Table Of ContentMAS207: E
LECTROMAGNETISM
HENK VAN ELST
AstronomyUnit
SchoolofMathematicalSciences
QueenMary,UniversityofLondon
MileEndRoad
LondonE14NS,UnitedKingdom
E–Mail: [email protected]
May24,2004
MAS207: ELECTROMAGNETISM
These lecture notes are only intended for the use of students studying
(cid:0)
the courseMAS207: Electromagnetism. Any distribution for any other
purposeisnotpermitted.
The selection of topics covered in these lecture notes is influenced by
(cid:0)
thefollowingtextbooks,lecturenotesandwebsites:
– Classical Electrodynamics, John D Jackson, Wiley & Sons, New
York,(cid:1) 1999.
– Electrodynamics(LecturesonTheoreticalPhysics,Vol.III),Arnold
Sommerfeld,AcademicPress,NewYork,1952.
– The Feynman Lectures on Physics Vol. II, Richard P Feynman,
RobertBLeighton,MatthewSands,Addison-Wesley,Reading(MA),
1964.
– FoundationsofElectromagneticTheory,JohnRReitz,FrederickJ
Milford,RobertWChristy,Addison-Wesley,Reading(MA),1979.
– AFirstCourseinGeneralRelativity,BernardFSchutz,Cambridge
UniversityPress,Cambridge,1985.
– Electromagnetism and Special Relativity, Reza Tavakol, Queen
Mary,UniversityofLondon,Lecturenotes,1984. Unavailable.
– Theorie B: Elektrodynamik, Gottfried Falk, University of Karls-
ruhe,Notestakeninlectures,SummerSemester1988[inGerman].
Unavailable.
– Theorie B:Elektrodynamik, Julius Wess, University of Karlsruhe,
LecturenotesbyThomasStrohm,SummerSemester1989[inGer-
man]. Unavailable.
– PhysikII:Elektrodynamik,FriedrichHerrmann,UniversityofKarls-
ruhe,Script,1997[inGerman]. AvailableonlineatURL:
www-tfp.physik.uni-karlsruhe.de/ didaktik/.
(cid:2)
– Wolfram Research’s World of Science, serviced by Eric W Weis-
stein,URL:scienceworld.wolfram.com.
– TheMacTutorHistoryofMathematicsarchive,URL:
www-history.mcs.st-andrews.ac.uk.
– MAS209:FluidDynamics,DavidBurgessandHenkvanElst,Queen
Mary,UniversityofLondon,Lecturenotes,1997–2001. Available
fromauthors.
– Relativity and Cosmology, Bernard Carr, Queen Mary, University
ofLondon,Lecturenotes,1989. Unavailable.
Copyright (cid:3)c 2002bytheSchoolofMathematicalSciences,QueenMary,
UniversityofLondon,HenkvanElst.
Contents
1 Introductoryremarks 1
1.1 FieldvariablesandMaxwell’sfieldequations . . . . . . . . . 1
1.2 Specialcases . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Integraltheoremsandbalanceequations . . . . . . . . . . . . 6
1.3.1 Gauß’andStokes’integraltheorems . . . . . . . . . . 6
1.3.2 Balanceequations . . . . . . . . . . . . . . . . . . . 6
1.3.3 Poynting’s theorem . . . . . . . . . . . . . . . . . . . 8
1.4 Maxwell’sfieldequationsinintegralform . . . . . . . . . . . 9
2 Mathematicaltechniques 13
2.1 VectorcalculusinCartesiancoordinates . . . . . . . . . . . . 13
2.2 Vectorcalculusinorthogonalcurvilinearcoordinates . . . . . 14
2.2.1 Cylindricalpolarcoordinates . . . . . . . . . . . . . . 15
2.2.2 Sphericalpolarcoordinates . . . . . . . . . . . . . . . 17
2.3 Vectoranalyticalidentities . . . . . . . . . . . . . . . . . . . 18
2.4 ConsequencesofGauß’integraltheorem . . . . . . . . . . . . 19
2.5 Dirac’sdeltafunction . . . . . . . . . . . . . . . . . . . . . . 20
2.6 Orthonormalfunctionexpansions. . . . . . . . . . . . . . . . 21
2.6.1 Fourierseriesexpansions . . . . . . . . . . . . . . . . 22
2.6.2 Fourierintegralrepresentations . . . . . . . . . . . . 22
3 Electrostatics 25
3.1 Coulomb’slawandsuperpositionprinciple . . . . . . . . . . 25
