④10CLASSICALELECTRODYNAMICSwherethelastterm OzoE(t,x)/at isthefamous displacement current.Thisterm was introduced, in a stroke of genius, by Maxwell in order tomake theright hand side of this equation divergence free when j(t,x) is assumed to rep-resentthe density of thetotal electriccurrent,which canbesplitupin“or-dinary"conduction currents,polarisation currentsand magnetisation currentsThe displacement current is an extra term which behaves like a current densityflowing in vacuum. As we shall see later, its existence has far-reaching phys-ical consequences as it predicts the existence of electromagnetic radiation thatcan carry energy and momentum over very long distances, even in vacuum.1.3.3ElectromotiveforceIf an electric field E(t,x) is applied to a conducting medium, a current densityj(t,x) will be produced in this medium. There exist also hydrodynamical andchemical processes which can create currents. Under certain physical condi-tions,andforcertainmaterials,onecansometimes assumealinearrelationshipbetweenthecurrentdensityjandE,calledOhm'slaw:(1.25)j(t,x) =oE(t,x)where is the electric conductivity (S/m).In the most general cases, for in-stance in an anisotropic conductor, is atensorWecanviewOhm'slaw,Equation(1.25)above,asthefirstterminaTaylorexpansionofthelawj[E(t,x)l.Thisgeneral lawincorporatesnon-lineareffectssuch asfrequencymixing.Examplesof mediawhich arehighly non-linear aresemiconductorsandplasma.Wedrawtheattentiontothefactthatevenincaseswhen the linear relation between E and j isa good approximation, we still haveto use Ohm's law with care. The conductivity is, in general, time-dependent(temporal dispersive media) but then it is often the case that Equation (1.25) isvalid for each individual Fourier component of the field.If the current is caused by an applied electric field E(t,x), this electric fieldwill exert work on the charges in the medium and, unless the medium is super-conducting, therewill be some energy loss.The rate at whichthis energy isexpended is j·E per unit volume. If E is irrotational (conservative),j willdecay away withtime.Stationary currentsthereforerequirethatan electricfield which corresponds to an electromotive force (EMF)is present.In thepresence of such a field EEMF, Ohm's law, Equation (1.25) above, takes theformj= o(Estat + EEMF)(1.26)Dao/CED/BOO)Draftreleased13thNovember2000at22:01.0由
“main” 2000/11/13 page 10 10 CLASSICAL ELECTRODYNAMICS where the last term ∂ε0E(t,x)/∂t is the famous displacement current. This term was introduced, in a stroke of genius, by Maxwell in order to make the right hand side of this equation divergence free when j(t,x) is assumed to represent the density of the total electric current, which can be split up in “ordinary” conduction currents, polarisation currents and magnetisation currents. The displacement current is an extra term which behaves like a current density flowing in vacuum. As we shall see later, its existence has far-reaching physical consequences as it predicts the existence of electromagnetic radiation that can carry energy and momentum over very long distances, even in vacuum. 1.3.3 Electromotive force If an electric field E(t,x) is applied to a conducting medium, a current density j(t,x) will be produced in this medium. There exist also hydrodynamical and chemical processes which can create currents. Under certain physical conditions, and for certain materials, one can sometimes assume a linear relationship between the current density j and E, called Ohm’s law: j(t,x) = σE(t,x) (1.25) where σ is the electric conductivity (S/m). In the most general cases, for instance in an anisotropic conductor, σ is a tensor. We can view Ohm’s law, Equation (1.25) above, as the first term in a Taylor expansion of the law j[E(t,x)]. This general law incorporates non-linear effects such as frequency mixing. Examples of media which are highly non-linear are semiconductors and plasma. We draw the attention to the fact that even in cases when the linear relation between E and j is a good approximation, we still have to use Ohm’s