On the possibility of a relativistic correction to the E and B fields around a current-carrying wire
OOn the possibility of a relativistic correction to the E and B fields around acurrent-carrying wire
Ron Folman ∗ Department of Physics, Ben-Gurion University of the Negev, Be’er Sheva 84105, Israel (Dated: November 3, 2018)It is well known that electric and magnetic fields may change when they are observed fromdifferent frames of reference. For example, the motion of a charged probe particle moving parallel toa current-carrying wire would be described by utilizing different electric or magnetic fields, dependingon from which frame of reference the system is observed and described. To describe the situation inall frames by utilizing the theory of relativity, one has to first describe the situation in one particularframe, and this choice in the case of a current-carrying wire is the topic of this paper. Specifically,I consider the question of in which frame the current carrying wire is neutral. The importance ofrelaxation processes is emphasized. As an example, I examine a specific alternative to the standardchoice, and consider its theoretical and experimental validity. An outcome of alternative approachesis that in the rest frame of a wire, running a current introduces also an electric field by giving riseto a minute charge. Present day experimental sensitivities, specifically those of cold ions, may beable to differentiate between the observable signatures predicted by the different approaches.
PACS numbers: 37.10.Gh, 32.70.Cs, 05.40.-a, 67.85.-d
The theory of electromagnetism [1] and the special the-ory of relativity [2] have been confirmed in an abundanceof experiments. In the following I describe an effect thatis a result of these two theories and which may have beenleft unnoticed in the experiments performed due to itsrelatively weak nature.The interplay between electric and magnetic fieldswhen one moves from one frame of reference to anotherstands at the base of our understanding of how the theoryof relativity affects these fields [3]. The special theory ofrelativity clearly explains how these fields are observedas changing as a function of the frame of observation,but in order to know what the fields are, the fields in oneparticular frame must first be determined by other the-oretical or experimental means. It is the description ofthis particular frame, usually chosen to be the rest frame,that is at the core of this paper.The dependence of an elementary charge (e.g. thatof an electron or a proton) on its velocity would havebeen observed in particle accelerators, in the neutralityof atoms or even when heating a metal, and has been in-vestigated by numerous methods ([3] and refs. therein).Specifically, experiments intended to measure a secondorder change of the charge with its velocity, seem to haveobserved no effect [4, 5]. It is therefore of common be-lief that at least in low energies, such a dependence hasbeen ruled out. However, it seems that the dependenceof charge density on the mean velocity (so-called drift ve-locity) in the rest frame of a wire has not been examinedthoroughly and therefore cannot be ruled out. In thispaper I explain why such a fundamental effect could beconsidered as possible and calculate the expected signal.(following the above, we shall assume the invariance ofthe elementary charge with respect to its velocity, mean-ing that if the total number of particles is conserved then the total charge is conserved.)More specifically, the E and B fields are a result of thecharge density ρ and the current density j . It is thesephysical observables that are analyzed henceforth. Asis well known, the 4-vector ( ρc, j ), c being the speed oflight, transforms according to the Lorentz transforma-tions when the observer moves from one frame to theother. However, as noted, these transformations mayonly tell us what the charge and the current are in aspecific frame if we have previously determined them inanother frame. It is common practice to assume that inthe rest frame of a wire (which we define here as the restframe of the nuclei or the protons that form a wire) acurrent does not give rise to a charge. This comes aboutfrom the assumption that the electrons behave as freeparticles and therefore accelerating them to some finitevelocity does not change the particle-particle distances.If indeed the effect we consider in this paper is real, thelatter assumption is not correct.The question of the charging of a current carrying wirein its rest frame, may be phrased in the following way:Albert Einstein, in his 1905 paper, wrote [6]: ”At v=V,all moving objects - viewed from the ”stationary” system- shrink into plane figures” (where V is the speed of light).It is therefore widely believed (although I am not sure ifthis has been verified) that in accelerators, when cloudsof particles with strong enough restoring forces, e.g. anensemble of nucleons in a nucleus, are accelerated to somefinite speed, the cloud length is Lorentz contracted thusproducing a pancake shaped cloud in the lab frame of theaccelerator [7]. In the case of a nucleus, this comes aboutfrom the fact that the restoring forces keep its sphericalshape in its rest frame. If such a pancake indeed exists,the question can then be asked as to whether such acontraction of an ”electron cloud” may occur in a current a r X i v : . [ phy s i c s . c l a ss - ph ] J un carrying wire thereby producing a net charge in the wirerest frame.Following this work [8], I became aware of numerousprevious theoretical presentations of similar ideas and thecriticism they received (see Refs. [9, 10] and referencestherein). In Appendix I we give an example of such atheory and the criticism it received. Contrary to someof the previous works (e.g. [11]), the analysis describedhere will adhere to standard electrodynamics. Also con-trary to some previous works which state, to my mindwithout sufficient proof, that the wire in its rest frameis indeed charged (e.g. [17] and more recently [18]), it isclaimed in this work that not enough is known to arriveat this conclusion. At the same time, it seems there is notheoretical or experimental proof in favor of the standardapproach which holds that the wire is neutral in its restframe.This paper goes beyond the previous presentations asit emphasizes the importance of internal dynamics, givingrise to restoring (relaxation) processes (as in the pancakeexample above), and the fact that only a three dimen-sional many-body covariant analysis of the system canbe expected to give a theoretical resolution regarding atwhich frame the equilibrium is at the ρ net = 0 point.If this happens in several frames, a situation not com-patible with relativity, the question most relevant maybe in which frame the restoring dynamics are faster. Inthis context we analyze in detail Maxwell’s equations andOhm’s law. This work also presents a new analysis ofthe experimental sensitivity required to observe the hy-pothesized effect and analyzes the feasibility of severalnovel experimental methods to make such an observa-tion. The analysis indicates that cold ions may serve asa probe which is sensitive enough to differentiate betweenthe different possibilities.Let us begin directly with pointing out briefly onecounter example to the standard approach, followingwhich it is discussed in a detailed and consistent man-ner. If one assumes (as is commonly done) a zero netcharge density for a current-carrying wire in its rest frame S ( ρ + = | ρ − | = ρ where ρ is the free electron den-sity per unit length in the material when no current isapplied), one must, due to Lorentz transformations, as-sume a charge density of ( γ − /γ ) ρ in frame S (cid:48) movingat the drift velocity of the electrons in frame S [where γ = (1 − v /c ) − ]. Namely, the wire, as observed in S (cid:48) ,becomes charged. One intuitive explanation for this ap-parent charging is that somehow the net charge densityis dependent on the velocity of the charges in S (cid:48) so that ρ (cid:48) + (cid:54) = | ρ (cid:48)− | . This cannot be the right explanation simplybecause the same velocity difference in S does not giverise to a similar difference of densities in frame S . As de-scribed below, while the common choice [ ρ + = | ρ − | in S and ( γ − /γ ) ρ in S (cid:48) ] is consistent with the special theoryof relativity, it may not be the only plausible applicationof the theory. For example, one could have applied the theory and the arguments in a symmetric way in S and S (cid:48) . As in S the electrons are moving and the positivecharges are stationary, one could perhaps hypothesize tohave in S a net charge of ρ net = ρ (1 − γ ) while in S (cid:48) one would expect ρ (cid:48) net = ρ ( γ − γ = (1 + v /c ), the latter hypothesis gives rise to asecond order effect in the electron velocity in the restframe of the wire.The difference between the two choices may perhaps bemade more apparent by the following ”exposition”: wesimulate a normal conductor with two infinite solid rods,one made of positive ions and one made of negative ions.As we are not interested in absolute charge but only incharge density (per unit length), the boundary conditionsare of no importance. While the positive rod is at restin S , a current in S is realized by moving the negativerod with a velocity equal to the electron