3.1.1 Gauß’law . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Electrostaticscalarpotential . . . . . . . . . . . . . . . . . . 27
3.3 Poisson’sandLaplace’s equations . . . . . . . . . . . . . . . 29
3.3.1 Formalsolutions . . . . . . . . . . . . . . . . . . . . 30
3.3.2 Boundaryconditionsanduniquenessofsolutions . . . 31
CONTENTS
3.3.3 BoundaryvalueproblemsandGreen’sfunctions . . . 32
3.4 Furtherremarkonboundaryconditions . . . . . . . . . . . . 33
3.5 Energydensityofelectrostaticfields . . . . . . . . . . . . . . 34
3.6 Methodofimages . . . . . . . . . . . . . . . . . . . . . . . . 36
3.6.1 Pointchargenearagroundedconductingplane . . . . 37
3.6.2 Pointchargenearagroundedconductingsphere . . . 38
3.6.3 Pointchargenearacharged,insulated,conductingsphere 40
3.6.4 DirichletGreen’sfunctionforaconductingsphere . . 41
3.6.5 Conductingsphereinauniformelectrostaticfield . . . 42
3.7 Separationofvariablesmethod . . . . . . . . . . . . . . . . . 44
3.7.1 Laplace’sequationinCartesiancoordinates . . . . . . 44
3.7.2 Laplace’sequationinsphericalpolarcoordinates . . . 46
3.8 Multipoleexpansionsandmultipolemoments . . . . . . . . . 53
4 Magnetostatics 57
4.1 Ampe`re’sforcelawandBiot–Savartlaw . . . . . . . . . . . . 57
4.1.1 Magneticfieldduetocurrent-carryingstraightwire . . 60
4.1.2 Ampe`re’slaw . . . . . . . . . . . . . . . . . . . . . . 60
4.2 Magnetostaticvectorpotential . . . . . . . . . . . . . . . . . 61
4.2.1 Gaugefreedom . . . . . . . . . . . . . . . . . . . . . 62
4.3 Remarkonboundaryconditions . . . . . . . . . . . . . . . . 63
4.4 Energydensityofmagnetostaticfields . . . . . . . . . . . . . 64
5 Time-dependentelectricandmagneticfields 67
5.1 Energy and linear momentum balance equations for electro-
magneticfields . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.2 Inhomogeneouswaveequations . . . . . . . . . . . . . . . . 70
5.3 Planewavesolutionsinvacuum . . . . . . . . . . . . . . . . 71
5.4 Electromagneticscalarandvectorpotentials . . . . . . . . . . 75
5.4.1 Gaugetransformations . . . . . . . . . . . . . . . . . 76
5.4.2 Retardedsolutions . . . . . . . . . . . . . . . . . . . 77
6 LorentzinvarianceofMaxwell’sfieldequations 79
6.1 Historicalremarks. . . . . . . . . . . . . . . . . . . . . . . . 79
6.2 Specialtheoryofrelativity . . . . . . . . . . . . . . . . . . . 81
6.3 Lorentztransformationsofelectromagneticfieldvariables . . 83
6.4 Lengthcontraction,timedilationandvelocity-compositionlaw 85
CONTENTS
6.5 Equationsofmotionforcompactbodies . . . . . . . . . . . . 85
CONTENTS
Chapter 1
Introductory remarks
Many phenomena and applications we encounter in our everyday lives are
grounded on electromagnetic interactions. Examples that spring to one’s
mindare: theswitchingonofanelectriclightbulb;thetransmissionofsignals
thatletuscommunicateviaradio,televisionormobilephones;theinnerwork-
ings of our PCs; or the generation of lightnings in a thunder storm. Electro-
magnetism(luckily)revealsitselftousasaninherentlylinearphenomenonof
Nature,whichmeansthatthelinearsuperpositionofelectromagneticfieldsis
possible. Thisisexploitedtoouradvantageinmanyofourelectromagneticap-
plications; e.g., many phone calls simultaneously going down the same wire,
or different TV stations broadcasting at the same time without interference.
Surprisingly, itseemsthatwehavelongstoppedwonderingabouttheseissues
whichappearsofamiliartous(orperhapsnot?).
