CH 33 Electromagnetic Waves - United States Naval Academy · 2016-07-08 · Page 6 4. Now let’s...

[SHIVOK SP212] March 17, 2016

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CH 33

Electromagnetic Waves

I. ElectromagneticWaves

A. Maxwell’sRainbow

1. Asthefigureshows,wenowknowawidespectrum(orrange)ofelectromagneticwaves:Maxwell’srainbow.Inthewavelengthscaleinthefigure,(andsimilarlythecorrespondingfrequencyscale),eachscalemarkerrepresentsachangeinwavelength(andcorrespondinglyinfrequency)byafactorof10.

2. _________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________.

3. VisibleSpectrum:

CH: 16 Review items:


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B. TheTravelingElectromagneticWave,Qualitatively

1. Someelectromagneticwaves,includingxrays,gammarays,andvisiblelight,are________________________________________fromsourcesthatareofatomicornuclearsize.Figure33‐3showsthegenerationofsuchwaves.AtitsheartisanLCoscillator,whichestablishesanangularfrequency____________________________________________.Chargesandcurrentsinthiscircuitvarysinusoidallyatthisfrequency.

2. Figure33‐4showshowtheelectricfieldandthemagneticfieldchangewithtimeasonewavelengthofthewavesweepspastthedistantpointPinthelastfigure;ineachpartofFig.33‐4,thewaveistravelingdirectlyoutofthepage.


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3. Atadistantpoint,suchasP,thecurvatureofthewavesissmallenoughtoneglectit.Atsuchpoints,thewaveissaidtobea_________________________________________.

4. Herearesomekeyfeaturesregardlessofhowthewavesaregenerated:

a) Theelectricandmagneticfieldsandarealwaysperpendiculartothedirectioninwhichthewaveistraveling.Thewaveisatransversewave.

b) Theelectricfieldisalways______________________tothemagneticfield.

c) Thecrossproduct______________________alwaysgivesthedirectioninwhichthewavetravels.

d) Thefieldsalwaysvary________________________________________.Thefieldsvarywiththesamefrequencyandare______________________witheachother.

5. Wecanwritetheelectricandmagneticfieldsassinusoidalfunctionsofpositionx(alongthepathofthewave)andtimet:

6. HereEmandB

maretheamplitudesofthefieldsand,andkarethe

angularfrequencyandangularwavenumberofthewave,respectively.

7. ________________________________________________________________________________________________________________________________________________________________________________________.

8. Thespeedofthewave(invacuum)isgivenbyc.

Its value is about 3.0 x108 m/s

9. TheratioofamplitudesoftheElectricandMagneticfieldsarealsorelatedtothespeedoflightasfollows:

(Eq 33‐4)

10. Themagnitudesofthefieldsateveryinstantandatanypointarerelatedby:

(Eq 33‐5)

Let’s now prove equations 33‐4 and 33‐5 with Calculus.


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C. TheTravelingElectromagneticWave,Quantitatively

1. Let’srepresentanElectromagneticwaveasinFig33‐5

2. ThedashedrectangleofdimensionsdxandhinFig.33‐6isfixedatpointPonthexaxisandinthexyplane.

3. Astheelectromagneticwavemovesrightwardpasttherectangle,themagneticfluxBthroughtherectanglechangesand—accordingtoFaraday’slawofinduction—inducedelectricfieldsappearthroughouttheregionoftherectangle.WetakeEandE+dEtobetheinducedfieldsalongthetwolongsidesoftherectangle.Theseinducedelectricfieldsare,infact,theelectricalcomponentoftheelectromagneticwave.

a) StartingwithMaxwell’sEquationthatrelatestheinducedelectricfieldtothechangingmagneticflux:


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b) Applyittoourdrawingsof33‐5and33‐6;thus:

c) Thefluxthroughtherectangleis:

d) Finding

e) Therforesubstituteandthus

f) SO (Eq33‐11)

g) ThusfromEqs33‐1and33‐2

h) SorewritingEq33‐11weget:

i) Finallyyoucanseethat

where = c (Eq33‐13)

If we divide Eq 33‐1 by 33‐2 and then substitute in Eq 33‐13 we get Eq 33‐5. You can prove on your own.


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4. Nowlet’sprovetheEquation33‐3usingCalculus.

a) ExaminetheMagneticfield

b)

Fig.33‐7Thesinusoidalvariationoftheelectricfieldthroughthisrectangle,located(butnotshown)atpointPinFig.33‐5b,Einducesmagneticfieldsalongtherectangle.TheinstantshownisthatofFig.33‐6:isdecreasinginmagnitude,andthemagnitudeoftheinducedmagneticfieldisgreaterontherightsideoftherectanglethanontheleft.

c) StartingwithMaxwell’sEquationthatrelatestheinducedmagneticfluxtothechangingelectricalfield:

d) Applyittoourdrawingof33‐7thus:

e) Theelectricfieldthroughtherectangleis:

f) Thismeans

g) So

h) Whichleadsto


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i) Justaswedidbeforesubstitutingweget:

j) Now

k) Whichfinallymeansthat

D. EnergyTransportandthePoyntingVector

1. TherateofenergytransportperunitareaiscalledthePoyntingVector.

2. InstantaneousEnergyflowrate.

3. AverageEnergyTransportedovertimeorIntensity

a)

(1) Remember that the average for Sin2f for any f is ½.