law with care. The conductivity σ is, in general, time-dependent (temporal dispersive media) but then it is often the case that Equation (1.25) is valid for each individual Fourier component of the field. If the current is caused by an applied electric field E(t,x), this electric field will exert work on the charges in the medium and, unless the medium is superconducting, there will be some energy loss. The rate at which this energy is expended is j · E per unit volume. If E is irrotational (conservative), j will decay away with time. Stationary currents therefore require that an electric field which corresponds to an electromotive force (EMF) is present. In the presence of such a field E EMF , Ohm’s law, Equation (1.25) above, takes the form j = σ(E stat +E EMF) (1.26) Downloaded from http://www.plasma.uu.se/CED/Book Draft version released 13th November 2000 at 22:01
E111.3ELECTRODYNAMICSTheelectromotiveforceisdefinedas (Estat +EEMF)-dl8=(1.27)where dl is a tangential line element of the closed loop C.1.3.4Faraday's lawof inductionIn Subsection 1.1.2 we derived the differential equations for the electrostaticfield. In particular, on page 4 we derived Equation (1.6) which states thatV × Estat(x)= 0 and thus that Estat is a conservative field (it can be expressedas a gradient of a scalarfield). This implies that the closed line integral of EstatinEquation(1.27)abovevanishesandthatthisequationbecomesd EEMF .dl8=(1.28)ICIthasbeenestablishedexperimentallythatanonconservativeEMFfieldisproduced in a closed circuit C if the magnetic flux through this circuit varieswith time. This is formulated in Faraday's law which, in Maxwell's general-ised form, readsdb E(t, x) dl =8(t,x) :Φm(t,x)dtJC(1.29)adB(t,x)-dS =SB(t,x)dtJsatwhere mis themagneticfux and S is the surfaceencircled by C whichcanbeinterpreted as ageneric stationary“loop"and not necessarilyasa conductingcircuit. Application of Stokes' theorem on this integral equation, transforms itintothedifferentialequation0V×E(t,x)=(1.30)-B(t,x)atwhich is valid for arbitrary variations in the fields and constitutes the Maxwellequation which explicitly connects electricity with magnetism.Anychangeof themagneticflux@mwill induceanEMF.Letusthereforeconsider the case, illustrated if Figure 1.3.4 on the following page, that the"loop" is moved in such a way that it links a magnetic field which varies duringthe movement.The convective derivative is evaluated according to the well-knownoperatorformulaDraftveed13thNoyper2000at22:01.Downloadedfromhttp://w/CED/BOO)①由①
“main” 2000/11/13 page 11 1.3 ELECTRODYNAMICS 11 The electromotive force is defined as E = ✠C (E stat +E EMF)· dl (1.27) where dl is a tangential line element of the closed loop C. 1.3.4 Faraday’s law of induction In Subsection 1.1.2 we derived the differential equations for the electrostatic field. In particular, on page 4 we derived Equation (1.6) which states that ∇×E stat(x) = 0 and thus that E stat is a conservative field (it can be expressed as a gradient of a scalar field). This implies that the closed line integral of E stat in Equation (1.27) above vanishes and that this equation becomes E = ✠C E EMF · dl (1.28) It has been established experimentally that a nonconservative EMF field is produced in a closed circuit C if the magnetic flux through this circuit varies with time. This is formulated in Faraday’s law which, in Maxwell’s generalised form, reads E(t,x) = ✠C E(t,x)· dl = − d dt Φm(t,x) = − d dt ☎S B(t,x)· dS = − ☎S dS· ∂ ∂t B(t,x) (1.29) where Φm is the magnetic flux and S is the surface encircled by C which can be interpreted as a generic stationary “loop” and not necessarily as a conducting circuit. Application of Stokes’ theorem on this integral equation, transforms it into the differential equation ∇×E(t,x) = − ∂ ∂t B(t,x) (1.30) which is valid for arbitrary variations in the fields and constitutes the Maxwell equation which explicitly connects electricity with magnetism. Any change of the magnetic flux Φm will induce an EMF. Let us therefore consider the case, illustrated if Figure 1.3.4 on the following page, that the “loop” is moved in such a way that it links a magnetic field which varies during the movement. The convective derivative is evaluated according to the wellknown operator formula Draft version released 13th November 2000 at 22:01. Downloaded from http://www.plasma.uu.se/CED/Book