The theoretical understanding of electromagnetic interactions and the math-
ematical formulation of electromagnetism (or electrodynamics) as a classi-
cal field theory of physics have been completed during the nineteenth cen-
tury, with major contributions by the French physicist Charles Augustin de
Coulomb(1736–1806), the Frenchmathematicianand physicistAndre´ Marie
Ampe`re (1775–1836), the English physicist Michael Faraday (1791–1867),
the GermanphysicistHeinrich Hertz (1857–1894),and, last but not least, the
ScottishmathematicianandphysicistJamesClerkMaxwell(1831–1879). The
aim of this course is to provide an introduction to the mathematical formula-
tion of electromagnetism and to discussmany examples—mostof them at-
tributedtoelectrostatics, magnetostatics, orvacuumfieldconfigurations—
that ariseas simplespecialsubcases and can be solved analytically. Contrary
to most text books, whereMaxwell’s field equations, a setof coupledlinear
first-orderpartialdifferentialequations,arederivedoneafteranotherfollowing
their historical development (inductive approach), we will here present them
rightatthebeginningandthenturntodiscussvarioussubcases,asmentioned
above(deductive approach). Thehopeisthatthiswillhelptodemystifythem
(andindeedanyothermathematicalmodelusedintheoreticalphysics)andbe-
comefamiliarwiththematan earlierstageinthecoursethanusual. Inwork-
ingwithMaxwell’sfieldequationswemakeuseofmanytoolsfromapplied
mathematicsandmathematicalphysics.
1.1 Field variables and Maxwell’s field equations
Electromagnetic fields can be sourcedby electric charges and electriccur-
1
2 MAS207:Electromagnetism2002
rents, but,aswewillseelatter, theycanalsoexistin“vacuo”, wherenosuch
sourcesarepresent. Electricchargecantakebothpositiveandnegativevalues.
The SI unit for electric charge is the Coulomb ( ). The charge carried by
(cid:0)(cid:2)(cid:1)
a single electron is . Its magnitude
(cid:3)(cid:5)(cid:4)(cid:7)(cid:6)(cid:9)(cid:8)(cid:10)(cid:0)(cid:12)(cid:11)(cid:14)(cid:13)(cid:12)(cid:15)(cid:17)(cid:16)(cid:18)(cid:0)(cid:20)(cid:19)(cid:21)(cid:19)(cid:12)(cid:22)(cid:21)(cid:22)(cid:24)(cid:23)(cid:26)(cid:25)(cid:28)(cid:27)(cid:17)(cid:29)(cid:31)(cid:30) (cid:0)!(cid:15)#"%$’&((cid:1) )*(cid:3)(cid:5)(cid:4)()
constitutes the basic unit of electric charge; in Nature electric charge is in-
deed quantised. Nevertheless, at a sufficiently macroscopic scale (compared
toatomicscales)electricchargecanbetreatedascontinuouslydistributeddue
tothesmallnessof .
)*(cid:3)(cid:5)(cid:4)()
The macroscopic field variables we use to describe the dynamics of elec-
tromagnetic fields and their interactions with continuous material media as
functionsof atimecoordinate andposition in Euclidian(flat)space, (cid:1) ,
+ , -
are:
(i) —theelectricfieldstrength,SIunit: ,alsoreferred
./(cid:6)0.1(cid:23)(cid:26)+324,5(cid:29) (cid:0)769;=8#<?>:
toas ,
(cid:0)A@(cid:31)(cid:23)(cid:26)B(cid:17)CEDF(cid:29)FGIH
(ii) — the magnetic field strength, SI unit: , also re-
J (cid:6)KJL(cid:23)(cid:26)+324,M(cid:29) (cid:0) ;(cid:20)69>8
ferredtoas ,
(cid:0)%NO(cid:23)(cid:26)PRQ4CTS(cid:28)(cid:29)
(iii) —theelectricexcitation, SIunit: > ,
UV(cid:6)0UW(cid:23)(cid:26)+F24,5(cid:29) (cid:0) <
:
(iv) —themagneticexcitation, SIunit: > ,
XY(cid:6)0XZ(cid:23)(cid:26)+324,5(cid:29) (cid:0) ;
:
(v) —theelectricchargedensity,SIunit: > ,and
[(cid:7)(cid:6)W[M(cid:23)(cid:26)+324,M(cid:29) (cid:0)
:]\
(vi) —theelectriccurrentdensity,SIunit: > .
