(Pre-and)PostTestsandSurveysAllengineeringstudentsarebeingtestedintheircorecoursesthisacademicyearatthebeginningofthesemesterandagainattheendofthesemester.Thesedatawillbeusedtoimproveyourlearningexperiences.•  Yourresponsestothesetests&surveyswillhavenobearingonyourcoursegrade.•  Iwillnotseeyourscores.•  Iwillonlytrackwhetherornotyoutookthetestsandsurveys.

•  Therearefourcomponentstothepreandposttestrequirements:•  Twoin-classexams;Twoon-linesurveys•  Anexamandasurveybothatthebeginningandattheendofthesemester.•  Ifyouparticipateinall4components,youwillget100%on2%ofyourgrade•  Ifyoumissanyofthe4components,youwillgetazerofor2%ofyourgrade.•  Secondin-classexamThursday,April26inlecture.

TakesurveyTODAYandsaveconfirmationpage.DeadlineMondayApril3011:59pm.https://rutgers.ca1.qualtrics.com/jfe/form/SV_2lVQrms4jn96hJX

•  Taketheexamsandsurveysseriously!Answerallpartstobestofyourabilities.•  Ifyoudonottaketheexamsandsurveysseriously,youwillgetazero.•  IfyoutaketheexamsorsurveysforsomeoneelseORyouasksomeonetotakethe

examsforyou,youwillnotonlygetazerobutmaybeseparatedfromtheuniversitybecauseyouhaveviolatedacademicintegrity. 1

Summerstudyopportunity:PlasmaphysicsUndergraduatePlasmaPhysicsWorkshop

July18through20,2018atPrincetonPlasmaPhysicsLabThisworkshoptargetsfirstandsecond-yearundergradsandintroducesthemtoplasmaphysicsfromanexperimentalandatheoreticalperspective.Theobjectiveisforparticipantstogainanintroductoryunderstandingofthefieldofplasmaphysicsandopportunitieswithin.Alltravel,boardingandmealswillbecoveredbyPPPL.Underrepresentedstudentsareespeciallyencouragedtoapply.

https://www.pppl.gov/education/science-education/programs/workshop-plasma-physics-undergraduates 2

AnnouncementsThisweek: courseevalhttps://sakai.rutgers.edu/portal/site/sirs•  Requiredposttest–ThursdayApril26inlecture

•  Ifmisslecture,limitedopportunityformakeup;elsezerofor2%ofgrade

•  Homework10dueThursdayApril26:Chapter31+32•  RecitationonFridayApril27:Chapter32.•  QuizonFridayApril27:Homework10,Lectures22+23+24

Finalexam:Wednesday,May9,20184:00to7:00PMinPhysicsLectureHall.

•  30multiplechoicequestions,≈15fromChapters30-32,≈15cumulativeChapters21-29

•  ReviewsessionTuesday,May84to6PM227SerinPhysics&Astro

Allexamsareclosed-book,nocalculatorsorotherelectronicdevicesallowed.

AnnouncementsFinalexam

•  Wednesday,May9,2018inlecture:4:00–7:00pm•  30multiplechoicequestions.≈15fromChapters30-32,≈15fromChapters21-29

•  ExamreviewTuesday,May8,4:00– 6:00pm227SerinPhysics&Astro

Allexamsareclosed-book,nocalculatorsorotherelectronicdevicesallowed.Allquestionswillbemultiplechoice.Forthefinalexam,youmaybringwithyouthree(3)"formulasheets",on8.5x11inchsheetsofpaper(OKtousebothsides)onwhichyoumayhandwriteanyformulaeordiagramsornotesorproblemsolutionsthatmightbehelpfultoyouduringtheexam.Informationonthesheetsmustbehandwritten,noattachmentsareallowed.Thenumericalvaluesofrelevantconstantswillbeprovidedtoyou.Youshouldbring#2pencilstotheexamsforthecomputerforms.