+12CLASSICALELECTRODYNAMICSdsB(x)dlB(x)FIGURE1.3:A loopC which moves with velocityv ina spatially varyingmagnetic field B(x) will sense a varying magnetic flux during the motion.da+v.V(1.31)dtatwhich follows immediately from the rules of differentiation of an arbitrarydifferentiable function f(t,x(t). Applying this rule to Faraday's law, Equa-tion (1.29) on the previous page, we obtainadB-dS=--B-(v.V)B.dS(1.32)8(t,x) =ds.dtJsotJs5During spatial differentiation v is to be considered as constant, and Equa-tion (1.15) on page 7 holds also for time-varying fields:V·B(t,x)= 0(1.33)(it is one of Maxwell's equations)so that, according to Equation(F.60)onpage 155,(1.34)VX(BXv)=(v.V)BDownloadedfromhttp://wSe/CED/BookDraft version released 13th November 2000 at 22:01.tnl:①由
“main” 2000/11/13 page 12 12 CLASSICAL ELECTRODYNAMICS B(x) dS v v dl C B(x) FIGURE 1.3: A loopC which moves with velocity v in a spatially varying magnetic field B(x) will sense a varying magnetic flux during the motion. d dt = ∂ ∂t +v ·∇ (1.31) which follows immediately from the rules of differentiation of an arbitrary differentiable function f(t,x(t)). Applying this rule to Faraday’s law, Equation (1.29) on the previous page, we obtain E(t,x) = − d dt ☎S B· dS = − ☎S dS· ∂ ∂t B− ☎S (v ·∇)B· dS (1.32) During spatial differentiation v is to be considered as constant, and Equation (1.15) on page 7 holds also for time-varying fields: ∇·B(t,x) = 0 (1.33) (it is one of Maxwell’s equations) so that, according to Equation (F.60) on page 155, ∇×(B×v) = (v ·∇)B (1.34) Downloaded from http://www.plasma.uu.se/CED/Book Draft version released 13th November 2000 at 22:01
田131.3ELECTRODYNAMICSallowing us to rewrite Equation (1.32) on the facing page in the following way)f eEMF.dl=-% /B.ds8(t,x)=dtJsJa(1.35)rd,B.dS- /vx(Bxv)-ds-JsatJSWith Stokes' theorem applied to the last integral, we finally geto.d EEMF dl = -8(t,x)= 4B.dS-(Bxv)dl(1.36)IsatJCor,rearrangingtheterms,a.[(EEMF -vxB)-dl = --B·dS(1.37)Jsatwhere EEMF is the field which is induced in the “loop," i.e., in the movingsystem. The use of Stokes’ theorem “backwards" on Equation (1.37) aboveyieldsa.Vx(EEMF _-vxB) = -B(1.38)atIn thefixed system, an observer measures the electric fieldE = EEMF-vxB(1.39)Hence,amoving observermeasures thefollowing Lorentz force ona chargeqqEEMF = qE+q(vxB)(1.40)corresponding to an “effective"electric field in the“loop" (moving observer)EEMF = E+(vxB)(1.41)Hence,we can conclude thatfora stationaryobserver,the Maxwell equationaVXE=-(1.42)atis indeed valid even if the“loop"is movingDraftverd13thNovper2000at22:01.Downloadedfromhttp://ww:/CED/BOO)④由①
“main” 2000/11/13 page 13 1.3 ELECTRODYNAMICS 13 allowing us to rewrite Equation (1.32) on the facing page in the following way: E(t,x) = ✠C E EMF · dl = − d dt ☎S B· dS = − ☎S ∂ ∂t B· dS− ☎S ∇×(B×v)· dS (1.35) With Stokes’ theorem applied to the last integral, we finally get E(t,x) = ✠C E EMF · dl = − ☎S ∂ ∂t B· dS− ✠C (B×v)· dl (1.36) or, rearranging the terms, ☎C (E EMF −v×B)· dl = − ☎S ∂ ∂t B· dS (1.37) where E EMF is the field which is induced in the “loop,” i.e., in the moving system. The use of Stokes’ theorem “backwards” on Equation (1.37) above yields ∇×(E EMF −v×B) = − ∂ ∂t B (1.38) In the fixed system, an observer measures the electric field E = E EMF −v×B (1.39) Hence, a moving observer measures the following Lorentz force on a charge q qE EMF = qE+q(v×B) (1.40) corresponding to an “effective” electric field in the “loop” (moving observer) E EMF = E+(v×B) (1.41) Hence, we can conclude that for a stationary observer, the Maxwell equation ∇×E = − ∂ ∂t B (1.42) is indeed valid even if the “loop” is moving. Draft version released 13th November 2000 at 22:01. Downloaded from http://www.plasma.uu.se/CED/Book
