Lorenz Gauge Condition and Its Relation to Far-Field Longitudinal Electric Wave
LLorenz Gauge Condition and Its Relation to Far-Field Longitudinal Electric Wave ∗ Altay Zhakatayev
Robotics Department, Nazarbayev University, Nur-Sultan, Kazakhstan (Dated: April 30, 2020)In this short paper, it is shown that Lorenz gauge condition leads to disappearance of long-rangelongitudinal electric wave emitted by an arbitrary electrical system. In any other gauge, longitudinalelectric wave would be non-negligible. Implications of the obtained results are discussed.
I. INTRODUCTION
Derivations of radiation of infinitesimal electricdipoles, linear traveling and standing wave antennas andother radiating systems are well known. In these deriva-tions, it is often shown that no far-field longitudinalelectric wave is emitted. However, in this paper, thiscommonly accepted point of view is challenged. Specif-ically, we show that in the general case and within theframework of classical electrodynamics, solutions yieldthe emission of long-range longitudinal electric wave.
II. THEORY
Our purpose in this paper is to show that Lorenz gaugecondition “destroys” long-range longitudinal electric fieldor wave. For starter, we will describe briefly fundamentalconcepts.For the derivation, spherical coordinate system is used,which has unit vector ˆr , ˆ θ , ˆ φ . r denotes arbitrary radius-vector with magnitude r , while t is time. ω is angularfrequency of the radiation, while k = ωc is “spatial” fre-quency or wave-number. All force and potentials fieldsare assumed to be dependent on r and t (e.i. E = E ( r , t ), B = B ( r , t ), A = A ( r , t ), V = V ( r , t )), but we will notwrite it explicitly in order to be concise. The term longi-tudinal electric field is used to denote the component ofelectric field E r along radial ˆr direction. A. Background
Maxwell equations have the following form ∇ · E = ρ(cid:15) , (1a) ∇ · B = , (1b) ∇ × E = − ∂ B ∂t , (1c) ∗ In some textbooks authors use the term “Lorentz” gauge con-dition. However, the correct name should be “Lorenz” gaugecondition, named after Ludvig Lorenz. ∇ × B = µ ( J + ∂ E ∂t ) . (1d)By utilizing vector caclulus identities, the vector poten-tial is introduced from (1b), while the scalar potential isdefined by using the vector potential in (1c). As a result,the following expression is obtained for the electric field E = −∇ V − ∂ A ∂t . (2)It is claimed that divergence of the vector potential canbe set to arbitrary value [1]. Different values correspondto different gauge conditions. The frequently utilizedgauge condition is called Lorenz gauge and it is definedas ∇ · A = − c ∂V∂t . (3)We will now switch to a so-called complex or j nota-tion. According to [2] Sec. 3.6, in the most general casethe far-field terms of vector and scalar potentials havethe following form A = ( A r ( θ, φ ) ˆr + A θ ( θ, φ ) ˆ θ + A φ ( θ, φ ) ˆ φ ) e − j ( kr − ωt ) r , (4a) V = V ( θ, φ ) e − j ( kr − ωt ) r . (4b)This can also be confirmed by checking the solution offar-field radiation of any electrodynamics system (shortdipoles, long traveling wave and standing wave antennas,circular antenna and etc.). In essence, (4) means thatangular ( θ , φ ) and radial ( r ) dependence of potentials canbe separated or decoupled. Our focus is not on how toget the vector and scalar potentials (4), but to investigatethe implications which follow from already having thesepotentials. B. Derivation in Lorenz Gauge