drift velocity ina normal conductor. If we accept the standard choice for S , an observer in S would claim to see no charge. Hisfriend is an observer in S ∗ , a frame in which both rodshave the same velocity (in opposite directions). As in S there is a current while there is no charge, Lorentz trans-formations do not allow for any other frame to observea charge of zero. This is well known to the observer in S and thus he communicates to his friend that he mustbe observing a non-zero charge, giving rise to a non-zeroelectric field. The friend, however, sees a completely sym-metrical world. He observes two infinite rods, built ex-actly the same way with the exception of an oppositecharge, going in opposite directions with equal velocity.As he believes the sign of the velocity vector should haveno effect on how charge changes with velocity, he con-cludes that this symmetrical world must give rise to acharge of zero, in contradiction to what his friend com-municated to him, and obviously in contradiction to thetheory of relativity. Of course, if the non standard choice[i.e. ρ net = ρ (1 − γ )] was observed in S , Lorentz trans-formations would have predicted a zero charge density inthe symmetrical world observed in S ∗ .Let us emphasize that the following detailed analysisof the situation in which the S ∗ frame noted above is theneutral frame, is given here as a mere possible counterexample to the standard approach. As in the real lifesituation of a current carrying wire the S ∗ frame is clearlynot symmetric (masses and interactions are different), itmay very well be that another frame eventually presentsitself as the unique frame in which ρ net = 0.In his famous lectures on physics Feynman explains indetail an example which concerns the electric and mag-netic fields produced by a current-carrying wire, as ob-served from different frames of reference [19]. FollowingFeynman’s formalism, I examine the above alternativeapproach to the problem at hand. If one is not able todetermine theoretically a single preferred option, thenone will have to resort to experiment.This paper is structured as follows: First I put for-ward in simple terms the difference between the two ap-proaches regarding their underlying arguments. I thendescribe in more detail the standard approach followedby the alternative approach. I conclude with the theoret-ical and experimental consequences due to the differencebetween the two approaches.Let me first describe the system we are addressing. Aspresented in Fig. 1, a current-carrying wire is placedin the lab frame ( S ). Due to the current, the electronshave some mean velocity v e . A positively charged probeparticle with charge q = +1 moves parallel to the wireaxis with velocity v p and has the same velocity as theelectrons so that v p = v e . It is well known that due toits charge and velocity in the magnetic field of the wire,the particle experiences a magnetic force, the Lorentzforce ( F L ), which is equal in this case to qv p B and di-rected away from the wire. The question addressed byFeynman is what would be the forces experienced by theprobe particle in its own frame of reference ( S (cid:48) ), the so-called particle rest frame, namely, in the frame where theprobe particle’s velocity is zero. While it is clear that themagnitude of the above magnetic force is zero in S (cid:48) , thequestion is if any other forces are created? Specifically,concerning forces that are perpendicular to v p , one wouldrequire, in order to be consistent with the theory of rel-ativity, that the sum of forces in S (cid:48) ( F (cid:48) ) would be largerthan the sum of forces in S ( F ) by the relativistic Lorentzfactor γ so that F (cid:48) = γF . This is so because the forceis the derivative of the momentum in time, and as we FIG. 1: The system of the wire and probe particle. The wireis stationary in the lab frame S and the particle is stationaryin its rest frame S’. The electric ( F C ) and magnetic ( F L )forces in both frames are presented. Note that in S’ there isno magnetic force as the particle does not move. Contrary tothe standard approach, in the alternative approach presentedin this work there is a Coulomb force also in S. do not expect the momentum in the direction transverseto the relative velocity of the frames (the latter being v p ) to change when moving from frame to frame, we ex-pect F (cid:48) ∆ t (cid:48) = F ∆ t . As the theory of relativity dictates∆ t = γ ∆ t (cid:48) we therefore require F (cid:48) = γF . Hence, al-though the magnetic force in S (cid:48) is zero, the total force in S (cid:48) cannot be zero.In his usual brilliant manner, Feynman simply explainsthat as the wire becomes shorter in frame S (cid:48) , the densityof the positive charges in the wire grows and this cre-ates an electric field which gives rise to a Coulomb force( F (cid:48) C ) which acts on the probe particle in S (cid:48) . However, togive rise to an electric field Feynman has to differentiatebetween the relativistic effect due to the frame changeas it acts on the positive and the negative charges. Noquantified physical argument is put forward to justify thespecific asymmetry of the ( γ − /γ ) factor leading to anon-zero charge density in S (cid:48) . In fact this factor is sim-ply a Lorentz transformation of the assumption of neu-trality in the rest frame of the positive charges. As willbe explained in the following, the latter assumption isequivalent to the assumption that no restoring forces areat work on the electrons such that the electrons’ inter-particle distance in some frame other than S is restoredto their original distance in the wire frame before the cur-rent was turned on. Put in simple words this means thatthe electrons behave as free particles. This may be called”The Standard Assumption”. As Feynman’s treatmentfollowing this ”assumption” is completely consistent withthe theory of relativity, he indeed finds that as required F (cid:48) = γF .In the alternative approach presented here we assumethat both the positive and negative charges have thesame restoring forces, so that they maintain their naturalinter-particle distances in their rest frame. Following theabove ”pancake” example, in both frames the density isincreased only for the moving particles due to Lorentzcontraction. This imposed symmetry may be called ”AnAlternative Assumption”. This approach also leads to F (cid:48) = γF and hence it is also consistent with the theoryof relativity. However, it leads to the charging of thecurrent-carrying wire in its rest frame, a surprising re-sult indeed. Thus, while both approaches are consistentwith relativity, they start with very different assumptionsand arrive at very different descriptions concerning thephysical situation in frame S .It seems likely to this author that both these assump-tions present the extreme edges of a wide spectrum ofpossibilities. What is common to all models occupyingthis spectrum is that all, but the one in the standardedge, predict the charging of the wire in its rest frame,in varying magnitudes.Let me now briefly describe the standard approach (inmy own abbreviated way). A detailed account may befound in Ref. [19]. First, note that the magnetic force F L experienced by the probe particle in S is F L = qv p B = qv p J πrc (cid:15) = qv p ρ v e πrc (cid:15) = qρ v πrc (cid:15) , (1)where r is the distance from the wire, ρ is the free elec-tron density (per unit length) in the material when nocurrent is applied, and I have taken (following Feynman’sexample) v = v e = v p .Next, Feynman calculates the Coulomb force F (cid:48) C in S (cid:48) by integrating the net charge along the wire axis and itselectric field at the position of the particle. Note thatFeynman takes the positive charge density to be ρ (cid:48) + = γρ and the negative charge density to be ρ (cid:48)− = − ρ /γ so that the net charge density is ( γ − /γ ) ρ = γ (1 − /γ ) ρ = γβ ρ , where β = v/c . The Coulomb forcecomes out to be F (cid:48) = F (cid:48) C = γβ ρ (cid:90) q π(cid:15) ( r + x ) r (cid:112) ( r + x ) dx = γβ ρ q π(cid:15) r = γF L = γF, (2)where, according to the standard approach, F and F (cid:48) arethe total forces in S and S (cid:48) , respectively.Let me now describe in detail the alternative approach.As noted, the assumption regarding the charge neutralityof the wire in S is replaced with a symmetry assumptionrequiring that ρ (cid:48) = − ρ , where ρ is the net charge den-sity (noted previously as ρ net ). For simplicity, we shallnot write the relevant parameters in units of ρ as inthe standard approach but rather in units of ρ ∗∗ = γρ ,where γ = (1 − v e /c ) − .To see exactly what the charge density should be inthe two frames, let us remind ourselves of the four vectortransformations. The vector ( ρc, j x ) transforms underthe Lorentz transformation matrix γ (cid:18) − β − β (cid:19) . We therefore see that ρ (cid:48) c = γρc − γβj . As we are arediscussing ρ and j in units of ρ ∗∗ (in which j is simply v e ),and as j is in the positive direction while the boost β from S to S (cid:48) is in the negative direction, we find ρ (cid:48) = γρ + γβ .Imposing symmetry, namely that ρ (cid:48) = − ρ , we find that ρ = (1 /γ − ρ ∗∗ and ρ (cid:48) = (1 − /γ ) ρ ∗∗ . Consequently,the net charge in S (cid:48) is now ( γ − ρ contrary to thestandard ( γ − /γ ) ρ . That the positive charge density in