The Longest Range Electric Field, and Gravity R.L. Collins, retired, e−mail [email protected] or [email protected] 6/16/1997

ABSTRACT The vector potential is A=(µo/4π)qv/r, from which B=∇×A. Upon any acceleration of the charge, one finds a long range electric field, E=− ∂A/∂t. The range of this 1/r field far exceeds that from a charge, which goes as 1/r2 or a dipole which goes as 1/r3. This E field differs from Hertz’ solution for an oscillating charge. Hertz’ finding of TEM waves from an oscillating charge is flawed by the implicit Galilean transformation of “retarded time”, t’=t−r/v. By relating all time to a clock sited at the source, the need for retarded time at the observer is avoided and one finds an E wave solution. The waves which result involve E and B fields, but only the E field is long range (1/r). These are classical waves which remain tied to the source, and carry away no energy. The electric field affects neutral bodies by the Stark effect, exchanging energy and leading to a weak force of attraction very like gravity. This suggests that gravity is founded in the electromagnetic force, and is not fundamental.

ACCELERATION OF CHARGE A proper accounting for the effects of accelerated charge is important in electromagnetism. Currently, there is controversy as to the consequences of the acceleration of charge. Classical electromagnetism is very firm on the point. As Jackson puts it, “The motion of charged particles in external force fields necessarily involves the emission of radiation whenever the charges are accelerated.” (1) By radiation, he means Hertzian

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(TEM) waves. The radiation described by classical physics has no explanation for the photoelectric effect nor the Compton effect, experiments which indicate that radiation consists of compact bundles of energy whose energy is inversely proportional to wavelength. Atomic and molecular physics depend on the stability of charges in motion, requiring acceleration of charge without loss of energy through radiation. Pragmatic physicists have learned to live with the problem, that acceleration of charge leads to EM radiation (except when it does not). This study reexamines the basis of the classical expectation of EM radiation from accelerated charges. It finds that the Hertzian solution is flawed, and raises doubts as to the validity of the TEM wave concept. The classical transverse electromagnetic (TEM) waves are solutions to the wave equations obtainable from Maxwell’s equations. Maxwell’s equations take on a very simple form in a vacuum: (2) ∇•E = 0

∇×E = −∂B/∂t [1]

∇•B = 0

∇×B = µ0ε0 ∂E/∂t

From the two “curl” equations, one can obtain the wave equations: ∇2E = µ0ε0 ∂2E/∂t2

∇2B = µ0ε0 ∂2B/∂t2

and

[2]

An obvious solution consists of TEM waves, in which the amplitudes of the oscillating fields are everywhere the same throughout an infinite plane perpendicular to the direction of propagation. The resulting velocity is “c”, predicted correctly by 1/√µ0ε0. This TEM solution is unwieldy, extending throughout all space. If the solution is restricted to a finite region, the two divergence laws create insuperable problems. And, light can easily be confined to a narrow beam as in a flashlight or a laser. At beam’s edge, one must find charges to terminate the lines of electric field and also magnetic

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monopoles to terminate the lines of magnetic field. Or, dare one think it, there must exist “longitudinal” components of the fields contrary to the TEM solution. Radiation sources are always finite in size, while the TEM solution is infinitely wide. The bridge between these was discussed by Hertz. In 1879, Hertz derived mathematically spherical TEM waves arising from an oscillating electric dipole. These waves evolve as they travel, with their radii of curvature increasing as ct. Eventually, they approach the Maxwellian TEM waves. He correctly predicted that energy should be radiated away into space. Let’s look again at the Hertz problem, in the light of the advances of physics since 1879. Reitz and Milford give a very readable account of this “classical radiation”, described next. (3) The oscillating dipole is taken as a charge, oscillating along the “z” axis at angular velocity ω. Far away from the dipole, i.e. for a radial distance large compared with twice the amplitude “l” of the moving charge, the vector potential points along the “z” axis and can be approximated as |A| = Az (r,t) = (µ0/4π)(1/r)(lI0)sin ω(t−r/v) [3] Retarded time, t−r/v, accounts for the time delay between events at the source and at the field point. The scalar potential, Φ, can be found from the Lorentz condition, div A + µ0ε0 ∂Φ/∂t = 0

[4]

Resolving the vector potential into spherical components, one obtains: Ar = (µ0/4π)(1/r)(lI0)cos θ sin ω(t−r/v) [5] Aθ = −(µ0/4π)(1/r)(lI0)sin θ sin ω(t−r/v) [6] Aφ = 0

[7]

