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Preface
Unit 1. Optics
Chapter1:The Nature of Light
Table of Contents
1.1The Propagation of Light
1.2The Law of Reflection
1.3Refraction
1.4Total Internal Reflection
1.5Dispersion
1.6Huygens’s Principle
1.7Polarization
Chapter2:Geometric Optics and Image Formation
2.1Images Formed by Plane Mirrors
2.2Spherical Mirrors
2.3Images Formed by Refraction
2.4Thin Lenses
2.5The Eye
2.6The Camera
2.7The Simple Magnifier
2.8Microscopes and Telescopes
Chapter3:Interference
3.1Young's Double-Slit Interference
3.2Mathematics of Interference
3.3Multiple-Slit Interference
3.4Interference in Thin Films
3.5The Michelson Interferometer
Chapter4:Diffraction
4.1Single-Slit Diffraction
4.2Intensity in Single-Slit Diffraction
4.3Double-Slit Diffraction
4.4Diffraction Gratings
4.5Circular Apertures and Resolution
4.6X-Ray Diffraction
4.7Holography
Unit 2. Modern Physics Chapter5:Relativity
5.1Invariance of Physical Laws
5.2Relativity of Simultaneity
5.3Time Dilation
5.4Length Contraction
5.5The Lorentz Transformation
5.6Relativistic Velocity Transformation
5.7Doppler Effect for Light
5.8Relativistic Momentum
5.9Relativistic Energy
Chapter6:Photons and Matter Waves
6.1Blackbody Radiation
6.2Photoelectric Effect
6.3The Compton Effect
6.4Bohr’s Model of the Hydrogen Atom
6.5De Broglie’s Matter Waves
Chapter7:Quantum Mechanics
7.1Wave Functions
7.2The Heisenberg Uncertainty Principle
7.3The Schrӧdinger Equation
7.4The Quantum Particle in a Box
7.5The Quantum Harmonic Oscillator
7.6The Quantum Tunneling of Particles through Potential Barriers
Chapter8:Atomic Structure
8.1The Hydrogen Atom
8.2Orbital Magnetic Dipole Moment of the Electron
8.3Electron Spin
8.4The Exclusion Principle and the Periodic Table
8.5Atomic Spectra and X-rays
8.6Lasers
Chapter9:Condensed Matter Physics
9.1Types of Molecular Bonds
9.2Molecular Spectra
9.3Bonding in Crystalline Solids
9.4Free Electron Model of Metals
9.5Band Theory of Solids
9.6Semiconductors and Doping
9.7Semiconductor Devices
9.8Superconductivity
Chapter10:Nuclear Physics
10.1Properties of Nuclei
10.2Nuclear Binding Energy
10.3Radioactive Decay
10.4Nuclear Reactions
10.5Fission
10.6Nuclear Fusion
10.7Medical Applications and Biological Effects of Nuclear Radiation
Chapter11:Particle Physics and Cosmology
11.1Introduction to Particle Physics
11.2Particle Conservation Laws
11.3Quarks
11.4Particle Accelerators and Detectors
11.5The Standard Model
11.6The Big Bang
11.7Evolution of the Early Universe
Appendix A:Units
Appendix B:Conversion Factors
Appendix C:Fundamental Constants
Appendix D:Astronomical Data
Appendix E:Mathematical Formulas
Appendix F:Chemistry
Appendix G:The Greek Alphabet
Index
PREFACE
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About University Physics
UniversityPhysics isdesignedforthetwo-orthree-semestercalculus-basedphysicscourse.Thetexthasbeendeveloped tomeetthescopeandsequenceofmostuniversityphysicscoursesandprovidesafoundationforacareerinmathematics, science,orengineering.Thebookprovidesanimportantopportunityforstudentstolearnthecoreconceptsofphysicsand understand how those concepts apply to their lives and to the world around them.
Due to the comprehensive nature of the material, we are offering the book in three volumes for flexibility and efficiency.
Coverage and scope
Our UniversityPhysics textbookadherestothescopeandsequenceofmosttwo-andthree-semesterphysicscourses nationwide.Wehaveworkedtomakephysicsinterestingandaccessibletostudentswhilemaintainingthemathematical rigorinherentinthesubject.Withthisobjectiveinmind,thecontentofthistextbookhasbeendevelopedandarranged toprovidealogicalprogressionfromfundamentaltomoreadvancedconcepts,buildinguponwhatstudentshavealready learnedandemphasizingconnectionsbetweentopicsandbetweentheoryandapplications.Thegoalofeachsectionis toenablestudentsnotjusttorecognizeconcepts,buttoworkwiththeminwaysthatwillbeusefulinlatercoursesand futurecareers.Theorganizationandpedagogicalfeaturesweredevelopedandvettedwithfeedbackfromscienceeducators dedicated to the project.