S (cid:48) is ρ (cid:48) + = γρ and the negative charge density is ρ (cid:48)− = ρ is a natural outcome under the assumption of symmetry,as the positive charges are moving and the electrons areat rest. In the same way, in S we now have a net chargeof (1 − γ ) ρ as the electrons are moving and the positive charges are at rest. Thus, our conclusion concerning thestate of the wire in S must be different from that ofstandard approach that the wire in S is not charged. Tomake sure that also the alternative approach is consistentwith the theory of relativity, let us now calculate theforces and see if we again find F (cid:48) = γF as required.In S we now expect to find both a Coulomb force anda magnetic force as depicted in Fig. 1. As the probeparticle is positively charged and as the wire is negativelycharged, and as the probe particle is moving in the samedirection as the wire electrons, the electric and magneticforces will be in directions opposite to each other. Notealso that in order to be consistent, the ρ appearing inthe magnetic force in S must also be boosted by γ as itis due to the electrons. The total force in S will thus be F = F L + F C = γρ β q π(cid:15) r +(1 − γ ) ρ (cid:90) q π(cid:15) ( r + x ) r (cid:112) ( r + x ) dx =( γβ + 1 − γ ) × qρ π(cid:15) r . (3)In frame S (cid:48) we find F (cid:48) = F (cid:48) L + F (cid:48) C = 0 + ( γ − × qρ π(cid:15) r , (4)and we therefore find F (cid:48) F = γ − γβ + 1 − γ = γ (5)as required.Let us now consider the theoretical consequences. Oneof the most important issues to clarify is how both ap-proaches deal with charge conservation, which we assumeto be correct. For example, as the alternative approachclaims that in S a current induces a charge density, theconclusion must be that for an infinite wire, an infinitecharge must be deposited. This is not very appealingindeed. However, this is also what happens in S (cid:48) in thestandard approach.As it is not very clear what charge conservation meansfor an infinitely long wire, and as it is not clear what anexperiment verifying any result concerning an infinitelylong wire should look like, we shall consider a current ina conducting loop of finite dimensions.Let us thus examine a loop in the shape of a squareof unit length, so that in essence there are two wires ofthe form noted above in the ˆ x direction (see Fig. 2), andbetween them, at their ends, vertical connections [20].It is clear that the total charge of the two horizontalwires A and B is zero in S for the standard approachand negative, 2(1 − γ ), for the alternative approach. Weof course expect that in S (cid:48) both approaches would leadto the same charge the loop had in S , as is verified inappendix II. This is of course to be expected as in rel-ativity the number of negative and positive charges isframe independent.But what about the situation in the alternative ap-proach where the mere fact that we ran a current in S does not adhere to charge conservation? This is perhapsthe most far reaching implication of the alternative ap-proach.In most experimental scenarios a current source (e.g.battery) must be embedded into the loop. An ad-hoc as-sumption may then be that the current source (acting asa charge reservoir) balances the hypothesized charging bybeing positively charged. Indeed, the previous analysis ofcharge conservation in the loop when changing betweendifferent frames, indicates, that the balance charge onthe current source does not need to change from frameto frame. This is the same in both approaches.However, in this work, as we aim at a feasible exper-imental configuration, we will not consider a loop witha current source, as this is known to give rise to a firstorder effect in the electron velocity [9, 21]. The lattercharging is a much stronger effect and would mask thesignal we are searching for [22].When there is no current source (e.g. one introduces acurrent by a varying magnetic flux), and the loop chargemust be preserved in S when inducing a current, twoprocesses may take place. First, as by moving from aninfinite wire to a finite loop without a charge reservoirwe have changed the boundary conditions of the prob-lem, it may be that the restoring forces are now suchthat the second order effect vanishes in the rest frame.Second, the current density may not be homogeneousthroughout the cross-section of the wire such that someparts of the cross-section are negatively charged (wherethe current runs) and some positively charged, so that FIG. 2: The current loop composed of two wires (presented)and vertical contacts at their ends (not presented). We alsopresent the two frames of reference: S is the lab frame inwhich the wire is stationary and S (cid:48) moves at the velocity ofthe electrons in wire A. integrating over the cross section of the wire retains thecharge neutrality. For example, if we examine the knownfact that superconducting currents run along the wireedges, the expected increase in charge density where thecurrent runs would mean that the center of the wire be-comes positively charged while the edges become nega-tively charged. One should note that such a scenario ofcourse creates electric fields which aim to counter-act theinhomogeneous charge distribution, but at least in thecase of a superconductor, these edge currents are stable.Also in a regular conductor, the well known pinch ef-fect, in which parallel currents attract each other therebycausing the electron density to be larger in the centerof the wire, is eventually balanced by electric fields tocreate a stable non-homogeneous current density. In theexperimental section of this paper, we will focus on super-conductor loops due to the high electron velocity. Highvelocities may also be found in other conductors such asGraphene [23].Let us now discuss the issue of internal dynamics.First, let us briefly remind ourselves of the importanceof such internal dynamics in relativity. John Bell, per-haps best remembered for his brilliant work known as”Bell’s inequalities”, has clearly stated that if two par-ticles (or space ships in his famous paradox [24]) are atrest in some system S with a distance L between them,and if in that system they are then accelerated with thesame force and for the same time, the distance betweenthem remains a constant. This must be so if relativity isto be consistent with the known laws of motion, namely, x ( t ) = x (0) + (cid:82) v ( t ) dt . So why are physicists, from theheavy ion collision community [7] or from other commu-nities (e.g. [25]), confident that they should see Lorentzcontraction in S ? The answer is also given by Bell’s para-dox. If the above two particles retain their distance fromone another as L , that distance in their new rest frame S (cid:48) must be γL as relativity demands a factor γ betweenlength measurements done in two different frames. So infact, if these two particles would represent the edges ofsome system (such as a solid rod) this system would feelstretched in its rest frame, thus giving rise to internaldynamics such as relaxation or restoring forces. If theserestoring forces shrink the system back to its equilibriumlength L in the new rest frame S (cid:48) , then, the system mustalso shrink in S . A system restored to a sphere in S (cid:48) (asin the heavy ion collision example) would thus look likea pancake in S .Applying the above insight to the current carryingwire, it seems the solution to the debate would be foundif one can decide if the electrons in the ion lattice arecloser to a system of isolated (non-interacting) particles,in which case their distance from one another is increasedin other frames (e.g. to γL in S (cid:48) ), or they are closer to aninteracting system with strong restoring forces in someother frame, in which case we would expect their inter-particle distance in S to shrink (e.g. to L/γ if their dis-tance in S (cid:48) is restored to L ) - once the current is turnedon. How then can we try and describe the internal dy-namics inside the wire, which we shall name ”longitu-dinal relaxation”? One such description of the internaldynamics is given by Ohm’s law, the latter being our nextstation.So far, we have considered a symmetric physical sys-tem of positively and negatively charged particles (forfinal remarks regarding the symmetric system see [26]).This is perhaps the situation in a plasma of electrons andpositrons (as well as a system of electrons and holes in asemiconductor), or perhaps, as noted in the introduction,in a wire made of two counter propagating solid rods, onemade of negative ions and one of positive ions. However,for the case of a normal current carrying wire, one needsto take into account the possible asymmetry caused bythe fact that the positive particles are much heavier andthey are held together by strong bonds which form astringent lattice [27]. One therefore needs to account forinternal dynamics in the presence of the above asymme-try. Perhaps the most adequate formalism available forthe movement of the electrons within a wire is Ohm’slaw. Let us therefore analyze Ohm’s law in the contextof the ideas presented in this paper. Could it be that thesimple form of Ohm’s law, (cid:126)j = σ (cid:126)E , is inconsistent withthe ρ (cid:54) = 0 suggested by the alternative approach?The covariant form of Ohm’s law is [28]: j α = σc F αβ u β + 1 c ( u β j β ) u α (6)where j α is the 