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B is the curl of the vector potential, A. It is apparent from inspection that only the φ−component of B is non−zero. Bφ = (1/r)(∂/∂r)(rAθ) − (1/r)(∂Ar/∂θ) [8] = (µ0/4π)(lI0/r)sin θ [(ω/v)cos ω(t−r/v) + (1/r)sin ω(t−r/v)] [9] Although only the long−range (far field, 1/r) case is of interest, we include for completeness the other components. The electric fields are more complex. We need the scalar potential Φ in order to calculate them, and this is found from [4]. We assume the source involves simple harmonic motion of a charge, q0, and I0 = − ωq0 connects the angular velocity of the dipole with the moving charge and the amplitude of the equivalent current. It follows that: Φ = (l/4πε0)(cos θ)(1/r)[(q0/r) cos ω(t−r/v) − (ωq0/v)sin ω(t−r/v)] [10] The evaluation of the components of E is then straightforward, but tedious: Er = −∂Φ/∂r − ∂Ar/∂t [11] = (1/4πε0)2lI0cos θ [{sin ω(t−r/v)}/(r2v) −{cos ω(t−r/v)}/(r3ω)] Eθ = −(1/r)∂Φ/∂θ − ∂Aθ/∂t [12] = −(1/4πε0)lI0sin θ [(1/ωr3 − ω/rv2)cos ω(t−r/v) − (1/r2v)sin ω(t− r/v)] Eφ = 0

[13]

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In the far−field case, the only remaining fields are Bφ and Eθ, each falling off as 1/r. They each vary with time as cos ω(t−r/v), and the Poynting vector indicates that there is an outward flow of energy. If one plots these fields, one finds they comprise wider and wider loops of diminishing intensity and each wavefront has radius of curvature “r=ct”. The problem with this approach lies with the seemingly innocuous concept of “retarded time”. The notion of retarded time is basically a Galilean concept, with time being everywhere the same. Since 1905, we have known the importance of observed time and have rejected simultaneity. To avoid the subtleties of SR, let’s place the clock (t) at the site of the oscillator. This clock can be as large as may be needed, so it can easily be read by the observer. Provided the speed of the vector potential is c, the observer finds: |A| = Az (r,t) = (µ0/4π)(1/r)(lI0)sin ωt [14] That is, the observer is looking backward in time just as astronomers do when examining nascent galaxies at large red shifts. The fact that the light takes the same time to reach the observer, so he can read the time, as does the radiation from the oscillator, means that there is now no need for “retarded time” in the equation for the vector potential. The vector potential seen by the observer (at r,t) corresponds precisely to the instantaneous motion of the source at the time indicated to the observer by the clock (at 0,t). An effect of this seemingly small change is that the 1/r term in the B field of Hertz’ solution disappears! The only B term surviving is Bφ = (1/r)(∂/∂r)(rAθ) − (1/r)(∂Ar/∂θ) [15] = (µ0/4π)(lI0/r2)sin θ sin ωt The transverse electric field does not involve a partial derivative with respect to “r”, and is unaffected by the deletion of the retarded time: 5

Eθ = −(1/r)∂Φ/∂θ − ∂Aθ/∂t [16] = −(1/4πε0)lI0sin θ [(1/ωr3 − ω/rv2)cos ωt − (1/r2v)sin ωt] However, the radial component now includes a 1/r component: Er = −∂Φ/∂r − ∂Ar/∂t [17] = (1/4πε0)lI0cos θ [{sin ωt}/(r2v) −{cos ωt}{2/(r3ω)+ω/(v2r)}] Notice that the magnitude of the longest range components of E is precisely equal to that of − ∂A/∂t, and the direction of E is always along the “z” axis. One of the stranger things about this solution is that the long range E field is no longer transverse to a radius vector from the source! Notice also that the long range (1/r) component of the transverse electric field is now out of phase with the longest range (1/r2) component of the transverse magnetic field. The sign of the Poynting vector hence oscillates with time and falls off as 1/r3. These E waves carry away no energy. The fields remain tied to the source, and may have any amplitude whatever. They are wholly classical, and contain no reference to Planck’s constant.

E WAVES Hertz’ solution is the basis for the TEM wave concept of light, and it is wrong. Stripped of the Galilean concept of time, the E waves can no longer be associated with the electromagnetic spectrum which covers such diverse phenomena as radio, light, and gamma rays. This finding is in total conformity with the fact that Planck’s constant is necessary for dealing with the electromagnetic spectrum, while E waves are wholly classical and may

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have any amplitude whatsoever. These E waves derive from the concept of the vector potential. Aside from its use as a mathematical aid for the calculation of the magnetic field, it was not immediately obvious that the vector potential corresponds to anything real. Aharanov and Bohm (4) startled a lot of physicists when they predicted correctly that the phase, of charged particles in motion, changes when immersed in a vector potential, even in regions where the B field vanishes. London (5) found good use for the vector potential in predicting the quantization of magnetic flux trapped within a superconducting loop, later found. It has been found useful in evaluating the amplitude of the oscillating magnetic flux which circulates about a confined elementary charged particle. This led to a finding that the amplitude is a multiple of h/4e, which suggests that photons carry away a quantum of magnetic flux, h/4e. (6) This vector potential has an extremely long range (1/r), and it oscillates with time in response to the source motion. The accelerations experienced by charges in an atom are seldom sinusoidal, typically peaking as an electron approaches the nucleus. In general, E = −∂A/∂t = − (µ0/4π)(1/r)qa [18] and it is obvious that the motion, of the charges which comprise the atoms of a neutral solid, must lead to time−varying long range electric fields. And so, the acceleration a of charge q leads to very long range electric fields, similar to those at the source but with amplitude diminishing as 1/r.