VOLUME I
Unit 1: Mechanics
Chapter 1: Units and Measurement
Chapter 2: Vectors
Chapter 3: Motion Along a Straight Line
Chapter 4: Motion in Two and Three Dimensions
Chapter 5: Newton’s Laws of Motion
Chapter 6: Applications of Newton’s Laws
Chapter 7: Work and Kinetic Energy
Chapter 8: Potential Energy and Conservation of Energy
Chapter 9: Linear Momentum and Collisions
Chapter 10: Fixed-Axis Rotation
Chapter 11: Angular Momentum
Chapter 12: Static Equilibrium and Elasticity
Chapter 13: Gravitation
Chapter 14: Fluid Mechanics
Unit 2: Waves and Acoustics
Chapter 15: Oscillations
Chapter 16: Waves
Chapter 17: Sound
VOLUME II
Unit 1: Thermodynamics
Chapter 1: Temperature and Heat
Chapter 2: The Kinetic Theory of Gases
Chapter 3: The First Law of Thermodynamics
Chapter 4: The Second Law of Thermodynamics
Unit 2: Electricity and Magnetism
Chapter 5: Electric Charges and Fields
Chapter 6: Gauss’s Law
Chapter 7: Electric Potential
Chapter 8: Capacitance
Chapter 9: Current and Resistance
Chapter 10: Direct-Current Circuits
Chapter 11: Magnetic Forces and Fields
Chapter 12: Sources of Magnetic Fields
Chapter 13: Electromagnetic Induction
Chapter 14: Inductance
Chapter 15: Alternating-Current Circuits
Chapter 16: Electromagnetic Waves
VOLUME III
Unit 1: Optics
Chapter 1: The Nature of Light
Chapter 2: Geometric Optics and Image Formation
Chapter 3: Interference
Chapter 4: Diffraction
Unit 2: Modern Physics
Chapter 5: Relativity
Chapter 6: Photons and Matter Waves
Chapter 7: Quantum Mechanics
Chapter 8: Atomic Structure
Chapter 9: Condensed Matter Physics
Chapter 10: Nuclear Physics
Chapter 11: Particle Physics and Cosmology
Pedagogical foundation
Throughout UniversityPhysics youwillfindderivationsofconceptsthatpresentclassicalideasandtechniques,aswell asmodernapplicationsandmethods.Mostchaptersstartwithobservationsorexperimentsthatplacethematerialina contextofphysicalexperience.Presentationsandexplanationsrelyonyearsofclassroomexperienceonthepartoflongtimephysicsprofessors,strivingforabalanceofclarityandrigorthathasprovensuccessfulwiththeirstudents.Throughout thetext,linksenablestudentstoreviewearliermaterialandthenreturntothepresentdiscussion,reinforcingconnections betweentopics.Keyhistoricalfiguresandexperimentsarediscussedinthemaintext(ratherthaninboxesorsidebars), maintainingafocusonthedevelopmentofphysicalintuition.Keyideas,definitions,andequationsarehighlightedin thetextandlistedinsummaryformattheendofeachchapter.Examplesandchapter-openingimagesofteninclude contemporaryapplicationsfromdailylifeormodernscienceandengineeringthatstudentscanrelateto,fromsmartphones to the internet to GPS devices.
AdditionalProblems applyknowledgeacrossthechapter,forcingstudentstoidentifywhatconceptsandequations areappropriateforsolvinggivenproblems.Randomlylocatedthroughouttheproblemsare UnreasonableResults exercisesthataskstudentstoevaluatetheanswertoaproblemandexplainwhyitisnotreasonableandwhat assumptions made might not be correct.
Challenge Problems extend text ideas to interesting but difficult situations. Answers for selected exercises are available in an Answer Key at the end of the book.
Additional resources
Student and instructor resources
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About the authors
Senior contributing authors
Samuel J. Ling, Truman State University
Dr.SamuelLinghastaughtintroductoryandadvancedphysicsforover25yearsatTrumanStateUniversity,whereheis currentlyProfessorofPhysicsandtheDepartmentChair.Dr.LinghastwoPhDsfromBostonUniversity,oneinChemistry andtheotherinPhysics,andhewasaResearchFellowattheIndianInstituteofScience,Bangalore,beforejoiningTruman. Dr.Lingisalsoanauthorof AFirstCourseinVibrationsandWaves,publishedbyOxfordUniversityPress.Dr.Linghas considerableexperiencewithresearchinPhysicsEducationandhaspublishedresearchoncollaborativelearningmethodsin physicsteaching.HewasawardedaTrumanFellowandaJepsonfellowinrecognitionofhisinnovativeteachingmethods. Dr. Ling’s research publications have spanned Cosmology, Solid State Physics, and Nonlinear Optics.
