The Geometric Representation of Electrodynamics by Exterior Differential Forms

Transcription

1 1 The Geometi Repesentation of letodynamis by xteio Diffeential Foms Pete Russe Institute fo Nanoeletonis, Tehnishe Univesität Münhen isst. 21, D Munih, Gemany bstat The aim of this ontibution is to show that eletomagneti theoy an be simplified and laified by geometi fomulation. The geometi fomulation of Maxwell s theoy aomplished by exteio diffeential alulus intodues geometial imagey and suppots physial undestanding. xteio diffeential alulus has simple and onise ules fo omputation. The objets of exteio diffeential alulus have a lea geometial signifiane and the laws of eletomagnetis assume a simple and elegant fom. v I. INTRODUCTION Maxwell s equations establish the analyti and oneptual famewok of eletomagnetism [1], [2]. While Maxwell lealy has stated the impotane of the displaement uent and has the temendous ahievement of having ompleted the eletomagneti theoy, he has not pesented his equations in the fom we use them now. Maxwell s oiginal fomulation of eletomagnetism onsisted of 20 equations in 20 vaiables. Heaviside intodued veto alulus with ul and divegene opeatos to efomulate 12 of these 20 equations into fou veto equations and gave the Maxwell s equations today s ommon epesentation [3]. letomagneti theoy meges physial, mathematial, and geometial ideas. In this omplex envionment eative thinking and the onstution of new onepts is suppoted by imagey as povided by geometi models [4]. lthough Heaviside s notation has been a onsideable oneptual step fowad in the fomulation of Maxwell s theoy it suffes fom dawbaks suh as fo example not popely distinguishing pola vetos fom axial vetos and salas fom pseudo-salas. Moden diffeential geomety, based on the wok of Hemann Günte Gassmann and Élie Catan suppots a pitoial way of thinking that is helpful in developing the sientist s intuition. In 1844, Hemann Günte Gassmann published his book Die lineale usdehnungslehe, ein neue Zweig de Mathematik [5], in whih he developed the idea of an algeba in whih the symbols epesenting geometi entities suh as points, lines, and planes ae manipulated using etain ules. Gassmann intodued what is now alled exteio algeba, based upon the exteio podut a b = b a. 1 Based on Gassmann s exteio algeba Élie Catan [6] developed the exteio alulus. xteio alulus has poven to be the natual language of field theoy sine it yields simple and ompat mathematial fomulae, and simplifies the solution of Fig. 1. Path of integation a fo the definition of the voltage and b fo the definition of the uent. field theoetial poblems. Futhemoe it establishes a diet onnetion to geometial images and povides additional physial insight [7] [11]. xteio alulus as a mathematial tool fo eletomagnetis has aleady been disussed by Deshamps and Wanik [12], [13]. In [14] the autho of this pape has applied the exteio alulus in an intodutoy level ouse on eletomagnetis fo miowave engineeing. Hehl and Obukhov have published an intodution into eletodynamis based on exteio alulus [15], and Lindell has published a ompehensive teatment of diffeential foms in eletomagnetis [16]. II. DIFFRNTIL FORMS Sala and veto fields may be epesented by exteio diffeential foms. Diffeential foms ae an extension of the veto onept. The use of diffeential foms does not mean to give up the veto onept and its physial intepetations. The diffeential fom epesentation supplies additional physial insight in addition to the onventional veto pitue. The enegy W 21 equied to move a patile with eleti hage q in an eleti field with omponents x, y, and z, fom point x 1 to x 2 is obtained by the integal W 21 t = q x2 x 1 x x, y, z, tdx+ + y x, y, z, tdy + z x, y, z, tdz. 2 Figue 1a shows the path of integation fo the definition of the voltage v 21 fom node 2 to node 1. The line integal sums up the pojetion of the field veto on the vetoial path element. The ontibution of the integand is popotional to the podut of the magnitudes of field veto with the infinitesimal path element and the osine of the angle enlosed between them. The eleti and magneti fields ae epesented

