2.3. MAGNETOSTATICS
li
Particularlysimpleexpressionsareobtainedforplanarcurrentloops,i.e. incaseswherethe
curveγliesinaplane.Introducingcoordinatessuchthattheunitvectornormaltotheplane
n=e
3
coincideswiththecoordinatevectore
3
,wethenobtainfortheithcomponentofthe
magneticmoment
m
i
=−
I
2c
e
i
·
γ
ds×x=−
I
2c
γ
e
i
·(ds×x)=−
I
2c
γ
ds·(x×de
i
).
ApplicationofStokeslawthenyieldsm
i
=−
I
2c
S(γ)
dσe
3
·(∇×(x×e
i
)). Notingthat
e
3
·(∇×(x×e
i
))=−2δ
3i
,weobtain
m=
AI
c
e
3
,
i.e. amagneticmomentperpendiculartotheplanesupportingthecurrentloopandpropor-
tionaltotheareaoftheloop.
 Imagine e asystem of f point particlesatcoordinates s x
i
whereeachparticle carriescharge
q
i
,isofmassm
i
,andmoveswithvelocityv
i
. Inthiscase,j=
i
v
i
q
i
δ(x−x
i
). Equation
(2.57)reducesto
1
2c
i
q
i
x
i
×v
i
=
1
2c
i
q
i
m
i
l
i
,
(2.61)
wherel =x×(mv) istheangularmomentumcarried by apointparticle. . [Onlyparts s of
themagneticmomentcarriedbygenuineelementaryparticlesareduetotheirorbitalangular
momentuml.Asecondcontributionstemsfromtheir‘intrinsic’angularmomentumorspin.
Specifically,foranelectronatrest,m=2.002×
e
2cm
S,where|S|=1/2istheelectronspin
andthepre–factor2.002isknownastheg–factoroftheelectron.]
2.3.3 Magneticforces
Tounderstandthe physicalmeaningoftheconnections(currents fields)derivedabove,letus
exploreanintuitivelyaccessiblequantity,viz.themechanicalforcescreatedbyacurrentdistribution.
Assumingthatthechargedensityofthemobilechargecarriersinthewireisgivenbyρ,andthatthese
chargesmovewithavelocityv,thecurrentdensityinthewireisgivenbyj=ρv.
19
Comparison
withtheLorentzforcelaw(1.6)showsthatinamagneticfieldthewirewillbesubjecttoaforce
density = c
−1
ρv×B. Specifically,forathin n wire of cross section dA, , and d carrying acurrent
I=ρdAv(visthecomponentofv=ve
alongthewire.)theforceactingonasegmentoflength
dlwillbedF=fdAdl=c
−1
ρdAdl(ve
)×B=c
−1
Ids×dB.Summarizing,theforceactingona
lineelementdsofawirecarryingacurrentIisgivenby
df=
I
c
ds×B.
(2.62)
Inthefollowing,wediscussafewapplicationsoftheprototypicalforce–formula(2.62):
19
TheamountofchargeflowingthroughasmallsurfaceelementdσduringatimeintervaldtisdQ=dσρn·vdt.
DividingbydtwefindthatthecurrentthroughdσisgivenbyI
=dσn·(ρv)whichmeans thatj=ρvisthe
currentdensity.
Convert pdf to web form - Convert PDF to html files in C#.net, ASP.NET MVC, WinForms, WPF application
How to Convert PDF to HTML Webpage with C# PDF Conversion SDK
embed pdf into web page; pdf to web converter
Convert pdf to web form - VB.NET PDF Convert to HTML SDK: Convert PDF to html files in vb.net, ASP.NET MVC, WinForms, WPF application
PDF to HTML Webpage Converter SDK for VB.NET PDF to HTML Conversion
pdf to html conversion; convert from pdf to html
lii
CHAPTER2. THESTATICLIMIT
Forcesoncurrentloops
Considerasingle closed currentloopγ carryingacurrentI. . Thecurrentflowwillgeneratea
magneticfield(2.54)whichwillinturnactonthelineelementsoftheloop,asdescribedby(2.62).