^W(cid:6)0^_(cid:23)(cid:26)+324,M(cid:29) (cid:0) <‘;
:
It should be emphasised that idealisations such as field strengths at a point
mustbeviewedasamathematicalconstructthatpermitsadescriptionofelec-
tromagnetic phenomena at the macroscopic level. They may fail to have a
meaningatthemicroscopiclevel.
Maxwell’s field equations for macroscopic field configurations are given as
twosetsofpartialdifferentialequationswhicharelinearinthemacroscopic
fieldvariables:
(a)thetimeevolutionequationsfor and ,
U J
a
U (1.1)
a (cid:8)Lbc(cid:30)dX (cid:6) (cid:8)e^
+
a
J f (1.2)
a bV(cid:30)d. (cid:6) gh2
+
which could be referred to as Ampe`re–Maxwell law and Faraday’s law, re-
spectively, and
MAS207:Electromagnetism2002 3
(b)theinitialvalueconstraintequationsfor and ,
U J
(1.3)
(cid:15) (cid:6) (cid:0) (cid:6) b(cid:3)(cid:2)IU(cid:9)(cid:8) [
U (cid:1)
(1.4)
(cid:15) (cid:6) (cid:0) (cid:6) b(cid:3)(cid:2)IJ 2
J (cid:1)
which could be referred to as Gauß’ law (named after the German mathe-
maticianandastronomerCarlFriedrichGauß,1777–1855)andzeromagnetic
chargeslaw,respectively.
Itcanbeshownthattheconstraintequationsarepreservedintime,i.e.,
a
(cid:0) (1.5)
U
a (cid:6) (cid:15)
+
a
(cid:0) (1.6)
J
a (cid:6) (cid:15) 2
+
provided thecondition(thecontinuityequation)
a
[ f (1.7)
(cid:15)O(cid:6) a b(cid:4)(cid:2)(cid:20)^
+
holds. Itisan empiricalresultthatforallelectriccharge andcurrentdistribu-
tionsoccurringinNaturethisisindeedthecase. Thisresultisbetterknownas
theconceptofconservationofelectriccharge.
To understand why, conventionally, one maintains a distinction between the
macroscopic field variables and on the one hand and and on the
. U J X
other, thefollowingcrudeguidelinemaybehelpful. Oneinterprets and
U X
to be sourced by, respectively, unbounded macroscopic charges and currents
in continuous material media only (represented through and ), while
[ ^ .
and are interpreted to be sourced by unbounded macroscopic charges and
J
currentsaswellasboundedmicroscopiccharges andcurrents(thetotaleffect
ofthelatteroftenbeingrelatedtothefieldvariables and whichwewill
(cid:5) (cid:6)
introducebelow).
The fundamental electric and magneticfields are and , as they (a) deter-
. J
minethemechanicalforce onapointcharge movingwithvelocity ina
(cid:7) (cid:3) (cid:8)
givenelectromagneticfieldaccordingtoLorentz’sforcelaw
f (1.8)
(cid:7) (cid:6)0(cid:3) (cid:23) . (cid:8) (cid:30)eJ (cid:29) 2
(b) can be derived from electromagnetic potentials, as we will discuss in
chapter5,and(c)becauseofthewaytheybehaveunderLorentztransforma-
tions between different inertial reference frames, as we will discuss in chap-
ter6.