2 2 D W z z J x J x x Fig. 2. Geometi epesentation of a a 1-fom, b a 2-fom, a 3-fom. x y y Fig. 4. The oientation of an aea. Fig. 3. One-fom with ending sufaes. by 1-foms and H espetively. The ommon physial intepetation of the eleti field is elated to the foe on a point-like unit hage. This foe pitue yields in a natual way to the veto epesentation and to the visualization of the eleti field via field lines. nothe viewpoint is to onside the enegy of a hage moved though the field. We an visualize the field via the hange of the enegy of a test hage moved though the field. This enegy pitue is moe elated to diffeential foms. Figue 2a shows the epesentation of the field via the sufaes of onstant test hage enegy o onstant eleti potential espetively. Fo an eletostati field the sufaes assoiated with the one-fom ae equipotentials. The voltage between two points 2 and 1 is given by v 21 = In geneal the sufaes of a 1-fom also may end o meet eah othe. We point out that the dimension of the diffeential fom is V and H has the dimension. The diffeential foms and H expess the hanges of the eleti and magneti potentials ove an infinitesimal path element. Figue 3 shows a situation we enounte in time vaiable fields. In the ente of the stutue the field intensity is highe than at its edges. In this ase the integal 3 will depend on the path fom x 1 to x 2 and we annot assign a sala potential to the field. The ile on the integal symbol denotes the integation ove a losed bounday. Figue 1b shows the path of integation fo the definition of the uent i. The elation between the dietion of efeene fo the uent and the oientation of the path of integation is shown in Figue 1b. The uent is ounted positive if its dietion oinides with the dietion of efeene. Diffeential foms ae essentially the expessions unde an integation symbol. In ode to intodue the onept of exteio diffeential foms let us onside the uent i flowing in x-dietion though the sufae in Figue 4a. To ompute the uent we have to integate the x-omponent J x of the uent density ove the sufae in the yz-plane i = J x dy dz. 4 If we integate a uent density ove an aea we have to onside the oientation of the aea. If in Figue 4a the uent density J x is positive, the uent i also will be positive. Inveting the dietion of J x will yield a negative uent. This invesion may be pefomed by mioing the oodinates with espet to the yz-plane. How do we know whethe a sufae integal is positive o negative? The answe is: We have to define a positive oientation. positive oiented o ighthanded Catesian oodinate system is speified as follows: If we ae looking in z-dietion on the xy-plane the x-axis may be otated lokwise by 90 into the y-axis. In Figue 4a the veto omponent J x is pointing in positive oientation. In Figue 4b the oodinate system as well as the veto field wee otated by 180 aound the z-axis. Physially nothing has hanged. Howeve in the left figue, the veto pointing towads the obseve is positive, wheeas in the ight figue the veto pointing away fom the obseve is positive. xteio diffeential foms allow to epesent the oientation of a oodinate system. We intodue the exteio podut o wedge podut dy dz with the popety dy dz = dz dy. 5 n exteio diffeential fom is the exteio podut of diffeential foms. xteio diffeential foms onsisting wedge poduts of two diffeentials o sums of suh poduts ae alled two-foms. We may deide eithe dy dz = dy dz o dy dz = dy dz. Deiding dy dz = dy dz 6 assigns to dy dz the positive oientation and to dz dy the negative oientation. The integal 4 an now be witten in the oientation-independent fom i = J x dy dz. 7 n exteio diffeential fom of ode p is alled a p- fom. In n-dimensional spae the ode of a diffeential fom may assume values 0...n. Table I summaizes some p-foms desibing field quantities. In diffeential fom notation a lea distintion between salas, pseudosalas, pola vetos and axial vetos is made. Salas ae epesented by 0-foms, pseudosalas by 3-foms, pola vetos by 1-foms and axial vetos by 2-foms. Fo a p-fom U and a q-fom V the ommutation elation is U V = 1 p+q+1 V U. 8 The uent flowing in a onduto vaies though the osssetion. We desibe the flow of the uent by a uent