Onemaythuswonderwhethertheloopexertsanetforceonitself.Integratingover(2.62),wefind
thatthisforceisgivenby
F=
γ
df=
I
c
γ
ds×B
(2.54)
=
I
2
c
2
γ
γ
ds×(ds
×(x−x
))
|x−x
|
3
=
=
I
2
c
2
γ
γ
ds
(ds·(x−x
))
|x−x
|
3
(x−x
)ds·ds
|x−x
|
3
=
=−
I
2
c
2
γ
ds·
γ
ds
·∇
1
|x−x
|
=0.
Thesecondterminthesecondlinevanishesbecauseitchangessignundercoordinateinterchange
x↔x
. Inthethirdlineweusedthat(x−x
)|x−x
|
−3
=−∇|x−x
|
−1
andthattheintegral
ofthegradientofafunctionoveraclosedloopvanishes. Weconcludethataclosedcurrentloop
doesnotexertanetforceontoitself.
I
1
I
2
x
y
F
Figure2.2:
Twoinfinitewiresexertingaforceoneachother
However,distinctcurrentdistributionsdo,ingeneral,actbymagneticforcesontoeachother.
Bywayofexample,considertwoinfiniteparallelwireskeptatadistancedandcarryingcurrents
I
1
andI
2
,resp. Choosingcoordinatesasshowninthefigureandparameterizingthelineelements
asds
1,2
=dx
1,2
e
x
,weobtaintheforceF
12
wire#2exertsonwire#1as
F
12
=
I
1
I
2
c
2
dx
1
dx
2
e
x
×e
x
×(e
x
(x
1
−x
2
)−e
y
d)
|(x
1
−x
2
)
2
+d
2
|
3
)=
=e
y
I
1
I
2
d
c
2
dx
1
dx
2
1
(x
2
2
+d
2
)
3/2
=e
y
2I
1
I
2
c
2
d
dx
1
,
whereinthesecondlineweused
−∞
dx(x
2
+d
2
)
−3/2
=2/d
2
. Reflectingtheirinfinitelength,the
wiresexertaninfiniteforceoneachother.However,theforceperunitlength
dF
dx
=
2I
1
I
2
c2d
e
y
isfinite
anddependsinversequadraticallyonthedistancebetweenthewires.Theforceisattractive/repulsive
forcurrentsflowinginthesame/oppositedirection. (Oneofthemoreprominentmanifestationsof
theattractivemagneticforcesbetweenparallelcurrentflowsisthephenomenonofcurrentimplosion
inhollowconductors.)
C#: How to Determine the Display Format for Web Doucment Viewing
RasterEdge web document viewer for .NET can convert PDF, Word, Excel into Bitmap, as well as SVG files at the same time, and then render image form to show
embed pdf into website; changing pdf to html
C#: How to Add HTML5 Document Viewer Control to Your Web Page
the necessary resources for creating web document viewer addCommand(new RECommand("convert")); _tabFile.addCommand new UserCommand("pdf"); _userCmdDemoPdf.
convert pdf to html for online; convert pdf into webpage
2.3. MAGNETOSTATICS
liii
Forcesonlocalcurrentdistributions
Consideraspatiallylocalizedcurrentdistributioninanexternalmag-
B
m
neticfield. Wewishtocomputetotalforceactingonthedistribution
F=
d
3
xf=
1
c
d
3
xj×B.
(2.63)
Choosingtheoriginofthecoordinatesystemsomewhereinsidethecurrent
distribution,andassumingthatthemagneticfieldvariesslowlyacrossthe
extentofthecurrentflow,onemayTaylorexpandBintheLorentzforce
formula:
F=
1
c
d
3
xj(x)×[B(0)+x·∇B(0)+...].
(x·∇Bisavectorwithcomponentsx
i
i
B
j
.) Theauxiliaryidentity(2.56)impliesthatthefirst
contributiontotheexpansionvanish. Theithcomponentoftheforceisthusgivenby
F
i
1
c
d
3
x
ijk
j
j
x
l
l
B
k
(2.56)
=
1
2c
d
3
x
ijk
mjl
(j×x)
m
l
B
k
=
=
1
2c
d
3
x(δ
kl
δ
im
−δ
km
δ
il
)(j×x)
m
l
B
k
=
1
2c
d
3
x((j×x)
i
k
B
k
−(j×x)
k
i
B
k
)=
=∂
i
|
x=0
(m·B(x)),
or
F=∇|
x=0
(m·B(x)),
(2.64)
whereinthesecondlineweused∇·B=∂
l
B
l
=0Thus,(a)theforceisstrongestifthemagnetic
momentisalignedwiththemagneticfield,and(b)proportionaltotherateatwhichtheexternal
magneticfieldvaries.Specifically,forauniformfield,noforcesact.
