Displacement induced electric force and natural self-oscillation of a free electron
aa r X i v : . [ phy s i c s . c l a ss - ph ] J un Displacement induced electric force and natural self-oscillation of a free electron
Zhixian Yu
Department of Physics and Astronomy, University of New Mexico, Albuquerque NM 87131 USA andCollege of Physics Science, Qingdao University, Qingdao 266071 China
Liang Yu ∗ Laboratory of Physics, Jiaming Energy Research, Qingdao 266003 China (Dated: October 8, 2018)We show that a kind of displacement induced temporary electric force of a single point chargecan be derived by using Maxwell stress analysis. This force comes from the variation of the charge’selectric intensities that follow Coulomb’s inverse square law, and it is a kind of displacement depen-dent temporary restoring force. We also show the possible existence of natural self-oscillation of afree electron which is driven by this restoring self-force of its own electric fields.
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INTRODUCTION
The distance dependence of the electrostatic force ofCoulomb’s inverse square law has been tested in highprecision through different methods, see for example thediscussion of Jackson[1] and experiment of Williams etal[2].In this paper we want to show that some temporarydisplacement dependent electric force may be generatedin relation to the Coulomb’s law. This is a kind of restor-ing electric self-force. It is induced by any displacementof the electric charge which changes the distribution ofits electric fields that follow Coulomb’s law.
THE VARIATION OF DISTRIBUTION ANDMAGNITUDE OF ELECTRIC INTENSITY OF ACHARGE SUBJECT TO TEMPORARYDISPLACEMENT
For a particle with electric charge e at rest at origin O first, according to Coulomb’s law the distribution ofits spherically symmetric electric intensity E follows theinverse square law, that is E = er e r e , (1)where r e is the magnitude of position vector from theparticle to a field point. Suppose this charge is shiftedalong z axis by a displacement Z e in a time interval △ t and become stationary again at a position O e . Thus theelectromagnetic fields depending on its velocity and ac-celeration may be neglected first. By this displacement Z e , the electric intensity E of Eq. (1) will change inboth magnitude and distribution. During this time in-terval any variation signal of the electromagnetic fieldswill propagate first from the origin O to a spherical shellwith radius c ∆ t as shown in Fig. 1, where c is velocity oflight. At a position P on this spherical shell, the distanceto origin O is the radius r = c ∆ t and to the new origin O e is [ r e ]. The relation between [ r e ] and r = c ∆ t is that[ r e ] = ( c ∆ t ) + Z e − c ∆ tZ e cos θ, (2)where θ is the angle between r and z axis. Now the elec-tric field outside this spherical shell does not change, themagnitude of the electric intensity on the outer boundaryof this surface at P is E out = er = e ( c ∆ t ) . (3)The magnitude of the electric intensity on the innerboundary of this surface at P is E in = e [ r e ] . (4)Since by Eq. (2) [ r e ] = ( c ∆ t ) , these two electric in-tensities of Eq. (3) and Eq. (4) are not equal. Butaccording to the requirement of div E = 0 in charge freespace, they should be equal in magnitude and direction.Although this requirement is applied for the total elec-tric field, it is also correct for the Coulomb field alonehere. Since E out of Eq. (3) is the primary value, it mustremain unchanged, thus E in should be modified to fulfilthe requirement of div E = 0, that is to change E in to E in = e [ r e ] · [ r e ] ( c ∆ t ) = er e M F = e ( c ∆ t ) , (5)where M F is a kind of modification function. From Eq.(5) it is defined as
