Active Dipoles, Electric Vector Potential and Berry Phase
DDual Aharanov-Bohm Berry Phase and the Electric Vector Potentialdue to the Generation of Electricity through Permanent Bound and FreeCharge Polarization
Michael E. Tobar, Raymond Y. Chiao, and Maxim Goryachev ARC Centre of Excellence for Engineered Quantum Systemsand ARC Centre of Excellence for Dark Matter Particle Physics,Department of Physics, University of Western Australia,35 Stirling Highway, Crawley, WA 6009, Australia. School of Natural Sciences, University of California Merced,5200 N. Lake Rd, Merced, CA 95343, USA (Dated: January 7, 2021)To understand the creation of electromagnetic energy (or a photonic degree of freedom) from anexternal energy source, an electromotive force must be generated, capable of separating positiveand negative charges. The separation of charges (free or bound) may be modelled as a permanentpolarization, which has a non-zero electric vector curl, created by an external force per unit charge,sometimes referred as an impressed electric field. The resulting system forms a physical dipole inthe static case, or a Hertzian dipole in the time dependent case. This system is the electrical dualof the magnetic solenoid described by a magnetic vector potential and excited by an electrical cur-rent. Correspondingly, the creation of an electric dipole, from the forceful separation of positiveand negative charge, may be described by an electric flux density, which exhibits an electric vectorpotential and a magnetic current boundary source, within the frame work of two-potential theorywithout the need for the existence of the magnetic monopole. From this result we derive the Dualelectric Aharanov-Bohm (DAB) Berry phase and make the conjecture that it should be equivalentto the geometric phase that is described in modern electric polarization theory, which also describesthe nature of the permanent polarization of a ferroelectric. This work gives a formal meaning to theelectric vector potential that defines the DAB geometric phase, and determines that a permanentpolarization has both a scalar and vector potential component, and we show that it must be con-sidered to fully describe the nature of a physical electric dipole, which inevitably is a generator ofelectricity. Additionally, we show that Faraday’s and Ampere’s law may be derived from the timerate of change of the Aharanov-Bohm phase and the DAB phase respectively, independent of theelectromagnetic gauge.
I. Introduction
The magnetic Aharonov-Bohm (AB) effectis a phenomenon where a charged particle’swave function is effected by the magnetic vec-tor potential, (cid:126)A , despite both the electric andmagnetic field being zero [1]. Underlying thiseffect is the general concept of geometric orBerry phase [2] apparent in many areas ofphysics [3] and not restricted to quantum me-chanics, which includes optics [4, 5], condensedmatter physics [6, 7], fluid mechanics [8], andso forth. Other related effects includes; 1)The Aharonov-Casher effect [9–13], which de-scribes the effect of neutral particles with mag- netic moments, effected by an isolated staticpositive or negative electric charge. The iso-lated electric monopole charge distribution cre-ates an effective charge vector potential expe-rienced by magnetic particles, and has beenmeasured using magnetic flux vorticies [11] orneutrons (with a dipole moment) [10]. Likethe AB effect the charge vector potential as-sociated with the Aharonov-Casher effect re-veals a geometric phase in a charge-vortex in-teraction [14]; 2) The He-McKellar-Wilkens ef-fect [15, 16], dual to the Aharonov-Casher ef-fect, which looks at the effect of neutral parti-cles with electric dipole moments (EDMs) in-duced by a magnetic monopole, and; 3) The a r X i v : . [ phy s i c s . c l a ss - ph ] J a n Figure 1: Illustration of the electric fieldgenerated by an idealised dipole made fromtwo point charges ± q ie separated by adisplacement vector (cid:126)L . There is an associatedexternal force per unit charge, (cid:126)E ie , whichsupplies the energy to seperate (and hencepolarize) the impressed free charges. Thenon-conservative nature means anelectromotive force (or voltage), is generated,resulting in a voltage output across the dipole.dual