each small enough to be absorbed by a single electron. } Yet, despite the particle picture of light it presents, the quantum theory needs the fre- } quency of the light to describe the photon energy.
Which theory are we to believe? A great many scientific ideas have had to be re- vised or discarded when they were found to disagree with new data. Here, for the first time, two different theories are needed to explain a single phenomenon. This situation 0) is not the same as it is, say, in the case of relativistic yetsus newtonian mechanics, where one turns out to be an approximation of the other. The connection between the wave Figure 2.14 (a) The wave theory and quantum theories of light is something else entirely. of light explains diffraction and
To appreciate this connection, let us consider the formation of a double-slit in- _*etference, which the quantum
PIS os Ons theory cannot account for. (b) The terference pattern on a screen. In the wave model, the light intensity at a place on quantum theory explains the pho- the screen depends on E*, the average over a complete cycle of the square of the in- — toelectric effect, which the wave Stantaneous magnitude E of the em waves electric field. In the particle model, this _ theory cannot account for.
68 Chapter Two ; intensity depends instead on Nhv, where N is the number of photons per second per unit area that reach the same place on the screen. Both descriptions must give the same value for the intensity, so N is proportional to E* If N is large enough, somebody looking at the screen would see the usual double-slit interference pat- tern and would have no reason to doubt the wave model. If N is small—perhaps so small that only one photon at a time reaches the screen—the observer would find a series of ‘apparently random flashes and would assume that he or she is watch- ing quantum behavior.
If the observer keeps track of the flashes for long enough, though, the pattern they form will be the same as when N is large. Thus the observer is entitled to conclude that the probability of finding a photon at a certain place and time depends on the value of E* there. If we regard each photon as somehow having a wave associated with it, the intensity of this wave at a given place on the screen détermines the likelihood that a photon will arrive there. When it passes through the slits, light is behaving as a wave does. When it strikes the screen, light is behaving as a particle does. Apparently light travels as a wave but absorbs and gives off energy as a series of particles.
We can think of light as having a dual character. The wave theory and the quan- tum theory complement each other, Either theory by itself is only part of the story and can explain only certain effects. A reader who finds it hard to understand how light can be both a wave and a stream of particles is in good company: shortly before his death, Einstein remarked that “All these fifty years of conscious brooding have brought me no nearer to the answer to the question, ‘What are light quanta?” The “true nature” of light includes both wave and particle characters, even though there is nothing in everyday life to help us visualize that.
2.5 X-RAYS They consist of high-energy photons
The photoelectric effect provides convincing evidence that photons of light can transfer energy to electrons. Is the inverse process also possible? That is, can part or all of the kinetic energy of a moving electron be converted into a photon? As it happens, the in- verse photoelectric effect not only does occur but had been discovered (though not understood) before the work of Planck and Einstein. .
In 1895 Wilhelm Roentgen found that a highly penetrating radiation of unknown nature is produced when fast electrons impinge on matter. These x-rays were soon found to travel in straight lines, to be unaffected by electric and magnetic fields, to pass readily through opaque materials, to cause phosphorescent substances to glow, and to expose photographic plates. The faster the original electrons, the more pene- trating the resulting x-rays, and the greater the number of electrons, the greater the in- tensity of the x-ray beam.
Not long after this discovery it became clear that x-rays are em waves. Electro- magnetic theory predicts that an accelerated electric charge will radiate em waves, and a rapidly moving electron suddenly brought to rest is certainly accelerated. Ra- diation produced under these circumstances is given the German name bremsstrahlung (“braking radiation”). Energy loss due to bremsstrahlung is more important for electrons than for heavier particles because electrons are more violently accelerated when passing near nuclei in their paths. The greater the energy of an electron’and the greater the atomic number of the nuclei it encounters, the more en- ergetic the bremsstrahlung.