4-current, u α = γ ( c, (cid:126)u ) is the 4-velocityof the medium and F αβ is the field strength tensor.This may also be written as (recalling that j = j and j i = − j i for i = 1 − g αβ j β = σF αβ β β + β α β β j β (7)where g αβ is the metric tensor and β α = u α /c = γc ( c, (cid:126)u ).In Appendix III we show that this reduces for (cid:126)u = 0 tothe known form of (cid:126)j = σ (cid:126)E . We also show in the appendixthat if u (cid:28) c , one finds to first order the current density (cid:126)j = ρ(cid:126)β + σ ( (cid:126)E + (cid:126)β × (cid:126)B ), which is the familiar form of theHall effect.However, most relevant for our discussion is the charge,namely, α = 0. One then easily finds that Eq. 7 reducesto ρ − γ ( γρ − γ(cid:126)u · (cid:126)j ) = σγ(cid:126)u · (cid:126)E (8)In the rest frame of the wire, where (cid:126)u = 0 and γ = 1,one finds the identity 0 = 0 for any charge density ρ .This means that Ohm’s law puts no constraints on whatthe charge density of the wire will be in the rest frame ofthe wire. Indeed, if we run a current through a charged capacitor plate (parallel to its plane) it will obey Ohm’slaw although the wire is charged. To conclude, it seems asif the alternative approach does not contradict Ohm’s lawin the simple way examined above, and more elaboratetests of the internal dynamics are required if one is totry and theoretically differentiate between the differentapproaches.Let us now also analyze the situation in the directiontransverse to the current, namely within the wire crosssection, and attempt to analyze the processes dominatingthe ”transverse relaxation”. To begin with, one shouldnote that together with the continuity equation Ohm’slaw (in the wire rest frame) gives ∇ ( σ E ) = ( ∇ σ ) · E + σ ∇ · E = ∇ · j = − ˙ ρ, (9)which together with Gauss’s law ∇ · E = ρ/(cid:15) gives:( ∇ σ ) · E + ρσ/(cid:15) = − ˙ ρ, (10)and as long as σ is uniform one finds˙ ρ = − ρσ/(cid:15) → ρ ( t ) = ρ (0) e − σt/(cid:15) , (11)where this calculation does not hold for the surface where ∇ σ (cid:54) = 0. This seems to indicate that any charge in thebulk of the wire would decay to zero.The above calculation should be met with some scepti-cism. First, the (cid:15) /σ time scale of roughly 10 − secondsseems to indicate that the above model is not complete.Further doubt should arise when one notes there is nolength scale in the equations. Such a time scale meansfor example that a wire with a radius smaller than theradius of a single atom would already require the elec-trons to move at a speed faster than that of light! Moreso, one would expect, for example, that there would beconsiderable difference in this time scale depending onwhether the wire radius is much larger or much smallerthan the electron mean free path of a few nano-meters.Furthermore, the idea that the spatial distribution of theextra charge would be infinitely thin (as is implied by thecharging of the surface) would give rise to a very highelectrostatic energy arising from the electron-electron re-pulsion. A more complete model would thus give a finitewidth distribution as a function of the charge. A similarresult should come when one takes into account the finitetemperature of the electrons from which one would as-sume that the thermal velocity would cause diffusion ofthe electrons against any tendency to concentrate themat a specific location.A more complete description should include the mag-netic force ev × B pushing the electrons towards the cen-ter of the wire (which may be termed a self inducedHall effect or a pinch as in plasma physics). One maycombine between Eqs. 9-11 and the above, by utilizing j = σ ( E + v × B ) and the continuity equation, assumingthe conductivity is homogeneous, and noting that − ˙ ρ = ∇ · j = ∇ · σ ( E + v × B ) = ρσ/(cid:15) + σ ∇ · v × B = ρσ/(cid:15) − σ v · ( ∇ × B ) . (12)This gives rise in a steady state ( ˙ ρ = 0) to ρ = (cid:15) v ·∇ × B . The conclusion of Eq. 11 is thus not necessarilyvalid.In appendix IV we analyze the role of the Lorentz forcein different frames. It is shown that if we assume thebulk of the wire in its rest frame has no internal trans-verse electric field, the Lorentz force in other frames istypically not zero. Hence, if the Lorentz force is consid-ered as the only force acting on the electrons and thusthe only force responsible for the steady state, a zerotransverse electric field in S does not lead to a transversesteady state in other frames. If in S there is a transverseelectric field which leads to a steady state by equalingthe above magnetic force, then some form of non zerotransverse charge density distribution in S must result.Whether integrating over such a transverse distributionshould give an overall neutrality - is another question.Let us emphasize that Lorentz transformations shouldexist for every point in the wire cross section. Hence, asthe transverse charge distribution changes with time inone frame, it should also change in time in other frames,and as the transverse distribution reaches a steady statein one frame its boosted distribution in another frame(boosting as usual parallel to the current or wire axis)should also present a steady state. Once this steadystate is reached, there is no transverse current and j = σ ( E + v × B ) = σ E (cid:107) . It remains to be seen if these trans-verse processes determine in some way what the allowedcharging in the rest frame is or what frame is the neutralframe.To conclude the theoretical analysis we have made, wemay say the following: as a current is made to run in awire, electrons are accelerated in the rest frame of thewire to their drift velocity. If they are to be consideredas an ensemble of free particles, kinematics demands thattheir particle-particle distances would not change and noextra net charge density would appear. This requires thecharging of the wire in other frames. If the electrons ina wire are subjected to significant longitudinal restoringforces, then the wire may be restored to neutrality fasterin some other frame, in which case, the wire in its restframe will have to be charged. Any solution would alsoneed to take into account relaxation forces in the trans-verse direction. Thus a full three dimensional many-bodycovariant analysis is required.Let us now examine the experimental implications withsome detail. Let us start with a simple straight copperwire, being part of a square loop circuit, assuming the extra charge is provided by the current source. Cop-per has a density of 8 . . . . In 1 mole ofcopper there are 6 . × atoms. Therefore in 1m ofcopper there are about 8 . × atoms. Copper has onefree electron per atom, so the electron density is equalto n = 8 . × electrons per m . Let us assume atypical laboratory situation in which we have a currentof I = 1 Ampere in a wire of 1mm diameter (i.e. a crosssection area of A = 7 . × − m ). The drift velocitycan therefore be calculated to be v e = InAq = 1(8 . × . × − . × − )= 9 . × − m/s (13)As β = v/c = 9 . × − / . × = 3 . × − ,we can now calculate the Lorentz factor γ to be equal toabout γ = 1 (cid:112) − β ≈ β ≈ . × − (14)In a 1m wire we expect to have 8 . × × . × − = 6 . × electrons, so that the excess chargepredicted by the alternative approach would be that ofapproximately 3 × − electrons per meter or ρ net = − × − C/m. This is perhaps the strongest expectedsignal, namely that a positively charged probe particleshould be attracted to a current-carrying wire, even whenthe particle is at rest relative to the wire.It is very hard to answer the question of whether ornot previous experiments should have observed such anattraction. Typically, charged particles in the vicinityof a current-carrying wire would be mostly affected bythe magnetic Lorentz force. Following Eq. 3 and theapproximation of γ in Eq. 14, one finds that the ratio R between the electric force and the magnetic force is R = F C F L = − v e c / ( v e v p c + v e v p c ) . (15)This ratio equals approximately half for small veloci-ties and when v e = v p and may therefore seem to pointto a significant and observable effect. However, usingthe value for v e calculated above and taking into account m p v p = K B T , one finds that even for a heavy probeparticle (e.g. an atomic mass of 100), the particle wouldhave to be at a temperature of 0 . R , is already at the 0 . | e | next to a normal conductorwould be very small.Such low velocities of probe particles, and indeed where v e = v p , may be found in a parallel current-carryingwire configuration (e.g. in the setup by which the Am-pere standard is determined). However, the overall wirecharge is again the very small quantity calculated aboveand the magnetic Lorentz force acting on all the mov-ing charges in the probe wire would again dominate. Letus calculate the forces involved explicitly. The magneticLorentz force between two 1m wires is µ π d I I which isfor 1A and 1m distance 2 × − N. On the other hand,the expected electric field E(r) is (Eq. 3): E ( r ) = (1 − γ ) × ρ π(cid:15) r ≈ ρ net π(cid:15) r (16)where ρ net = − β ρ . At 1m distance, the electric field isabout E = 9 × − V/m.If the second parallel wire has the same current andtherefore the same induced charge of ρ net , the electricforce between the wires will be on the order of 10 − N,much smaller than the magnetic Lorentz force which theexperiment is measuring. One