Field of a moving locked charge in classical electrodynamics
aa r X i v : . [ phy s i c s . c l a ss - ph ] J u l Field of a moving locked charge in classical electrodynamics
Alexander J. Silenko ∗ Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China,Bogoliubov Laboratory of Theoretical Physics,Joint Institute for Nuclear Research, Dubna 141980, Russia,Research Institute for Nuclear Problems,Belarusian State University, Minsk 220030, Belarus (Dated: July 17, 2020)
Abstract
The paradox of a field of a moving locked charge (confined in a closed space) is consideredand solved with the use of the integral Maxwell equations. While known formulas obtained for instantaneous fields of charges moving along straight and curved lines are fully correct, measurablequantities are average electric and magnetic fields of locked charges. It is shown that the average electric field of locked charges does not depend on their motion. The average electric field ofprotons moving in nuclei coincides with that of protons being at rest and having the same spatialdistribution of the charge density. The electric field of a twisted electron is equivalent to the fieldof a centroid with immobile charges which spatial distribution is defined by the wave function ofthe twisted electron.
Keywords: classical electromagnetism; beams in accelerators; twisted electron ∗ Electronic address: [email protected]
1t has been shown in Ref. [1] that the derivation of an electric field of a moving lockedcharge may lead to paradoxes. Such charges are confined in closed spaces like electrons inatoms, protons in nuclei, and charged particles in storage rings. It follows from the Lorentztransformations that the electric (and magnetic) field created by a charge significantly de-pend on its motion. In particular, the Lorentz transformation of the scalar potential has theform Φ = Φ p − β = γ Φ , β = v c , (1)where Φ is the scalar potential in the rest frame and v and γ are the velocity and Lorentzfactor in the laboratory frame. When the particle moves along the x axis, the correspondingLorentz transformation of the electric field is given by [2–4] E x = E x , E y = E y p − β = γE y , E z = E z p − β = γE z , (2)where E and E denote the electric field in the rest and laboratory frames, respectively.Certainly, electrodynamics presents general formulas for instantaneous electric and magneticfields defined by the Li´enard–Wiechert potentials [2, 4]. The instantaneous fields of a chargemoving along a circle have been given in the books [3, 4] (see also Ref. [5]).The differences between Φ and Φ and E and E condition a difference between electricfields in the laboratory and rest frames [see Eqs. (5) and (6) below] and creates an illusion ofa distinction between electric fields of a moving locked charge and the corresponding chargeat rest. In particular, this situation takes place for protons in nuclei and particles and nucleiin storage rings (specifically, for beam–beam interactions) [1]. It seems that the electric fieldshould be bigger in absolute value than the field of the corresponding point charge. Naiveaveraging Φ and E over a charge trajectory leads to the relations Φ = Φ and E = E which strongly contradict to atomic and nuclear physics. This consideration clearly showsthe presence of the paradox.We should mention that the analyzed paradox differs from the Schiff paradox [4, 6] andsome other paradoxes explained in Ref. [4]. In particular, the Schiff paradox is caused by adifference between the Maxwell equations in the rotating frame and the Minkowski space.Beam–beam interactions are important for electric-dipole-moment experiments in storagerings. In these experiments, one can simultaneously use two beams consisting of particlesmoving in the clockwise and counterclockwise directions with the same momentum [7, 8].2hen the two beams are separated in space [7], particles of one beam create a verticalelectric field and a radial magnetic one acting on particles of another beam. However, it hasbeen shown in Ref. [1] that their electric fields are defined by the Coulomb law even in thecase of high velocities of nucleons and particles. In the present study, we give a more clearexplanation of this paradox.This explanation uses the two integral Maxwell equations: I E · d S = 4 π Z ̺dV , (3) I B · d l = 4 πc Z (cid:18) j + 14 π ∂ E ∂t (cid:19) · d S . (4)In the first equation, S is a closed surface limiting the volume V and ̺ and j are the chargeand current densities. In the second equation, l is a closed line limiting the surface S .The key point of the present explanation is an independence of the average charge den-sity of charged particles, nucleons, and nuclei moving along a definite trajectory from thevelocity of their motion, v . This independence is conditioned by the charge conservationlaw. Evidently, the total charge does not depend on the velocity. Therefore, ̺ = ̺ and j = ̺ v = ̺ v in any point of the particle (nucleon, nucleus) trajectory. Here ̺ is theaverage charge density of slowly (nonrelativistically) moving particles and nuclei. The useof Eqs. (3) and (4) simplifies an analysis of distinctive features of the fields of locked charges.Let us first compare fields created by beams of charged particles or nuclei in a storage ringand in a free space. We can suppose the beam radii to be rather small as compared with thedistance to the beams. In this case, the