E14CLASSICALELECTRODYNAMICS1.3.5Maxwell'smicroscopicequationsWearenowabletocollecttheresultsfromtheaboveconsiderationsandfor-mulate the equations ofclassical electrodynamics valid for arbitraryvariationsintimeandspaceofthecoupledelectricandmagneticfieldsE(t,x)andB(t,x)The equations arep(t,x)V·E= (1.43a)E0OB(1.43b)VXE+drV·B=0(1.43c)OE(1.43d)×B-E01021= μoj(t,x)In theseequationsp(t,x)represents thetotal,possibly bothtimeand spacede-pendent, electric charge, i.e., free as well as induced (polarisation) charges,and j(t,x) represents the total, possibly both time and space dependent, elec-tric current, i.e., conduction currents (motion of free charges) as well as allatomistic(polarisation,magnetisation)currents.Asthey stand,theequationsthereforeincorporatetheclassical interactionbetweenall electricchargesandcurrents in the system andarecalled Maxwell's microscopic equations.An-other name often used for them is the Maxwell-Lorentz equations. Togetherwiththeappropriateconstitutiverelations,whichrelatepandjtothefieldsandthe initial andboundary conditions pertinenttothephysical situation athand, they form a system of well-posed partial differential equations whichcompletely determine E and B1.3.6Maxwell'smacroscopicequationsThe microscopic field equations (1.43)provide a correct classical picture forarbitraryfieldand sourcedistributions,includingbothmicroscopicand macro-scopic scales.However,formacroscopic substances it is sometimes conveni-ent to introduce new derived fields which represent the electric and magneticfields in which, in an average sense, the material properties of the substancesare already included. These fields are the electric displacement D and the mag-netisingfield H.In themostgeneral case,these derivedfields arecomplicatednonlocal,nonlinearfunctionalsoftheprimaryfieldsEandB:(1.44a)D = D[t,x; E,B]H = H[t,x; E, B](1.44b)Downloade/CED/BOOKDraft vn released 13th November2000 at 22:01.0://0由
“main” 2000/11/13 page 14 14 CLASSICAL ELECTRODYNAMICS 1.3.5 Maxwell’s microscopic equations We are now able to collect the results from the above considerations and formulate the equations of classical electrodynamics valid for arbitrary variations in time and space of the coupled electric and magnetic fields E(t,x) and B(t,x). The equations are ∇·E = ρ(t,x) ε0 (1.43a) ∇×E+ ∂B ∂t = (1.43b) ∇·B = 0 (1.43c) ∇×B−ε0µ0 ∂E ∂t = µ0j(t,x) (1.43d) In these equations ρ(t,x) represents the total, possibly both time and space dependent, electric charge, i.e., free as well as induced (polarisation) charges, and j(t,x) represents the total, possibly both time and space dependent, electric current, i.e., conduction currents (motion of free charges) as well as all atomistic (polarisation, magnetisation) currents. As they stand, the equations therefore incorporate the classical interaction between all electric charges and currents in the system and are called Maxwell’s microscopic equations. Another name often used for them is the Maxwell-Lorentz equations. Together with the appropriate constitutive relations, which relate ρ and j to the fields, and the initial and boundary conditions pertinent to the physical situation at hand, they form a system of well-posed partial differential equations which completely determine E and B. 1.3.6 Maxwell’s macroscopic equations The microscopic field equations (1.43) provide a correct classical picture for arbitrary field and source distributions, including both microscopic and macroscopic scales. However, for macroscopic substances it is sometimes convenient to introduce new derived fields which represent the electric and magnetic fields in which, in an average sense, the material properties of the substances are already included. These fields are the electric displacement D and the magnetising field H. In the most general case, these derived fields are complicated nonlocal, nonlinear functionals of the primary fields E and B: D = D[t,x;E,B] (1.44a) H = H[t,x;E,B] (1.44b) Downloaded from http://www.plasma.uu.se/CED/Book Draft version released 13th November 2000 at 22:01