3 J 3 TBL I DIFFRNTIL FORMS y z x dy dz x Fig. 5. fx, t x, t = x dx + y dy + z dz Hx, t = H x dx + H y dy + H z dz Dx, t = D x dy dz + D y dz dx + D z dx dy Bx, t = B x dy dz + B y dz dx + B z dx dy J x, t = J x dy dz + J y dz dx + J z dx dy Sx, t = S x dy dz + S y dz dx + S z dx dy Qx, t = ρ dx dy dz W ex, t = W e dx dy dz W mx, t = W m dx dy dz Cuent flow. density veto field Jx = [J x x, J y x, J z x] T. The uent I is flowing though a tube fomed by the uent density field lines going though the bounday of the aea as shown in Figue 5. Figue 2b shows the tube epesentation of a two-fom. The two-fom is visualized by a bundle of tubes aying the uent. The uent density is invesely popotional to the oss-setional aea of the tubes. Figue 6b shows the tube epesentations of the fundamental two-fom dy dz. If the sufae is an abitaily oiented uved sufae in thee-dimensional spae and the uent density veto has the x-, y- and z-omponents J x, J y and J z, we have to pefom the integation ove i = J x dy dz + J y dz dx + J z dx dy. 9 The fist tem of the integand onens the integation of the x-omponent of the uent density ove the pojetion of the sufae on the yz-plane and so foth. Let us intodue the uent density fom J by the exteio diffeential fom J = J x dy dz + J y dz dx + J z dx dy. 10 The uent i may be expessed in a ompat notation as the integal of the diffeential fom J i = J. 11 The dimension of the uent density diffeential fom J is. The eleti flux density fom D has the dimension s, and the magneti flux density fom B has the dimension Vs. These diffeential foms epesent the uent o the flux though an infinitesimal aea element. Fig. 6. The fundamental a 1-fom, b 2-fom, 3-fom in Catesian oodinates. The eleti hage q is given by the volume integal ove the eleti hage density ρ. Figue 2 shows the gaphi visualization of a thee-fom by subdividing the volume into ells. The ell volume is invesely popotional to the hage density. Fo the eleti hage density we may intodue a thee-fom, the so-alled hage density fom Q = ρ dx dy dz. 12 The hage density fom Q with the dimension s epesents the hage in an infinitesimal volume element. We obtain the hage q by pefoming the volume integal ove the theefom Q: q = Q. 13 Figue 6 shows the fundamental 1-, 2- and 3-foms in Catesian oodinates. III. MXWLL S QUTIONS IN INTGRL FORM V The integal fom of Maxwell s equations is given by: H = d D + J, mpèe s Law 14 dt = d B, Faaday s Law 15 dt B = 0, Magneti Flux Continuity V V D = V 16 Q. Gauss Law 17 mpèe s law elates uent to magneti field. It states that the sum of ondution and displaement uents though an aea equals the magneti tension aound the bounday. Faaday s law elates the magneti flux to the eleti field. It says that the time deivative of the magneti flux though an aea equals the eleti tension aound the bounday. Both equations ae tied togethe via the onstitutive equations elating flux densities to field intensities D = ǫ, 18 B = µ H. 19 The sta opeato o Hodge opeato is defined by f = f dx dy dz, 20a x dx + y dy + z dz = x dy dz + y dz dx + z dx dy. 20b

4 4 Veto Diffeential Opeato TBL II DIFFRNTIL OPRTORS xteio Diffeential Opeato gad f df ul d div B db ul gad f = 0 d df = 0 div ul = 0 d d = 0 div gad f d df o d df ul ul d d o d d Fig. 7. a ea with bounday and b volume V with bounday V. The sta opeato has the popety = In n-dimensional spae the sta opeato elates a p-fom to a n p-fom. The sta opeato depends on the meti oodinate system. xteio diffeential fom epesentation of veto fields is fully ovaiant. xteio deivation is independent fom the metis of oodinate system and spae. The metis only has to be onsideed when a p-fom is mapped to a n p-fom. We also note that mateial paametes ou at the same plae as meti paametes. IV. TH XTRIOR DRIVTIV The exteio deivative du of an exteio diffeential fom U is given by d U = dx i U. 22 x i i The podut ules fo exteio diffeentiation ae: d U + V = d U + d V, 23a d U V = The exteio deivatives of p-foms ae 0-fom: dfx = 