M F = [ r e ] ( c ∆ t ) = 1 + Z e ( c ∆ t ) − Z e cos θc ∆ t . (6)This kind of modification may also be seen from the re-lation of the solid angles d Ω and d Ω e subtending to anarea element ds at position P from the origins at O and O e respectively as shown in Fig. 2. For the two solidangles we have d Ω = dsr = ds ( c ∆ t ) , (7) FIG. 1. The relations between r = c ∆ t and [ r e ]. d Ω e = ds [ r e ] , (8)and following Eq. (6) we have d Ω d Ω e = [ r e ] r = [ r e ] ( c ∆ t ) = M F. (9)Since the number of electric lines within solid angle d Ω e should be the same as that within d Ω, the area density ofelectric lines which represents the electric intensity willchange by the ratio
M F . This modification now not onlymodifies the electric intensity at position P , but shouldalso be down to the whole solid angle d Ω e . Thus we maywrite the electric intensity within the solid angle d Ω e as E e = er e M F. (10)Then when the charge is subject to a temporary displace-ment Z e during time interval ∆ t , the electric intensitiesshould be changed totally according to Eq. (10) withinthe spherical surface of radius r = c ∆ t . MAXWELL STRESS ANALYSIS OF THEDISPLACEMENT DEPENDENT ELECTRICFORCE ON A CHARGE SUBJECT TO ATEMPORARY DISPLACEMENT
It is known that electromagnetic force on a systemmay be calculated from the Maxwell stress on the sur-face that encloses this system. We see from above thatfor a particle with charge e and subject to a temporarydisplacement Z e , the distribution and magnitude of itselectric intensity follows the modified Coulomb’s law asEq. (10). Now we want to show that this particle willget a displacement dependent electric force generated byits own electric field, and this can be calculated from theMaxwell stress surrounding this particle. Now we maytake a spherical surface with center at O e and radius a FIG. 2. The relation between solid angles d Ω e and d Ω. surrounding this particle, and according to Eq. (10), theelectric intensity at this surface E a as shown in Fig. 3 is E a = ea M F. (11)According to the theory of Maxwell stress, the electricforce on this particle may be calculated from the stress onthe above-mentioned surface. For any electric intensityin space there is a corresponding Maxwell stress tensor T as[3] T = 14 π [ EE − I ( E )] , (12)where I is the unit second rank tensor, the elements of T are T ij = 14 π [ E i E j − I ij E ] , (13)its force element dF i on an area element ds at the surfaceis dF i = X j =1 T ij ds j , (14)where ds j is the j component of ds . The force F i on thatsurface is F i = Z dF i . (15)On the spherical surface surrounding the charged parti-cle, the distribution of electric intensity is Eq. (11) withdirections along the normal of the surface element ds .Using Eqs. (13), (14) and (15), the total force along z axis on the closed spherical surface is F z = I ( T zz ds z + T zx ds x + T zy ds y )= I π
12 ( E z − E x − E y ) ds z , (16)since the last two terms integrate to zero owing to theirsymmetry. Here ds z = a sin θ e cos θ e dθ e dφ and we may take θ ≈ θ e when Z e is small. Substituting M F of Eq. (6) into Eq. (16) and using E z = E a cos θ e , E x = E a sin θ e cos φ and E y = E a sin θ e sin φ , we have F z = I π e a [1 + Z e ( c ∆ t ) − Z e cos θ e c ∆ t ] (cos θ e − sin θ e cos φ − sin θ e sin φ ) a sin θ e cos θ e dθ e dφ = I e a [1 + Z e ( c ∆ t ) + 4 Z e cos θ e + 2 Z e ( c ∆ t ) − Z e cos θ e c ∆ t − Z e cos θ e ( c ∆ t ) ](2 cos θ e −
1) cos θ e sin θ e dθ e . (17) FIG. 3. The distribution of electric intensities and electricfield lines between two spherical surface with radii r = c ∆ t and r e = a . Here we integrate φ from 0 to 2 π . The odd terms of cos θ e integrate to zero. Since Z e is small, the important termof above integration is the first order term of Z e , thosehigher order terms can be neglected, thus we have F z = I e a ( − Z e cos θ e c ∆ t )(2 cos θ e −