beams can be considered as finite or infinite thinfilaments. The electric field of an infinite filament at rest is defined by E = E r r , E = 2 τr , (5)where r is the distance to the filament and τ is the charge of the unit of its length. Whenthe filament moves, its electric field is given by E = 2 τ γr , (6)where γ is the Lorentz factor. This significant difference has been noted in Ref. [1].The motion of charges creates also an electric current which magnetic field is given by B = 2 τ γ β × r r . (7)3hen a filament is finite but its length satisfies the condition l ≫ r and observation pointsare far from the ends of the filament, the electric and magnetic fields are also defined byEqs. (6) and (7). Certainly, Eqs. (3) and (4) lead to the same results because of the Lorentzcontraction of the filament length. Therefore, the fields nearby the moving filament areproportional to the Lorentz factor γ under the above-mentioned conditions. This situationtakes place for charged filaments in both the free space and a storage ring (if a curvature of abeam trajectory can be neglected). We should underline that all results previously obtainedfor instantaneous fields of charges moving along straight and curved lines are fully correct.However, only average electric and magnetic fields of locked charges are measurable.It has been ascertained in Ref. [1] that the average electric and magnetic fields createdby a charged beam in a storage ring are defined by the following equations: E = 2 τ r r , (8) B = 2 τ β × r r . (9)This paradox has a clear explanation. The Lorentz contraction of the length of themoving filament increases the local charge density γ times. This effect takes also place forparticle bunches in a storage ring, and their instantaneous fields satisfy Eqs. (6) and (7).Nevertheless, the average charge density of charges confined in a closed space is constantand cannot be changed by their motion. As a result, the average fields of moving chargesare defined by Eqs. (8) and (9) for average quantities. The latter equation follows fromthe fact that the average linear densities of the charge and current ( τ and i , respectively)satisfy the relation i = τ v . (10)Certainly, the average value of the quantity ∂ E / ( ∂t ) is zero.To expound this situation and to check a dependence of the average electric and magneticfields on the velocity of the finite filament, let us specify that the length of the filament atrest is l . While the field of the relativistically moving filament is equal to E = γE , itslength is correspondingly shorter ( l = l /γ ). The average field of the moving filament isdefined by its linear charge density when the filament charge, Q = τ l , is extended on thebeam circumference C . Therefore, τ = Q/C = τ l/C = τ and i = τ v = τ v .4n quantum mechanics, one can consider stationary states of a charged particle in thestorage ring. For such states, the charge density is proportional to the square of a waveeigenfunction and is almost independent of the azimuth φ defined relative to the center ofthe storage ring. In any stationary state, the particle is extended on the whole beam pathand a passage to the average charge density is well substantiated. Average values (i.e.,expectation values) of the particle momentum in the Cartesian coordinates are equal to zeroin any stationary state. Particles and nuclei in storage rings are usually (but not always)uniformly distributed along the beam trajectory, e.g., due to the momentum spread. In thiscase, the charge density is stationary, uniform, and independent of the beam velocity. Thefields need not be averaged and satisfy Eqs. (8) and (9). Even if the charged particle issingle, one can consider its stationary states in the storage ring. In any stationary state, thefields are stationary and are defined by Eqs. (8) and (9). In the classical limit, one passesto classical wave theory and describes particles and nuclei by de Broglie waves. The use ofsuch a description in the stationary state can also simplify the explanation of the paradox ofthe electric and magnetic fields created by a moving particle or nucleus in the storage ring.The corresponding de Broglie wave fills the whole beam path and its momentum is equal tozero ( P = p = 0), while P φ = 0. We can use the integral Maxwell equations (3) and (4) andcan check the independence of the electric field strength of the particle/nucleus velocity forany closed surface S containing the storage ring.We can conclude that passing to the charge and current densities sheds light on theorigin of the paradox investigated. When the charged filament or the single charge moves,its electric field increases as compared with the static case due to the length contraction.This effect also takes place for the closed filament and charge but only for periods of timewhen these sources are close to the observation point. Averaged electric fields of movingsources do not depend on their velocities because their averaged charge densities remain thesame.We can now consider the electric field of a proton moving in a nucleus. Certainly, themoving proton undergoes the Lorentz contraction and its charge density increases γ times.As a result, the scalar potential and the electric field strength of its field increase accordinglyEqs. (1) and (6). However, the average charge density of the proton, ̺ , does not dependon its motion in a fixed closed space. Indeed, the alternative statement ̺ = γ ̺ is absurdbecause its integration leads to a noninvariance of the proton charge.5herefore, the average electric field of protons moving in nuclei, in agreement with Ref.