may then wish to simplycharge a disconnected piece of wire as a probe to thehypothesized induced charge in the current-carrying wire.Namely, to achieve via a macroscopic system v p = 0.Let us assume this probe charge to be q p = 1C and seewhat forces we may expect. The expected electric forcewill be F = Eq p ≈ − N. On the other hand, the probecharge will induce a rather strong dipole in the current-carrying wire (as the internal field in the metallic wire,perpendicular to its longitudinal axis, is zero, we neglectthe polarization of the atoms). An upper limit on thedipole d per unit length may be estimated as the diameterof the wire (1mm) multiplied by the charge of the freeelectrons per unit length which we have estimated aboveto be 6 . × . × − ≈ C/m. As the forceis simply the field gradient times the dipole, we find theforce per unit length of the current-carrying wire to be(for a point like probe and without geometrical factors): F = 10 × − × π(cid:15) ddr q p r = 102 π(cid:15) r (17)which is in the closest area to the probe (i.e. distance of1m) of the order of F = 1 . × N, which is obviouslyhuge compared to the induced electric force we would liketo measure. In appendix V we show that a lower limitgives 7 × N.However, if we choose q p = e , the above lower limitvalue for the force reduces to about 10 − N while the pre-dicted electrical force is q p E = 1 . × − × × − ≈ − N, not very much smaller. This may enable a rel-ative measurement although the absolute values are ex-tremely small. As the electric field of the wire scales as r while the dipole force scales as r − r for the upperand lower limits respectively, reducing the distance willnot help.One may also wish to try and measure the interactionbetween the electric field induced by the hypothesizedcharging of the current-carrying wire and the polarizationit induces in atoms. In appendix VI, I explain why this isa difficult experiment to perform and consequently whyprevious experiments may have overlooked the effect. Inappendix VII, I explicitly calculate the magnitude of thissignal.The above shows that the effect is quite elusive and itstands to reason that experiments so far have overlookedit. I present in the following an initial feasibility study ofone idea that may enable the observation of the effect.For the sake of this example, let us assume that thedistribution of charges in a superconducting wire whichis part of a persistent current loop, follows the currentdensity distribution. Namely, that the excess in negativecharge is taken from nearby parts of the wire throughwhich current does not run, so that integrating over thecross section of the wire, charge neutrality is maintained.Such a charge distribution may be observable by coldions. Inhomogeneous charge distributions in supercon-ductors from other effects [31] may also be possible andshould of course be analyzed.As the mean velocity of electrons along the wire axisof a superconductor may be orders of magnitude higherthan the typical electron drift velocity in normal con-ductors (simply because the cross section in which thesuper current runs is very small), one would expect thatdue to the alternative approach, there would be an ex-cess of electrons in the edges of the wire. As noted, suchan excess would create an excess of positive charges inthe center of the wire. The exact current distributionin a superconductor and the calculation of the velocityis not a trivial matter [32] but let us make a simplifiedestimate. A superconductor may carry a current den-sity of 10 A/cm (or 10 − A/ µ m ), similar to that of anormal wire. Hence, a superconducting strip of 100 µ mwidth and 1 µ m thickness can carry 1A of current, similarto our example copper wire above. However, due to theconcentration of current in the edges, one may roughlyestimate that this 1A of current utilizes a wire cross sec-tion of about 1 µ m , six orders of magnitude less thanin the above copper wire. This determines v e and β tobe six orders of magnitude larger than previously. In-deed, the literature discusses velocities as high as 1km/s[33]. Following Eq. 16, while using the same free electrondensity as in copper with the reduced cross section, theelectric field also becomes six orders of magnitude larger.Hence, at 10 µ m distance from the wire, one would expectan electric field of about 1V/m. This field should reversesign when moving from the edge of the wire to its center.The effect of such a field should be observable if the forceof Eq it applies on the ion is similar or at least not muchsmaller in magnitude relative to the force applied on theion by the trapping potential. The latter, in the harmonicapproximation, is simply kx where k = mω and x maybe estimated as the ground state size (cid:112) (cid:126) /mω . Indeed,for a cold ion trap of 10MHz frequency, both forces are onthe order of 10 − N. Recently, a force of about 10 − Ninduced by an electric field of 1 . /d , where d is the ion-surface distance, taking the ion-surface distance to 100 − µ m would only reduce thesignal electric field by one to two orders of magnitude,while reducing the masking fields by considerably more;third, one may oscillate the super current in directionor amplitude to form an oscillating force, and this couldhelp purify the signal [36]. Finally, the ion should be in astate without a magnetic moment as the force due to themagnetic field induced by the 1A current, is significant.Noting the above considerations, it seems reasonable thatadequate experimental parameter values may be found soas to ensure the visibility of the hypothesized effect.To summarize, this paper did not attempt to give a fi-nal answer, but rather to lay down in a consistent manner(using known electrodynamics and charge conservation)the fundamental aspects of the problem, while introduc-ing several insights beyond previous works concerning re-laxation processes and experimental feasibility.As an example of a possible alternative to the stan-dard approach, I have presented a different approach asto how the physical situation concerning the electric andmagnetic fields in the stationary frame S of a current-carrying wire, may be determined. While typically onedetermines the state of the system to be neutral in S (by assuming electrons are a free ensemble and thereforetheir particle-particle distances do not change upon ac-celeration), and makes use of the theory of relativity tocalculate the state in other frames, one possible alter-native approach begins with a physical symmetry basedargument requiring that the restoring forces at work onboth positive and negative charges are equal, and uti-lizes relativistic transformations to arrive at a differentdescription of frame S .In the symmetry based argument leading to neutralityin the ”middle man” frame S ∗ , we modeled a current car-rying wire as made of two counter propagating solid rods,one made of negative ions and one of positive ions, andwhere current comes from their relative speeds. It seemsthere would be a consensus that the system is completelysymmetric and the charge density should be zero. Hence, if it is indeed found that the total charge density is zeroin S or some frame other than S ∗ (in which the two rodshave the same velocity), the theoretical analysis shouldbe able to explain quantitatively what the difference isbetween the latter rod model and the real-life system ofa current carrying wire in its rest frame and in the framewhere it is neutral.The main message of this work is three fold: First,it seems that both theoretical and experimental state-ments made so far, have not been able to sufficientlyprove in favor of one resolution or another. Second, itseems that the rules of relativity alone may not be suffi-cient to solve the debate. A theoretical attempt to decidewhich approach is favorable, should take into account in-ternal restoring (relaxation) processes, both transverseand longitudinal, in order to find out in which framethe equilibrium is at the ρ net = 0 point. If this hap-pens in several frames, a situation not compatible withthe known transformations between frames, the questionmost relevant may be in which frame the restoring dy-namics are faster. Only a full three dimensional many-body covariant analysis could perhaps decisively bringa theoretical resolution. The answer may very well besystem and preparation dependent and will thus haveto take into account specific boundary conditions. Thethird message is that an experimental feasibility studyindicates that cold ions may serve as a probe which issensitive enough to differentiate between different possi-bilities.It may eventually be found that theory does not favora specific approach [40], and choosing between them ex-perimentally will perhaps be a formidable, yet as we haveshown - possible, task. ACKNOWLEDGEMENTS
I am sincerely thankful to Daniel Rohrlich, YonathanJapha, Ferdinand Schmidt-Kaler, Michael Gedalin andValery Dikovsky for their critical review of this work,and for helpful discussions. Special thanks to Ashok K.Singal, who was kind enough to repeat his argumentsafter nearly 20 years. Finally, I deeply thank my dearfriend and colleague Carsten Henkel with whom I havehad many enlightening discussions. This work wassupported by the German-Israeli fund (GIF) as part ofa project dedicated to cold ions near charge and currentcarrying structures.