1-fom: dux = 2-fom: f x + U x + d U V + 1 deg U U d V.23b f f dx + ydy + z dz, Uz y Uy z z Uz x Uy x Ux y dy dz dz dx dx dy, dvx = Vx x + Vy y + Vz z dx dy dz, 3-fom: dqx = 0. V. POINCRÉ S LMM fom V fo whih dv = 0 is said to be losed, and a fom V fo whih V = du is said to be exat. Fo diffeential foms the statement V = du implies dv = 0. The elation dd U = 0 24 may be veified easily. In onventional veto notation this oesponds to ulgad = 0 and div ul = 0. ll exat foms ae losed. Howeve it may also be shown, that all losed foms ae exat. Poinaé s lemma states dv = 0 V = du. 25 VI. TH STOKS THORM In 14 and 15 line integals ove the bounday of the sufae ae elated to sufae integals ove. Figue 7a shows the elation between the oientation of the aea and the bounday. The line integal ove the losed ontou is alled iulation. In 16 and 17 the sufae integals ae pefomed ove the bounday V of the volume V. Figue 7b shows the oientation of the bounday sufae V. The Stokes theoem elates the integation of a p-fom U ove the losed p-dimensional bounday V of a p + 1-dimensional volume V to the volume integal of du ove V via U = d U. 26 V This summaizes the Stokes theoem and the Gauss theoem of onventional veto notation. VII. MXWLL S QUTIONS IN LOCL FORM pplying Stokes theoem to the integal fom of Maxwell s equations 14 to 17 we obtain the diffeential epesentation of Maxwell s equations: V d H = D + J, mpèe s Law 27 t d = B, t Faaday s Law 28 d B = 0, Magneti Flux Continuity 29 d D = Q. Gauss Law 30 Figue 8 shows the gaphial epesentation of Maxwell s equations afte Deshamps [12]. Faaday s Law Magneti Flux Continuity mpèe s Law Gauss Law Φ Constitutive Relations / t ǫ µ H / t B 0 D / t J 0 Q / t 0 Fig. 8. Gaphial epesentation of Maxwell s equations. 0-fom 1-fom 2-fom 3-fom

5 5 D D H S H Fig. 9. D. The exteio podut of the field fom and the flux density fom Fig. 10. The Poynting fom S as the podut of the field foms and H. VIII. NRGY ND POWR The eleti and the magneti enegy densities ae epesented by the 3-foms W e = 1 2 D = 1 2 xd x + y D y + z D z dx dy dz, 31a W m = 1 2 H B = 1 2 H xb x + H y B y + H z B z dx dy dz. 31b Figue 9 visualizes the exteio podut of the field one-fom and the flux density two-fom D. The esulting enegy density thee-foms W e and W m ae visualized by the subdivision of the spae into ells as shown in Figue 9. Multiplying mpèe s law fom the left with and Faaday s law fom the ight with H, we obtain This yields dh = t D + J, 32 d = t B H. 33 d H = t D H t B J. 34 This equation an be bought into the fom d H = 1 t 2 D H B J. 35 The powe loss density p L x, t with the oesponding diffeential fom is given by P L = p L x, tdx dy dz 36 P L = σ. 37 Due to the impessed uent density J 0, a powe pe unit of volume P 0 = J 0 38 is added to the eletomagneti field. Intoduing the Poynting diffeential fom S = H 39 and inseting 31a, 31b, 37 and 38 into 35 yields the diffeential fom of Poynting s theoem: d S = t W e t W m P L + P Fig. 11. Segmentation of a losed stutue. Figue 10 visualizes the Poynting two-fom as the exteio podut of the eleti and magneti field one-foms and H. The potential planes of the eleti and magneti fields togethe fom the tubes of the Poynting fom. The distane of the eleti and magneti potential planes exhibit the dimensions V and espetively. The oss setional aeas of the flux tubes have the dimension V. The powe flows though these Poynting flux tubes. Integating 40 ove a volume V and tansfoming the integal ove S into a sufae integal ove the bounday V, we obtain the integal fom of Poynting s Theoem: S = P 0 d W e d W m P L. 41 V V dt V dt V V IX. TLLGN S THORM Figue 11 shows the segmentation of an eletomagneti stutue into diffeent egions R l sepaated by boundaies B lk. The egions R l may ontain any eletomagneti substutue. In a netwok analogy the two-dimensional manifold of all bounday sufaes B lk epesents the onnetion iuit, wheeas the subdomains V l ae epesenting the iuit elements. Tellegen s theoem states fundamental elations between voltages and uents in a netwok and is of onsideable vesatility and geneality in netwok theoy [17]. The field fom of Tellegen s theoem may be deived dietly fom Maxwell s equations [18] and is given by x, t H x, t = V The integation is pefomed ove both sides of all bounday sufaes. lso the integation ove finite volumes filled with ideal eleti o magneti ondutos gives no ontibution to