Jeff Sanny, Loyola Marymount University
Dr.JeffSannyearnedaBSinPhysicsfromHarveyMuddCollegein1974andaPhDinSolidStatePhysicsfromthe UniversityofCalifornia–LosAngelesin1980.HejoinedthefacultyatLoyolaMarymountUniversityinthefallof1980. Duringhistenure,hehasservedasdepartmentChairaswellasAssociateDean.Dr.Sannyenjoysteachingintroductory physicsinparticular.Heisalsopassionateaboutprovidingstudentswithresearchexperienceandhasdirectedanactive undergraduate student research group in space physics for many years.
William Moebs, Formerly of Loyola Marymount University
Salameh Ahmad, Rochester Institute of Technology–Dubai
John Aiken, University of Colorado–Boulder
Raymond Benge, Terrant County College
Gavin Buxton, Robert Morris University
Erik Christensen, South Florida State College
Clifton Clark, Fort Hays State University
Nelson Coates, California Maritime Academy
Herve Collin, Kapi’olani Community College
Carl Covatto, Arizona State University
Alejandro Cozzani, Imperial Valley College
Danielle Dalafave, The College of New Jersey
Nicholas Darnton, Georgia Institute of Technology
Ethan Deneault, University of Tampa
Kenneth DeNisco, Harrisburg Area Community College
Robert Edmonds, Tarrant County College
William Falls, Erie Community College
Stanley Forrester, Broward College
Umesh Garg, University of Notre Dame
Maurizio Giannotti, Barry University
Bryan Gibbs, Dallas County Community College
Lynn Gillette, Pima Community College–West Campus
Mark Giroux, East Tennessee State University
Matthew Griffiths, University of New Haven
Alfonso Hinojosa, University of Texas–Arlington
Steuard Jensen, Alma College
David Kagan, University of Massachusetts
Sergei Katsev, University of Minnesota–Duluth
Jill Leggett, Florida State College–Jacksonville
Alfredo Louro, University of Calgary
James Maclaren, Tulane University
Ponn Maheswaranathan, Winthrop University
Seth Major, Hamilton College
Oleg Maksimov, Excelsior College
Aristides Marcano, Delaware State University
James McDonald, University of Hartford
Ralph McGrew, SUNY–Broome Community College
Paul Miller, West Virginia University
Tamar More, University of Portland
Farzaneh Najmabadi, University of Phoenix
Richard Olenick, The University of Dallas
Christopher Porter, Ohio State University
Liza Pujji, Manakau Institute of Technology
Baishali Ray, Young Harris University
Andrew Robinson, Carleton University
Aruvana Roy, Young Harris University
Gajendra Tulsian, Daytona State College
Adria Updike, Roger Williams University
Clark Vangilder, Central Arizona University
Steven Wolf, Texas State University
Alexander Wurm, Western New England University
Lei Zhang, Winston Salem State University
Ulrich Zurcher, Cleveland State University
1 | THE NATURE OF LIGHT
Figure 1.1 Due to total internal reflection, an underwater swimmer’s image is reflected back into the water where the camera is located. The circular ripple in the image center is actually on the water surface. Due to the viewing angle, total internal reflection is not occurring at the top edge of this image, and we can see a view of activities on the pool deck. (credit: modification of work by “jayhem”/Flickr)
Chapter Outline
1.1 The Propagation of Light
1.2 The Law of Reflection
1.3 Refraction
1.4 Total Internal Reflection
1.5 Dispersion
1.6 Huygens’s Principle
1.7 Polarization
Introduction
Ourinvestigationoflightrevolvesaroundtwoquestionsoffundamentalimportance:(1)Whatisthenatureoflight,and(2) howdoeslightbehaveundervariouscircumstances?AnswerstothesequestionscanbefoundinMaxwell’sequations(in ElectromagneticWaves(http://cnx.org/content/m58495/latest/) ),whichpredicttheexistenceofelectromagnetic wavesandtheirbehavior.Examplesoflightincluderadioandinfraredwaves,visiblelight,ultravioletradiation,andX-rays. Interestingly,notalllightphenomenacanbeexplainedbyMaxwell’stheory.Experimentsperformedearlyinthetwentieth centuryshowedthatlighthascorpuscular,orparticle-like,properties.Theideathatlightcandisplaybothwaveandparticle characteristics is called wave-particle duality, which is examined in Photons and Matter Waves. Inthischapter,westudythebasicpropertiesoflight.Inthenextfewchapters,weinvestigatethebehavioroflightwhenit interacts with optical devices such as mirrors, lenses, and apertures.