Appendix I: A proposal from the 1990s
Following this work [8], I became aware of a previouspresentation of a similar idea two decades ago [11] and thecriticism it received [12]. In this paper we therefore re-visit the idea and introduce several novelties: first, whilethe previous work was criticized as being ”unpalatable”[13] as it used theoretical and textual language beyond0what is common practice, this work analyzes the prob-lem within the established and accepted language andlaws of physics (e.g. conservation laws). For example,the previous work states that standard electrodynamicsis incorrect [14], while this work does not support sucha claim. Furthermore, the significant difference betweenthe previous formulation of the problem as well as theproposed solution, and the present one, results in differ-ent experimental predictions.Let us note in some detail five differences between thepresent and the previous work: • In [14] it is claimed that the standard electrody-namics theory is incorrect: ”We have proved thatin SET (standard electrodynamics) the law of in-variance of charge is not well founded”. The presentpaper makes no such claim. In this paper I claimSET is just as founded in both of the competingalternatives. Namely, I claim that there are severalLorentz invariant options to choose from. • In [14] new and perhaps debatable theoretical con-structions such as ”Lorentz invariant charges” aredefined, whereas in this paper only the standard el-ements of electrodynamics are used, and it is shownthat the total charge of a current loop is conservedbetween frames in both theories. In the criticismmade by [13] it is said: ”We find Ivezic’s modi-fied definitions of the charge invariance and of thecharge neutrality themselves to be unpalatable”.Indeed while the previous proposal aims at chargeneutrality for a wire in all frames, the present pro-posal makes no such claim. • In [15] it is claimed that there is no electric forcebetween two parallel current carrying wires in therest frame. In this paper it is claimed that this forceis not zero. Because Ref. [15] utilizes both Lorentzcontraction of the wire and Lorentz enhancement ofthe electron density at the same time, the two ef-fects cancel each other and the neutral wire cannotfeel the electric field of the other wire. I believe thatin [14, 15] there is confusion between the role of thetwo Lorentz effects. In the criticism made by Ref.[13] it is said: ”But in order to measure the totalcharge in a section, he simultaneously employs twodifferent stretches of the conductor length in anygiven reference frame, one for the ions and anotherfor the electron subsystem”. The same confusion isapparent in Eq. (3) of Ref. [15], where the mag-netic force between two parallel wires is calculated. • Due to the same confusion, when in Ref. [16] thesituation in a stationary superconducting loop isaddressed, it is said that: ”This causes the moving-electron subsystem to shrink to a smaller length inthe laboratory frame...the electric field caused by N moving electrons situated on a smaller, contracted,ring...”. On the contrary, in this paper, no Lorentzcontraction of the ring in which the electrons moveis hypothesized; only the contraction of the meandistance between electrons. What is assumed isthat as the current is concentrated at the edges ofthe superconductor, these edges will become nega-tive while the center will become positively charged.As in the previous item, this is another experimen-tal prediction that is different from that presentedin [14–16]. • Contrary to the previous work, this paper does notrequire that the basic law of charge conservationbe broken, as it does not claim that a bare loopbecomes charged when current is made to flow.It is therefore quite obvious that the formulation ofthe problem and the proposed solution presented in thispaper are very different than those presented in the early1990s.In addition, this paper adds several new consider-ations. First, the previous work did not consider theasymmetry of the system due to the fact that theprotons or positively charged centers that form thewire’s solid lattice are much heavier than the conductingelectrons. This work considers this through an analysisof Ohm’s law. Most importantly, this work emphasizesthe importance of longitudinal and transverse relaxationforces. Second, this work analyzes the feasibility of novelexperimental techniques in observing the hypothesizedeffect.
Appendix II: Charge conservation
Let us describe what happens in S (cid:48) in the standardapproach. The vector ( ρ net c, j x ) transforms under theLorentz transformation matrix we have presented earlierso that in the standard approach, as ρ net = 0 in S forany wire, one finds ρ (cid:48) net = − βj x γ/c . As j x for wiresA and B in S is equal and opposite, it is clear that inany other frame the net charges on the two wires wouldbe the same but with opposite sign, and so the totalcharge of zero is conserved in the standard approach.For completeness I also arrive at this conclusion by adirect calculation, presented below, where, as expected,the calculation reveals that the standard choice leads in S (cid:48) to an equal but opposite charge density in the twowires, so that the total charge in the complete loop ismaintained at zero.Similarly, the alternative approach also leads in S (cid:48) tothe same charge observed in S . This is also presentedbelow. The total charge density on both wires comes outto be 2( γ − γ ) and taking into account Lorentz contrac-tion, the total charge on the two wires is 2(1 − γ ) whichis equal to their charge in S . The vertical connectionsremain unchanged.1For a direct calculation, let us first remind ourselveswhy ρ (cid:48) = ργ (cid:48) /γ where ρ and ρ (cid:48) are the apparent chargedensities, and where γ and γ (cid:48) are the Lorentz factorsdue to the particles’ velocity relative to the two framesof reference. Let us take the Lorentz transformation ofthe electric field stating E (cid:48)⊥ = γ ∗ ( E ⊥ + v × B ⊥ ) [3] (inthe case where β is perpendicular to E ), where γ ∗ is dueto the velocity v = βc between the frames, and apply itto two oppositely charged parallel plates lying in the xzplane and moving with velocity v x = v = β c . Let usalso note that all Lorentz factors are due to movement inthe x direction. The above transformation is now E (cid:48) y = γ ∗ ( E y − vB z ) and as B z = ρv /c (cid:15) , the transformationtakes the form E (cid:48) y = ρ (cid:48) /(cid:15) = γ ∗ ρ/(cid:15) (1 − vv /c ). Hence ρ (cid:48) /ρ = γ ∗ (1 − vv /c ). As γ (cid:48) = γ γ ∗ (1 − ββ ) one findsthat ρ (cid:48) /ρ = γ (cid:48) /γ or ρ (cid:48) = ργ (cid:48) /γ .In the standard approach, the charge density of thefirst wire (A) in S (cid:48) is proportional to ( γ − /γ ). Thepositive charges in the second wire (B) on the other sideof the loop have exactly the same velocity relative to S (cid:48) asthose in wire A and so their charge density is proportionalto γ just as it is in wire A. In S (cid:48) , the electrons in wireB have a velocity of 2 v/ (1 + β ) relative to the electronsin wire A, and so one may describe the situation of theelectrons in wire B as having γ = 1 in their rest frameand γ (cid:48) = (1 + β ) / (1 − β ) in S (cid:48) . As ρ = − /γ , theircharge density in S (cid:48) should be proportional to − γ (cid:48) /γ .Hence the charge density in wire B on the other side ofour loop is simply proportional to γ − γ (cid:48) /γ = 1 /γ − γ which is just the negative of the charge density in wireA. Consequently, the total charge on the two wires in thestandard approach is zero also in S (cid:48) .In the alternative approach the charge density of wireA in S (cid:48) is γ − ρ ). Again, also the positivecharges on wire B would experience a γ factor just as itis in wire A. In S (cid:48) , the electrons in wire B have again avelocity of 2 v/ (1 + β ) relative to the electrons in wireA, and so again γ (cid:48) = (1 + β ) / (1 − β ) and their chargedensity should thus be proportional to − γ (cid:48) . Hence thecharge density in wire B is simply γ − γ (cid:48) . Adding thecharge density of the two wires one finds that the totalcharge density is 2( γ − γ ), and dividing by the Lorentzcontraction one finds that the total charge is the sameas it was in S . Appendix III: Covariant Ohm’s law