Whiletheforceonamagneticmomentreliesonthepresenceofafieldgradient,evenaconstant
fielddoesexertafinitetorque
N=
d
3
xx×f=
1
c
d
3
xx×(j×B).
(2.65)
Approximatingthefieldbyitsvalueattheorigin,
N
i
=
1
c
d
3
x
ijk
klm
x
j
j
l
B(0)
m
=
1
c
d
3
x(δ
il
δ
jm
−δ
im
δ
lj
)x
j
j
l
B(0)
m
=
=
1
c
d
3
x(x
j
j
i
B(0)
j
−x
j
j
j
B(0)
i
)
(2.56)
=
1
2c
d
3
x
lji
(x×j)
l
B(0)
j
=
=(m×B(0))
i
.
Thus,thetorqueactingonthemoment,
N=m×B(0),
(2.66)
isperpendiculartoboththeexternalfieldandthemomentvector. Itactssoastoalignthefield
andthemoment.
C# PDF: How to Create PDF Document Viewer in C#.NET with
to images or svg file; Free to convert viewing PDF print designed PDF document using C# code; PDF document viewer can be created in C# Web Forms, Windows
how to convert pdf file to html document; convert pdf form to html
C# PDF Convert to SVG SDK: Convert PDF to SVG files in C#.net, ASP
In some situations, it is quite necessary to convert PDF document into SVG image format. indexed, scripted, and supported by most of the up to date web browsers
convert pdf to html link; converting pdf to html email
liv
CHAPTER2. THESTATICLIMIT
C# PDF Converter Library SDK to convert PDF to other file formats
C#.NET can manipulate & convert standard PDF developers to conduct high fidelity PDF file conversion C#.NET applications, like ASP.NET web form application and
convert pdf to url link; convert pdf to web page online
C# Image: How to Integrate Web Document and Image Viewer
RasterEdgeImagingDeveloperGuide8.0.pdf: from this user manual, you can find the detailed instructions and Now, you may add a new Web Form to your web project.
convert pdf to html file; convert pdf to webpage
Chapter3
Electrodynamics
3.1 Magneticfieldenergy
AsapreludetoourdiscussionofthefullsetofMaxwellequations,letusaddressaquestionwhich,in
principle,shouldhavebeenansweredinthepreviouschapter:Whatistheenergystoredinastatic
magneticfield? In n section2.2.2, , the e analogous questionforthe electricfield was answered in a
constructivemanner:wecomputedthemechanicalenergyrequiredtobuildupasystemofcharges.
Itturnedoutthatthe answercould be formulated entirelyin termsoftheelectric field,without
explicitreferencetothechargedistribution creatingit. . Bysymmetry,onemightexpectasimilar
prescriptiontoworkinthemagneticcase.Here,onewouldaskfortheenergyneededtobuildupa
currentdistributionagainstthemagneticfieldcreatedbythoseelementsofthecurrentdistribution
thathavealreadybeenputinplace.Amomentsthought,however,showsthatthisstrategyisnot
quite asstraightforwardlyimplementedasintheelectric case: : nomatterhowslowly y wemove a
‘currentloop’in anmagneticfield,anelectricfieldactingonthechargecarriersmaintainingthe
currentintheloopwillbeinduced—theinductionlaw. Workwillhavetobedonetomaintaina
constantcurrentanditisthisworkfunctionthatessentiallyenterstheenergybalanceofthecurrent
distribution.
Tomakethispicturequantitative,consideracurrentloopcarry-
I
B
ingacurrentI. Wemaythinkofthisloopasbeingconsecutively
builtupbyimportingsmallcurrentloops(carryingcurrentI)from
infinity(see thefigure.) ) The e currentsflowingalong adjacentseg-
ments of f these loops s will eventually y cancel l so o that t only y the net
currentI flowingaroundtheboundaryremains.Letus,then,com-
putetheworkthatneedstobedonetobringinoneoftheseloops
frominfinity.