1.1 | The Propagation of Light
Learning Objectives
By the end of this section, you will be able to:
• Determine the index of refraction, given the speed of light in a medium
• List the ways in which light travels from a source to another location
Thespeedoflightinavacuum c isoneofthefundamentalconstantsofphysics.Asyouwillseewhenyoureach Relativity, itisacentralconceptinEinstein’stheoryofrelativity.Astheaccuracyofthemeasurementsofthespeedoflightimproved, itwasfoundthatdifferentobservers,eventhosemovingatlargevelocitieswithrespecttoeachother,measurethesame valueforthespeedoflight.However,thespeedoflightdoesvaryinaprecisemannerwiththematerialittraverses.These facts have far-reaching implications, as we will see in later chapters.
The Speed of Light: Early Measurements
ThefirstmeasurementofthespeedoflightwasmadebytheDanishastronomerOleRoemer(1644–1710)in1675.He studiedtheorbitofIo,oneofthefourlargemoonsofJupiter,andfoundthatithadaperiodofrevolutionof42.5haround Jupiter.Healsodiscoveredthatthisvaluefluctuatedbyafewseconds,dependingonthepositionofEarthinitsorbitaround the Sun. Roemer realized that this fluctuation was due to the finite speed of light and could be used to determine c RoemerfoundtheperiodofrevolutionofIobymeasuringthetimeintervalbetweensuccessiveeclipsesbyJupiter. Figure 1.2(a)showstheplanetaryconfigurationswhensuchameasurementismadefromEarthinthepartofitsorbitwhereit isrecedingfromJupiter.WhenEarthisatpoint A,Earth,Jupiter,andIoarealigned.Thenexttimethisalignmentoccurs, Earthisatpoint B,andthelightcarryingthatinformationtoEarthmusttraveltothatpoint.Since B isfartherfromJupiter than A,lighttakesmoretimetoreachEarthwhenEarthisat B.Nowimagineitisabout6monthslater,andtheplanets arearrangedasinpart(b)ofthefigure.ThemeasurementofIo’speriodbeginswithEarthatpoint A′ andIoeclipsedby Jupiter.ThenexteclipsethenoccurswhenEarthisatpoint B′ ,towhichthelightcarryingtheinformationofthiseclipse musttravel.Since B′ isclosertoJupiterthan A′ ,lighttakeslesstimetoreachEarthwhenitisat B′ .Thistimeinterval betweenthesuccessiveeclipsesofIoseenat A′ and B′ isthereforelessthanthetimeintervalbetweentheeclipsesseen at A and B.Bymeasuringthedifferenceinthesetimeintervalsandwithappropriateknowledgeofthedistancebetween JupiterandEarth,Roemercalculatedthatthespeedoflightwas 2.0×108 m/s, whichis33%belowthevalueaccepted today.
Figure 1.2 Roemer’s astronomical method for determining the speed of light. Measurements of Io’s period done with the configurations of parts (a) and (b) differ, because the light path length and associated travel time increase from A to B (a) but decrease from A′ to B′ (b).
whilethepulsestraveleddowntothemirrorandback.Knowingtherotationalspeedofthewheel,thenumberofteethon thewheel,andthedistancetothemirror,Fizeaudeterminedthespeedoflighttobe 3.15×108 m/s, whichisonly5% too high.
Figure 1.3 Fizeau’s method for measuring the speed of light. The teeth of the wheel block the reflected light upon return when the wheel is rotated at a rate that matches the light travel time to and from the mirror.
Today,thespeedoflightisknowntogreatprecision.Infact,thespeedoflightinavacuum c issoimportantthatitis accepted as one of the basic physical quantities and has the value
c =2.99792458×108 m/s≈3.00×10
where the approximate value of 3.00×108 m/s is used whenever three-digit accuracy is sufficient.
where v is the observed speed of light in the material. Sincethespeedoflightisalwayslessthan c inmatterandequals c onlyinavacuum,theindexofrefractionisalways greaterthanorequaltoone;thatis, n ≥1 Table1.1 givestheindicesofrefractionforsomerepresentativesubstances. Thevaluesarelistedforaparticularwavelengthoflight,becausetheyvaryslightlywithwavelength.(Thiscanhave
importanteffects,suchascolorsseparatedbyaprism,aswewillseein Dispersion.)Notethatforgases, n iscloseto 1.0.Thisseemsreasonable,sinceatomsingasesarewidelyseparated,andlighttravelsat c inthevacuumbetweenatoms. Itiscommontotake n =1 forgasesunlessgreatprecisionisneeded.Althoughthespeedoflight v inamediumvaries considerably from its value c in a vacuum, it is still a large speed. Medium
Table 1.1Index of Refraction in Various Media For light with a wavelength of 589 nm in a vacuum
Example 1.1
Speed of Light in Jewelry
Calculate the speed of light in zircon, a material used in jewelry to imitate diamond.