In this appendix we briefly examine the covariant formof Ohm’s law.Starting with, g αβ j β = σF αβ β β + β α β β j β (18)where g αβ is the metric tensor, j β is the 4-current, β α = u α /c = γc ( c, (cid:126)u ) is the 4-velocity of the medium and F αβ is the field strength tensor. Taking α (cid:54) = 0 one finds (cid:126)j − γ (cid:126)β ( γρ − γ (cid:126)β · (cid:126)j ) = σγ ( (cid:126)E + (cid:126)β × (cid:126)B ) (19)If (cid:126)u = 0 one finds (cid:126)j = σ (cid:126)E . If | (cid:126)u | (cid:28) c , andone takes the first order in γ ≈ β , one finds (cid:126)j tot = (cid:126)j e + (cid:126)j m = ρ(cid:126)β + σ ( (cid:126)E + (cid:126)β × (cid:126)B ), where j e and j m are the dominant terms in the so-called Galilean electric( E (cid:29) cB ) and magnetic ( cB (cid:29) E ) limits. Appendix IV: The Lorentz force in differentframes
Here, as an example, I calculate the Lorentz force onan electron inside a current carrying wire, in the direc-tion perpendicular to the current, if there is no transverseelectric field in the wire rest frame. The boost betweenframes is in the direction of the current. If we expect,in other frames, to reach a steady state in the trans-verse charge distribution, the transverse Lorentz force F ⊥ , must be zero (if we assume it is the only force actingon the electrons).Let us denote the wire axis as ˆ x so that the currentdensity is j ˆ x (the total current is I). The wire has cylin-drical symmetry and its radius is R . For simplicity (andhopefully without loss of generality) we look at an elec-tron situated on the ˆ y axis having a distance 0 < r < R from the center of the wire, so that the probe point is r ˆ y ( z = 0). This electron is moving with the current at adrift velocity. We will work in cgs units.We first note that in the wire rest frame the electricfields at the probe point are (we denote x, y, z as 1 , , E = ConstE = 0 E = 0 (20)where E drives the current and E , = 0 because thereis no charge inside the wire and due to Gauss’s law.Similarly, the magnetic fields are: B = 0 B = 0 B = µ π Ir/R (21)where from Ampere’s law (cid:82) (cid:126)B · (cid:126)dl = µ I c with I c /I = πr /πR . As (cid:126)I = I ˆ x , the velocity of our probe electron is (cid:126)v e = v e ( − ˆ x ) and we find that the Lorentz force ev e × B is away from the surface and towards the center of the wirei.e. in the direction of − ˆ y . This is to be expected as whatwe have calculated is analogous to the attraction betweentwo parallel currents. It is however an interesting resultin the fact that the Lorentz force works against the chargetendency to concentrate at the edges.Let us now find the electric and magnetic fields in otherframes. We denote V (or β = V /c ) as the boost velocity.2Lorentz transformations of the fields are: E (cid:48) = E = ConstE (cid:48) = γ ( E − βB ) = − γβ µ π Ir/R E (cid:48) = γ ( E + βB ) = 0 (22)and B (cid:48) = B = 0 B (cid:48) = γ ( B + βE ) = 0 B (cid:48) = γ ( B − βE ) = γ µ π Ir/R . (23)As the total Lorentz force is F = q ( E + c v × B ) wehave F ⊥ = e ( E (cid:48) + 1 c v (cid:48) e × B (cid:48) ) = − eµ π IrR γ ( v (cid:48) e c + β ) , (24)where v (cid:48) e is the electron velocity in the new frame.According to relativity v (cid:48) e = ( v e − V ) / (1 − v e V /c ) andso for the Lorentz force to be zero we need to demand − c v e − V − v e V /c = VcORv e = 0 . (25)This outcome means that if we have current and wewant a steady state in all frames, we must have transverseelectric fields in the wire rest frame.What happens when our assumption E = 0 is notmade? From symmetry arguments one may concludethat there is a radial electric field all around the cross-section of the wire. Then, if we introduce a closed sphereof surface A into the bulk of the wire and recall that E is homogeneous, we may use the integral form of Gauss’slaw (cid:15) (cid:82) E · dA = ρ to conclude that the region internalto the sphere is charged.Assuming E ⊥ (cid:54) = 0, the conclusion of a non uniformcharge density can be made quantitative if a steady stateexists in the rest frame, namely (up to factors of c ), E ⊥ = − v × B . Since ∇· E (cid:107) = 0 and v is constant, we may write: ρ = (cid:15) ∇ · E ⊥ = (cid:15) v · ( ∇ × B ) . (26)where the last equality comes from the fact that ρ = (cid:15) ∇ · E ⊥ = − (cid:15) ∇ · v × B = − (cid:15) B · ∇ × v + (cid:15) v · ∇ × B [29]. ∇ × v = 0 is termed an ”irrotational flow” (noturbulence) and may also come from the assumption that v is proportional to E ( j = σE ) and that ∇× E is the timederivative of the magnetic field which is zero in steadystate.As Ampere’s law implies ∇× B = µ J = µ ρ − v , where ρ − is the charge density of the free electrons in the steadystate, we have: ρ = ρ + + ρ − = (cid:15) µ ρ − v = ρ − v /c , (27) namely ρ − = − ρ + / (1 − v /c ) = − γ ρ + , which meansthat there is charging, or at least a non-uniform trans-verse charge density. Appendix V: A lower limit on the induceddipole of a wire
We have made a rough estimate for an upper limit ofthe dipole of a metallic wire. This estimate is perhaps toolarge as it takes the maximal possible value independentof the size of the inducing charge. Let us now derive alower limit by making use of the known induced dipolein a grounded metallic sphere. We simulate our 1mmdiameter wire by a line of spheres with radius a = 0 . x i . Weplace the charge on the z axis at z . According to Eq.2.6 in Jackson [37] the force between a probe charge q p and a single sphere is expected to be: | F | = q p π(cid:15) aq p r i r i (1 − a r i ) ] (28)where r i = (cid:112) x i + z is the distance between the probecharge and the center of the sphere, the second ratio is theimage charge, and the last ratio is simply the Coulombforce factor of one over distance square, where the dis-tance is calculated between the probe charge and the im-age charge in the sphere.If we omit terms of a/r i with a high power, as we as-sume a (cid:28) z , and we also position spheres symmetricallyalong the x axis so that only the vertical force componentcounts, we find the total force to be: F tot ≈ Σ i q p a π(cid:15) a r i z r i = Σ i q p π(cid:15) az r i ≈ (cid:90) ∞−∞ dx a q p π(cid:15) az r i . (29)We then find that F tot ≈ q p z π(cid:15) (cid:90) ∞−∞ dx ( x + z ) = q p π(cid:15) z (cid:90) ∞−∞ du (1 + u ) == q p (cid:15) z . (30)Putting in a probe charge of 1C at a distance of 1m,we find a force of 7 × N, only about two orders ofmagnitude smaller than our upper limit value. Notethat in this approximation the force is not dependent onthe diameter of the wire.
