Electricity, Magnetism & Optics - The University of Alabama

282 Pages·2009·22.65 MB·English

Patrick R. LeClair
Electricity, Magnetism & Optics
– an algebra-based introduction –
October12,2009
The University of Alabama
Tuscaloosa,Alabama

Contents
PartI Relativity
1 Relativity .................................................................. 3
1.1 FramesofReference..................................................... 3
1.2 MovingFramesofReference.............................................. 5
1.2.1 LackofaPreferredReferenceFrame................................. 6
1.2.2 RelativeMotion .................................................. 7
1.2.3 InvarianceoftheSpeedofLight..................................... 9
1.2.4 Principlesofspecialrelativity....................................... 10
1.3 ConsequencesofRelativity ............................................... 11
1.3.1 LackofSimultaneity .............................................. 11
1.3.2 TimeDilation .................................................... 12
1.3.3 LengthContraction................................................ 18
1.3.4 Timeandpositionindifferentreferenceframes ........................ 21
1.3.5 AdditionofVelocitiesinRelativity .................................. 24
1.3.6 Space-timeIntervals............................................... 28
1.4 Mass,Momentum,andEnergy ............................................ 33
1.4.1 RelativisticMomentum ............................................ 33
1.4.2 RelativisticEnergy................................................ 34
1.4.3 RelativisticMass ................................................. 37
1.5 Problems .............................................................. 38
PartII ElectricityandMagnetism
2 ElectricForcesandFields.................................................... 43
2.1 PropertiesofElectricCharges ............................................. 43
2.2 InsulatorsandConductors ................................................ 44
2.2.1 ChargingbyConduction ........................................... 45
2.2.2 ChargingbyInduction ............................................. 47
2.3 Coulomb’sLaw ......................................................... 48
2.4 TheElectricField ....................................................... 50
2.4.1 ElectricFieldLines ............................................... 51
2.4.2 Whathappenswhenwehavetwochargestogether? .................... 52
2.5 ConductorsinElectrostaticEquilibrium..................................... 53
2.6 FaradayCages .......................................................... 55
2.7 ThevandeGraaffGenerator .............................................. 56
2.8 Gauss’Law ............................................................ 58
2.8.1 ElectricFlux ..................................................... 58
2.8.2 Gauss’LawasaConservationLaw .................................. 60
2.8.3 Example:TheFieldAroundaSphericalChargeDistribution............. 60
2.8.4 Example:TheFieldAboveaFlatConductor .......................... 62
2.8.5 Example:TheFieldInsideandOutsideaHollowSphericalConductor..... 63
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vi Contents
2.8.6 Example:TheFieldDuetoaLineofCharge .......................... 65
2.9 Miscellanea ............................................................ 66
2.10 Problems .............................................................. 67
3 ElectricalEnergyandCapacitance............................................ 71
3.1 ElectricalPotentialEnergy................................................ 71
3.2 ElectricPotential........................................................ 74
3.2.1 ElectricPotentialandPotentialEnergyduetoPointCharges ............. 75
3.2.2 EnergyofaSystemofCharges...................................... 76
3.3 Potentialsandchargedconductors.......................................... 81
3.4 EquipotentialSurfaces ................................................... 82
3.5 PotentialDifferenceSourcesasCircuitElements ............................. 83
3.6 Capacitance ............................................................ 83
3.6.1 Parallel-PlateCapacitors ........................................... 84
3.6.2 Energystoredincapacitors ......................................... 85
3.6.3 CapacitorsasCircuitElements...................................... 87
3.6.4 CombinationsofCapacitors ........................................ 88
3.6.5 Capacitorswith(non-conducting)stuffinside.......................... 92
3.7 DielectricsinElectricFields .............................................. 94
3.8 Problems .............................................................. 98
4 CurrentandResistance...................................................... 103
4.1 ElectricCurrent ......................................................... 103