Solution
Rearranging the equation n = c/v for v gives us
v = c n.
Theindexofrefractionforzirconisgivenas1.923in Table1.1,and c isgivenin Equation1.1.Enteringthese values in the equation gives
Significance
Thisspeedisslightlylargerthanhalfthespeedoflightinavacuumandisstillhighcomparedwithspeedswe normallyexperience.Theonlysubstancelistedin Table1.1 thathasagreaterindexofrefractionthanzirconis diamond.Weshallseelaterthatthelargeindexofrefractionforzirconmakesitsparklemorethanglass,butless than diamond.
1.1
CheckYourUnderstanding Table1.1 showsthatethanolandfreshwaterhaveverysimilarindicesof refraction. By what percentage do the speeds of light in these liquids differ?
The Ray Model of Light
Youhavealreadystudiedsomeofthewavecharacteristicsoflightinthepreviouschapteron ElectromagneticWaves (http://cnx.org/content/m58495/latest/) .Inthischapter,westartmainlywiththeraycharacteristics.Therearethree waysinwhichlightcantravelfromasourcetoanotherlocation(Figure1.4).Itcancomedirectlyfromthesourcethrough emptyspace,suchasfromtheSuntoEarth.Orlightcantravelthroughvariousmedia,suchasairandglass,totheobserver. Lightcanalsoarriveafterbeingreflected,suchasbyamirror.Inallofthesecases,wecanmodelthepathoflightasa straight line called a ray.
Figure 1.4 Three methods for light to travel from a source to another location. (a) Light reaches the upper atmosphere of Earth, traveling through empty space directly from the source. (b) Light can reach a person by traveling through media like air and glass. (c) Light can also reflect from an object like a mirror. In the situations shown here, light interacts with objects large enough that it travels in straight lines, like a ray.
Experimentsshowthatwhenlightinteractswithanobjectseveraltimeslargerthanitswavelength,ittravelsinstraightlines andactslikearay.Itswavecharacteristicsarenotpronouncedinsuchsituations.Sincethewavelengthofvisiblelightis lessthanamicron(athousandthofamillimeter),itactslikearayinthemanycommonsituationsinwhichitencounters objectslargerthanamicron.Forexample,whenvisiblelightencountersanythinglargeenoughthatwecanobserveitwith unaided eyes, such as a coin, it acts like a ray, with generally negligible wave characteristics. Inallofthesecases,wecanmodelthepathoflightasstraightlines.Lightmaychangedirectionwhenitencountersobjects (suchasamirror)orinpassingfromonematerialtoanother(suchasinpassingfromairtoglass),butitthencontinuesin astraightlineorasaray.Theword“ray”comesfrommathematicsandheremeansastraightlinethatoriginatesatsome
point.Itisacceptabletovisualizelightraysaslaserrays.The raymodeloflight describesthepathoflightasstraightlines. Sincelightmovesinstraightlines,changingdirectionswhenitinteractswithmaterials,itspathisdescribedbygeometry andsimpletrigonometry.Thispartofoptics,wheretherayaspectoflightdominates,isthereforecalled geometricoptics. Twolawsgovernhowlightchangesdirectionwhenitinteractswithmatter.Thesearethe lawofreflection,forsituations inwhichlightbouncesoffmatter,andthe lawofrefraction,forsituationsinwhichlightpassesthroughmatter.Wewill examine more about each of these laws in upcoming sections of this chapter.
1.2 | The Law of Reflection
Learning Objectives
By the end of this section, you will be able to:
• Explain the reflection of light from polished and rough surfaces
• Describe the principle and applications of corner reflectors
Wheneverwelookintoamirror,orsquintatsunlightglintingfromalake,weareseeingareflection.Whenyoulookata pieceofwhitepaper,youareseeinglightscatteredfromit.Largetelescopesusereflectiontoformanimageofstarsand other astronomical objects.
The law of reflection states that the angle of reflection equals the angle of incidence, or
Thelawofreflectionisillustratedin Figure1.5,whichalsoshowshowtheangleofincidenceandangleofreflectionare measured relative to the perpendicular to the surface at the point where the light ray strikes.
Figure 1.5 The law of reflection states that the angle of reflection equals the angle of incidence— θr = θi The angles are measured relative to the perpendicular to the surface at the point where the ray strikes the surface.