1) cos θ e sin θ e dθ e , (18)thus F z = − e a Z e c ∆ t . (19)This displacement dependent F z is independent of thepositive or negative sign of charge e since it’s propor-tional to e . This is a kind of displacement dependentrestore force induced by the variations of the charge’s ownelectric intensity which follows Coulomb’s inverse squarelaw. The magnitude of this force is dependent on theradius a of the spherical surface boundary of the charge.Here this force is also a temporary force. After the time interval ∆ t , as time goes on, ∆ t increases to ∆ t + t , where t is the additional time, then Eq. (19) will change to F z = − e a Z e c (∆ t + t ) . (20)When t becomes large, F z reduces to zero.This problem cannot be treated by usual methods suchas the time dependent Coulomb’s law[4], the Lienard-Wiechart potential[5] or the Feynman’s formula[6] fora charge undergoing an arbitrary translation motion,since the variations of internal structure of the electricfields are not taken into account in these methods. Al-though the retarded time calculations are used in thesemethods, the historical conditions of the electric fieldsare ignored. The problem here is a kind of heredi-tary electromagnetism[7]. We do not use the integro-differential equation of functional analysis for this prob-lem but the transit effect of displacement is treated insimilar way[7]. Similar analysis about the radiation ofelectric charge induced by its acceleration via the vari-ation of electric field lines was given by others[8]. Ourdiscussions here emphasize only on the effect of displace-ment. The effect of velocity dependent magnetic fields isneglected and the equation of static cases is used as Eq.(14). POSSIBLE EFFECT OF THE DISPLACEMENTDEPENDENT RESTORE FORCE ON THEDYNAMICS OF A FREE ELECTRON
The displacement dependent restoring force of Eq.(19) is a kind of self-induced force. For a single freeelectron with charge e and mass m e which is not subjectto any external force, this self-induced force will affect itsdynamical motion. We may take M r as the radius a above, where M is an undetermined numerical constant, r is the classical radius of electron which is defined as[6] r = e m e c ≈ . × − cm. (21)Substituting M r into Eq. (19), we get F z = − e M r ) Z e c ∆ t . (22)For an electron possessing self-sustained harmonic oscil-lation with angular frequency ω , the time interval ∆ t may be taken as π/ω , which is half of its period. Thenwe have F z = − e M r ) ωZ e cπ = − m e cωZ e M πr , (23)since m e = e / ( r c ) according to Eq. (21). If we take2 c/ (15 M πr ) = ω , then F z = − m e ω Z e , (24)which is a standard equation of harmonic oscillation ofthe free electron. This displacement dependent restoreforce implies that the electron may have a kind of natu-ral self-oscillation as the “mono-electron oscillator” in aPenning trap of the experiment of Dehmelt et al[9].The oscillating electron will have oscillating staticfields, velocity dependent and acceleration dependentfields, which may store energy and exchange energy be-tween each other[10, 11]. They need not always radiateout their energy through acceleration dependent electro-magnetic fields if these fields have standing wave modesolutions[12, 13]. ∗ [email protected] [1] J. D. Jackson, Classical Electrodynamics , 3rd ed. (JohnWiley and Sons, 1998) p. 8.[2] E. H. Williams, J. E. Faller, and H. A. Hill, Phys. Rev.Lett. , 721 (1971).[3] J. D. Jackson, Classical Electrodynamics , 1st ed. (JohnWiley and Sons, 1962) p. 193.[4] D. J. Griffiths and M. A. Heald, Am. J. Phys. , 111(1991).[5] W. K. H. Panofsky and M. Philips, Classical Electricityand Magnetism (Addison-Wesley, 1962) Chap. 19, p. 341.[6] R. P. Feynman, R. B. Leighton, and M. Sands,
The Feyn-man lectures on physics , Vol. 2 (Addison-Wesley, 1964)Chap. 21.[7] V. Volterra,
Theory of functionals and of integraland integro-differential equations (Addison-Wesley, 1959)Chap. VI, p. 194.[8] H. C. Ohanian, Am. J. Phys. , 170 (1980).[9] D. Wineland, P. Ekstorm, and H. Dehmelt, Phys. Rev.Lett. , 1279 (1973).[10] J. B. Marion, Classical Electromagnetic Radiation , 2nded. (Academic Press, 1980) Chap. 8, p. 285.[11] L. Mandel, J. Opt. Soc. Am. , 1011 (1972).[12] H. Bateman, The Mathematical Analysis of Electrical andOptical Wave Motion on the Basis of Maxwell’s Equa-tions (Dover, 1955) Chap. 3, p. 36.[13] R. B. Adler, L. J. Chu, and R. M. Fano,