4.2 GettingCurrenttoFlow .................................................. 104
4.3 DriftVelocityandCurrent ................................................ 106
4.4 ResistanceandOhm’sLaw ............................................... 107
4.4.1 DriftVelocityandCollisions........................................ 108
4.4.2 Current,ElectricField,andVoltage .................................. 110
4.4.3 Resistance ....................................................... 111
4.4.4 ResistorsasCircuitElements ....................................... 112
4.4.5 ResistivityofMaterials ............................................ 114
4.4.6 VariationofResistancewithTemperature ............................. 115
4.5 ElectricalEnergyandPower .............................................. 116
4.6 Problems .............................................................. 117
5 Direct-CurrentCircuits...................................................... 119
5.1 SourcingVoltage........................................................ 119
5.2 SourcingCurrent........................................................ 122
5.3 CombinationsofResistors ................................................ 124
5.3.1 ResistorsinSeries ................................................ 124
5.3.2 ResistorsinParallel ............................................... 125
5.3.3 Example:aComplexResistorCombination ........................... 127
5.4 CurrentandVoltageMeasurementsinCircuits ............................... 129
5.4.1 MeasuringVoltage ................................................ 129
5.4.2 MeasuringCurrent ................................................ 130
5.5 Kirchhoff’sRulesandComplexdcCircuits.................................. 132
5.5.1 Example:analyzingasimpleparallelorseriescircuit ................... 133
5.5.2 Example:analyzingacomplexcircuit ................................ 134
5.6 RCCircuits............................................................. 136
5.7 Miscellaneous .......................................................... 138
5.8 Problems .............................................................. 139
Contents vii
6 Magnetism ................................................................. 145
6.1 MagneticFieldsandForces ............................................... 145
6.1.1 TheMagneticForce............................................... 147
6.1.2 MagnetismasaConsequenceofRelativity............................ 148
6.1.3 MagneticFieldofaLong,StraightWire .............................. 151
6.1.4 Handedness...................................................... 153
6.2 Ampe`re’sLaw .......................................................... 154
6.2.1 Ampe`re’sandGauss’Laws......................................... 155
6.3 TheMagneticFieldinVariousSituations.................................... 156
6.3.1 MotionofaChargedParticleinaMagneticField ...................... 156
6.3.2 MagneticForceonaCurrent-CarryingConductor ...................... 159
6.3.3 MagneticForceBetweenTwoParallelConductors ..................... 160
6.3.4 TorqueonaCurrentLoop .......................................... 162
6.3.5 MagneticFieldsofCurrentLoopsandSolenoids....................... 163
6.4 PermanentMagneticMaterials ............................................ 165
6.4.1 Non-permanentmagneticmaterials .................................. 167
6.4.2 Electromagnets................................................... 167
6.4.3 PermeabilityandMagnetsonYourFridge ............................ 168
6.5 Problems .............................................................. 170
7 InducedVoltagesandInductance ............................................. 171
7.1 InducedVoltagesandMagneticFlux ....................................... 171
7.2 Faraday’sLawofInduction ............................................... 172
7.3 Inductance ............................................................. 173
7.3.1 MutualInductance ................................................ 173
7.3.2 SelfInductance................................................... 174
7.4 Transformers ........................................................... 178
7.5 VoltageInducedbytheMotionofaConductorinaFIeld ...................... 179
7.5.1 Eddycurrentbrakes ............................................... 181
7.6 Generators ............................................................. 182
7.7 Asummaryofsorts...................................................... 183
8 acCircuitsandElectromagneticwaves ........................................ 185
8.1 ResistorsinanacCircuit ................................................. 185
8.2 CapacitorsinacCircuits.................................................. 187
8.3 InductorsinacCircuits................................................... 189
8.4 Filters ................................................................. 190