Weexpecttoseereflectionsfromsmoothsurfaces,but Figure1.6 illustrateshowaroughsurfacereflectslight.Sincethe lightstrikesdifferentpartsofthesurfaceatdifferentangles,itisreflectedinmanydifferentdirections,ordiffused.Diffused lightiswhatallowsustoseeasheetofpaperfromanyangle,asshownin Figure1.7(a).People,clothing,leaves,and wallsallhaveroughsurfacesandcanbeseenfromallsides.Amirror,ontheotherhand,hasasmoothsurface(compared withthewavelengthoflight)andreflectslightatspecificangles,asillustratedin Figure1.7(b).WhentheMoonreflects from a lake, as shown in Figure 1.7(c), a combination of these effects takes place.
Figure 1.6 Light is diffused when it reflects from a rough surface. Here, many parallel rays are incident, but they are reflected at many different angles, because the surface is rough.
Figure 1.7 (a) When a sheet of paper is illuminated with many parallel incident rays, it can be seen at many different angles, because its surface is rough and diffuses the light. (b) A mirror illuminated by many parallel rays reflects them in only one direction, because its surface is very smooth. Only the observer at a particular angle sees the reflected light. (c) Moonlight is spread out when it is reflected by the lake, because the surface is shiny but uneven. (credit c: modification of work by Diego Torres Silvestre)
Figure 1.8 (a) Your image in a mirror is behind the mirror. The two rays shown are those that strike the mirror at just the correct angles to be reflected into the eyes of the person. The image appears to be behind the mirror at the same distance away as (b) if you were looking at your twin directly, with no mirror.
1.9 A light ray that strikes two mutually perpendicular reflecting surfaces is reflected back exactly parallel to the direction from which it came.
Manyinexpensivereflectorbuttonsonbicycles,cars,andwarningsignshavecornerreflectorsdesignedtoreturnlight inthedirectionfromwhichitoriginated.Ratherthansimplyreflectinglightoverawideangle,retroreflectionensures highvisibilityiftheobserverandthelightsourcearelocatedtogether,suchasacar’sdriverandheadlights.TheApollo astronautsplacedatruecornerreflectorontheMoon(Figure1.10).LasersignalsfromEarthcanbebouncedfromthat corner reflector to measure the gradually increasing distance to the Moon of a few centimeters per year.
Figure
Figure 1.10 (a) Astronauts placed a corner reflector on the Moon to measure its gradually increasing orbital distance. (b) The bright spots on these bicycle safety reflectors are reflections of the flash of the camera that took this picture on a dark night. (credit a: modification of work by NASA; credit b: modification of work by “Julo”/Wikimedia Commons)
Workingonthesameprincipleastheseopticalreflectors,cornerreflectorsareroutinelyusedasradarreflectors(Figure 1.11)forradio-frequencyapplications.Undermostcircumstances,smallboatsmadeoffiberglassorwooddonotstrongly reflectradiowavesemittedbyradarsystems.Tomaketheseboatsvisibletoradar(toavoidcollisions,forexample),radar reflectors are attached to boats, usually in high places.
Figure 1.11 A radar reflector hoisted on a sailboat is a type of corner reflector. (credit: Tim Sheerman-Chase)
Asacounterexample,ifyouareinterestedinbuildingastealthairplane,radarreflectionsshouldbeminimizedtoevade detection. One of the design considerations would then be to avoid building 90° corners into the airframe.
1.3 | Refraction
Learning Objectives
By the end of this section, you will be able to:
• Describe how rays change direction upon entering a medium
leavesthetank,andinthiscase,itcantraveltwodifferentpathstogettoyoureyes.Thechangingofalightray’sdirection (looselycalledbending)whenitpassesthroughsubstancesofdifferentrefractiveindicesiscalled refraction andisrelated tochangesinthespeedoflight, v = c/n .Refractionisresponsibleforatremendousrangeofopticalphenomena,fromthe action of lenses to data transmission through optical fibers.
Figure 1.12 (a) Looking at the fish tank as shown, we can see the same fish in two different locations, because light changes directions when it passes from water to air. In this case, the light can reach the observer by two different paths, so the fish seems to be in two different places. This bending of light is called refraction and is responsible for many optical phenomena. (b) This image shows refraction of light from a fish near the top of a fish tank.
Figure1.13 showshowarayoflightchangesdirectionwhenitpassesfromonemediumtoanother.Asbefore,theangles aremeasuredrelativetoaperpendiculartothesurfaceatthepointwherethelightraycrossesit.(Someoftheincidentlight isreflectedfromthesurface,butfornowweconcentrateonthelightthatistransmitted.)Thechangeindirectionofthelight raydependsontherelativevaluesoftheindicesofrefraction(ThePropagationofLight)ofthetwomediainvolved.In thesituationsshown,medium2hasagreaterindexofrefractionthanmedium1.Notethatasshownin Figure1.13(a),the directionoftheraymovesclosertotheperpendicularwhenitprogressesfromamediumwithalowerindexofrefraction toonewithahigherindexofrefraction.Conversely,asshownin Figure1.13(b),thedirectionoftheraymovesaway fromtheperpendicularwhenitprogressesfromamediumwithahigherindexofrefractiontoonewithalowerindexof refraction. The path is exactly reversible.