6 6 these integals. The pime and double pime denote the ase of a diffeent hoie of soues and a diffeent hoie of mateials filling the subdomains. lso the time agument may be diffeent in both ases. X. TH LCTROMGNTIC POTNTILS The Maxwell s equations ae a system of twelve oupled sala patial diffeential equations. The intodution of eletomagneti potentials allows a systemati solution of the Maxwell s equations [19] [21]. We ae distinguishing between sala potentials and veto potentials. fte solution of the wave equation fo a potential, all field quantities may be deived fom this potential. Due to 29, i.e. d B = 0 the magneti flux density is fee of divegene. Theefoe B may be epesented as the exteio deivative of an one-fom : B = d. 43 The oesponding veto field is alled the magneti veto potential. Inseting 43 into the seond Maxwell s quation 28 yields d + t = oding to Poinaé s lemma, the exteio deivative of the one-fom inside the bakets vanishes, we may expess this one-fom as the exteio deivative of the sala potential Φ and obtain = d Φ. 45 t The negative sign of Φ has been hosen due to the physial onvention in defining potentials. Wheeas in eletostatis the eleti field may be omputed fom a sala potential Φ, in the ase of apidly vaying eletomagneti fields, we also need the veto potential. The potentials and Φ ae not defined in an unambiguous way. dding the gadient of a sala funtion Ψ to the veto potential does not influene the magneti indution B. The eleti field also emains unhanged, if and Φ togethe ae tansfomed in the following way: 1 = + d Ψ, 46 Φ 1 = Φ Ψ t. 47 This tansfomation is alled a gauge tansfomation. The onefom may be defined in an unambiguous way, if we ae pesibing its exteio deivative. Inseting 43 and 45 into the fist Maxwell s equation 27 yields d d +µǫ 2 t 2 +µσ t +µ d Inseting 45 and 18 into 30 yields ǫ Φ t + σφ = µj d d Φ + d t = 1 ǫ Q. 49 Sine we may hoose the exteio deivative of abitaily, we an make use of this option in ode to deouple the diffeential equations fo and Φ. We impose the so-alled Loentz ondition given by d + µ ǫ t Φ + σφ = Togethe with 48 and 49 we obtain the equations d d d d µǫ 2 t 2 µσ t = µj 0 51 d d Φ µǫ 2 t 2 Φ µσ t Φ = 1 ǫ Q. We define the ovaiant deivative by the opeato 52 d U = 1 deg U+1 d U. 53 We intodue the Laplae opeato defined by = d d + d d. 54 pplying the Laplae opeato to a zeo-fom Φ and an onefom espetively yields Φ = d d Φ 55 = d d d d. 56 With the Laplae opeato we an wite 51 and 52 as µǫ 2 t 2 µσ t = µj 0, 57 Φ µǫ 2 t 2 Φ µσ t Φ = 1 ǫ Q. 58 The field intensities and H deived fom and Φ satisfy the fou Maxwell s quations 27 to 30. The equations 57 and 58 ae alled wave equations, sine thei solutions desibe popagating waves. quation 57 is a veto wave equation, wheeas 58 is a sala wave equation. It is possible to deive both potentials x, and Φx, t fom one veto, the so-alled eleti Hetz veto Π e x, t. We intodue the eleti Hetz diffeential fom Π e = Π ex dx + Π ey dy + Π ez dz. 59 The Loentz ondition 50 is fulfilled, if and Φ ae deived fom the Hetz fom Π e via = µǫ t Π e + µσ Π e, 60 Φ = dπ e. 61 Inseting 60 into 57, we obtain µ ǫ t Π + σ e µǫ 2 t 2 Π e µσ t Π e = µ J Fo J 0 = 0, i.e. without impessed uent soues, we obtain the homogeneous wave equation Π e µǫ 2 t 2 Π e µσ t Π e = 0. 63

