E51.2MAGNETOSTATICSdr110FIGURE1.2:Ampere'slawdescribeshowasmallloopC,carryingastatic electric current J through its tangential line element dl located atx,experiencesa magnetostaticforcefrom a small loop C, carryingastatic electric current J' through the tangential line element dl' located atx.The loops can have arbitrary shapes as long as they are simple andclosed.located at x'. According to Ampere's law this force is, in vacuum, given by theexpressiondl'x(x-x')HOJJ'dixF(x)= 5Ix-x'34元(1.8)HoJJ"f. dlxarxv4元JcJcHere dl and dl' are tangential line elements of the loops C and C',respectively,and, in SI units, μo = 4π× 10-7 ~ 1.2566× 10-6 H/m is the vacuum permeab-ility. From the definition of o and μo (in SI units) we observe that1074元c (F/m) × 4 × 10-7 (H/m) = 2(s2/m2)(1.9)CO=which is auseful relation.At first glance, Equation (1.8) above appears to be unsymmetric in termsof the loops and therefore to be a force law which is in contradiction withNewton's third law. However, by applying the vector triple product “bac-cab"Draft version releaSed13thNovenmber2000at 22:01.Downloadedfromhttp://ww-plasu.se/CED/Boo)①由①
“main” 2000/11/13 page 5 1.2 MAGNETOSTATICS 5 O C dl J J 0 C 0 x−x 0 x dl 0 x 0 FIGURE 1.2: Ampère’s law describes how a small loop C, carrying a static electric current J through its tangential line element dl located at x, experiences a magnetostatic force from a small loop C 0 , carrying a static electric current J 0 through the tangential line element dl 0 located at x 0 . The loops can have arbitrary shapes as long as they are simple and closed. located at x 0 . According to Ampère’s law this force is, in vacuum, given by the expression F(x) = µ0 JJ 0 4π ✠C ✠C0 dl× dl 0 ×(x−x 0 ) |x−x 0 | 3 = − µ0 JJ 0 4π ✠C ✠C0 dl× ✝dl 0 ×∇ ✁ 1 |x−x 0 | ✂✡✞ (1.8) Here dl and dl 0 are tangential line elements of the loops C and C 0 , respectively, and, in SI units, µ0 = 4π×10−7 ≈ 1.2566×10−6 H/m is the vacuum permeability. From the definition of ε0 and µ0 (in SI units) we observe that ε0µ0 = 107 4πc 2 (F/m) ×4π×10−7 (H/m) = 1 c 2 (s2 /m2 ) (1.9) which is a useful relation. At first glance, Equation (1.8) above appears to be unsymmetric in terms of the loops and therefore to be a force law which is in contradiction with Newton’s third law. However, by applying the vector triple product “bac-cab” Draft version released 13th November 2000 at 22:01. Downloaded from http://www.plasma.uu.se/CED/Book
6CLASSICALELECTRODYNAMICSformula (F.54) on page 155, we can rewrite (1.8) in thefollowing wayμoJJF(x) = -dl.di4元(1.10)HoJJ'X-xdl.dl4元JcJc-xRecognising thefact that the integrand in thefirst integral is an exactdifferential so that this integral vanishes, we can rewrite the force expression, Equa-tion (1.8) on the previous page, in the following symmetric wayoJJ"x-x'F(x) =dl-dl(1.11)4元x-xThis clearly exhibits the expected symmetry in terms of loops C and C'1.2.2Themagnetostatic fieldIn analogy with the electrostatic case, we may attribute the magnetostatic in-teraction to a vectorial magnetic field Bstat. I turns out that Bstat can be definedthroughaxdBstat(x) (1.12)4元1x-xwhich expresses the small element dBstat(x) of the static magnetic field setup at the field point x by a small line element dl' of stationary current J' atthe sourcepointx'.TheSI unitforthemagneticfield,sometimes calledthemagneticfluxdensityormagneticinduction,isTesla (T)If we generalise expression (1.12) to an integrated steady state current dis-tribution j(x), we obtain Biot-Savart's law:x-xμoBstat(x) = j(x)xd'x4元/Jx-xT3(1.13)=-AO. j(x)xv(d'xIx-xl4元／ComparingEquation(1.5)onpage3withEquation(1.13),weseethatthereex-ists a close analogy between theexpressions for Estat and Bstat but that they dif-fer in their vectorial characteristics. With this definition of Bstat, Equation(1.8)on theprevious pagemaywewritten dl× Bstat(x)(1.14)F(x) =,nreleased13thNovember2000at22:01Daa/CBD/BoO)Draftn-f1?由