Figure 1.13 The change in direction of a light ray depends on how the index of refraction changes when it crosses from one medium to another. In the situations shown here, the index of refraction is greater in medium 2 than in medium 1. (a) A ray of light moves closer to the perpendicular when entering a medium with a higher index of refraction. (b) A ray of light moves away from the perpendicular when entering a medium with a lower index of refraction.
Theamountthatalightraychangesitsdirectiondependsbothontheincidentangleandtheamountthatthespeedchanges. Forarayatagivenincidentangle,alargechangeinspeedcausesalargechangeindirectionandthusalargechange inangle.Theexactmathematicalrelationshipisthe lawofrefraction,orSnell’slaw,aftertheDutchmathematician WillebrordSnell(1591–1626), who discovered it in 1621. The law of refraction is stated in equation form as
Here n1 and n2 aretheindicesofrefractionformedia1and2,and θ1 and θ2 aretheanglesbetweentheraysandthe perpendicularinmedia1and2.Theincomingrayiscalledtheincidentray,theoutgoingrayiscalledtherefractedray,and the associated angles are the incident angle and the refracted angle, respectively.
Snell’sexperimentsshowedthatthelawofrefractionisobeyedandthatacharacteristicindexofrefraction n couldbe assignedtoagivenmediumanditsvaluemeasured.Snellwasnotawarethatthespeedoflightvariedindifferentmedia,a key fact used when we derive the law of refraction theoretically using Huygens’s principle in Huygens’s Principle.
Example 1.2
Determining the Index of Refraction
Findtheindexofrefractionformedium2in Figure1.13(a),assumingmedium1isairandgiventhattheincident angle is 30.0° and the angle of refraction is 22.0° .
Strategy
Theindexofrefractionforairistakentobe1inmostcases(anduptofoursignificantfigures,itis1.000). Thus, n1 =1.00 here.Fromthegiveninformation, θ1 =30.0° and θ2 =22.0°. Withthisinformation,the only unknown in Snell’s law is n2, so we can use Snell’s law to find it.
Solution
From Snell’s law we have
Entering known values, n2 =1.00sin30.0° sin22.0° = 0.500 0.375 =1.33.
Significance
Thisistheindexofrefractionforwater,andSnellcouldhavedetermineditbymeasuringtheanglesand performingthiscalculation.Hewouldthenhavefound1.33tobetheappropriateindexofrefractionforwaterin allothersituations,suchaswhenaraypassesfromwatertoglass.Today,wecanverifythattheindexofrefraction is related to the speed of light in a medium by measuring that speed directly.
Example 1.2. Also by measurement, confirm that the angle of reflection equals the angle of incidence.
Example 1.3
A Larger Change in Direction
Supposethatinasituationlikethatin Example1.2,lightgoesfromairtodiamondandthattheincidentangle is 30.0° . Calculate the angle of refraction θ2 in the diamond.
Strategy
Again,theindexofrefractionforairistakentobe n1 =1.00 ,andwearegiven θ1 =30.0° .Wecanlookup theindexofrefractionfordiamondin Table1.1,finding n2 =2.419 .TheonlyunknowninSnell’slawis θ2 , which we wish to determine.
Solution
Solving Snell’s law for sinθ2 yields
Entering known values, sinθ2 = 1.00 2.419sin30.0°=(0.413)(0.500)=0.207.
The angle is thus θ2 =sin−1(0.207)=11.9°.
Significance
Forthesame 30.0° angleofincidence,theangleofrefractionindiamondissignificantlysmallerthaninwater (11.9° ratherthan 22.0° —see Example1.2).Thismeansthereisalargerchangeindirectionindiamond.The causeofalargechangeindirectionisalargechangeintheindexofrefraction(orspeed).Ingeneral,thelarger the change in speed, the greater the effect on the direction of the ray.
1.2
CheckYourUnderstanding In Table1.1,thesolidwiththenexthighestindexofrefractionafter diamondiszircon.Ifthediamondin Example1.3 werereplacedwithapieceofzircon,whatwouldbethe new angle of refraction?
1.4 | Total Internal Reflection
Learning Objectives
By the end of this section, you will be able to:
• Explain the phenomenon of total internal reflection
• Describe the workings and uses of optical fibers
Figure 1.14 (a) A ray of light crosses a boundary where the index of refraction decreases. That is, n2 < n1. The ray bends away from the perpendicular. (b) The critical angle θc is the angle of incidence for which the angle of refraction is 90°. (c) Total internal reflection occurs when the incident angle is greater than the critical angle.