Particle Properties of Waves 69
A rere
Wilhelm Konrad Roentgen _are accelerated in a vacuum by an electric field, and it was (1845-1923) was born in Lennep, the impact of these electrons on the glass end of the tube that Germany, and studied in Holland —_ produced the penetrating “x” (since their nature was then and Switzerland. After periods at unknown) rays that caused the salt to glow, Roentgen said of several German universities, _ his discovery that, when people heard of it, they would say, Roentgen became professor of “Roentgen has probably gone crazy.” In fact, x-rays were an physics at Wurzburg where, on immediate sensation, and only two months later were being November 8, 1895, he noticed used in medicine. They also stimulated research in new di- that a sheet of paper coated with rections; Becquerel’s discovery of radioactivity followed within barium platinocyanide glowed _a year. Roentgen received the first Nobel Prize in physics in when he switched on a nearby ‘1902. He refused to benefit financially from his work and died cathode-ray tube that was entirely in poverty in the German inflation that followed the end of covered with black cardboard. Ina cathode-ray tube electrons World War I.
nn eer
In 1912 a method was devised for measuring the wavelengths of x-rays. A dif- fraction experiment had been recognized as ideal, but as we recall from physical optics, the spacing between adjacent lines on a diffraction grating must be of the same order of magnitude as the wavelength of the light for satisfactory results, and gratings cannot be ruled with the minute spacing required by x-rays. Max von Laue realized that the wavelengths suggested for x-rays were comparable to the spacing between adjacent atoms in crystals. He therefore proposed that crystals be used to diffract x-rays, with their regular lattices acting as a kind of three-dimensional grat- ing. In experiments carried out the following year, wavelengths from 0.013 to 0.048 nm were found, 10°“ of those in visible light and hence having quanta 10* times as energetic.
Electromagnetic radiation with wavelengths from about 0.01 to about 10 nm falls into the category of x-rays. The boundaries of this category are not sharp: the shorter- wavelength end overlaps gamma rays and the longer-wavelength end overlaps ultravi- olet light (see Fig, 2.2),
Figure 2.15 is a diagram of an x-ray tube. A cathode, heated by a filament through which an electric current is passed, supplies electrons by thermionic emission. The high potential difference V maintained between the cathode and a metallic tar- get accelerates the electrons toward the latter. The face of the target is at an angle relative to the electron beam, and the x-rays that leave the target pass through the
Evacuated
{ith
Figure 2.15 An x-tay tube. The higher the accelerating voltage V, the faster the electrons and the shorter the wavelengths of the x-Tays.
70
Chapter Two
In modern x-ray tubes like these, circulating oil carries heat away “from the target and releases it to the outside air through a heat exchanger. The use of x-rays as a diagnostic tool in medicine is based upon the different extents to which different tissues absorb them. Because of its calcium con- tent, bone is much more opaque to x-rays than muscle, which in turn is more opaque than fat. To enhance contrast, “meals” that con- tain barium are given to patients to better display their digestive sys- tems, and other compounds may be injected into the bloodstream to enable the condition of blood ves- sels to be studied,
Relative intensity
ie) 0.02 0.04 0.06 0.08. 0.10 Wavelength, nm.
Figure 2.16 X-ray spectra of tungsten at various accelerating potentials.
side of the tube. The tube is evacuated to permit the electrons to get to the target unimpeded.
‘As mentioned earlier, classical electromagnetic theory predicts bremsstrahlung when electrons are accelerated, which accounts in general for the x-rays produced by an x-ray tube. However, the agreement between theory and experiment is not satisfactory in cer- tain important respects. Figures 2.16 and 2.17 show the x-ray spectra that result when tungsten and molybdenum targets are bombarded by electrons at several different accel- erating potentials. The curves exhibit two features electromagnetic theory cannot explain:
1 In the case of molybdenum, intensity peaks occur that indicate the enhanced pro-
duction of x-rays at certain wavelengths. These peaks occur at specific wavelengths for each target material and originate in rearrangements of the electron structures of the
12
10
‘Tungsten, 35 ky, ie
Relative intensity an
0 0.02 0.04 0.06 0.08 0.10 Wavelength, nm
Figure 2.17 X-ray spectra of tungsten and molybdenum at 35 kV accelerating potential.