[1], coincides with that of protons being at rest and having the same spatial distributionof the charge density. In particular, the electric field of a spherically symmetric nucleus isexactly defined by the Coulomb law in agreement with the real situation.The results obtained bring a rather important conclusion. The nucleus mass is affectedby a motion of protons and neutrons but the electric field of the nucleus is independentof their motion and can be obtained by a summation of electric fields of charges (protons) at rest . The electric and magnetic fields of a moving nucleus are defined by the Lorentztransformation from its rest frame and coincide with fields of a nucleus with the same chargedensity but with charges at rest.The same conclusions are valid for atoms and other closed systems. Amazingly, they arealso applicable for charged twisted (vortex) particles while they can be modeled by wavebeams or wave packets moving with certain velocities. Twisted particles possess an intrinsicorbital angular momentum (OAM). Twisted electron beams with large intrinsic OAMs (upto 1000 ~ ) have been recently obtained [9]. Since twisted electrons possess large magneticmoments [see Eq. (11) below], their discovery has opened new possibilities in the electronmicroscopy and investigations of magnetic phenomena (see Refs. [10–17] and referencestherein). The twisted electron is not a pointlike particle. Its spatial distribution is localizedin the transversal plane. Such an electron can be modeled by a charged centroid beingan extended object with a charge distribution defined by the wave function of the twistedelectron [10, 18]. The centroid as a whole is characterized by a center-of-charge radius vectorand by a kinetic momentum. An intrinsic motion of partial charges takes place. The twistedelectron possesses a tensor magnetic polarizability [19] and a measurable (spectroscopic)electric quadrupole moment [19, 20] and its interactions with external electric and magneticfields lead to nontrivial effects (see, e.g., Refs. [18, 19, 21–23]). The effective (kinematic)mass of a twisted electron manifesting in experiments is bigger than the electron rest mass[24, 25]. The effective mass corresponds to the total electron energy in the rest frame ofthe charged centroid. This energy is equal to the sum of the rest energy and the kineticenergy of a hidden motion of the electron and is quantized [25]. The inertial and kinematicmasses of the twisted electron always coincide [25]. Beams of free twisted electrons can bepresented as ensembles of plane waves and, certainly, their motion is not accelerated.The model of the charged centroid simplifies a consideration of an average electric field6reated by the twisted electron. Our analysis shows that this field (like the electric field ofatomic electrons) is equivalent to the field of immobile charges which spatial distribution isdefined by the wave function of the twisted electron. The total charge of these immobilecharges is equal to e .We can conclude that the analysis of the average electric field created by the twistedelectron confirms its modeling by the charged centroid with the kinematic and inertial massescoinciding with each other and different from the electron rest mass. In the centroid restframe , this field is equivalent to the above-mentioned field of immobile charges.A motion of the centroid is not the only source of the magnetic field of the twistedelectron. Any charge motion creates an electric current. Therefore, the intrinsic motion ofpartial charges forming the centroid conditions both the OAM L and the orbital magneticmoment µ L . The universal connection between the orbital magnetic moment µ L and theOAM L of a free pointlike particle has the form [26] µ L = e L E in any frame. Here E is the total energy of the twisted electron. The magnetic moment isalso contributed by the spin. If the Dirac spin magnetic moment of the electron is takeninto account and the small anomalous magnetic moment is disregarded, the total magneticmoment reads [18] µ = µ L + µ s = e ( L + 2 s )2 E . (11)In summary, we have considered the paradox of a field of a moving locked charge statedin Ref. [1] and have definitively solved it. Known formulas [2–4] describing instantaneous electric and magnetic fields of charges moving along straight and curved lines are fullycorrect. However, measurable quantities are average electric and magnetic fields of lockedcharges. The average charge density of charged particles, nucleons, and nuclei moving alonga definite trajectory does not depend on the velocity of their motion. With the use of theintegral Maxwell equations, we have shown that the average electric field of locked chargesdoes not depend on their velocity either. This is the main result obtained in the presentpaper. We have analyzed the electric fields of protons moving in nuclei and of the twistedelectron. The former field coincides with that of protons being at rest. The latter fieldis equivalent to the field of a centroid with immobile charges which spatial distribution isdefined by the wave function of the twisted electron.7 cknowledgements
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