8.5 ElectromagneticWaves................................................... 192
8.5.1 Electromagneticfieldsofacceleratingcharges ......................... 192
8.5.2 ProductionofElectromagneticWavesbyanAntenna ................... 192
8.5.3 PropertiesofElectromagneticWaves................................. 193
8.5.4 EnergytransferredbyEMwaves .................................... 194
8.5.5 TheEMspectra................................................... 196
PartIII Optics
9 ReflectionandRefractionofLight ............................................ 199
9.1 TheNatureofLight...................................................... 199
9.1.1 WavePacketsandWave-ParticleDuality.............................. 200
9.2 ReflectionofLight ...................................................... 201
9.2.1 TheRayApproximation ........................................... 201
9.2.2 TheLawofReflection ............................................. 201
9.3 RefractionofLight ...................................................... 202
9.3.1 Snell’sLaw ...................................................... 204
9.3.2 DispersionandPrisms ............................................. 205
9.3.3 Rainbows........................................................ 207
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9.4 TotalInternalReflection.................................................. 208
9.4.1 Fiberoptics ...................................................... 209
9.4.2 Multi-Touchscreens............................................... 210
10 Mirrors .................................................................... 213
10.1 FlatMirrors ............................................................ 213
10.1.1 Imageformation.................................................. 213
10.1.2 RayDiagrams.................................................... 215
10.1.3 ConventionsforRayDiagrams...................................... 216
10.1.4 Handedness...................................................... 217
10.2 SphericalMirrors........................................................ 217
10.2.1 ConcaveMirrors.................................................. 217
10.2.2 ConvexSphericalMirrors .......................................... 221
10.3 RayDiagramsforMirrors ................................................ 222
10.4 ParabolicMirrors........................................................ 223
11 Lenses ..................................................................... 227
11.1 Sphericalrefractingsurfaces .............................................. 227
11.1.1 FlatRefractingSurfaces ........................................... 230
11.2 SphericalLenses ........................................................ 230
11.3 Typesofsphericallenses ................................................. 234
SolutionstoProblems............................................................ 237
Bibliography.................................................................... 276
References.................................................................. 276
Part I
Relativity

Chapter 1
Relativity
Thedifferencebetweentruthandfictionisthatfictionhasto
makesense.–MarkTwain
AbstractNearlyallofthemechanicalphenomenaweobservearounduseverydayhavetodowith
objectsmovingatspeedsrathersmallcomparedtothespeedoflight.TheNewtonianmechanicsyou
learnedinpreviouscourseshandledthesecasesextraordinarilywell.Asitturnsout,however,New-
tonianmechanicsbreaksdowncompletelywhenanobject’sspeedisnolongernegligiblecompared
to the speed of light. Not only does Newtonian mechanics fail in this situation, it fails spectacu-
larly, leading to a variety of paradoxical situations. The resolution to these paradoxes is given by
thetheoryofrelativity,oneofthemostsuccessfulandaccuratetheoriesinallofphysics,whichwe
willintroduceinthischapter.Natureisnotalwayskind,however,andtheconsequencesofrelativity
seem on their face to flout common sense and our view of the world around us. We are used to
the notion that our position changes with time when we are in motion, but relativity implies that
passage of time itself changes when we are in motion. Nevertheless, we shall see that relativity is
aninescapableconsequenceofafewsimpleprinciplesandexperimentalfacts.Moreover,asitturns
out,thisnewdescriptionofnatureiscriticalforproperlyunderstandingelectricityandmagnetism,
optics,andnuclearphysics...mostoftherestofthiscourse!
1.1 FramesofReference
Describingmotionproperlyusuallyrequiresustochooseacoordinatesystem,andanoriginfrom
whichtomeasureposition.Whythisissoismoreclearwhenweconsiderthedifferencebetween
distanceanddisplacement.Forexample,wecansaythatapersonmovesthroughadisplacementof
10meters,∆x=10m,inaparticulardirection,e.g.,totheright.Thisdoesnotdescribetheposition
ofthepersonatall,onlythechangeinthatperson’spositionoversometimeinterval.