Aharonov-Bohm (DAB) effect, which as-sociates a Berry phase with an EDM or a per-manent polarization (macroscopic collection ofEDMs), such as that exhibited by an electret[17–19] or ferroelectet[20] due to an electric vec-tor potential.In the strict sense of duality, the DAB ex-periment requires monopoles to measure theDAB effect. However, the DAB geometric phaseshould be equivalent to the known one discov-ered in the 1990s, due to the spontaneous per-manent polarization of a ferroelectric [20], orthe permanent polarization of an electret in gen-eral [18, 20–23], and a magnetic monopole wasnot necessary to prove the existence of this al-ready widely accepted geometric phase.Classically a permanent polarization consistsof equal and opposite charges, ± q ie , displacedby a finite distance, (cid:126)L , to create a macro-scopic EDM, (cid:126)d = q ie (cid:126)L , where the vector di-rection is defined from − q ie to + q ie as shownin Fig.1 (with net charge = 0). To displacethe charges an external impressed force per unitcharge is required to seperate positive and nega-tive charges in the induction process. This con- cept is the basis of generating electrical powerfrom an external energy source, which suppliesa non-conservative electromotive force [24–26],allowing a voltage to exist across positively andnegatively charged terminals. This means anexternal force in the opposite direction of theCoulomb force is required to keep the charges instatic equilibrium, otherwise they will acceler-ate towards each other. At large distances fromthe dipole, the electric field appears as an idealdipole field determined by the EDM, (cid:126)d = q ie (cid:126)L .The ideal dipole exist only in the limit as (cid:126)L → q ie → ∞ . In contrast, for distances close tothe separated charges a real physical dipole haselectromagnetic structure.The ideal oscillating time dependent dipole iscommonly known as a Hertzian dipole, and inthe quasi static limit, | (cid:126)r | < λ ( λ is the wave-length of the radiation), the electrostatic nearfield dominates. Within the dipole, the volt-age and current oscillating out of phase as reac-tive power (no work is done) driven by an elec-tric vector potential [24]. In contrast, externalto the dipole, the electric field can be describeby either an electric scalar or vector potential,as the field is capable of doing work on a testcharge, but also exist as a reactive near field(or fringing field) due to the unusual boundarycondition between the outside and inside of theelectric dipole.In the case that the medium is an insula-tor, we consider a macroscopic bound chargedipole known as an electret [24]. An electret ex-hibits a quasi-permanent polarization (or morecorrectly a metastable state, which can lastfor years) in the absence of an applied elec-tric field. Such materials are commonly usedfor energy harvesting and electricity generation[27–32]. Moreover, modern polarization theoryintroduced in the 1990s [21–23] has shown thatthe general definition of the polarization was notsolely calculable from bulk characteristics of thevolume of bound charge, and that a change ofpolarization only had physical meaning if it wasquantified by using a Berry phase. This tech-nique has been very successful in first-principlesstudies of ferroelectric materials [20, 33].In this work we use the fact that the per-manent polarization vector has both a non-zerocurl and divergence. Thus, can be defined asa combination of a scalar and vector potential.Importantly, the electric vector potential gives anon-zero tangential surface term, which at theboundary can be viewed as an effective mag-netic current [24], a entity related to a Berryphase. Furthermore, we find that the time rateof change of the Berry phase leads to the deriva-tion of Ampere’s Law (magnetomotive force),and the rate of change of the AB phase leadsto the derivation of Faraday’s law (electromo-tive force). This is consistent with prior work,which derives motive forces from the Aharonov-Bohm and Aharonov-Casher effects [34]. II. Static Macroscopic Electric Dipole