Particle Properties of Waves 71
In a CT (computerized tomography) scanner, a series of x-ray exposures of a patient taken from different directions are combined by a computer to give cross-sectional images of the parts of the body being examined. In effect, the tissue is sliced up by the gomputer on the basis of the x-ray exposures, and any desired slice can be displayed. This technique enables an abnormality to be detected and its exact location established, which might be impossible to do from an ordinary x-ray picture. (The word tomogra- phy comes from tomos, Greek for “cut.”)
target atoms after having been disturbed by the bombarding electrons. This phenom- enon will be discussed in Sec. 7.9; the important thing to note at this point is the pres- ence of x-rays of specific wavelengths, a decidedly nonclassical effect, in addition to a continuous x-ray spectrum.
2 The x-rays produced at a given accelerating potential V vary in wavelength, but none has a wavelength shorter than a certain value Amin. Increasing V decreases Amin. At a - particular V, Amin is the same for both the tungsten and molybdenum targets. Duane and Hunt found experimentally that Amin is inversely proportional to V; their precise relationship is
.24 X 10° : X-ray production Amin = a Vim (2.12)
The second observation fits in with the quantum theory of radiation. Most of the electrons that strike the target undergo numerous glancing collisions, with their energy going simply into heat. (This is why the targets in x-ray tubes are made from high- melting-point metals such as tungsten, and a means of cooling the target is usually em- ployed.) A few electrons, though, lose most or all of their energy in single collisions with target atoms. This is the energy that becomes x-rays,
X-tays production, then, except for the peaks mentioned in observation 1 above, represents an inverse photoelectric effect. Instead of photon energy being transformed into electron KE, electron KE is being transformed into photon energy. A short wave- length means a high frequency, and a high frequency means a high photon energy hy.
72
Chapter Two
Since work functions are only a few electronvolts whereas the accelerating poten- tials in x-ray tubes are typically tens or hundreds of thousands of volts, we can ignore the work function and interpret the short wavelength limit of Eq. 2.12) as corre- sponding to the case where the entire kinetic energy KE = Ve of a bombarding elec- tron is given up to a single photon of energy h¥max. Hence
he . Ve = hvmax = ae he _ 1.240 x 107° Ain = Ve = eye a Vv “m
which is the Duane-Hunt formula of Eq. (2.12)—and, indeed, the same as Eq. (2.11) except for different units. It is therefore appropriate to regard x-ray production as the inverse of the photoelectric effect.
Sn ota EERE REEERmmnammmmatnttll
Example 2.3
Find the shortest wavelength present in the radiation from an x-tay machine whose accelerat- ing potential is 50,000 V.
Solution From Eq. (2.12) we have
1.24X10°V-m = to 2 248 x 10! m= 0: Armin 500x10°V 2.48 X 10H m = 0.0248 nm
This wavelength corresponds to the frequency
c 3.00 X 10° m/s max = oa = 1.21 x 10” Pm 248 X10 m He
2.6 X-RAY DIFFRACTION
How x-ray wavelengths can be determined
A crystal consists of a regular array of atoms, each of which can scatter em waves. The mechanism of scattering is straightforward. An atom in a constant electric field be- comes polarized since its negatively charged electrons and positively charged nucleus experience forces in opposite directions. These forces are small compared with the forces holding the atom together, and so the result is a distorted charge distribution equivalent to an electric dipole. In the presence of the alternating electric field of an em wave of frequency v, the polarization changes back and forth with the same fre- quency v. An oscillating electric dipole is thus created at the expense of some of the energy of the incoming wave. The oscillating dipole in turn radiates em waves of fre- quency v, and these secondary waves go out in all directions except along the dipole axis. (In an assembly of atoms exposed to unpolarized radiation, the latter restriction does not apply since the contributions of the individual atoms are random.)
In wave terminology, the secondary waves have spherical wave fronts in place of the plane wave fronts of the incoming waves (Fig. 2.18). The scattering process, then,
Particle Properties of Waves
73
Scattered ZA waves Incident Unscattered waves waves
i
Figure 2.18 The scattering of electromagnetic radiation by a group of atoms. Incident plane waves are reemitted as spherical waves.
involves atoms that absorb incident plane waves and reemit spherical waves of the same frequency.