7 7 The field intensities and H follow fom 19, 43, 45, 60 and 61: H = d d Π e µǫ 2 t 2 Π e µσ t Π e, 64 = d ǫ t Π e + σ Π e. 65 Subtating fom 64 the wave equation 63 we obtain = ddπ e fo J 0 = Let us now onside the lossless ase with impessed uent soues. In this ase it is helpful to use the impessed eleti polaization M e0 x, t instead of the impessed uent density J 0 x, t. The oesponding diffeential fom is M e0 = M ex dy dz + M ey dz dx + M ez dx dy. 67 The impessed eleti polaization fom M e0 is elated to an impessed eleti uent J 0 via t M e0 = J By this way it follows fom 62 Π e µǫ 2 t t 2 Π e = 1 ǫ t M e0 fo σ = 0 69 by integation ove t we obtain Π e µǫ 2 t 2 Π e = 1 ǫ M e0 fo σ = Sine the soue of the Hetz veto field is an impessed eleti polaization, the Hetz veto also is alled the eleti polaization potential. Fom the solution of 70 we obtain and H via 64 and 65. Fom 64 and 70 we obtain = d dπ e 1 ǫ M e0. 71 In the geneal ase J 0 0 and σ 0 we obtain an equation ontaining time deivatives up to thid ode. This diffiulty an be avoided by using the fequeny domain epesentation. XI. CURVILINR COORDINTS In exteio alulus the field equations may be fomulated without efeene to a speifi oodinate system. Depending on the poblem the hoie of a speifi oodinate system may simplify the poblem solution onsideably. We intodue an othogonal uvilinea oodinate system u = ux, y, z, v = vx, y, z, w = wx, y, z. 72 The oodinate uves ae obtained by setting two of the thee oodinates u, v, w onstant. Coodinate sufaes ae defined by setting one of the thee oodinates onstant. In an othogonal oodinate system in any point exept singula points of the spae the thee oodinate uves ae othogonal. The same holds fo the thee oodinate sufaes going though any point. The diffeentials dx, dy, dz by the diffeentials du, dv, dw ae elated to dx = x x x du + dv + dw, 73 u v w dy = y y y du + dv + dw, 74 u v w dz = z z z du + dv + dw. 75 u v w The ules fo tansfomation of the Catesian basis two-foms dx dy, dy dz, dz dx and the Catesian basis thee-fom dx dy dz follow dietly fom the above equations by applying the ules of the exteio podut. Using the meti oeffiients g 1, g 2 and g 3 g 2 1 = x u x u, g2 2 = x v x v, we intodue the unit one-foms g2 3 = x w x w 76 s 1 = g 1 du, s 2 = g 2 dv, s 3 = g 3 dw. 77 The integal of s 1 = g 1 du along any path with v and w onstant yields the length of the path. In a iula ylindi oodinate system, defined by x = osφ, y = sin φ, 78 z = z, the unit diffeential foms ae s 1 = d, s 2 = dφ, s 3 = dz. 79 In a spheial oodinate system, defined by x = sinθ osφ, y = sinθ sinφ, 80 z = osθ, the unit diffeential foms ae s 1 = d, s 2 = dθ, s 3 = sinθdφ. 81 Fo the uvilinea unit diffeentials the Hodge opeato as defined in 20a and 20b is f = fs 1 s 2 s 3, u s 1 + v s 2 + w s 3 = u s 2 s 3 + v s 3 s 1 + w s 1 s 2, u s 2 s 3 + v s 3 s 1 + w s 1 s 2 fs 1 s 2 s 3 = f. = u s 1 + v s 2 + w s 3, XII. TIM-HRMONIC LCTROMGNTIC FILDS 82 Fo the desiption of time-hamoni eletomagneti fields the intodution of phasos is useful. We an desibe a timehamoni eleti field by the phaso diffeential fom 0 x that yields the time dependent diffeential fom x, t = R { xe jωt}. 83

8 8 Fo time-hamoni eletomagneti fields we an wite the Maxwell s equations as d H = jω ǫ + M e0, 84 d = jω µ H + M m0, 85 whee ǫ is the omplex pemittivity, µ is the omplex pemeability and the two-foms M e0 and M m0 epesent the impessed eleti and magneti polaizations. Impessed eleti polaization M e0 and equivalent impessed eleti uent J 0 ae elated by J 0 = jωm e0. 86 To ompute the eletomagneti field we fist ompute eleti Hetz fom Π e and/o the magneti Hetz fom Π m whih satisfy the Helmholtz equation: Π e + ω 2 µǫπ e = 1 ǫ M e0, 87 Π m + ω 2 µǫ Π m = 1 µ M m0. 88 The eleti and magneti field foms and H an be deived fom the eleti and magneti Hetz foms: = d d Π e + ω 2 µǫπ e jωµ d Π m, 89 H = jωǫ d Π e + d dπ m + ω 2 µǫ Π m. 90 XIII. TH GRN S FUNCTION The solution of the veto field poblem fo a unit point-like veto soue is given by the dyadi Geen s funtion with the omponents G ij elating the ith omponent of the exited field at a point x to the jth omponent of the exiting soue at a point x [22], [23]. We define the Geen s double one-fom G = G 11 dxdx + G 12 dxdy + G 13 dxdz 91 + G 21 dydx + G 22 dydy + G 23 dydz + G 31 dzdx + G 32 dzdy + G 33 dzdz. Double one-foms ae diffeential foms epesenting dyadis [24], [25]. Unpimed diffeentials dx i and pimed diffeentials dx j ommute, i.e. they may be intehanged without hanging the sign. With the Geen s funtion we an solve the Helmholtz equation fo any soue distibution by onsideing