Consider,first,anordinarypointparticlekeptata(mechanical)
potentialU.Therateatwhichthispotentialchangesiftheparticle
changesitspositionisd
t
U=d
t
U(x(t))=∇U·d
t
x=−F·v,where
Fistheforceactingontheparticle. Specifically,forthecharged
particlesmovinginourprototypicalcurrentloops,F=qE,where
Eistheelectricfieldinducedbyvariationofthemagneticfluxthroughtheloopasitapproaches
frominfinity.
lv
C# Image: Save or Print Document and Image in Web Viewer
this. During the process, your web file will be automatically convert to PDF or TIFF file and then be printed out. Please
batch convert pdf to html; export pdf to html
C# PDF Convert to Jpeg SDK: Convert PDF to JPEG images in C#.net
and quickly convert a large-size multi-page PDF document to a group of high-quality separate JPEG image files within .NET projects, including ASP.NET web and
how to convert pdf into html; converting pdf into html
lvi
CHAPTER3. ELECTRODYNAMICS
Nextconsideraninfinitesimalvolumeelementd
3
xinsidetheloop. Assumingthatthecharge
carriers move at avelocity v, , the e charge ofthevolume elementisgiven by q q = = ρd
3
x and rate
ofitspotentialchange by d
t
φ= −ρd
3
xv·E= −d
3
xj·E. Wemaynowusetheinductionlaw
to express theelectric field in terms ofmagnetic quantities:
ds·E = = −c
−1
δS
dσn·d
t
B =
−c
−1
d
t
δS
dσn·d
t
(∇×A) = −c
−1
ds·d
t
A. Since e this holds regardless of f the geometry
of the e loop, we have E = −c
−1
d
t
A, where e d
t
A is the change e in vector potential l due e to the
movementoftheloop.Thus,therateatwhichthepotentialofavolumeelementchangesisgiven
byd
t
φ=c
−1
d
3
xj·d
t
A. Integratingovertimeandspace,wefindthatthetotalpotentialenergyof
theloopduetothepresenceofavectorpotentialisgivenby
E=
1
c
d
3
xj·A.
(3.1)
Althoughderivedforthespecificcaseofacurrentloop,Eq.(3.1)holdsforgeneralcurrentdistribu-
tionssubjecttoamagneticfield. (Forexample,forthecurrentdensitycarriedbyapointparticle
at x(t), j = = qδ(x−x(t))v(t), , we e obtain E E = = qv(t)·A(x(t)), , i.e. . the e familiar r Lorentz–force
contributiontotheLagrangianofachargedparticle.)
Now,assumethatweshifttheloopatfixedcurrentagainstthemagneticfield. Thechangein
potentialenergycorrespondingtoasmallshiftisgivenbyδE=c
−1
d
3
xj·δA,where∇×δA=δB
denotesthechangeinthefieldstrength.Usingthat∇×H=4πc
−1
j,werepresentδEas
δE=
1
d
3
x(∇×H)·δA=
1
d
3
x
ijk
(∂
j
H
k
)δA
i
=
=−
1
d
3
xH
k
ijk
j
δA
i
=
1
d
3
xH·δB,
whereintheintegrationbypartswe notedthatdue tothespatialdecay ofthe fieldsnosurface
termsatinfiniteyarise.DuetothelinearrelationH=µ
−1
0
B,wemaywriteH·δB=δ(H·B)/2,
i.e. δE=
1
δ
B·E. Finally,summingoverallshiftsrequiredtobringthecurrentloopinfrom
infinity,weobtain
E=
1
d
3
xH·B
(3.2)
forthemagneticfieldenergy. Notice(a)thatwehaveagainmanagedtoexpresstheenergyof
thesystementirelyintermsofthefields,i.e.withoutexplicitreferencetothesourcescreatingthese
fieldsand(b)thestructuralsimilaritytotheelectricfieldenergy(2.20).
3.2 Electromagneticgaugefield
ConsiderthefullsetofMaxwellequationsinvacuum(E=DandB=H),
∇·E = = 4πρ,
∇×B−
1
c
∂t
E =
c
j,
∇×E+
1
c
∂t
B = = 0,
∇·B = = 0.