For a dipole, the standard text book exam-ple assumes unphysical point charges, shown infig.1, with a scalar potential given by, V = q ie π(cid:15) (cid:18) r + − r − (cid:19) = q ie ( r − − r + )4 π(cid:15) r + r − , (1)where, in spherical coordinates ( r, φ, θ ), r ± = r + L ∓ Lr cos θ. (2)Calculating the potential difference between thetwo point charges from eqn.(1), we derive aninfinite voltage! In other words, an infinite im-pressed force per unit charge (cid:126)E ie is required tokeep two point charges separated at a finite dis-tance, which highlights the unphysical nature ofthe point charge dipole.A better approximation is to assume idealsurface charges, so q ie = σ ie πa e as shown in fig.2,so the force is spread over an area. Such per-manent electric dipoles occurs in bound charge(ideal electret) and free charge (battery or elec-tricity generation) systems [24]. In this work weanalyse the dipoles in terms of potentials, andshow that an electric vector potential drives thecreation of electricity. We also define the sepa- Figure 2: A physical dipole of oppositelypolarity surface charges, ± σ ie , where q ie = σ ie πa e , and a e , is the effective radius thatthe charge is spread over. The external forceper unit charge, (cid:126)E ie , is finite and supplies theenergy to seperate (and hence polarize) theimpressed free charges. The voltage outputacross the dipole can be shown to be due to aneffective azimuthal magnetic surface currentboundary source, which modifies Faraday’slaw, given by (cid:126)J im = − (cid:126) ∇ × (cid:126)E ie . For a constantvalue of (cid:126)E ie , the effective magnetic current ison the radial surface so (cid:126)κ im = (cid:126)J im δ ( r − a e )(Weber convention for magnetic current). Theseparated free charges then generate aconservative electric field, (cid:126)E , inside andoutside the voltage source.ration of free charge by a polarization vector, (cid:126)P ie = (cid:15) (cid:126)E ie = σ ie ˆ z, (3)which is related to the impressed force [24]. Incontrast, for a bound charge system with a per-manent dipole is an electret, and assuming alinear electronic susceptibility, (cid:126)P ie = (cid:15) (cid:126)E ie = σ ie ˆ z, (4)where (cid:15) r is the dielectric constant, (cid:15) = (cid:15) (cid:15) r and σ ie represents impressed free or bound chargerespectively. In these cases an effective mag-netic current surface density exists at the ra-dial boundary of the dipole and acts as a sourceterm, which has been shown to be given by [24], (cid:126)κ im = − σ ie (cid:15) ˆ φ, (5)in the Weber convention. Next we consider thegeneral time dependent case. III. Quasi-Static Time DependentMacroscopic Hertzian Dipole; Fieldsand Potentials
Maxwell’s equations for an ideal electricitygenerator with impressed bound or free charge( (cid:15) = (cid:15) ) volume density, ρ ie , has been shown tobe given by [24] (Weber convention), (cid:126) ∇ · (cid:126)E = ρ ie (cid:15) and (cid:126) ∇ · (cid:126)E ie = − ρ ie (cid:15) , (6) (cid:126) ∇× (cid:126)B − (cid:15)µ ∂ (cid:126)E∂t = µ ( (cid:126)J ie + (cid:126)J f ); (cid:126)J ie = (cid:15) ∂ (cid:126)E ie ∂t , (7) (cid:126) ∇ · (cid:126)B = 0 , (8) (cid:126) ∇ × (cid:126)E + ∂ (cid:126)B∂t = 0 and (cid:126) ∇ × (cid:126)E ie = − (cid:126)J im . (9)or in terms of the total electric field, (cid:126)E T by (cid:126) ∇ · (cid:126)E T = 0 , (10) (cid:126) ∇ × (cid:126)B − (cid:15)µ ∂ (cid:126)E T ∂t = (cid:126)J f , (11) (cid:126) ∇ · (cid:126)B = 0 , (12) (cid:126) ∇ × (cid:126)E T + ∂ (cid:126)B∂t = − (cid:126)J im , (13)with the following constitutive relations (cid:126)E T = (cid:126)E ie + (cid:126)E (14) Here, (cid:126)J f in the lossless case has zero divergence,since ρ f = 0, and (cid:126)J im also has zero divergencesince ρ im = 0. (cid:126)J im exists on the radial boundaryof the dipole, and drives the impressed electricfield, (cid:126)E ie , by the left hand rule and also setsthe boundary condition for the parallel compo-nents of the fields on the radial boundary. Herethe ∂ (cid:126)B∂t term in eqn.