Appendix VI: The experimental challenge of in-duced polarization in the electric field of a wire
Atom chips are devices in which isolated ultra coldatoms (in vacuum) are typically trapped in magnetictraps, microns away from the surface of the chip.3Current-carrying wires on the surface of the chip pro-vide the trapping magnetic fields. The magnetic fieldsinteract with the magnetic moment of the neutral atoms(see our review [38]).In a recent experiment we have carried out with anatom chip, we have also induced an electric field whichinteracts with the induced polarizability of the atoms, sothat the neutral atoms interact with both fields simulta-neously [39].In general, aside from simple situations such as be-tween capacitor plates, electric fields are quite hard toaccurately engineer as they very much depend on wherethe ground is or in general, what are the potential sur-faces in the vicinity of the charge. In our atom chip workwhere we combined electric and magnetic fields to cre-ate a lattice of traps, we have concentrated thousands ofelectrons on our surface electrodes in the vicinity of theneutral atoms. The charging and relatively high elec-tric field were achieved by utilizing a capacitor config-uration on the surface of the chip. The typical param-eters of the setup we have used were 100 − − F coming from an electrodearea of 2 × µ m and an electrode distance of a fewtens of microns. The electric force is attractive and at-tempts to pull the atoms towards the surface while themagnetic force creates a barrier against such a crash.The magnetic force is simply F m = µ B × dB/dr , where µ B = 1 . × Hz/G is the Bohr magneton and thetypical magnetic gradient at an atom-surface distance of50 − µ m is 1 − − N. As the electric force was seen to changethe magnetic potentials but on the other hand was notable to crash the atoms to the chip surface, it is clearthat its magnitude was only about 10 − N or less.As noted, the charging was made possible due toa capacitor configuration. The capacitor also createdquite a strong field of about 10 V/m, which gave riseto a potential modulation of about 100 µ K in the atommagnetic trapping potential. Having in a similar config-uration (e.g. a current-carrying wire close to a groundpotential) an excess of 10 − electrons, as expected bythe alternative approach, would give rise to a field whichis smaller by 6 − − Appendix VII: Possible experiments involvinginduced polarization
Let us now explicitly calculate the electric force actingon a neutral atom next to a current-carrying wire (in SIunits). We use the previous calculations of the hypothe-sized field induced by a wire carrying 1A of current. Thefield we found in Eq. 16 was E ( r ) = 9 × −
12 1 r V/m.As the electric potential is just V ( r ) = − αE ( r ),where α is the atomic polarizability which is typically5 × − Cm /V and αE is the induced dipole, we ex- pect a force of: F = − α dE ( r ) dr = 81 × − . × − r (31)which, at an atom-distance of 1 µ m, should amount toabout 2 × − N, an extraordinarily small number. Thepotential modulation (in degrees Kelvin), mentioned inappendix VI and equal to ∆ T = − αE ( r ) /k B , where k B = 1 . × − J/K is the Boltzman factor, would alsobe much too small to be detectable, even with ultra coldatoms.Let us make the same calculation for the induced dipoleof a neutral metallic wire of unit length. In Eq. 17 wehave found that the induced dipole d of such a wire isbounded from above by d = 1Cm per unit length. Forthe purposes of this example, we use this value as ourestimate for the dipole strength. Using the same E(r)noted above we find the force on a piece of wire of unitlength to be: F = d dE ( r ) dr = 9 × − r (32)which, at a distance of say 1m (the Ampere standardsetup), amounts to about 10 − N, four orders of magni-tude smaller than the force the Ampere standard setupwas meant to measure. However, moving the wires to adistance of 1mm (again, current is running in only onewire), the force would now have a magnitude of 10 − N,and this may be measurable. However, while the up-per limit used here scales as q , the lower limit (as calcu-lated in appendix V) scales as q , thus giving rise for theexpected 10 − electrons to a difference of 10 − in themagnitude of the force. ∗ Electronic address: [email protected][1] James Clerk Maxwell, A dynamical theory of the electro-magnetic field, Philosophical Transactions of the RoyalSociety of London , 459512, (1865). (This article ac-companied a December 8, 1864 presentation by Maxwellto the Royal Society.)[2] Albert Einstein, Zur Elektrodynamik bewegter K¨orper,Annalen der Physik , 891 (1905). English translation:On the Electrodynamics of Moving Bodies, by GeorgeBarker Jeffery and Wilfrid Perrett (1923); Another En-glish translation On the Electrodynamics of Moving Bod-ies by Megh Nad Saha (1920).[3] John David Jackson, Classical Electrodynamics (3rd edi-tion), John Wiley and Sons Inc. (1999), Sections 11.9-11.10.[4] D. F. Bartlett and B. F. L. Ward,Is an electron’s chargeindependent of its velocity?, Phys. Rev. D , 3453(1977).[5] C.S. Kenyon and W.F. Edwards, Test of current-dependentelectricfields, Physics Letters A , 391(1991). ρ = 0 is not a unique solution for therest frame of the wire and that the Lorentz transforma-tions could be, for example, applied symmetrically, cameto my mind while teaching in 2009-2010 an undergradu-ate course in electromagnetism. 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Deacon et al., Cyclotron Resonance study of theelectron and hole velocity in graphene monolayers, arxiv0704.0410 (2007).[24] John Stewart Bell, Speakable and unspeakable in quan-tum mechanics, Cambridge: Cambridge University Press(1987). See also Bells spaceship paradox in wikipedia andreferences therein.[25] Aleksandar Gjurchinovski, Reflection of light from a uni-formly moving mirror, American Journal of Physics ,1316 (2004).[26] Imagine if you will the replacement of a normal metallicwire by two solid rods, one made of positive ions and theother of negative ions. A current carrying wire may thenbe simulated by simply moving the negative rod with thedrift velocity v e in the frame S of the positive rod. Nowlet us construct a frame S ∗ moving at velocity V relativeto S , in which both rods move in opposite directions withthe same velocity. Due to Lorentz transformations, thisvelocity will not exactly be v e / u (cid:48) = ( u − V ) / (1 − uV /c ) the reader may easily convincehimself that V is simply the solution of the quadraticequation v e /c × V + 2 × V + v e = 0). As the situationin S ∗ is completely symmetric, it does not matter if a ve-locity affects charge density or not; it is clear that whatever the physics is, the total charge density is zero. Theonly assumption required is that the sign of the velocityvector does not change the charge density in magnitudeor sign. Utilizing the rule which has been presented inappendix II, namely ρ (cid:48) = ργ (cid:48) /γ , where ρ and ρ (cid:48) are theapparent charge densities, and where γ and γ (cid:48) are theLorentz factors due to the particles’ velocity relative tothe two frames of reference, one easily finds that trans-forming to S , the charge density there is proportional to1 − γ as predicted by the alternative approach.[27] Another type of asymmetry which should perhaps be con-sidered lies in the fact that only for the solid lattice ofthe positive charges may it be claimed that there existsa frame in which all the particles in the closed loop areat rest. No such frame exists for the electrons, when thecurrent is non-zero. However, this is exactly the situa-tion in a ring accelerator and nevertheless we expect to see Lorentz contraction of a nucleus everywhere along itscircumference. In this sense, the loop wire may be viewedas a ring accelerator. 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