“main” 2000/11/13 page 6 6 CLASSICAL ELECTRODYNAMICS formula (F.54) on page 155, we can rewrite (1.8) in the following way F(x) = − µ0 JJ 0 4π ✠C ✠C0 ✝dl·∇ ✁ 1 |x−x 0 | ✂✟✞ dl 0 − µ0 JJ 0 4π ✠C ✠C0 x−x 0 |x−x 0 | 3 dl· dl 0 (1.10) Recognising the fact that the integrand in the first integral is an exact differential so that this integral vanishes, we can rewrite the force expression, Equation (1.8) on the previous page, in the following symmetric way F(x) = − µ0 JJ 0 4π ✠C ✠C0 x−x 0 |x−x 0 | 3 dl· dl 0 (1.11) This clearly exhibits the expected symmetry in terms of loops C and C 0 . 1.2.2 The magnetostatic field In analogy with the electrostatic case, we may attribute the magnetostatic interaction to a vectorial magnetic field B stat . I turns out that B stat can be defined through dB stat(x) def ≡ µ0 J 0 4π dl 0 × x−x 0 |x−x 0 | 3 (1.12) which expresses the small element dB stat(x) of the static magnetic field set up at the field point x by a small line element dl 0 of stationary current J 0 at the source point x 0 . The SI unit for the magnetic field, sometimes called the magnetic flux density or magnetic induction, is Tesla (T). If we generalise expression (1.12) to an integrated steady state current distribution j(x), we obtain Biot-Savart’s law: B stat(x) = µ0 4π ☎V j(x 0 )× x−x 0 |x−x 0 | 3 d 3 x 0 = − µ0 4π ☎V j(x 0 )×∇ ✁ 1 |x−x 0 | ✂ d 3 x 0 (1.13) Comparing Equation (1.5) on page 3 with Equation (1.13), we see that there exists a close analogy between the expressions for E stat and B stat but that they differ in their vectorial characteristics. With this definition of B stat , Equation (1.8) on the previous page may we written F(x) = J ✠C dl×B stat(x) (1.14) Downloaded from http://www.plasma.uu.se/CED/Book Draft version released 13th November 2000 at 22:01
+71.2MAGNETOSTATICSIn order to assess the properties of Bstat, we determine its divergence andcurl.Taking the divergence of both sides ofEquation (1.13)on thefacing pageand utilising formula (F.61) on page 155, we obtainMo.d3xV.Bstat(x) = j(x)×V4元-0/V) [V×j(x)]dx4元J(x-x(1.15)+% [ (x) [>×(d3xIx-x'l4元／=0where the first term vanishes because j(x') is independent of x so that V×j(x')=0, and the second term vanishes since, according to Equation (M.78)onpage 175, V×[Vα(x)] vanishes for any scalar field a(x),Applying the operator“bac-cab"rule, formula (F.67)on page 155, the curlof Equation (1.13)on theprecedingpage can be writtenV×Bsta(x) = -% v× / j(x)×v(d3xx-x4元poj(x)2dx(1.16)4元X-XH[G(x).Vj"d'x4元，Ix-In the first of the two integrals on the right hand side, we use the representationof the Dirac delta function Equation (M.73) on page 174, and integrate thesecond one by parts, by utilising formula (F.59) on page 155 as follows:d3x[i(x').V'JV'["-j(x)] "Y3 j(x)ax,d3x=α/i()最() ds-/[()](x-xl(1.17)Then we note that the first integral in the result, obtained by applying Gauss'stheorem, vanishes when integrated over a large sphere far away from the loc-alised source j(x'), and that the second integral vanishes because V-j= O forDraft ve13thNcer2000at22:01.Downloadedfron1/0①
“main” 2000/11/13 page 7 1.2 MAGNETOSTATICS 7 In order to assess the properties of B stat , we determine its divergence and curl. Taking the divergence of both sides of Equation (1.13) on the facing page and utilising formula (F.61) on page 155, we obtain ∇·B stat(x) = − µ0 4π ∇· ☎V j(x 0 )×∇ ✁ 1 |x−x 0 | ✂ d 3 x 0 = − µ0 4π ☎V ∇ ✁ 1 |x−x 0 | ✂ ·[∇×j(x 0 )]d 3 x 0 + µ0 4π ☎V j(x 0 )· ✝∇×∇ ✁ 1 |x−x 0 | ✂✡✞ d 3 x 0 = 0 (1.15) where the first term vanishes because j(x 0 ) is independent of x so that ∇ × j(x 0 ) ≡ 0, and the second term vanishes since, according to Equation (M.78) on page 175, ∇×[∇α(x)] vanishes for any scalar field α(x). Applying the operator “bac-cab” rule, formula (F.67) on page 155, the curl of Equation (1.13) on the preceding page can be written ∇×B stat(x) = − µ0 4π ∇× ☎V j(x 0 )×∇ ✁ 1 |x−x 0 | ✂ d 3 x 0 = − µ0 4π ☎V j(x 0 )∇ 