A monochromatic beam of x-rays that falls upon a crystal will be scattered in all di- rections inside it. However, owing to the regular arrangement of the atoms, in certain directions the scattered waves will constructively interfere with one another while in others they will destructively interfere. The atoms in a crystal may be thought of as defining families of parallel planes, as in Fig. 2.19, with each family having a charac- teristic separation between its component planes. This analysis was suggested in 1913 by WL Bragg, in honor of whom the above planes are called Bragg planes.
The conditions that must be fulfilled for radiation scattered by crystal atoms to un- dergo constructive interference may be obtained from a diagram like that in Fig. 2.20. A beam containing x-rays of wavelength A is incident upon a crystal at an angle @ with a family of Bragg planes whose spacing isd. The beam goes past atom A in the first plane and atom B in the next, and each of them scatters part of the beam in random directions. Constructive interference takes place only between those scattered rays that are parallel and whose paths differ by exactly A, 2A, 3A, and so on. That is, the path difference must be nA, where n is an integer. The only rays scattered by A and B for which this is true are those labeled I and II in Fig. 2.20.
The first condition on I and II is that their common scattering angle be equal to the angle of incidence @ of the original beam. (This condition, which is independent
Figure 2.19 Two sets of Bragg planes in a NaCl crystal.
74 Chapter Two
Path difference
=2dsin® \
°
O°
The interference pattern pro- Figure 2.20 X-ray scattering from a cubic crystal. duced by the scattering of x-rays
from ions in a crystal of NaCl. The 7
bright spots correspond to the di-
ections TeX d ‘ : i ‘ aieegra ane alee aati of wavelength, is the same as that for ordinary specular reflection in optics: angle of
interfere constructively. The cubic incidence = angle of reflection.) The second condition is that
pattern of the NaCl lattice is sug- :
gested by he fourfold symmetry 2d sin 9 = nd n=1,2,3,... (2.13) of the pattern. The large central spot is due to the unscattered
aay beam, since ray Il must travel the distance 2d sin @ farther than ray 1. The integer n is the
order of the scattered beam.
The schematic design of an x-ray spectrometer based upon Bragg’ analysis is shown in Fig. 2.21. A narrow beam of x-rays falls upon a crystal at an angle @, and a detector is placed so that it records those rays whose scattering angle is also @. Any x-rays reach- ing the detector therefore obey the first Bragg condition. As 6 is varied, the detector
at spi Detector ~
N N X-rays \
AY Crystal ;
/
Path of / Collimators: detector, al
Figure 2.21 X-ray spectrometer.
Particle Properties of Waves
will record intensity peaks corresponding to the orders predicted by Eq. (2.13). If the spacing d between adjacent Bragg planes in the crystal is known, the x-ray wavelength A may be calculated.
2.7 COMPTON EFFECT
Further confirmation of the photon model
According to the quantum theory of light, photons behave like particles except for their lack of rest mass. How far can this analogy be carried? For instance, can we consider a collision between a photoh and an electron as if both were billiard balls?
Figure 2.22 shows such a collision: an x-ray photon strikes an electron (assumed to be initially at rest in the laboratory coordinate system) and is scattered away from its original direction of motion while the electron receives an impulse and begins to move. We can think of the photon as losing an amount of energy in the collision that is the same as the kinetic energy KE gained by the electron, although actually separate photons are involved. If the initial photon has the frequency v associated with it, the scattered photon has the lower frequency pv’, where
Loss in photon energy = gain in electron energy hy — hv' = KE (2.14)
.
From Chap. 1 we recall that the momentum of a massless particle is related to its energy by the formula s
E= pe (1.25)
Since the energy of a photon is hy, its momentum is
(2.15)
Photon momentum p=
Incident photon E=hu > p=hule \9 N _ E= Vm s pe? peoet = P (a) Scattered ‘ Pee : psin€ electron ®
Figure 2.22 (a) The scattering of a photon by an electron is called the Compton effect. Energy and momentum are conserved in such an event, and as a result the scattered photon has less energy (longer wavelength) than the incident photon. (b) Vector diagram of the momenta
and their components of the incident and scattered photons and the scattered electron.
76 Chapter Two
ii c “nicdche OO Ene
Momentum, unlike energy, is a vector quantity that incorporates direction as well as magnitude, and in the collision momentum must be conserved in each of two mutually perpendicular directions. (When more than two bodies participate in a collision, momentum must be conserved in each of three mutually perpendicular directions.) The directions we choose here are that of the original photon and one perpendicular to it in the plane containing the electron and the scattered photon (Fig. 2.22).