(3.3)
C# TIFF: C#.NET Code to Create Online TIFF Document Viewer
We still demonstrate how to create more web viewers on PDF and Word documents at the DLL into your C#.NET web page, you may create a new Web Form (Default.aspx
convert pdf form to html form; how to convert pdf to html code
3.2. ELECTROMAGNETICGAUGEFIELD
lvii
Asinprevioussectionswewilltrytouseconstraintsinherenttotheseequationstocompactifythem
toasmallersetofequations.Asinsection2.3,theequation∇·B=0implies
B=∇×A.
(3.4)
(However,wemaynolongerexpectAtobetimeindependent.)Now,substitutethisrepresentation
intothelawofinduction: ∇×(E+c
−1
t
A)=0. ThisimpliesthatE+c
−1
t
A=−∇φcanbe
writtenasthegradientofascalarpotential,or
E=−∇φ−
1
c
t
A.
(3.5)
We have, , thus, managed d to represent the electromagnetic fields as in terms of derivatives of a
generalized four–componentpotential A= {A
µ
}= (φ,−A). . (The e negative sign multiplying A
hasbeenintroducedforlaterreference.) SubstitutingEqs.(3.4)and(3.5)intotheinhomogeneous
Maxwellequations,weobtain
−∆φ−
1
c
t
∇·A=4πρ,
−∆A+
1
c2
2
t
A+∇(∇·A)+
1
c
t
∇φ=
c
j
(3.6)
Theseequationsdonotlookparticularlyinviting. However,asinsection2.3wemayobservethat
thechoiceofthegeneralizedvectorpotentialAisnotunique;thisfreedomcanbeusedtotransform
Eq.(3.6)intoamoremanageableform:Foranarbitraryfunctionf:R
3
×R→R,(x,t)→f(x,t).
The transformation A A → → A+∇f f leaves s the magnetic field unchanged while the electric field
changesaccordingtoE→E−c
−1
t
∇f.If,however,wesynchronouslyredefinethescalarpotential
asφ→φ−c
−1
t
f,theelectricfield,too,willnotbeaffectedbythetransformation.Summarizing,
thegeneralizedgaugetransformation
A →
A+∇f,
φ →
φ−
1
c
t
f,
(3.7)
leaves the electromagnetic fields (3.4) and (3.5) unchanged. . (In n the fourcomponent shorthand
notationintroducedabove,thegaugetransformationassumeseasy–to–memorize theformA
µ
A
µ
−∂
µ
f,wherethefour–derivativeoperator{∂
µ
}=(∂
0
,∇)andx
0
=ctasbefore.)
Thegaugefreedommaybeusedtotransformthevectorpotentialintooneofseveralconvenient
representations.OfparticularrelevancetothesolutionofthetimedependentMaxwellequationsis
theso–calledLorentzgauge
∇·A+
1
c
t
φ=0.
(3.8)
Thisequation,too,affordsacompactfour–vectornotation:forageneralvector{v
µ
}=(v
0
,v)we
defineavector{v
µ
}≡(v
0
,−v),i.e. aan n objectwith ‘raised components’thatdiffersfromthe
onewith‘loweredcomponents’byasignchangeinthespace–likesector. Usingthisnotation,the
Lorentzconditionassumestheform∂
µ
A
µ
=0. Itisalwayspossibletosatisfythisconditionbya
suitablegaugetransformation. Indeed,ifA
does notobey theLorentzcondition,wemaydefine
A
µ
=A
µ
−∂
µ
f toobtain0
!
=∂
µ
A
µ
=∂
µ
A
µ
−∂
µ
µ
f.Ifwechosefsoastosatisfytheequation
lviii
CHAPTER3. ELECTRODYNAMICS
µ
f
µ
=∂
µ
A
µ
,theLorentzequationissatisfied.Expressedintermsofspaceandtimecomponents,
thislatterequationassumestheform
∆−
1
c
2
2
t
f=−
∇·A
+
1
c
t
φ
,
i.e. awaveequationwithinhomogeneity−(∇·A+c
−1
t
φ). Weshallseemomentarilythatsuch
equationscanalwaysbesolved,i.e. animplementationoftheLorentzgaugeconditionispossible.