For the time evolution equations (1.1) and (1.2) amongst Maxwell’s field
equations to become a closed system of partial differential equations, they
4 MAS207:Electromagnetism2002
needtobesupplementedbyconstitutiverelationsoftheform
(1.9)
U (cid:6) U(cid:1)(cid:0).e29J(cid:3)(cid:2)
(1.10)
X (cid:6) X(cid:4)(cid:0).d29J(cid:3)(cid:2)A2
andageneralisedOhm’slaw(namedaftertheGermanphysicistGeorgSimon
Ohm,1789–1854)
(1.11)
^ (cid:6) ^(cid:5)(cid:0).d29J(cid:3)(cid:2)A(cid:11)
The determination of the precise form of these relations for a given macro-
scopicmaterialmediumisahighlycomplexmatterandtypicallyfallsintothe
realm of condensed matter physics and solid state physics. To a first ap-
proximation,whichisconvenientformostpracticalpurposes,onecanassume
continuousmediato have linear electricandmagneticpropertiesandnegligi-
bledispersionandlossessothatlinearconstitutive relationsoftheform
f (1.12)
U (cid:6) (cid:6)(cid:8)(cid:7) . (cid:5)
(cid:0) (1.13)
X (cid:6) (cid:9) J (cid:8) (cid:6)
(cid:7)
can be used. The new field variables thus introduced are and
(cid:5) (cid:6) (cid:5) (cid:23)(cid:26)+F24,A(cid:29)
whicharecalled,respectively,theelectricpolarisationandthe
(cid:6) (cid:6) (cid:6) (cid:23)(cid:26)+324,M(cid:29)
magnetisationof a given macroscopicmaterial medium. Theconstants
(cid:6) (cid:7) (cid:6)
> < ,thepermittivityofvacuum,and ,
(cid:9)
(cid:23) (cid:0)(cid:20)G(cid:5)(cid:25)(cid:11)(cid:10)(cid:13)(cid:12)(cid:15)(cid:14)(cid:5)(cid:29) (cid:30)1(cid:0)!(cid:15)(cid:11)(cid:16) (cid:7) (cid:6)W(cid:25)(cid:11)(cid:10)1(cid:30) (cid:0)!(cid:15)#"(cid:17)(cid:16) 69>8#<:
the permeability of vacuum, are related to the speed of light in vacuum,
698#:
, through . Ohm’s law (1826) takes the linear
(cid:9)
(cid:12) (cid:6) (cid:16)(cid:21)(cid:27)(cid:21)(cid:27)(cid:17)(cid:19)(cid:12)(cid:27)(cid:21)(cid:16)I(cid:25)(cid:19)(cid:18)(cid:21)(cid:20) :; (cid:6)(cid:8)(cid:7) (cid:7) (cid:6)(cid:22)(cid:12)(cid:5)"(cid:23)(cid:14)
form
(1.14)
^W(cid:6)(cid:25)(cid:24)_. 2
withthepositiveconstant knownastheelectricconductivity.
(cid:24)
Assumingaknownelectricchargedensity andaknownelectriccurrentden-
[
sity , and using the particular constitutive relations (1.12) and (1.13), we
^
can rewrite Maxwell’s field equations (1.1)–(1.4) to yield (using the special
relativisticallymoreappropriatetimecoordinate )
(cid:12)3+
a (cid:27) a
. G(cid:26)(cid:12) (cid:9) f (cid:23)(cid:28)(cid:12) (cid:5)e(cid:29) f (1.15)
a (cid:8)Lb (cid:30)dJ (cid:6) (cid:8) (cid:7) ^ a bc(cid:30) (cid:6)(cid:30)(cid:29)
(cid:12)3+ (cid:12)3+
a
J f (1.16)
a bV(cid:30)e. G(cid:26)(cid:12) (cid:6) g
(cid:12)3+
(cid:0) (1.17)
(cid:15) (cid:6) (cid:0) (cid:6) b(cid:4)(cid:2)(cid:5).hG(cid:26)(cid:12) (cid:8) (cid:0)(cid:31)(cid:12)3[ (cid:8)Lb(cid:4)(cid:2)(cid:28)(cid:23)(cid:28)(cid:12) (cid:5)d(cid:29)(cid:19)(cid:2)
. (cid:1)
(cid:6) (cid:7) (cid:12) (cid:14)
(1.18)
(cid:15) (cid:6) (cid:0) (cid:6) b (cid:2)(cid:20)J (cid:11)
J
Thisistheformwewillusethroughoutthiscourse. Thetheoryofpartialdif-
ferential equations states that for given (known) sources , ,
(cid:12)3[M(cid:23)(cid:28)(cid:12)3+F24,A(cid:29) ^O(cid:23)(cid:28)(cid:12)3+324,5(cid:29)
and ,thereexistuniquesolutions and
(cid:12) (cid:5) (cid:23)(cid:28)(cid:12)3+F24,A(cid:29) (cid:6) (cid:23)(cid:28)(cid:12)3+F24,5(cid:29) .e(cid:23)(cid:28)(cid:12)3+324,M(cid:29)?G(cid:26)(cid:12) J (cid:23)(cid:28)(cid:12)F+324,M(cid:29)