(2) Theenergydensityu(=)withinanelectricfield,canbewrittenas:


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b) VariationofIntensitywithDistance

(1) TheintensityI(powerperunitarea)measuredatthespheremustbe

4. SampleProblem:A10‐kWradiostationradiatessphericalelectromagneticwaves.Themaximumvalue(amplitude)ofthewave’soscillatingelectricfieldatadistanceof5.0kmfromthestationisclosestto:

A. 3.6 V/m B. 0.16 V/m C. 0.56 V/m D. 1.6 V/m E. 16 V/m

Show all work


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E. RadiationPressure

1. Electromagneticwaveshavelinearmomentumandthuscanexertapressureonanobjectwhenshiningonit.

2. Duringtheintervalt,theobjectgainsanenergyUfromtheradiation.Iftheobjectisfreetomoveandthattheradiationisentirelyabsorbed(takenup)bytheobject,thenthemomentumchangepisgivenby

3. Iftheradiationisentirelyreflectedbackalongitsoriginalpath,themagnitudeofthemomentumchangeoftheobjectistwicethatgivenabove,or

4. Since anditfollowsthat

5. Finally,theradiationpressureinthetwocasesare


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6. SampleProblems:

a) Thenexttwoproblemsdealwithanelectromagneticwaveinamaterialwheretheelectricfieldhasay‐componentonlyand,inSIunits,isgivenbyEy=(40.0V/m)sin[(1.4x107m‐1)x–(4.2x1015rad/s)t)]wherexisinmetersandtisinseconds.

(1) ThewavelengthanddirectionoftravelofthewaveareclosesttoA. 449 nm in the positive x direction. B. 449 nm in the negative x direction. C. 333 nm in the positive x direction. D. 333 nm in the positive x direction. E. 282 nm in the positive x direction.

Show all work/Explain:

(2) Iftheelectromagneticwaveisfullyreflectedbyasurface,theradiationpressureisclosestto

A. 2.03 x 10-8 N/m2

.

B. 3.45x 10-7 N/m2

.

C. 1.42 x 10-8 N/m2

.

D. 7.08 x 10-9 N/m2

.

E. 5.63 x 10-9 N/m2

.

Show all work


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F. Polarization

1. Polarized______________________________________________________________________.

a) Diagram‐Figure33‐9ashowsanelectromagneticwavewithitselectricfieldoscillatingparalleltotheverticalY‐axis.

2. Polarizedrandomlyorunpolarized


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(1) IntensityofUnpolarizedLight

(a) IftheintensityoforiginalunpolarizedlightisIo,then

theintensityoftheemerginglightthroughthepolarizer,I,ishalfofthat.

3. Polarizingsheets

a) Wecantransformunpolarizedvisiblelightintopolarizedlightbysendingitthroughapolarizingsheet,asshownbelow.

(1) __________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________.


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4. IntensityofPolarizedLight

a) Supposenowthatthelightreachingapolarizingsheetisalreadypolarized.

b) Figure33‐12showsapolarizingsheetintheplaneofthepageandtheelectricfieldofsuchapolarizedlightwavetravelingtowardthesheet(andthuspriortoanabsorption).

c) WecanresolveEintotwocomponentsrelativetothepolarizingdirectionofthesheet:parallelcomponentE

yistransmittedbythesheet

andperpendicularcomponentEzisabsorbed.Sinceqistheangle

betweenandthepolarizingdirectionofthesheet,thetransmittedparallelcomponentis

d) Since,then

(1) ___________________________________________________________________________________________________________________________________________________________________________________________________________________________________.


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e) Asshowninfigure33‐13,oftenunpolarizedlightwillbesentthroughatleasttwosheets.Thefirstsheetisoftencalledapolarizer,andtheadditionalsheetsarecalledanalyzers.

f) Ifthetwosheetsareparallelallthelightpassedbythefirstisalsopassedbythesecond.Ifthesheetsareperpendicular(thesheetsaresaidtobecrossed),nolightispassedbythesecondsheet(figure33‐14).


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g) Soobviouslythereisthecasewheretheyarenotparallelorperpendicular.Thefollowingisasampleproblemonhowthiscaseishandled.

(1) Solution:

(a) Theanglebetweenthedirectionofpolarizationofthelightincidentonthefirstpolarizingsheetandthepolarizingdirectionofthatsheetis1=70°.IfI0istheintensityoftheincidentlight,thentheintensityofthelighttransmittedthroughthefirstsheetis:

(b) Thedirectionofpolarizationofthetransmittedlightmakesanangleof70°withtheverticalandanangleof2=20°withthehorizontal.2istheangleitmakeswiththepolarizingdirectionofthesecondpolarizingsheet.Consequently,thetransmittedintensityis:

In the figure on the left, a beam of light, with Intensity

43W/m2 and polarization parallel to the y axis, is sent

into a system of two polarizing sheets with polarizing

directions at angles of q1=70± and q2=90± to the Y‐axis. What is the Intensity of the light transmitted by the

two‐sheet system?