(13) can be identified as amagnetic displacement current and (cid:126)J f can onlyexist if an external circuit is coupled to the idealelectricity generator[27–32] or the generator isnon-ideal with an effective internal resistance.The modified form of these equations meansin general an electric vector potential, (cid:126)C , can beintroduced, along with the electric scalar po-tential, V , and the magnetic vector potential, (cid:126)A . The possible existence of an electric vec-tor potential and a magnetic scalar potentialhas been postulated to exist through the dualof Maxwell’s equations being excited by mag-netic monopoles and magnetic currents[35–38]and is known as two-potential theory. More-over, the electrical engineering community havealso shown that the dual of Maxwell’s equationmay be excited by non-conservative electromag-netic systems or electricity generators [24–26],without the need for monopoles to exist. Thus,from two-potential theory, and given there isno magnetic scalar field in the system we aredescribing, we may write the potential of thedefined fields in eqns. (6)-(14) as, (cid:126)B = − µ ∂ (cid:126)C∂t + (cid:126) ∇ × (cid:126)A (15) (cid:126)E = − (cid:126) ∇ V − ∂ (cid:126)A∂t (16) (cid:126)E ie = (cid:126)P ie (cid:15) = (cid:126) ∇ V − (cid:15) (cid:126) ∇ × (cid:126)C ; (17) (cid:126)E T = (cid:126)D T (cid:15) = − (cid:15) (cid:126) ∇ × (cid:126)C − ∂ (cid:126)A∂t (18)Note, the field that experiences the “pure” vec-tor potential is (cid:126)E T = (cid:126)D T (cid:15) = (cid:126)P ie /(cid:15) + (cid:126)E , for boththe free and bound system.Inside the dipole the polarization field, (cid:126)P ie ,exists without any applied electric field, withboth vector and scalar potential components,with the scalar component exactly equal and op-posite to the scalar potential of the (cid:126)E field, con-sistent with eqn. (6). Meanwhile, (cid:126)E ie and (cid:126)E T have the same vector curl and hence the samecomponent of electric vector potential, whilesatisfying the constitutive given by eqn. (14).Outside the both the free and bound chargedipole, (cid:126)E ie = 0, which means from eqn. (17), (cid:126) ∇ V = 1 (cid:15) (cid:126) ∇ × (cid:126)C (19)This then gives two ways to describe the electricfield outside the dipole with either an electricvector or scalar potential. In the quasi staticlimit the solution is dominated by the electro-static near field of the dipole, which is reac-tive with the impressed current and voltage outof phase [24]. Thus, the electric field can bethought as a continuation of the same vector po-tential within the dipole, with the electric fieldgiven by the right hand rule, source from themagnetic current at the boundary, as shown inFig. 1. On the other hand, the electric fieldcan do work on a test particle, and in this casetakes the form of an equivalent scalar potentialas given by eqn. (19).Now by substituting the fields given in eqns.(15) and (18) back into the electric and mag-netic Gauss’ law we obtain ∂ ( (cid:126) ∇ · (cid:126)A ) ∂t = 0; ∂ ( (cid:126) ∇ · (cid:126)C ) ∂t = 0 , (20)so the divergence of the vector potentials mustbe time independent. Then by substituting ei-ther (16) or (17) into Gauss’ law, and using (20)we obtain, ∇ V = − ρ ie (cid:15) (21)Substituting, (15) and (18) into Faraday’s law we obtain, (cid:126) ∇ × (cid:126) ∇ × (cid:126)C + µ (cid:15) ∂ (cid:126)C∂t = (cid:15) (cid:126)J im , (22)then by substituting, (15) and (18) into Am-pere’s law we obtain, (cid:126) ∇ × (cid:126) ∇ × (cid:126)A + µ (cid:15) ∂ (cid:126)A∂t = µ (cid:126)J f . (23)It is well known that there is more than one setof potentials that can generate the same fields,given that (cid:126) ∇ × (cid:126) ∇ × (cid:126)C = − (cid:126) ∇ (cid:126)C + (cid:126) ∇ ( (cid:126) ∇ · (cid:126)C )to simplify we chose the gauge where the diver-gence of the vector potentials are zero (CoulombGauge), so we obtain, (cid:3) (cid:126)C = − (cid:15) (cid:126)J im , (24)and (cid:3) (cid:126)A = − µ (cid:126)J f , (25)Thus, we have successfully calculated the po-tentials in terms of the impressed sources, (cid:126)J im and ρ ie as well as any free current in the sys-tem, (cid:126)J f . For the lossless system with no load, ∇ · (cid:126)J f = 0. Note, that the impressed current, (cid:126)J ie = (cid:15) ∂ (cid:126)E ie ∂t , in our presentation is not considereda source term, as it is described as a polariza-tion current, which can either be from free orbound charge, impressed by the external forceper unit charge, (cid:126)E ie . IV. Berry Phase of a Macroscopic ElectricDipole