Describingpositionitselfrequiresustochoosefirstacoordinatesystem(suchascartesian,spher-
ical,etc.),andalsoanoriginforthiscoordinatesystemtodefineour“zero”position.Theessential
difference is that displacement is independent of the coordinate system we choose, but position is
not.Withoutchoosingacoordinatesystem,wecanonlysaythatthepersonhasrun10minacertain
timeinterval,movingfromx tox .
i f
As a concrete example, consider Fig. 1.1a. This illustrates a person moving 10m to the right,
whichperfectlydescribesadisplacement∆x.Wewillchooseanx−ycartesiancoordinatesystem,
whichwewillcallO,withitsoriginattheperson’sstartingpoint.Inthissystem,wecandescribe
theinitialandfinalpositionsPO andPO inthiscoordinatesystemasPO=(0,0)andPO=(x ,0)=
i f i f f
(∆x,0). This is shown in Fig. 1.1b. The displacement is the same as it was without a coordinate
system.Inthischapterwewillusetheconventionthatsuperscriptsrefertothecoordinatesystemin
whichthequantityinquestionwasmeasured.
What happens if we if we instead choose a different coordinate system O0, Fig. 1.1c, identical
toOexceptthatitsoriginisshifteddownwardbyy andtotheleftbyx?Nowtheinitialandfinal
i i
positionsofthepersonarePO0=(x,y)andPO0=(x ,y ).Still,thedisplacement∆xisthesame,as
i i i f f f
youcaneasilyverify.NomatterwhetherweobservethepersonfromtheOorO0system,wewould
describethesamedisplacement,eventhoughtheactualpositionsarecompletelydifferent.
3
4 1 Relativity
(a)
∆x=10m
(b) y O
x
∆x=x −0=x
(0,0) (xf,0) f f
∆y =0
(c) x
i
∆x! =∆x
yi (xi,yi) (xf,yf) ∆y! =0
!
y
O!
x!
Fig.1.1 Displacementisindependentofthecoordinatesystemwechoose,butpositionisnot.(a)Withoutchoosing
acoordinatesystem,wecanonlysaythatthepersonhasrun10minacertaintimeinterval,movingfromxitoxf.(b)
Ifwechooseanx−ycoordinatesystemOcenteredwithitsoriginontheperson’sstartingpointxi,wecandescribe
theinitialandfinalpositionsasPiO=(xi,0)andPfO=(xf,0).Thedisplacementisthesame.(c)Ifwechooseanew
coordinatesystemO0,identicaltoOexceptshifteddownwardbyyi andtotheleftbyxi,nowtheinitialandfinal
positionsarePiO0=(xi,yi)andPfO0=(xf,yi).Still,thedisplacementisthesame.
Inspecialrelativity,thissimplesituationnolongerholds-observersindifferentcoordinatesys-
temsdonotnecessarilydescribeeventhesamedisplacement,muchlessthesameposition.Fortu-
nately, the corrections of special relativity to the Newtonian mechanics you have already learned
areonlyappreciableatveryhighvelocities(non-negligiblecomparedtothespeedoflight),andfor
mosteverydaysituationsourusualintuitionisstillvalid.
Inanycase,particularlythosecaseswhererelativisticeffectsareimportant,itiscruciallyimpor-
tantthatwespecifyinwhichcoordinatesystemquantitieshavebeenmeasured.Wewillcontinueto
dothiswithasuperscriptofsomesorttospecifythecoordinatesystem,andasubscriptofsomesort
tofurtherdescribewhatisbeingmeasuredwithinthatsystem.Whenweonlyhavetwoframes,like
theexampleabove,wewilloftenjustuseaprime(0)totellthemapart.Inthepreviousexample,this
meanswewoulduseP0 insteadofPO0,andjustP insteadofPO.Itseemspedanticnow,butcareful
f f i i
bookkeepingistheonlythingsavingusfromterribleconfusionlater!
Coordinatesystemnotationexamples:
xO =xOfinalxpositionofanobjectmeasuredintheOcoordinatesystem
final f
vO0 ≡v0 velocityofacarmeasuredintheO0coordinatesystem
car car
PO0≡P0=(x0,y0)finalpositionofanobjectmeasuredintheO0coordinatesystem
f f f i
Finally, a word on terminology. In relativity, it is common to use “reference frame” in place of
“coordinatesystem,”tomakeexplicitthefactthatourcoordinatesystemandoriginarethepointof
referencefromwhichwemeasurephysicalquantities.Wewillusebothphrasingsinterchangeably
fromhereonout.