IntheLorentzgauge,theMaxwellequationsassumethesimplifiedform
∆−
1
c
2
2
t
φ = = −4πρ,
∆−
1
c
2
2
t
A = = −
c
j.
(3.9)
Incombinationwiththegaugecondition(3.8),Eqs.(3.9)arefullyequivalenttothesetofMaxwell
equations(3.3). Again,thefour–vectornotationmaybeemployedtocompactifythenotationstill
further.With∂
µ
µ
=−∆+c
−2
2
t
andj
µ
=(cρ,−j),thepotentialequationsassumetheform
ν
ν
A
µ
=
c
j
µ
,
µ
A
µ
=0.
(3.10)
Beforeturningto the discussion of thesolution of these equationsa few generalremarks are in
order:
 TheLorentzconditionistheprevalentgaugechoiceinelectrodynamicsbecause(a)itbrings
theMaxwellequationsintoamaximallysimpleformand(b)willturnoutbelowtobeinvariant
underthemostgeneralclassofcoordinatetransformations,theLorentztransformationssee
below. (ItisworthwhiletonotethattheLorentzconditiondoesnotunambiguouslyfixthe
gaugeofthevectorpotential. Indeed,wemayaddtoanyLorentzgaugevectorpotentialA
µ
afunctionA
µ
→ A
µ
+f
µ
satisfying the homogeneous waveequation∂
µ
µ
f = = 0without
alteringthecondition∂
µ
A
µ
=0.) Othergaugeconditionsfrequentlyemployedinclude
 theCoulomb b gaugeorradiationgauge∇·A=0(employedearlierinsection2.3.) ) The
advantageofthisgauge isthatthe scalarMaxwell equation assumesthesimpleformofa
Poissonequation∆φ(x,t)=4πρ(x,t)whichissolvedby φ(x,t)=
d
3
x
ρ(x,t)/|x−x
|,
i.e. byaninstantaneous’Coulombpotential’(hencethenameCoulombgauge.) Thisgauge
representationhasalsoprovenadvantageouswithinthecontextofquantumelectrodynamics.
However,wewon’tdiscussitanyfurtherinthistext.
3.3 Electromagneticwavesinvacuum
AsawarmuptoourdiscussionofthefullproblemposedbythesolutionofEqs.(3.9),weconsider
thevacuumproblem,i.e. asituationwherenosourcesarepresent,j
µ
=0. Takingthecurlofthe
secondofEqs.(3.9)andusing(3.4)wethenfind
∆−
1
c
2
2
t
B=0.
3.3. ELECTROMAGNETICWAVESINVACUUM
lix
Similarly,addingtothegradientofthefirstequationc
−1
timesthetimederivativeofthesecond,
andusingEq.(3.5),weobtain
∆−
1
c
2
2
t
E=0,
i.e.invacuumboththeelectricfieldandthemagneticfieldobeyhomogeneouswaveequations.
3.3.1 Solutionofthehomogeneouswaveequations
ThehomogeneouswaveequationsareconvenientlysolvedbyFouriertransformation. Tothisend,
wedefineafour–dimensionalvariantofEq.(2.10),
˜
f(ω,k)=
1
(2π)
4
d
3
xdtf(t,x)e
−ik·x+iωt
,
(3.11)
f(t,x)=
d
3
kdωf(ω,k)e
ik·x−iωt
,
The onedifferencetoEq.(2.10)is thatthe sign–convention in the exponentofthe(ω/t)–sector
ofthetransformdiffersfromthatin the (k,x)–sector. . We e nextsubjectthe homogeneous wave
equation
∆−
1
c
2
2
t
ψ(t,x)=0,
(whereψismeanttorepresentanyofthecomponentsoftheE–orB–field)tothistransformation
andobtain
k
2
ω
c
2
ψ(ω,k)=0.