Since we have defined a simple macroscopicpolarization of bound or free charge with re-spect to a 3D electric vector potential (cid:126)C , wemight believe a Berry phase exists similar tomodern polarization theory [7, 39]. In this case,the electric vector potential is a 3D Berry con-nection, and the Berry curvature is the field, (cid:126)D T = (cid:15) (cid:126)E T = (cid:126)P ie + (cid:15) (cid:126)E , given by eqn. (18).In fact, the electric dipole is dual to the mag-netic dipole, which was used in the original ABthought experiment, so on this premises a DABelectric effect should exist, and has been consid-ered previously [17–19].First lets consider semi-classically the wellknown AB magnetic Berry phase of a long cylin-drical electromagnetic solenoid, ∆ φ B AB , andwith the use of eq. (15) we can show, φ B AB = q (cid:126) ˛ P (cid:126)A · d(cid:126)l = q (cid:126) ˆ S ∇ × (cid:126)A · d(cid:126)S = q (cid:126) ˆ S (cid:126)B · d(cid:126)S + µ q (cid:126) ˆ S ∂ (cid:126)C∂t · d(cid:126)S. (26)Here, the closed path, P , of integration of themagnetic vector potential on the LHS of eqn.(26) encloses the surface, S , in which the mag-netic flux flows, with the first term on the RHSthe static contribution to the AB geometricphase, while the second term adds the time de-pendent term. For the static case if we consider, P as the path at the mid point of the solenoidaround the the electric current boundary, theminimum value of enclosed magnetic flux willbe given by the flux quantum, Φ = h/ (2 e ),so that ´ S (cid:126)B · d(cid:126)S = n Φ for a superconductingsystem with n Cooper pairs ( q = 2 e ). In con-trast, for a normal conductor with free electrons( q = e ), ´ S (cid:126)B · d(cid:126)S = 2 n Φ (measured by Webbet. al. [40]). Thus, in general the static ABphase in both the superconducting and normalconducting case is given by, φ B AB = 2 nπ .Now we consider in analogy the DAB elec-tric phase φ E AB , and with the use of eq.(18) weobtain, φ E AB = 1 q ˛ P (cid:126)C · d(cid:126)l = 1 q ˆ S ∇ × C · d(cid:126)S = 1 q ˆ S (cid:126)D T · d(cid:126)S + (cid:15)q ˆ S ∂ (cid:126)A∂t · d(cid:126)S. (27)Here, the closed path, P , of integration of theelectric vector potential on the LHS of eqn. (27)encloses the surface, S , in which the electricflux flows. Thus, in analogy, the first termon the RHS gives the static DAB geometricphase, while the second gives the general time dependent term. For the static case the geo-metric phase depends on the enclosed electricflux, Φ E = ´ S (cid:126)D T · d(cid:126)S , which for a path, P , atthe mid point of the magnetic current bound-ary, should be equal to the quantum of electriccharge, q = e , for a single electron system or, q = 2 e , for a paired electron system. Theseequations should be valid for both bound-chargeand free-charge permanently polarized systems,which generate electricity.Considering modern polarization theorybased on Berry phase, the definition of polar-ization was developed through the crystal lat-tice surface and volume charge distributions,which are necessarily related to the electricscalar potential. The theory does not calcu-late the geometric phase of a permanent po-larization through a vector potential relatedto an effective surface magnetic current, re-quired for non-conservative electrodynamic sys-tems, which generate electrical power, such asan electret [24]. However, because the vectorcurl of (cid:126)P ie and hence (cid:126)D T is non-zero as given byeqn. (17) and (18) an equivalent definition ordescription should be possible through the elec-tric vector potential, with a magnetic currentboundary source.As discussed by Vanderbuilt [39], modern po-larization theory is based on the heuristic re-placement of the position vector, (cid:126)r → i ∇ (cid:126)k , bythe (cid:126)k -derivative operator. Thus, Berry phaseis considered in momentum space rather thatposition space, and the polarization is quan-tised, so that (cid:126)P → (cid:126)P + ∆ (cid:126)P ie , corresponds to φ E AB → φ E AB + 2 π [23, 39]. In contrast, ourapproach allows us to relate the same quanta ofpolarization to the DAB electric phase in po-sition space, without the need for consideringperiodic unit cells (or a supercell) in (cid:126)k -space.In a similar way, Onoda et.al. [20] have de-scribed the topological nature of polarizationand charge pumping [41] in ferroelectrics usingan analogy to