Let us briefly demonstrate that the derivation in [2]Sec. 3.6, which proves that far-field longitudinal electricwave does not exist, is valid only for one specific gauge.Similar derivations are performed in [3] Sec. 1.13 and1.15. In the complex notation, Lorenz gauge conditionbecomes ∇· A = − jωV /c , from which we can obtain the a r X i v : . [ phy s i c s . c l a ss - ph ] A p r scalar potential and substitute it into (2). This wouldresult in the following expression for the electric field: E = − j c ω ∇ ( ∇ · A ) − jω A . (5)The vector potential from (4) is utilized to find ∇ · A = − A r jk e − j ( kr − ωt ) r , (6)where only the term with first power in r in the denom-inator is selected. This is done because we are analyz-ing far-field approximation and so terms decreasing fasterthen 1 /r can be neglected. Next let us find ∇ ( ∇ · A ) = − A r k e − j ( kr − ωt ) r ˆr , (7)where now only the radial term is selected, because weare interested in the longitudinal electric field. By sub-stitution of (7) into (5), then equating the radial termsfrom the right side of (5) to the radial component of thetotal electric field E = E r ˆr + E θ ˆ θ + E φ ˆ φ , we can find E r = − jωA r e − j ( kr − ωt ) r (1 − k c ω ) = 0 . (8)Thus, we can wrongly conclude that the long-range lon-gitudinal electric field is zero. However, this is due to thefact that (5) is valid only in the Lorenz gauge, while (2)is more general expression valid in any gauge. C. Derivation for Arbitrary Gauge
Let us now use (2) instead of (5). For that, gradientof the scalar potential (4) can be found ∇ V = − jkV e − j ( kr − ωt ) r ˆr , (9)where again we left only the most significant term alongthe radial direction. Time differential of the radial com-ponent of the vector potential is ∂A r ∂t ˆr = jωA r e − j ( kr − ωt ) r ˆr . (10)If (9) and (10) are substituted into (2), then we get forradial component E r = jk ( V − A r c ) e − j ( kr − ωt ) r (cid:54) = 0 . (11)It can be observed that in the general case longitudinalelectric field is nonzero. It becomes null only if kV = A r ω or V = A r c . Let us demonstrate that this identity is validonly in the Lorenz gauge. Time differential of the scalarpotential is ∂V∂t = jV ω e − j ( kr − ωt ) r . (12) If (12) and (6) are put into (3), then we get jV ωc e − j ( kr − ωt ) r = A r jk e − j ( kr − ωt ) r ⇒ V ωc − A r k = 0 . (13)After simple algebraic manipulations, we get from thelast equation V − A r c = 0. Thus, it is clear that underLorenz gauge condition (13), longitudinal electric field(11) vanishes. However, in the general case or in anyother gauge, where V − A r c (cid:54) = 0, E r is nonzero.Let us give two examples. For classical electric dipolewith charge separation distance d and wavelength λ , un-der the assumptions d (cid:28) λ (cid:28) r , the solutions for emittedscalar potential and radial component of the vector po-tential are [1] [Sec. 11.1] V = − q π(cid:15) ωdrc sin ( ω ( t − rc )) cos θ, (14a) A r = − q π(cid:15) ωdrc sin ( ω ( t − rc )) cos θ. (14b)Thus it is clear that for the classical dipole V − A r c =0 and as expected E r = 0. However, for the electricdipole under the assumptions λ (cid:28) d (cid:28) r and with thecurrent wave flowing in positive ˆz direction, the followingsolutions are obtained [4] V = − q π(cid:15) r (cid:32) cos( ω ( t − rc )) (cid:16) cos( ωd c (2 − cos θ )) − cos( ωd c cos θ )+11 + cos θ (cos( ωd c ) − cos( ωd c (2 + cos θ )))+11 − cos θ (cos( ωd c cos θ ) − cos( ωd c )) (cid:17) +sin( ω ( t − rc )) (cid:16) sin( ωd c cos θ ) − sin( ωd c (2 − cos θ ))+11 + cos θ (sin( ωd c (2 + cos θ )) − sin( ωd c ))+11 − cos θ (sin( ωd c ) − sin( ωd c cos θ )) (cid:17)(cid:33) , (15a) A r = − q π(cid:15) rc (cos θ ) · (cid:32) cos( ω ( t − rc )) (cid:16)
11 + cos θ (cos( ωd c ) − cos( ωd c (2 + cos θ )))+11 − cos θ (cos( ωd c cos θ ) − cos( ωd c )) (cid:17) +sin( ω ( t − rc )) (cid:16)