the solution as a ontinuous supeposition of pointlike soues. The Helmholtz equation 87 fo a point-like soue at x is Gx, x + k 2 Gx, x = 1 ǫ Ix, x. 92 The Laplae opeato only ats upon x and not on x sine in the Helmholtz equation x is the spae vaiable of the field whee x is the fixed loation of the soue. The identity kenel is given by Ix, x = δx x dxdx + dy dy + dz dz. 93 With the identity kenel we an map any one-fom U and any two-fom V fom the soue spae to the obsevation spae, Fig. 12. Hetzian dipole of length h. ϕ i.e., the espetive fom is mapped in itself and the pimed diffeentials ae eplaed by unpimed diffeentials. We obtain Ix, x Ux = Ux, 94a ϑ Ix, x Vx = Vx. 94b The pime at the integal symbol denotes that the integation is pefomed ove the pimed oodinates only. Fo a soue embedded in homogeneous isotopi spae the solution of 92 is given by Gx, x = e jk x x 4πǫ x x dxdx + dy dy + dz dz. 95 Fo details see [14]. Multiplying the Helmholtz equation 92 fom the ight with the impessed eleti polaization twofom M e0 x, integating ove dx dy dz and applying 94b we obtain afte ompaison with 87 the solution Π e x = Gx, x M e0 x. 96 This integation has been pefomed ove the pimed oodinates of the soue distibution. Fo the integation the pimed ae the vaiables and the unpimed oodinates denoting the loation of the point of obsevation ae the fixed paametes. XIV. PRIODIC SPHRICL WVS s an example we disuss the impulsive spheial wave emitted fom a Hetzian dipole unde impulsive exitation. The Hetzian dipole is a wie of length h with unifom uent i 0 t impessed Figue 12. In time domain the impessed polaization m e0 t and the impessed uent i 0 t ae elated via i 0 t = d dt m e0t. 97 Sine the uent in the dipole is flowing in z-dietion the impessed eleti polaization is M e0 x = M e0z xdx dy. 98 Inseting this and 95 into 96 yields an eleti Hetz fom with a z omponent only, Π e x = Π ez xdz, 99

9 9 with e jk x x Π ez x = 4πǫ x x M e0z x dx dy dz. 100 Fo h small ompaed with wavelength this yields Π ez x = e jk 4πǫ with = x. This yields in time-domain Π ez x, t = 1 4πǫ M e0z x dx dy dz 101 M e0zx, t dx dy dz. 102 Integating the eleti polaization fom M e0 ove the volume V of the Hetzian dipole yields t M e0z x dx dy dz = hm e0t = h i 0 t whee it is the uent though the Hetzian dipole and m e0 t the polaization due to this uent. The timedependent eleti Hetz fom fo the Hetzian dipole oiented in z dietion is Π e x, t = h 4πǫ 0 m e0 t dz. 104 Using 65 and 66 and onsideing that M e0 x, t vanishes outside the onduto, we an ompute x, t and Hx, t: 0 θ Δt Δt t = 2Δt t = 6Δt t = 10Δt /Δt Fig. 13. Wave pulse: a Pulse wavefoms, b Radial dependene of the wave pulse. Hx, t = d ǫ t Π ex, t, 105a x, t = d dπ e x, t. 105b Using 105a yields H = h 4π [ 1 2 m e0 t + 1 m e0 t ] sin θ sin θdφ. 106 The magneti field only exhibits a φ omponent H φ = h [ 1 4π 2 m e0 t + 1 m e0 t ] sin θ. 107 The eleti field fom is = h {[ 1 4πǫ 0 3 m e0 t m e0 + 2 [ 1 3 m e0 t ] sin θ d θ m e0 t t m e0 t ] osθ d The eleti field exhibits the θ- and -omponents θ = h [ 1 2πǫ 0 3 m e0 t m e0 t m e0 = h [ 1 4πǫ 0 3 m e0 }. 108 t ] osθ, 109 t m e0 t ] sinθ. 110 Fig. 14. Nea field of the Hetzian dipole unde pulse exitation. s an example we onside a wave pulse emitted fom a Hetzian dipole exited by a uent pulse. In Figue 13a the dipole uent pulse it = m t of width 2 t, its integal ove time mt and its time deivative m t ae depited. Figue 13b shows the time evolution of θ, 0, 0, t. The wave font of width 2 t mainly depends on m t and m t. In the fa-field egion, defined by t the tems popotional to 1/ in θ and H φ exhibit the double pulse shape speified by m t. The enegy onneted with this tem is onstained within the shell of width 2 t at the wave-font and tanspoted into the infinity. This is the adiated pat of the field. The eleti and magneti fafield time wave-foms θ and H φ of the wave pulse ae popotional to the time deivative of the diving uent it of the dipole. The nea field pats of the eleti and magneti field popotional to m t // 2 also ae onfined to the wave font shell of width 2 t. This pat of the wave font is aying the eletomagneti enegy fo building up the nea-field. It leaves behind the wave-font an eleti field popotional to mt // 3. This field behind the wavefont oesponds to the eletostati field exited by a stati