The initial photon momentum is hv/c, the scattered photon momentum is hv'/c, and the initial and final electron momenta are respectively 0 and p. In the original photon direction
Initial momentum = final momentum
RY ga cosh + pcos 8 (2.16) c t :
and perpendicular to this direction
Initial momentum = final momentum t O= Hsing — psn 0 (2.17)
The angle ¢ is that between the directions of the initial and scattered photons, and 6 is that between the directions of the initial photon and the recoil electron. From Eqs. (2.14), (2.16), and (2.17) we can find a formula that relates the wavelength difference between initial and scattered photons with the angle ¢ between their directions, both of which are readily measurable quantities (unlike the energy and momentum of the recoil electron). The first step is to multiply Eqs. (2.16) and (2.17) by ¢ and rewrite them as
pe cos 6 = hy — hy’ cos }
pe sin @ = hv’ sin b By squaring each of these equations and adding the new ones together, the angle @ is eliminated, leaving
prc? = (hv)* ~ 2(hv)(hv") cos @ + Cho’ (2.18) Next we equate the two expressions for the total energy of a particle
E=KE + mc (1.20) E= Vai + pe (1.24)
from Chap. | to give
(KE + me®)? = mc? + pc? pec? = KE* + 2mc? KE
Particle Properties of Waves
Since
KE = hp — hy’
we have
pe = (hv)? — 2(hv)(hv’) + (hy')? + 2me(hy — hy’) (2.19)
Substituting this value of p’c? in Eq. (2.18), we finally obtain 2me*(hv ~ hy’) = 2(hv)\(hy’)(1 — cos ¢) (2.20)
This relationship is simpler when expressed in terms of wavelength A. Dividing Eq. (2.20) by 2h? ¢, mefv pt ve
Te = ale Satie ¢)
h\e c cc and so, since p/c = 1/A and v'fc = 1A’,
mefl 1\_ l-cos¢ A 7 AA!
Compton effect M-A= ~a — cos 4) (2.21)
.
Equation (2.21) was derived by Arthur H. Compton in the early 1920s, and the phe- nomenon it describes, which he was the first to observe, is known as the Compton effect. It constitutes very strong evidence in support of the quantum theory of radiation.
Equation (2.21) gives the change in wavelength expected for a photon that is scat- tered through the angle ¢ by a particle of rest mass m. This change is independent of the wavelength A of the incident photon. The quantity
h Compton wavelength Ac= yrs (2.22)
is called the Compton wavelength of the scattering particle. For an electton Ac = 2.426 X 107 m, which is 2.426 pm (1 pm = 1 picometer = 107! m). In terms of Ac, Eq. (2.21) becomes
Compton effect A’ -2A=Ac(1 — cos b) (2.23)
The Compton wavelength gives the scale of the wavelength change of the incident photon. From Eq. (2.23) we note that the greatest wavelength change possible corre- sponds to ¢ = 180°, when the wavelength change will be twice the Compton wave- length Ac. Because Ac = 2.426 pm for an electron, and even less for other particles owing to their larger rest masses, the maximum wavelength change in the Compton effect is 4.852 pm. Changes of this magnitude or less are readily observable only in x-rays: the shift in wavelength for visible light is less than 0.01 percent of the initial wavelength, whereas for x-rays of A = 0.1 nm it is several percent. The Compton effect is the chief means by which x-rays lose energy when they pass through matter.
7
78 Chapter Two
renee A
Arthur Holly Compton (1892- 1962), a native of Ohio, was edu- cated at College of Wooster and Princeton. While at Washington University in St. Louis he found that x-rays increase in wavelength when scattered, which he ex- plained in 1923 on the basis of the quantum theory of light. This work convinced remaining doubters of the reality of photons.
After receiving the Nobel Prize in 1927, Compton, now at the University of Chicago, studied cosmic rays and helped es- tablish that they are fast charged particles (today known to be atomic nuclei, largely protons) that circulate in space and are not high-energy gamma rays as many had thought. He did this by showing that cosmic-ray intensity varies with latitude, which makes sense only if they are ions whose paths are influenced by the earth’s magnetic field. During World War II Compton was one of the leaders in the development of the atomic bomb.