Typesofquestions:LikeI-clickers,simplenumbers,formulaeStudy:Homework,Iclickers,examplesintextbook,collaborative+pre-RECOldexams:http://www.physics.rutgers.edu/~cizewski/227_s2018/Physics-227-s2018-old-exams.htm

FreetutoringviaMSLC:https://rlc.rutgers.edu/services/peer-tutoring

Lecture24TuesdayApril24,2018

Chap32continued:ElectromagneticWavesEnergy,momentumRadiationpressureStandingwaves

5

ModelofE-M

6

http://www.surendranath.org/GPA/Waves/EMWave/EMWave.html

SummaryChapter32E-MwavesMaxwell’sequations=>electromagneticwaves•  Travelingatspeedoflightinvacuum

•  Transverse;directionofpropagation

•  Sinusoidal

7

c = 1ε0µ0

= 3×108 m/s

!E ×!B

!E(x,t) �= y

∧

Emax cos(kx −ωt)!B(x,t) �= z

∧

Bmax cos(kx −ωt)

Emax = cBmax

•  Wavemotionisfunctionofposition(x)andtime(t)

Wherev=phasevelocityofthewaveX

y

Z

SinusoidalWaves

8

MechanicalWaves-review

y(x,t)= Acos(kx −ωt)

k =2πλ

ω = vk

E-MWavespropagatingin+xdirection!E(x,t) �= y

∧

Emax cos(kx −ωt)!B(x,t) �= z

∧

Bmax cos(kx −ωt)

Emax = cBmax

IclickerquestionTheelectricpartofanelectromagneticwavemovinginthe+zdirectionisgivenby

WhichofthefollowingstatementsabouttheycomponentofthemagneticfieldpartisCORRECT?

9

Ex = Asin(kz −ωt), Ey = Ez = 0

(A) By ∝ cos(ky−ωt)

(B) By ∝ cos(kz −ωt)

(C) By ∝ sin(kz −ωt)

(D) By = 0

(E) By isgiven bysomeother expression.

IclickeranswerTheelectricpartofanelectromagneticwavemovinginthe+zdirectionisgivenby

WhichofthefollowingstatementsabouttheycomponentofthemagneticfieldpartisCORRECT?

10

Ex = Asin(kz −ωt), Ey = Ez = 0

(A) By ∝ cos(ky−ωt)

(B) By ∝ cos(kz −ωt)

(C) By ∝ sin(kz −ωt)

(D) By = 0

(E) By isgiven bysomeother expression.

SummaryChapter32E-MwavesMaxwell’sequations=>electromagneticwaves•  Travelingatspeedoflightinvacuum

•  Transverse;directionofpropagation

•  Sinusoidal

Next:EnergyandmomentumofE-Mwaves

11

c = 1ε0µ0

= 3×108 m/s

!E ×!B

!E(x,t) �= y

∧

Emax cos(kx −ωt)!B(x,t) �= z

∧

Bmax cos(kx −ωt)

Emax = cBmax

EMEnergy(flow)inEMwaveEMwavescarryenergy,intheelectricandmagneticfields•  Recallenergydensities

12

u = uE +uB =12ε0E

2 +1

2µ0

B2

B =Ec= ε0µ0E

EnergydensityofB-field=EnergydensityofE-fieldButE(position,time)=>u(position,time)

u =12ε0E

2 +1

2µ0

ε0µ0E( )2= ε0E

2 = u

c = 1ε0µ0

EMEnergy(flow)inEMwave•  EnergyinvolumedV=areaxdistancetraveledintimedt

•  Energyflowperunittimeperunitarea(invacuum)=S

•  S“points”indirectionofenergyflow=directionofpropagationoftheEMwave

Ø ThePoyntingvector•  Units:power/unitarea•  BecauseE,B(position,time)=>S(position,time)

13

dU = udV = ε0E2(Acdt)

S =1AdUdt

= ε0cE2 = ε0cE(cB)=

EBµ0

!S =

1µ0

!E ×!B

c = 1ε0µ0

EMEnergyflow/intensityinEMwaveAssumesinusoidalwavetravelingin+xdirection

•  Direction=+xdirection•  Magnitude

•  TimeaverageofS:•  Timeaverageofcos=0•  IntensityI=Sav

14

!S(x,t)=

1µ0

y∧

Emax cos(kx −ωt)× z∧

Bmax cos(kx −ωt)⎡⎣⎢

⎤⎦⎥

!E(x,t) �= y

∧

Emax cos(kx −ωt)!B(x,t) �= z

∧

Bmax cos(kx −ωt)

S =1µ0

EmaxBmax cos2(kx −ωt)=

1µ0

EmaxBmax 1+ cos2(kx −ωt)[ ]

I = Sav =EmaxBmax

µ0

=E2

max

2µ0c=12ε0cE

2max

!S =

1µ0

!E ×!B

c = 1ε0µ0

Imaginethatwhenyouswitchonyourlamp,youincreasetheintensityoflightshiningonyourtextbookbyafactorof16.Bywhatfactordoestheaverageelectricfieldstrengthinthislightincrease?