10 10 Fig. 15. Fa field of the Hetzian dipole unde pulse exitation. XV. CONCLUSION dvantages ove onventional veto alulus makes the exteio diffeential foms an ideal famewok fo teahing eletomagnetis. It yields a lea and easy epesentation of the theoy and thows light upon the physis behind the fomalism. xial and pola vetos as well as salas and pseudosalas ae lealy distinguished. Rules fo omputation follow in a most natual way fom the notation. The tanslation of fomulae fom the diffeential fom notation to onventional veto notation not only is easy but also suppots undestanding of onventional veto notation. Diffeential fom notation is inheently fully ovaiant and simplifies dealing with uved oodinate systems. dipole. Figue 14 shows the eleti field in a meidional plane. In the fa-field we obtain the appoximate diffeential foms, θ, t = µ 0h m e0 t sin θ dθ, 111 4π h m e0 t H, θ, t = sin θ sin θ dφ 112 4π and the oesponding field omponents θ, θ, t = µ 0h 4π h H φ, θ, t = 4π m e0 m e0 t t sin θ, 113 sin θ. 114 Fom this it follows that the atio of eleti and magneti field in the fa-field is given by the wave impedane µ0 Z F0 =. 115 ǫ 0 The fa-field is depited in Figue 15. Fom 39, 111 and 112 we obtain the Poynting fom S fo the fa-field S, θ, t = 1 2 H = Z F0h π 2 2 m e0 t sin 3 θ dθ dφ. 116 In the fa-field the omplex Poynting veto exhibits only a adial omponent S, θ, t = Z F0h π m e0 t sin 2 θ. 117 The powe P, t adiated fom the Hetzian dipole though a spheial sufae with adius in the fa-field is obtained by integating S ove this sufae P, t = π θ=0 2π φ=0 S, θ, t. 118 RFRNCS [1] J. C. Maxwell, Teatise on letiity and Magnetism, vol. 1. New Yok: Oxfod Univesity Pess, [2] J. C. Maxwell, Teatise on letiity and Magnetism, vol. 2. New Yok: Oxfod Univesity Pess, [3] H. Giffiths, Olive Heaviside, in Histoy of Wieless T. K. Saka, R. Mailloux, and.. Oline, eds., pp , Hoboken, New Jesey: Wiley & Sons, 1 ed., [4]. I. Mille, Imagey in Sientifi Thought. Boston: Bikhäuse, [5] H. Gassmann and L. Kannenbeg, New Banh of Mathematis: The usdehnungslehe of 1844 and Othe Woks. Chiago: Open Cout Publishing, [6]. Catan, Les systèmes difféentielles extéieus. Pais: Hemann, [7] H. Flandes, Diffeential Foms. New Yok: ademi Pess, [8] W. L. Buke, pplied Diffeential Geomety. Cambidge: Cambidge Univesity Pess, [9] P. Bambeg and S. Stenbeg, Couse in Mathematis fo Students in Physis 2. Cambidge: Cambidge Univesity Pess, [10] T. Fankel, The Geomety of Physis. Cambidge: Cambidge Univesity Pess, [11] S. Weintaub, Diffeential Foms - Complement to Veto Calulus. New Yok: ademi Pess, [12] G. Deshamps, letomagnetis and diffeential foms, Poeedings of the I, pp , June [13] K. F. Wanik, R. Selfidge, and D. nold, Teahing eletomagneti field theoy using diffeential foms, I Tans. duation, vol. 40, pp , Feb [14] P. Russe, letomagnetis, Miowave Ciuit and ntenna Design fo Communiations ngineeing. Boston: teh House, [15] F. W. Hehl and Y. N. Obukov, Foundations of Classial letodynamis. Boston Basel Belin: Bikhäuse, [16] I. V. Lindell, Diffeential Foms in letomagnetis. New Yok: I Pess, [17] B. Tellegen, geneal netwok theoem with appliations, Philips Reseah Repots, vol. 7, pp , [18] P. Penfield, R. Spene, and S. Duinke, Tellegen s theoem and eletial netwoks. Cambidge, Massahusetts: MIT Pess, [19] J.. Statton, letomagneti Theoy. New Yok: MGaw-Hill, [20] R. F. Haington, Time Hamoni letomagneti Fields. New Yok: MGaw-Hill, [21] J.. Kong, letomagneti Wave Theoy. Wiley-Intesiene, [22] R. S. lliott, letomagnetis - Histoy, Theoy, and ppliations. New Yok: I Pess, [23] R.. Collin, Field Theoy of Guided Waves. New Yok: I Pess, [24] G. de Rham, Diffeentiable Manifolds. New Yok: Spinge, [25] K. F. Wanik and D. nold, letomagneti geen funtions using diffeential foms, J. letomagn. Waves and ppl., vol. 10, no. 3, pp , We obtain fom 117 and 118 P, t = Z F0h π 2 m e0 t. 119