Evidently,thesolutionψ mustvanish h forallvalues (ω,k),exceptforthoseforwhich the factor
k
2
−(ω/c)
2
=0. Wemaythuswrite
ψ(ω,k)=c
+
(k)δ(ω−kc)+c
(k)δ(ω+kc),
wherec
±
∈Carearbitrarycomplexfunctionsofthewavevectork.Substitutingthisrepresentation
into the inverse transform,we obtain the general solution n of the scalar r homogeneous wave
equation
ψ(t,x)=
d
3
k
c
+
(k)e
i(k·x−ckt)
+c
(k)e
i(k·x+ckt)
.
(3.12)
The general solution is obtained by y linearsuperposition n of elementary plane e waves, e
i(k·x∓ckt)
,
whereeachwaveisweightedwithanarbitrarycoefficientc
±
(k). Theelementaryconstituentsare
called waves becauseforanyfixedinstanceofspace,x/time,tthey harmonicallydepend on the
complementaryargumenttime,t/positionvector,x. Thewavesareplanarinthesensethatforall
points in theplanefixedbytheconditionx·k= = const. thephase e ofthewaveisidentical,i.e.
thesetofpoints k·x= = const. definesa‘wavefront’perpendiculartothewavevectork. . The
spacingbetweenconsequtivewavefrontswiththesamephasearg(exp(i(k·x−ckt))isgivenby
∆x=2π
k≡λ
,whereλisthewavelengthofthewaveandλ
−1
=2π/kitswavenumber.Thetemporal
oscillationperiodofthewavefrontsissetby2π/ck.
lx
CHAPTER3. ELECTRODYNAMICS
Focusingonafixedwavevectork,wenextgeneralizeourresultstothevectorialproblemposed
bythehomogeneouswaveequations.SinceeverycomponentofthefieldsEandBissubjecttoits
ownindependentwaveequation,wemaywritedowntheprototypicalsolutions
E(x,t)=E
0
e
i(k·x−ωt)
,
B(x,t)=B
0
e
i(k·x−ωt)
,
(3.13)
whereweintroducedtheabbreviationω=ckandE
0
,B
0
∈C
3
areconstantcoefficientvectors.The
Maxwellequations∇·B=0and(vacuum)∇·E=0implytheconditionk·E
0
=k·B
0
=0,i.e.the
coefficientvectorsareorthogonaltothewavevector. Wavesofthistype,oscillatinginadirection
perpendiculartothewavevector,arecalledtransverse waves. . Finally,evaluatingEqs.(3.13)on
thelawofinduction∇×E+c
−1
t
B=0,weobtaintheadditionalequationB
0
=n
k
×E
0
,i.e.
thevectorB
0
isperpendiculartobothkandE
0
,andofequalmagnitudeasE
0
.Summarizing,the
vectors
k⊥E⊥B⊥k,
|B|=|E|
(3.14)
formanorthogonalsystemandBisuniquelydeterminedbyE(andviceversa).
Atfirstsight,itmayseemthatwehavebeentoliberalinformulating
k
E
B
thesolution(3.13):whilethephysicalelectromagneticfieldisarealvector
field,thesolutions(3.13)aremanifestlycomplex.Thesimplesolutionto
thisconflictistoidentify ReEand d ReBwiththephysicalfields.
1
3.3.2 Polarization
Inthefollowingwewilldiscussanumberofphysicallydifferentrealizationsofplaneelectromagnetic
waves.SinceBisuniquelydeterminedbyE,wewillfocusattentiononthelatter.Letuschoosea
coordinatesystemsuchthate
3
k.Wemaythenwrite
E(x,t)=(E
1
e
1
+E
2
e
2
)e
ik·x−iωt
,
whereE
i
=|E
i
|exp(iφ
i
). DependingonthechoiceofthecomplexcoefficientsE
i
,anumberof
physicallydifferentwave–typescanbedistinguished.
Linearlypolarizedwaves
Foridenticalphasesφ
1
2
=φ,weobtain
ReE=(|E
1
|e
1
+|E
2
|e
2
)cos(k·x−ωt+φ),
i.e.avectorfieldlinearlypolarizedinthedirectionofthevector|E
1
|e
1
+|E
2
|e
2
.
Info.
Linearpolarizationisahallmarkofmanyartificiallightsources,e.g. laserlightisusuallylinearly
polarized.Likewise,theradiationemittedbymanyantennaeshowsapproximatelylinearpolarization.
——————————————–
Documents you may be interested
Documents you may be interested