2 ✁ 1 |x−x 0 | ✂ d 3 x 0 + µ0 4π ☎V [j(x 0 )·∇ 0 ]∇ 0 ✁ 1 |x−x 0 | ✂ d 3 x 0 (1.16) In the first of the two integrals on the right hand side, we use the representation of the Dirac delta function Equation (M.73) on page 174, and integrate the second one by parts, by utilising formula (F.59) on page 155 as follows: ☎V [j(x 0 )·∇ 0 ]∇ 0 ✁ 1 |x−x 0 | ✂ d 3 x 0 = xˆk ☎V ∇ 0 · ☛ j(x 0 ) ✝ ∂ ∂x 0 k ✁ 1 |x−x 0 | ✂✟✞✌☞ d 3 x 0 − ☎V ✍ ∇ 0 ·j(x 0 )✎ ∇ 0 ✁ 1 |x−x 0 | ✂ d 3 x 0 = xˆk ☎S j(x 0 ) ∂ ∂x 0 k ✁ 1 |x−x 0 | ✂ · dS− ☎V ✍ ∇ 0 ·j(x 0 )✎ ∇ 0 ✁ 1 |x−x 0 | ✂ d 3 x 0 (1.17) Then we note that the first integral in the result, obtained by applying Gauss’s theorem, vanishes when integrated over a large sphere far away from the localised source j(x 0 ), and that the second integral vanishes because ∇ ·j = 0 for Draft version released 13th November 2000 at 22:01. Downloaded from http://www.plasma.uu.se/CED/Book
E8CLASSICALELECTRODYNAMICSstationary currents (no charge accumulation in space). The net result is simply× Bstat(x) = μo[. j(x')8(x -x')d'x = μoj(x)(1.18)1.3 ElectrodynamicsAs we saw in the previous sections, the laws of electrostatics and magneto-statics can be summarised in two pairs of time-independent, uncoupled vectordifferentialequations,namelytheequationsofclassicalelectrostaticsp(x)VEstat(x)=(1.19a)E0V×Estat(x)= 0(1.19b)and theequations ofclassical magnetostaticsV·Bstat(x) = 0(1.20a)× Bstat(x) = μoj(x)(1.20b)Since there is nothing a priori which connects Estat directly with Bstat, we mustconsider classical electrostatics and classical magnetostatics as two independ-ent theories.However, when we include time-dependence, these theories are unifiedinto one theory,classical electrodynamics.This unification of the theories ofelectricity and magnetism is motivated by two empirically established facts:1.Electric charge is a conserved quantity and current is a transport of elec-tric charge. This fact manifests itself in the equation of continuity and,as a consequence, in Maxwell's displacement current.2. A change in the magnetic flux through a loop will induce an EMF elec-tric field inthe loop.This isthe celebratedFaraday's law of induction3The famous physicist and philosopher Pierre Duhem (1861-1916) once wrote:"The whole theory of electrostatics constitutes a group of abstract ideas andgeneral propositions,formulated in the clear and concise language of geometryand algebra, and connected with one another by the rules of strict logic. Thiswhole fully satisfies the reason of a French physicist and his taste for clarity,simplicity and order..-/CED/BOO)Draft version released 13th November 2000 at 22:01.DownloadO:④由D
“main” 2000/11/13 page 8 8 CLASSICAL ELECTRODYNAMICS stationary currents (no charge accumulation in space). The net result is simply ∇×B stat(x) = µ0 ☎V j(x 0 )δ(x−x 0 )d 3 x 0 = µ0j(x) (1.18) 1.3 Electrodynamics As we saw in the previous sections, the laws of electrostatics and magnetostatics can be summarised in two pairs of time-independent, uncoupled vector differential equations, namely the equations of classical electrostatics3 ∇·E stat(x) = ρ(x) ε0 (1.19a) ∇×E stat(x) = 0 (1.19b) and the equations of classical magnetostatics ∇·B stat(x) = 0 (1.20a) ∇×B stat(x) = µ0j(x) (1.20b) Since there is nothing a priori which connects E stat directly with B stat , we must consider classical electrostatics and classical magnetostatics as two independent theories. However, when we include time-dependence, these theories are unified into one theory, classical electrodynamics. This unification of the theories of electricity and magnetism is motivated by two empirically established facts: 1. Electric charge is a conserved quantity and current is a transport of electric charge. This fact manifests itself in the equation of continuity and, as a consequence, in Maxwell’s displacement current. 