Iclickerquestion

15

A.  Factorof256B.  Factorof16C.  Factorof4D.  Factorof2E.  Factorof1

I = Sav =EmaxBmax

µ0

=E2

max

2µ0c=12ε0cE

2max

Imaginethatwhenyouswitchonyourlamp,youincreasetheintensityoflightshiningonyourtextbookbyafactorof16.Bywhatfactordoestheaverageelectricfieldstrengthinthislightincrease?

Iclickeranswer

16

A.  Factorof256B.  Factorof16C.  Factorof4D.  Factorof2E.  Factorof1

I = Sav =EmaxBmax

µ0

=E2

max

2µ0c=12ε0cE

2max

Imaginethatwhenyouswitchonyourlamp,youincreasetheintensityoflightshiningonyourtextbookbyafactorof16.Ifyourbookisr=1mfromyourlamp,howintenseisthelightsource?(i.e.,whatisthetotalenergyflow/unittimeoutofthelamp?)

Powerofthesource

17

I = Sav =E2

max

2µ0c

P =!Sav •d

!A"∫ = I(4πr2 )

I =P

4πr2

Radpressureexample

18

•  Cometdusttails

http://hildaandtrojanasteroids.net/comettails.jpg

Electromagneticwavescarrymomentum•  EMradiationismadeupof“particles”=photonsthatcarryenergyandmomentum•  Momentumdensity:

•  Averagerateofmomentumtransfer/area:wheredV=Acdt

Momentumflow&pressureinEMwave

19

p =energy

c

dpdt

1A=Savc=Ic

dpdV

=1cd(energy)

dV=ε0E

2

c=ε0EcB

c= ε0EB

µ0

µ0

=Sc2

dpdV

=Sc2

c = 1ε0µ0

!S =

1µ0

!E ×!B

E = cB

Electromagneticwavescarrymomentum•  Averagerateofmomentumtransfer/area:

Pressure=force/unitarea=(dp/dt)/unitarea•  AmountofpressuredependsuponwhetherEMwaveis

•  Totallyabsorbed

•  Totallyreflected(changeinmomentumdoubled)

Momentumflow&pressureinEMwave

20

dpdt

1A=Savc=Ic

prad =Savc=Ic

prad =2Savc

=2Ic

StandingEMwavesConsiderlinearlypolarizedEMwavetravelingin-xdirectiononaperfectconductor•  Emustbezeroeverywhereonthey-z

conductingplane•  ButincidentE≠0iny-zplane•  NetEfield=0=incident+reflected

oscillatingE-field

21

Ey(x,t)�= Emax cos(kx +ωt)− cos(kx −ωt)[ ]

Bz(x,t)�= Bmax −cos(kx +ωt)− cos(kx −ωt)[ ]

cos(A±B)= cosAcosB ∓ sinAsinB

Ey(x,t)�= −2Emax sin kx sinωt

Bz(x,t)�= −2Bmax coskx cosωt

StandingEMwavesinacavity•  Standingwaves

•  Ehastobezeroatx=0andx=L•  Definesnormalmodes

22

Ey(x,t)�= −2Emax sin kx sinωt

Bz(x,t)�= −2Bmax coskx cosωt

L

https://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters_2017_Jan_1/quantum_theory_origins/index.html

λn =2Ln

n = 1,2,3...

fn =cλn

= nc2L

StandingEMwaves•  Standingwaves

•  Atx=0,Ealwayszero•  E=0wheresinkx=0Ø NodalplanesofE

•  AntinodalplanesofE(whenE=max)= NodalplanesofBwhencoskx=0•  NoteE(t)issinωtandB(t)iscosωt

Ø EandBfieldsareoutofphasewheninastandingwave23

Ey(x,t)�= −2Emax sin kx sinωt

Bz(x,t)�= −2Bmax coskx cosωt

x = 0,λ2, λ,

3λ2

...

kx = 0, π , 2π ...

x =λ4,3λ4,5λ4

...

kx =π2,3π2,5π2...

StandingEMwaves•  Standingwaves

•  Atx=0,Ealwayszero•  E=0wheresinkx=0•  NodalplanesofE

•  AntinodalplanesofE(whenE=max)= NodalplanesofBwhencoskx=0•  NoteE(t)issinωtandB(t)iscosωt

Ø EandBfieldsareoutofphasewheninastandingwave24

Ey(x,t)�= −2Emax sin kx sinωt

Bz(x,t)�= −2Bmax coskx cosωt

x = 0,λ2, λ,

3λ2

...

kx = 0, π , 2π ...

x =λ4,3λ4,5λ4

...

kx =π2,3π2,5π2...