Example 2.4
X-rays of wavelength 10.0 pm ate scattered from a target. (a) Find the wavelength of the x-rays scattered through 45°. (b) Find the maximum wavelength present in the scattered x-rays. (c) Find the maximum kinetic energy of the recoil electrons.
Solution
(a) From Eq. (2.23), A’ — A = AcQ — cos ¢), and so a
AL =A +Ac(1 — cos 45°) = 10.0 pm + 0.293Ac = 10.7 pm
(b) A! — A is a maximum when (1 ~ cos $) = 2, in which case
A =A + 2Ac = 10.0 pm + 4.9 pm = 14.9 pm
(© The maximum recoil kinetic energy is equal to the difference between the energies of the incident and scattered photons, so
where A’ is given in (b). Hence
KEmax = h(v — v') = ie( + a x)
A A’
KEmax
= 6.54 X 1075
which is equal to 40.8 keV.
__ 6.626 X 107*J - 33.00 X 10° m/s) ( 1 1 )
107}? m/pm 10.0 pm 14.9 pm
TS LL
The experimental demonstration of the Compton effect is straightforward. As in Fig. 2.23, a beam of x-rays of a single, known wavelength is directed at a target, and the wavelengths of the scattered x-rays are determined at various angles @. The results, shown in Fig. 2.24, exhibit the wavelength shift predicted by Eq. (2.21), but at each angle the scattered x-rays also include many that have the initial wavelength. This is not hard to understand. In deriving Eq. (2.21) it was assumed that the scattering par- ticle is able to move freely, which is reasonable since many of the electrons in matter
8 ‘ Particle Properties of Waves 79
Scattered
x-ray Y,
> Unscattered
x-ray Source of Collimators / monochromatic. 7 x-rays Path of spectrometer - - oa
Figure 2.23 Experimental demonstration of the Compton effect,
Relative intensity Relative intensity
Wavelength Wavelength
‘
Relative intensity
Relative intensity
Wavelength Wavelength
Figure 2.24 Experimental confirmation of Compton scattering. The greater the scattering angle, the greater the wavelength change, in accord with Eq, (2.21).
are only loosely bound to their parent atoms. Other electrons, however, are very tightly bound and when struck by a photon, the entire atom recoils instead of the single elec- tron. In this event the value of m to use in Eq. (2.21) is that of the entire atom, which is tens of thousands of times greater than that of an electron, and the resulting Comp- ton shift is accordingly so small as to be undetectable.
2.8 PAIR PRODUCTION
Energy into matter
As we have seen, in a collision a photon can give an electron all of its energy (the pho- toelectric effect) or only part (the Compton effect). It is also possible for a photon to materialize into an electron and a positron, which is a positively charged electron. In this process, called pair production, electromagnetic energy is converted into matter.
80 Chapter Two
ee) Electron
Positron
Figure 2,25 In the process of pair production, a photon of sufficient energy materializes into an elec- tron and a positron.
No conservation principles are violated when an electron-positron pair is created near an atomic nucleus (Fig. 2.25). The sum of the charges of the electron (q = —e) and of the positron (q = +e) is zero, as is the charge of the photon; the total energy, including rest energy, of the electron and positron equals the photon energy; and lin- ear momentum is conserved with the help of the nucleus, which carries away cae ae photon momentum for the process to occur. Because of its relatively enormous the nucleus absorbs only a negligible fraction of the photon energy. (Energy and lin- ear momentum could not both be conserved if pair production were to occur in em y space, so it does not occur there.)
Bubble-chamber photograph of electron-positron pair formation. A magnetic field perpendicular to the page caused the electron and positron to move in opposite curved paths, which are spirals be- cause the particles lost energy as they moved through the chamber. In a bubble chamber, a liquid (here, hydrogen) is heated above its normal boiling point under a pressure great enough to keep it liquid. The pressure is then released, and bubbles form around any ions present in the resulting un- stable superheated liquid. A charged particle moving through the liquid at this time leaves a track