magnetostatics, by introducinga vector field with a Berry phase as a linearresponse of the covalent part of polarization,which has incorporated a generalization of theBorn charge tensor. In principle this type ofdescription should be equivalent to a polariza-tion with a non-zero curl and an electric vec-tor potential as introduced in this work, so thecalculation may be done in position space. Asimilar strategy has also been previously pre-sented in [42, 43], and suggests the magneticcurrent boundary source carries the crystal mo-mentum. If we examen the unit cell of a super-cell in modern polarization theory, this meansthe minimum electric flux should be related by∆Φ E = ´ S ∆ (cid:126)P ie · d(cid:126)S , which is equal to 2 e fora spin paired non-magnetic system and e for aspin polarized system [23, 39], so in general thestatic DAB phase is a Berry phase given by, φ E AB = 2 nπ . V. Motive Force Equations from the TimeDependence of Aharonov-Bohm Phase
Previously an equivalence between theAharonov-Bohm effect of a solenoid and theAharonov-Casher effect of a charged rod hasbeen demonstrated, where the time-dependentAharonov-Casher phase was shown to inducea motive force via the SU(2) spin gauge field[34]. In a similar way to the time dependenceAharonov-Bohm effect derives Faraday’s law,responsible for electromagnetic induction andthe electromotive force (emf). Here we showthat the time dependence of the DAB phase de-rives Ampere’s law, the equation responsible formagnetomotive force (mmf).First we consider the time rate of change ofeqn.(26) and combining it with (15) we obtain, − (cid:15) ˛ P ∇ × (cid:126)C · d(cid:126)l − ˛ P (cid:126)E T · d(cid:126)l = ∂∂t ˛ S (cid:126)B · d(cid:126)S + µ (cid:15) ˛ S ∂ (cid:126)C∂t · d(cid:126)S, (28)which becomes, E = ˛ P (cid:126)E T · d(cid:126)l = − ∂∂t ˛ S (cid:126)B · d(cid:126)S − (cid:15) ˛ S (cid:0) ∇ × ∇ × (cid:126)C + µ (cid:15) ∂ (cid:126)C∂t (cid:1) · d(cid:126)S = − ∂ Φ B ∂t − ˛ S (cid:126)J im · d(cid:126)S (29)Here, E , is defined as the electromotive force(emf), then from eqn.(29) we obtain, E T = − I m enc = − ˛ S (cid:126)J im · d(cid:126)S = E + ∂ Φ B ∂t , (30)which is Faraday’s law [24]. Here, I m enc is theenclosed effective current boundary source, and E T ,the voltage across a dipole or total emf .Next we consider the time rate of change ofeqn.(27) and combining it with (18) we obtain, − µ ˛ P ∇ × (cid:126)A · d(cid:126)l + 1 µ ˛ P (cid:126)B · d(cid:126)l = (cid:15) ∂∂t ˛ S (cid:126)E T · d(cid:126)S + (cid:15) ˛ S ∂ (cid:126)A∂t · d(cid:126)S, (31)which becomes, F = 1 µ ˛ P (cid:126)B · d(cid:126)l = (cid:15) ∂∂t ˛ S (cid:126)E T · d(cid:126)S +1 µ ˛ S (cid:0) ∇ × ∇ × (cid:126)A + µ (cid:15) ∂ (cid:126)A∂t (cid:1) · d(cid:126)S = (cid:15) ∂∂t ˛ S (cid:126)E T · d(cid:126)S + ˛ S (cid:126)J f · d(cid:126)S, (32)which is the integral form of Ampere’s law [24].Here, F , is defined as the magnetomotive force(mmf), then by rearranging eqn.(32) we obtain, F T = I f enc = ˛ S (cid:126)J f · d(cid:126)S = F − ∂ Φ E ∂t , (33)Here, F T = I f enc = N × I , is the enclosed elec-trical current boundary source of a magneticdipole or inductor coil with N turns.Figure 3: Field and potential plots for acylindrical dipole of either free or boundcharge with AR=1. A)
2D vector plot of thenormalized electric flux density (cid:126)D T σ ie at y = 0,in the ( r − z ) plane, calculated from eqns. (40)and (41). B)
2D vector plot of the normalizedelectric field (cid:15) (cid:126)Eσ ie at y = 0, in the ( r − z ) plane,calculated from eqn. (39). C)
2D colourdensity plot of the normalized electric scalarpotential (cid:15)Vσ ie at y = 0, in the ( r − z ) plane,calculated from eqn. (35). D)
3D vector plotof the normalized electric vector potential, (cid:126)Cσ ie . E)
2D vector plot of the normalized electricvector potential, at z = 0, in the ( r − φ ) plane,one can see the electric vector potential ismaximum at the radial boundary where themagnetic current exists. VI. Electronic Properties MacroscopicCylindrical Dipole
In this section we assume a static (or quasi-static approximation), to analyse the electronicproperties of a cylindrical electric dipole of vary- Figure 4: Not to scale field and potential plotsfor a cylindrical dipole of either free or boundcharge.