11 + cos θ (sin( ωd c (2 + cos θ )) − sin( ωd c ))+11 − cos θ (sin( ωd c ) − sin( ωd c cos θ )) (cid:17)(cid:33) . (15b)As a result, V − A r c = − q π(cid:15) r (cid:32) cos( ω ( t − rc )) (cid:16) cos( ωd c (2 − cos θ )) − cos( ωd c )+1 − cos θ θ (cos( ωd c ) − cos( ωd c (2 + cos θ ))) (cid:17) +sin( ω ( t − rc )) (cid:16) sin( ωd c ) − sin( ωd c (2 − cos θ ))+1 − cos θ θ (sin( ωd c (2 + cos θ )) − sin( ωd c )) (cid:17)(cid:33) (cid:54) = 0 , (16)Therefore, for electric dipole system where charge sepa-ration distance is non-negligible, V − A r c (cid:54) = 0 and as aconsequence E r (cid:54) = 0. Indeed, it can be verified rathereasily that if (15) are utilized in (2), then we get non-zero longitudinal electric wave, but if (15) are used in(5), then due to exact cancellation of all terms, E r = 0.Plot of (16) as a function of θ and ωd c and for the timewhen ω ( t − rc ) = π is shown in Fig. 1. This plot lookssimilar if time is selected such that ω ( t − rc ) = 0 or for anyother time value. It can be observed that apart of tworegions ωd c ≈ θ ≈ V − A r c is substantially dif-ferent than null. We note that applying the infinitesimalelectric dipole assumptions ( d (cid:28) λ (cid:28) r ) or Lorenz gaugecondition lead to V − A r c ≈
0, i.e. lead to disappearanceof E r . Similar to our result, it was shown in [5] that theidentity V = cA x , where A x is the component of the vec-tor potential along the direction of propagation of planewave, is responsible for disappearance of the longitudi-nal electric field. The author also writes that the Fouriertransform, when applied to the Lorenz gauge condition,results in the identity V = cA x .Finally, we can also show that the far-field transverseelectric field is caused primarily by the vector potential,while the scalar potential does not contribute to it. Thecomponent of the gradient of the scalar potential in thespherical coordinate system along transverse direction is ∇ V = ∂V∂θ e − j ( kr − ωt ) r ˆ θ, (17)where it can be observed that this expression decreasesfaster than r . As a result, the far-field transverse elec-tric field arises solely due to contribution of the vectorpotential E θ ˆ θ = − ∂A θ ∂t ˆ θ = jωA θ e − j ( kr − ωt ) r ˆ θ. (18)We can summarize by stating that the far-field transverseelectric field is caused by current sources only (vector po-tential), while the long-range longitudinal electric fieldis caused by contribution of current and charge sources(vector and scalar potentials), respectively. Similar con-clusion applies for ˆ φ direction. For example, for smallcircular current loop antenna, transverse electric field isalong ˆ φ , not ˆ θ . FIG. 1. Visualization of V − A r c as a function of θ and η = ωd c .This plot is generated for ω ( t − rc ) = π . III. DISCUSSION
Even though the presented derivations are simple, theirimplications are far-reaching. Three main points can behighlighted with respect to obtained results. • Firstly, in many textbooks it is claimed that there isso-called “gauge freedom” or gauge invariance [1, 6–9]. In other words, we can switch from one gaugeinto another and that would lead to the same elec-tric and magnetic fields. Our derivations challengethis viewpoint. If long-range longitudinal electricfield exist in all gauges except the Lorenz gauge,then gauge freedom is no longer valid. It meansthat if derivations are performed in the Lorenzgauge, then the obtained results would be withoutlongitudinal electric field. However, if derivationsare done in the Coulomb or Weyl gauges, then lon-gitudinal electric fields would appear. • The second point is the consequence of the first.Often it is claimed that the vector and scalar po-tentials are not real physical quantities, but are uti-lized as mathematical tools [9]. However, thereare recent theoretical studies which demonstratethat the vector and scalar potentials carry moreinformation than the electric and magnetic fields[10, 11]. Also there are papers where the vectorand scalar potential waves were experimentally de-tected [12–16]. All these studies imply that thevector and scalar potentials are real physical fields.That would also explain why “gauge freedom” isnot valid. The following example can be given withrespect to confusion coming from the vector andscalar potentials and their gauge freedom. In clas-sical mechanics, acceleration of any given object isabsolute for all inertial reference frames, while