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5. Whenunpolarizedlightispassedthroughtwopolarizingfiltersinsuccession,itsintensityisdecreasedby80%.Theangle,θ,betweenthetransmissionaxesofthefiltersis:

A. 78.5°. B. 63.4°. C. 26.6°. D. 36.9°. E. 50.8°.

Show all work

6. Alaserproducesunpolarizedlightwithanintensityof5.0W/cm2.ThelightpassesthroughthreesheetsofPolaroidfilmasshown.ThetransmissionaxisofthesecondPolaroidmakesa30anglewiththatofthefirst,andtheaxisofthethirdmakesa60Þanglewiththatofthesecond(and90anglewiththatofthefirst).TheintensityofthelightthatemergesfromthethirdPolaroidisclosestto: A. 0 B. 1.4 W/cm2

C. 1.1 W/cm2

D. 0.16 W/cm2

E. 0.47 W/cm2 Show all work

7. Lightcanbepolarizedbymeansotherthanpolarizingsheet…suchasbyscatteringorreflection.


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G. ReflectionandRefraction

a) LawofReflection

b) LawofRefraction(Snell’sLaw)

c) TableofIndexesofRefraction


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d) Effectsofdifferentindexmediums

(1) Ifn2isequalton1,thenq2=q1andthebeamcontinuesun‐deflected.

(2) Ifn2>n1,thenq2<q1andthebeamisbentawayfromtheun‐deflecteddirectiontowardthenormal.

(3) Ifn2<n1,thenq2>q1andthebeamisbentawayfromtheun‐deflecteddirectionandawayfromthenormal.

(4) ItisamemoryaidtothinktowardthemediumwiththeHIGHERindex.Seein(b)itbendsdownandin(c)itbendsup.

(5) RefractionCANNOTbendabeamsomuchthattherefractedrayisonthesamesideofthenormalastheincidentray!

e) ExampleProblem:

Light in a vacuum is incident on the surface of an unknown medium.

They Physics Lab student decides she can figure out the medium if

she knows the index of the unknown material. She measures the

angle of the light in the unknown material and gets 21.28 ±. In the vacuum the beam of light makes and angle of 32.00± with the normal

to the surface. In your opinion what is the material made of?


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(1) Solution:

Thelawofrefractionstates

(2) Wetakemedium1tobethevacuum,withn1=1and1=32.0°.Medium2istheunknown,with2=21.28°.

(3) Wesolveforn2:

(4) Solooking________________thematerialis_______________________.

H. ChromaticDispersion

1. Theindexofrefractionnencounteredbylightinanymediumexceptvacuumdependsonthewavelengthofthelight.

2. Thedependenceofnonwavelengthimpliesthatwhenalightbeamconsistsofraysofdifferentwavelengthstherayswillberefractedatdifferentanglesbyasurface;thatis,thelightwillbespreadoutbytherefraction.

3. ThespreadingofthelightiscalledChromaticDispersion.


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4. ChromaticDispersionofWhitelight

5. Rainbows


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I. TotalInternalReflection

1. Foranglesofincidencelargethanc,suchasforraysfandgabove,thereis

norefractedrayandallthelightisreflected;thiseffectiscalledTotalInternalReflection.

2. ciscalledthe_______________________:

3. Whichmeans


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J. PolarizationbyReflection

1.

2.

3.


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K. SampleproblemsforReflection/Refraction

1. Abeamoflighttravelinginairstrikesthesurfaceofasolutionofcornsyrupinwateratanangleof30tothevertical.Ifthebeamisrefractedatanangleof19tothevertical,thenthespeedofthelightinthecornsyrupsolutionisclosestto:

A. 1.5 x 108 m/s B. 1.7 x 108 m/s C. 2.0 x 108 m/s D. 2.3 x 108 m/s E. 2.6 x 108 m/s

Show all work

2. Aflatpieceofglass(withindexofrefraction1.50)hasalayerofethanol(withindexofrefraction1.36)floatingontopofit.Lighttravelingintheglassstrikestheglass‐ethanolsurface.Thecriticalanglefortotalinternalreflectionintheglassisclosestto:

A. 65 B. 59 C. 47 D. 55 E. 39

Show all work


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3. Alayerofwater(n=1.33)existsonaslabofglass(n=1.46).Alaserbeamintheglassisincidentontheglass‐waterinterface.Relativetotheperpendiculartotheinterface,thesmallestanglefortotalinternalreflectionisclosestto:

A. 57.3°. B. 65.6°. C. 40.1°. D. 42.3°. E. 74.8°.

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4. Theangleofincidence(relativetothenormaltothesurface)forwhichthelightreflectedfromthewater‐diamondsurfaceiscompletelypolarizedisclosestto:

A. 57.6º. B. 59.5º. C. 61.2º. D. 66.3º. E. 64.1º.

Show all work

CH 33 Electromagnetic Waves - United States Naval Academy · 2016-07-08 · Page 6 4. Now let’s...

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