Above
AR=10: A)
2D vector plot ofthe normalized electric flux density (cid:126)D T σ ie at y = 0, in the ( r − z ) plane, calculated fromeqns. (40) and (41). B)
2D vector plot of thenormalized electric field (cid:15) (cid:126)Eσ ie at y = 0, in the( r − z ) plane, calculated from eqn. (39). C)
2D colour density plot of the normalizedelectric scalar potential (cid:15)Vσ ie at y = 0, in the( r − z ) plane, calculated from eqn. (35). Below similar plots to above but withAR=0.1: D) (cid:126)D T σ ie at y = 0, in the ( r − z ) plane. E) (cid:15) (cid:126)Eσ ie at y = 0, in the ( r − z ) plane. F) (cid:15)Vσ ie at y = 0, in the ( r − z ) plane.ing aspect ratios ( AR = a e L ), in terms of theelectric fields and potentials as described in Sec-tion III, where L is the axial length, and a e theradius of the cylinder. Assuming a macroscopicdipole of the form shown in Fig.2, we vary theaspect ratio and calculate the electric scalar, V ,and vector, (cid:126)C , potentials, as well as the electricfield, (cid:126)E , and electric flux density, (cid:126)D , rangingfrom a flat pancake structure ( AR → ∞ ) toa long needle type structure ( AR → z = 0, from the centre of the electric dipole forvarious aspect ratios, compared to theinfinitely long dipole ( AR → r = 0, from thecentre of the electric dipole for various aspectratios, compared to the infinely wide dipole AR → ∞ . Here, the length of the dipole is, L ,where AR = a e L , so the end face of the dipoleare at z/L = ± .tor and density plots for some aspect ratios areshown in Fig.3 and Fig.4.Assuming a constant polarization of (cid:126)P ie = σ ie ˆ z within the boundaries of the cylindrical dipole,then there will be a constant impressed surfacecharge density at each axial end face of ± σ ie ,with a corresponding impressed surface mag-netic current density at the radial boundary( r = a e ) of value, (cid:15)(cid:126)κ im = − δ ( r − a e ) σ ie ˆ φ [24].The potentials and field can be calculated from Figure 6: Above: Normalized z component ofthe electric field, E z , versus normalized radialdistance, at z = 0, from the centre of theelectric dipole for various aspect ratios. Notefor the infinite dipole ( AR →
0) the electricfield is zero for all r . Below: Normalized z component of the electric flux density, D T z ,versus normalized axial distance, at r = 0,from the midpoint of the electric dipole forvarious aspect ratios. Note, for the infinitelywide dipole ( AR → ∞ ) D T z is zero for all z .Note the tangential E z field across the radialboundary of the dipole, at ra e = 1, iscontinuous, while the normal D T z field iscontinuous across the axial boundary at, zL = ± .the surface charge density and the surfacemagnetic current density using the followingequations:1) The electric scalar potential, V ( (cid:126)r ) = 14 π(cid:15) ¨ (cid:48) S σ ie (cid:0) (cid:126)r (cid:48) (cid:1) dA | (cid:126)r − (cid:126)r (cid:48) | , (34)so the normalized value in cylindrical coordi-0nates is given by, (cid:15)V ( (cid:126)r ) σ ie =14 π ˆ a e ˆ π δ ( z (cid:48) − L ) − δ ( z (cid:48) + L ) | (cid:126)r − (cid:126)r (cid:48) | r (cid:48) d φ (cid:48) d r (cid:48) . (35)2) The electric vector potential, (cid:126)C ( (cid:126)r ) = (cid:15) π ˆ (cid:48) S (cid:126)κ im (cid:0) (cid:126)r (cid:48) (cid:1) | (cid:126)r − (cid:126)r (cid:48) | d (cid:126)r (cid:48) , (36)so the normalized value in cylindrical coordi-nates is given by, (cid:126)C ( (cid:126)r ) σ ie = − ˆ φ π ˆ L − L ˆ π δ ( r (cid:48) − a e ) | (cid:126)r − (cid:126)r (cid:48) | d φ (cid:48) d z (cid:48) . (37)3) The electric field vector ( (cid:126)E = − (cid:126) ∇ V ) , (cid:126)E ( (cid:126)r ) = 14 π(cid:15) ¨ (cid:48) S σ ie (cid:0) (cid:126)r (cid:48) (cid:1) dA ( (cid:126)r (cid:48) − (cid:126)r ) ˆ (cid:126)r (cid:48) (38)so the normalized value in cylindrical coordi-nates is given by, (cid:15) (cid:126)E ( (cid:126)r ) σ ie =14 π ˆ a e ˆ π δ ( z (cid:48) − L ) − δ ( z (cid:48) + L )( (cid:126)r (cid:48) − (cid:126)r ) ˆ (cid:126)r (cid:48) r (cid:48) d φ (cid:48) d r (cid:48) . (39)4) The electric flux density may in principle becalculated through integrating over the mag-netic current from the relation (cid:126)D = − (cid:126) ∇ × C (eqn.18). However instead, since we have al-ready calculated (cid:126)E ( (cid:126)r ), we use the relation, (cid:126)D T ( (cid:126)r ) = (cid:15) (cid:126)E T ( (cid:126)r ) + (cid:126)P ie , which leads to the fol-lowing normalized values, (cid:126)D ( (cid:126)r ) σ ie = (cid:15) (cid:126)E ( (cid:126)r ) σ ie + ˆ z inside the dipole (40) (cid:126)D ( (cid:126)r ) σ ie = (cid:15) (cid:126)E ( (cid:126)r ) σ ie outside the dipole (41)Some interesting points come out of thesesimulations. For small aspect ratios ( AR → ± σ ie charges are separated by large distanceswith respect to the radius of the charge. Inthis case the electric field and scalar potentialbetween the charges approaches zero (see Fig.5and Fig.6), in this case the electric vector po-tential is the main component of emf genera-tion. The opposite occurs for large aspect ra-tios for pancake like structures. In this casethe total electric field, (cid:126)E T or electric flux den-sity (cid:126)D T approaches zero. For this case because (cid:126)E ie ≈ − (cid:126)E inside the dipole, the potential dif-ference between the axial end faces due to theelectric scalar potential is equivalent to the emf generated across the dipole, and the electricvector potential approaches zero. This findingis consistent with [24], which determined thatthe magnetic current boundary source best de-scribes the output voltage of an AC or DC gen-erator, rather than the electric field. VII. Discussion
A macroscopic time independent magneticdipole can in principle exist without loss as apersistent DC current in a superconducting wireloop or coil not requiring any energy or power.For this situation all parts of Faraday’s law ineqn. (30) are zero, as there is no voltage or emfrequired. The strength of the magnetic dipoledepends on the enclosed electrical current inthe loop. For a superconducting coil, a currentmay be trapped with the use of a persistentswitch, and the strength of the magnetic fieldwill depend on the applied mmf , F T = N I ,as given by Ampere’s law in eqn. (33). Thus,once trapped the mmf exists as stored energy, E m = LI ( L is the inductance of the loop orcoil), and no work is required to keep the dipoleenergised.The electromagnetic dual of the magneticdipole is the electric dipole. However, for theelectric dipole to exist an emf must be ap-1plied to force the separation of charges, contraryto the magnetic dipole, this charge separationrequires work, or an impressed force per unitcharge from an external energy source. Thus,it is apparent that a permanent EDM or anensemble of permanent EDMs is an electricitygenerator (i.e. an electret). The natural ten-dency is for the electric dipole to discharge ordecay and emit a photon [44], which means theelectric dipole is intrinsically metastable and areless common in nature, in the quantum mechan-ical picture this is related to the fact that EDMsviolate both the parity and time-reversal sym-metries. We have shown the voltage suppliedby the electric dipole is determined by the en-closed effective magnetic current at the tangen-tial boundary given by eqn (30). In this dualsystem the electric vector potential exists, andhas a geometric phase.The next question is, can we devise an experi-ment to measure this geometric phase in a simi-lar way to the well-known AB experiment? Anyexperiment will need a full quantum mechani-cal description to understand if it would work,and act on the interference fringes of a passing particle such as an electron or a particle withan electric or magnetic dipole moment [17, 19].From fig. 3, we notice the vector potential ismaximum just outside the rim of the dipole atthe centre, at this same place the electric fieldis minimum. Passing particles around differentdirections would be the dual of the original ABexperiment. Another way would be to configurean experiment which generates electricity in theregime dominated by the electric vector poten-tial, and confirm the voltage output, this has al-ready been undertaken with energy harvesters,where electricity is generated by a polarizationin the absence of an applied electric field [24]. A. Acknowledgements
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