itsvelocity is relative. However, the relative nature ofvelocity does not imply that it is not real physi-cal quantity. Velocity has different magnitudes indifferent inertial frames of reference, but at eachinertial frame it is real physical quantity. Radiusvector of an object is even more relative quantitythan velocity, because radius-vector not only de-pends on the inertial frame of reference, but fora given inertial frame, it also depends on the ori-gin of the frame. If for a given inertial frame, itsorigin is changed, then the position vector of an ob-ject also changes. However, again relative nature ofposition vector does not imply that it is just math-ematical tool used to solve mechanics problems. Iflogic of gauge freedom is applied in classical me-chanics, then we would conclude that accelerationis real physical quantity, because it manifests itselfin Newton’s laws, while velocity and position aresimple mathematical tools. The difference betweenthe vector potential in electrodynamics and veloc-ity, position vectors in mechanics is that the latterquantities can be measured, while there are yet notools and methods which would permit measure-ment of the former physical quantity. • Thirdly, contrary to established opinion, the ob-tained results show that far-field longitudinal elec-tric wave might be real. By using the Lorenz gaugecondition unintentionally, in some textbooks au-thors demonstrate that longitudinal electric wavesare negligible. For instance, Lorenz gauge conditionwas utilized to derive radiation of electric dipole in[6] [Sec. 14-3], [17] [Sec. 14.1], [18] [Sec. 16-8], [2][Sec. 3.6], [3] [Sec. 1.15], [19] [Sec. 2.2]. In light ofthe presented derivations, their conclusions mightneed to be revised. One possible reason which ex-plains why (5) is more frequently used in textbooksthan (2) is that evaluation of the vector potential issufficient in the former, while both scalar and vec-tor potentials are necessary in the latter case. Asa result, when deriving radiation fields of electricdipoles, magnetic dipoles or antennas, it might bemore tempting to use (5). However, shorter pathmight mislead and give only part of the answer, notthe full answer. [1] D. J. Griffiths,
Introduction to Electrodynamics (PrenticeHall, USA, 1999).[2] C. A. Balanis,
Antenna Theory, Analysis and Design , 3rded. (John Wiley and Sons, USA, 2005).[3] R. W. P. King and C. W. Harrison,
Antennas and Waves:a modern approach (The MIT Press, USA, 1969).[4] A. Zhakatayev and L. Tlebaldiyeva, ArXiv e-prints(2018), arXiv:1806.05060 [physics.class-ph].[5] G. Rousseaux, “On the physical meaning of the gaugeconditions of classical electromagnetism: the hydrody-namics analogue viewpoint,” (2003).[6] W. K. H. Panofsky and M. Phillips,
Classical Electric-ity and Magnetism , 2nd ed. (Addison-Wesley PublishingCompany, Inc., USA, 1962).[7] W. R. Smythe,
Static and Dynamics Electricity , 3rd ed.(Taylor & Francis, USA, 1989).[8] J. D. Jackson,
Classical Electrodynamics , 3rd ed. (JohnWiley and Sons, Inc., USA, 1999).[9] J. Franklin,
Classical Electromagnetism (Pearson Addi-son Wesley, USA, 2005).[10] H. Reiss, Journal of Modern Optics , 1371 (2012), pMID: 23105173.[11] H. R. Reiss, Journal of Physics B: Atomic, Molecular andOptical Physics , 075003 (2017).[12] C. Monstein and J. P. Wesley, EPL (Europhysics Letters) , 514 (2002).[13] R. K. Zimmerman, Modern Physics Letters B , 649(2011).[14] R. K. Zimmerman, Journal of Applied Physics ,044907 (2013).[15] M. Daibo, S. Oshima, Y. Sasaki, and K. Sugiyama, IEEETransactions on Magnetics , 1 (2015).[16] M. Daibo, S. Oshima, Y. Sasaki, and K. Sugiyama, IEEETrans. on Applied Superconductivity , 1 (2016).[17] H. C. Ohanian, Classical Electrodynamics (Allyn and Ba-con, Inc, USA, 1988).[18] J. R. Reitz and F. J. Milford,
Foundations of Electro-magnetic Theory (Addison-Wesley Publishing Company,Inc., USA, 1960).[19] W. L. Stutzman and G. A. Thiele,