2. A change in the magnetic flux through a loop will induce an EMF electric field in the loop. This is the celebrated Faraday’s law of induction. 3The famous physicist and philosopher Pierre Duhem (1861–1916) once wrote: “The whole theory of electrostatics constitutes a group of abstract ideas and general propositions, formulated in the clear and concise language of geometry and algebra, and connected with one another by the rules of strict logic. This whole fully satisfies the reason of a French physicist and his taste for clarity, simplicity and order. . . ” Downloaded from http://www.plasma.uu.se/CED/Book Draft version released 13th November 2000 at 22:01
+91.3ELECTRODYNAMICS1.3.1EquationofcontinuityLet j denote the electric current density (A/m2). In the simplest case it can bedefined asj=vp wherevis the velocity of the chargedensity.In general,jhasto be defined in statistical mechanical terms as j(t,x)= Zaqa /vfa(t,x,v)d3ywhere fa(t,x,v) is the (normalised) distribution function for particle species withelectrical chargeqa,The electric charge conservation law can be formulated in the equation ofcontinuityap() +V-j(t,x)= 0(1.21)atwhich states that the time rate of change of electric charge p(t,x) is balancedby a divergence in the electric current density j(t,x).1.3.2Maxwell'sdisplacementcurrentWe recall from the derivation of Equation (1.18) on the preceding page thatthere we used the fact that in magnetostatics -j(x)= O.In the case of non-stationary sources and fields, we must, in accordancewith the continuity Equa-tion (1.21), set v-j(t,x) =-0p(t,x)/ot. Doing so, and formally repeating thesteps in the derivation of Equation (1.18) on the preceding page, we wouldobtaintheformalresultjt,x)(x-)d +0%p(t,x)V"d3V×B(t,x)=μo 4元0t／Ix-xa=Mo(. )+ 0e0()(1,22)where, in the last step, we have assumed that a generalisation ofEquation(1.5)on page 3 to time-varying fields allows us to make the identification10p(t,x)vd'x=.p(t,x)>(d'xIx-x!4元atATSdE(t,x)at(1.23)Later,we will need to consider this formal result further.The result is Max-well's source equation for the B fieldd(1.24)V× B(t,x)= μo ( j(t,x)+are0E(tx)Draftved13thNoer2000at22:01.DownlnadedfCED/BO:/ /由
“main” 2000/11/13 page 9 1.3 ELECTRODYNAMICS 9 1.3.1 Equation of continuity Let j denote the electric current density (A/m2 ). In the simplest case it can be defined as j = vρ where v is the velocity of the charge density. In general, j has to be defined in statistical mechanical terms as j(t,x) = ∑α qα ✏ v fα(t,x,v)d 3 v where fα(t,x,v) is the (normalised) distribution function for particle species α with electrical charge qα. The electric charge conservation law can be formulated in the equation of continuity ∂ρ(t,x) ∂t +∇·j(t,x) = 0 (1.21) which states that the time rate of change of electric charge ρ(t,x) is balanced by a divergence in the electric current density j(t,x). 1.3.2 Maxwell’s displacement current We recall from the derivation of Equation (1.18) on the preceding page that there we used the fact that in magnetostatics ∇ ·j(x) = 0. In the case of nonstationary sources and fields, we must, in accordance with the continuity Equation (1.21), set ∇ ·j(t,x) = −∂ρ(t,x)/∂t. Doing so, and formally repeating the steps in the derivation of Equation (1.18) on the preceding page, we would obtain the formal result ∇×B(t,x) = µ0 ☎V j(t,x 0 )δ(x−x 0 )d 3 x 0 + µ0 4π ∂ ∂t ☎V ρ(t,x 0 )∇ 0 ✁ 1 |x−x 0 | ✂ d 3 x 0 = µ0j(t,x)+µ0 ∂ ∂t ε0E(t,x) (1.22) where, in the last step, we have assumed that a generalisation of Equation (1.5) on page 3 to time-varying fields allows us to make the identification 1 4πε0 ∂ ∂t ☎V ρ(t,x 0 )∇ 0 ✁ 1 |x−x 0 | ✂ d 3 x 0 = ∂ ∂t ✝− 1 4πε0 ☎V ρ(t,x 0 )∇ ✁ 1 |x−x 0 | ✂ d 3 x 0 ✞ = ∂ ∂t E(t,x) (1.23) Later, we will need to consider this formal result further. The result is Maxwell’s source equation for the B field ∇×B(t,x) = µ0 ✁ j(t,x)+ ∂ ∂t ε0E(t,x)✂ (1.24) Draft version released 13th November 2000 at 22:01. Downloaded from http://www.plasma.uu.se/CED/Book