Anti-node

Node

L =3λ2

•  WhatisdistancebetweennodalplanesofE-field?

•  NodalplanesofEfieldwhen

•  Distancebetweennodalplanes=λ/2 = 5 m4=2.5 m

StandingwavenodalplanesforEandBfields

x = 0,λ2, λ,

3λ2

...

GivenstandingEMwavewithfrequencyf=30MHzWavelength λ = v

f=3×108 m/s30×106 /s

= 10 m

•  NodalplanesofEfieldwhen

•  Distancebetweennodalplanes=λ/2 = 5 m

•  WhatisdistancebetweennodalplaneofE-fieldandnodalplaneofB-field?•  NodalplanesofBfieldwhen(alsoanti-nodalplanesofEfield)

•  DistancebetweennodalplanesofEandBfields=λ/4=2.5 m

StandingwavenodalplanesforEandBfields

x = 0,λ2, λ,

3λ2

...

GivenstandingEMwavewithfrequencyf=30MHzWavelength λ = v

f=3×108 m/s30×106 /s

= 10m

x =λ4,3λ4,5λ4...

IclickerquestionThedrawingshowsasinusoidalelectromagneticstandingwave.Whichofthefollowingstatementsabouttheaverage(averagedovereitherxort)PoyntingvectorinthiswaveisCORRECT?

27

A.  TheaveragePoyntingvectorinthiswavepointsalongthex-axis.B.  TheaveragePoyntingvectorinthiswavepointsalongthey-axis.C.  TheaveragePoyntingvectorinthiswavepointsalongthez-axis.D.  TheaveragePoyntingvectorinthiswaveiszero.

E.  NoneoftheabovestatementsabouttheaveragePoyntingvectorinthiswaveiscorrect.

Sav =<1µ0

!E ×!B >sinacosa =

12sin2a

IclickeranswerThedrawingshowsasinusoidalelectromagneticstandingwave.Whichofthefollowingstatementsabouttheaverage(averagedovereitherxort)PoyntingvectorinthiswaveisCORRECT?

28

A.  TheaveragePoyntingvectorinthiswavepointsalongthex-axis.B.  TheaveragePoyntingvectorinthiswavepointsalongthey-axis.C.  TheaveragePoyntingvectorinthiswavepointsalongthez-axis.D.  TheaveragePoyntingvectorinthiswaveiszero.

E.  NoneoftheabovestatementsabouttheaveragePoyntingvectorinthiswaveiscorrect.

Sav =<1µ0

!E ×!B >sinacosa =

12sin2a

IclickersolutionThedrawingshowsasinusoidalelectromagneticstandingwave.Whichofthefollowingstatementsabouttheaverage(averagedovereitherxort)PoyntingvectorinthiswaveisCORRECT?

29

Sav =<1µ0

!E ×!B >

Sav =<1µ0

!E ×!B >=<

1µ0

Emax sin kx sinωtBmax coskx cosωt >

=<1µ0

EmaxBmax sin kx coskx sinωt cosωt >

=<1µ0

EmaxBmax{12sin2kx

12sin2ωt}>= 0

sinacosa =12sin2a

SummaryChapter32•  Maxwell’sequationsinvacuum,freespaceØ EMwaves

•  Directionofpropagation

•  Sinusoidal

•  Power/unitarea=Poyntingvector

Ø IntensityofEMwave

•  RadiationpressureofEMwave

•  StandingEMwaves30

E = cB c = 1ε0µ0

= 3×108 m/s

!S =

1µ0

!E ×!B

!E ×!B

!E(x,t) �= y

∧

Emax cos(kx −ωt)!B(x,t) �= z

∧

Bmax cos(kx −ωt)

I = Sav =EmaxBmax

µ0

=12ε0cE

2max

prad (absorbed) =Savc=Ic

prad (reflected) =2Savc

=2Ic

SummaryChapter32(cont)StandingE-Mwaves•  EandB90°outofphase

•  NodeswhereE=0;antinodesB=max:

•  AntinodeswhereE=max;nodesB=0:

•  Standingwavesinacavity

31

Ey(x,t)�= −2Emax sin kx sinωt

Bz(x,t)�= −2Bmax coskx cosωt

kx =π2,3π2,5π2...

kx = 0, π , 2π ...

https://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters_2017_Jan_1/quantum_theory_origins/index.html

SeeyouonThursday

SummaryofthesemesterRequiredposttestThursdayApril26

FinalExamWednesday,May9

32