Snell’s law states the relationship between angles and indices of refraction. It is given by n1 sinθ1 = n2 sinθ2. Whentheincidentangleequalsthecriticalangle ⎛ ⎝θ1 = θc ⎞ ⎠ ,theangleofrefractionis 90° ⎛ ⎝θ2 =90°⎞ ⎠ .Notingthat sin90°=1, Snell’s law in this case becomes n1 sinθ1 = n2
The critical angle θc for a given combination of materials is thus
Totalinternalreflectionoccursforanyincidentanglegreaterthanthecriticalangle θc ,anditcanonlyoccurwhenthe secondmediumhasanindexofrefractionlessthanthefirst.Notethatthisequationiswrittenforalightraythattravelsin medium 1 and reflects from medium 2, as shown in Figure 1.14.
Example 1.4
Determining a Critical Angle
Whatisthecriticalangleforlighttravelinginapolystyrene(atypeofplastic)pipesurroundedbyair?Theindex of refraction for polystyrene is 1.49.
Strategy
Theindexofrefractionofaircanbetakentobe1.00,asbefore.Thus,theconditionthatthesecondmedium(air) has an index of refraction less than the first (plastic) is satisfied, and we can use the equation
to find the critical angle θc, where n2 =1.00 and n1 =1.49.
Solution
Substituting the identified values gives
Significance
Thisresultmeansthatanyrayoflightinsidetheplasticthatstrikesthesurfaceatananglegreaterthan 42.2° is totallyreflected.Thismakestheinsidesurfaceoftheclearplasticaperfectmirrorforsuchrays,withoutanyneed forthesilveringusedoncommonmirrors.Differentcombinationsofmaterialshavedifferentcriticalangles,but anycombinationwith n1 > n2 canproducetotalinternalreflection.Thesamecalculationasmadehereshows thatthecriticalangleforaraygoingfromwatertoairis 48.6° ,whereasthatfromdiamondtoairis 24.4° ,and that from flint glass to crown glass is 66.3°
1.3
CheckYourUnderstanding Atthesurfacebetweenairandwater,lightrayscangofromairtowaterand from water to air. For which ray is there no possibility of total internal reflection?
Inthephotothatopensthischapter,theimageofaswimmerunderwateriscapturedbyacamerathatisalsounderwater. Theswimmerintheupperhalfofthephotograph,apparentlyfacingupward,is,infact,areflectedimageoftheswimmer below.Thecircularripplenearthephotograph’scenterisactuallyonthewatersurface.Theundisturbedwatersurrounding itmakesagoodreflectingsurfacewhenviewedfrombelow,thankstototalinternalreflection.However,attheverytop edgeofthisphotograph,raysfrombelowstrikethesurfacewithincidentangleslessthanthecriticalangle,allowingthe camera to capture a view of activities on the pool deck above water.
Fiber Optics: Endoscopes to Telephones
Fiberopticsisoneapplicationoftotalinternalreflectionthatisinwideuse.Incommunications,itisusedtotransmit telephone,internet,andcableTVsignals. Fiberoptics employsthetransmissionoflightdownfibersofplasticorglass. Becausethefibersarethin,lightenteringoneislikelytostriketheinsidesurfaceatananglegreaterthanthecriticalangle and,thus,betotallyreflected(Figure1.15).Theindexofrefractionoutsidethefibermustbesmallerthaninside.Infact, mostfibershaveavaryingrefractiveindextoallowmorelighttobeguidedalongthefiberthroughtotalinternalrefraction. Rays are reflected around corners as shown, making the fibers into tiny light pipes.
Figure 1.15 Light entering a thin optic fiber may strike the inside surface at large or grazing angles and is completely reflected if these angles exceed the critical angle. Such rays continue down the fiber, even following it around corners, since the angles of reflection and incidence remain large.
Bundlesoffiberscanbeusedtotransmitanimagewithoutalens,asillustratedin Figure1.16.Theoutputofadevice calledanendoscopeisshownin Figure1.16(b).Endoscopesareusedtoexploretheinteriorofthebodythroughitsnatural orificesorminorincisions.Lightistransmitteddownonefiberbundletoilluminateinternalparts,andthereflectedlightis transmitted back out through another bundle to be observed.
Figure 1.16 (a) An image “A” is transmitted by a bundle of optical fibers. (b) An endoscope is used to probe the body, both transmitting light to the interior and returning an image such as the one shown of a human epiglottis (a structure at the base of the tongue). (credit b: modification of work by “Med_Chaos”/Wikimedia Commons)