APPLICATION OF THE E WAVES TO GRAVITY The existence of a long range electric field suggests a connection with gravity. For the simulation of the gravitational force, it is not necessary that 7

this electric field act directly to pull two neutral masses toward one another. Experience suggests that the mere exchange of energy lowers the energy of the combination, and is physically equivalent to a force of attraction existing at one as a consequence of the other’s presence. The Heitler−London theory of covalent bonding and the interchanges of pi mesons between nucleons come to mind. The essential element is that the transient electric field does some work on the receiving atom, work done at the expense of the sending atom. The process goes both ways, and so each neutral mass is sending and receiving energy at the same rate. The gravitational energy of a mass “m” in the presence of another mass M, separated by a distance r, is given by the law of universal gravitation: U = −GmM/r

[19]

This has been verified experimentally, but the value of G remains the least well measured of the fundamental constants. The precision is less than four digits, in accordance with the difficulty of measuring these tiny forces in a laboratory experiment. Nominally, G = 6.71 x 10 −11 in SI units. Next, we want a quantitative description of the coupling constant between the electric fields at the source and the electric fields at the receiving mass. E = −∂A/∂t = −(µo/4π)qa/r [20] where E is measured a distance “r” from the source, qa. The source acceleration, a, is given the charge “q” in response to the force delivered by the electric field, F. a = Fq/m

[21]

and so, [20] can be written as E = −∂A/∂t = −(µo/4π)qFq/mr 8

[22] In order to get an idea of the magnitude of the electric fields involved, we now invoke the “classical radius” concept of a charge, i.e. the radius “b” such that all its rest mass energy resides in the electric field emanating from a sphere of that radius: mc2 = (1/4πε0)q2/b

[23]

m = (µo/4π)q2/b

[24]

Substitution in [21] leads to the desired result: E = −∂A/∂t = − F(b/r)

[25]

That is, the transient electric field present at a distance “r” is equal to the negative of the transient electric field at the source, modified by b/r. Keep in mind that it is only because charge is accelerated by the electric field F that the E field is created per [25]. For a proton, b=1.53x10−18 and for an electron, b=2.8x10−15 meters. The effect of an electric field on a neutral atom has been much studied, and is known as the Stark effect. (7) The primary effect is a splitting of the term levels, the details depending on spin states and coupling. The major point is that energy is exchanged between the source (the electric field) and the atom (which may gain or lose energy). The magnitude of the Stark effect for hydrogen is 1.273x10−31 joules per v/m. The source fields are quite respectable. The maximum electric field to which an “s” electron is subject is that existing just before the electron enters the nucleus. An electron at its diameter “b” distance from the nucleus is subject to an electric field of the order of 1021 v/m. For two hydrogen atoms at a separation of 1 meter, the peak transient electric fields at one due to the

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presence of the other are reduced by the b/r term to perhaps 106 v/m. This lasts but a very short time as the electron nears the nucleus, the order of 10− 24 seconds. Since the Stark effect can only be resolved at voltages exceeding 107 v/m, it is clear that the effect is rather weak and that only the linear Stark effect need be considered. Further, the Stark effect is a d.c. effect and represents the equilibrium after all transient effects have damped out. The maximum (d.c.) effect on energy exchange, between two hydrogen atoms is of the order of 10−31 joules per v/m. The actual effect will be less, by many orders of magnitude, because of the very short duration of these electric fields. This may be compared with the gravitational potential energy of one hydrogen atom at a distance of 1 meter from another, which is

U = −GmM/r = −18.6 x 10−65 joules [26] The gravitational potential energy is so tiny that these Stark effect energy exchanges appear large by comparison. The efficiency of this Stark effect is unknown, but need not be much to account for gravity.

DISCUSSION As was mentioned in the text following [2], the TEM wave concept of electromagnetic radiation poses enormous problems. In view of the corrected Hertz solution, there are no TEM waves anyway. The alternate description of electromagnetic radiation involves photons, compact assemblies of EM fields which move at the speed of light and which encapsulate energy inversely proportional to size (as measured by wavelength). Much remains to be discovered about the precise nature of photons, but the energy content can be understood if one accepts that all photons are similar (except for size) and each carries the same magnetic

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flux. (8) Should we accept as real the existence of E waves, derived from the vector potential? There is relatively little known about the vector potential. We have no instrumentation capable of measuring it, although it seems likely that its speed is c. That it can affect the phase of a moving charge is shown by the Aharanov−Bohm effect, even when the path is free of any B field. (4) It is remarkable that the vector potential seems able to penetrate a superconductor, since flux quantization within a loop of superconductor depends on matching up the phases of charges after they have circulated along a path which includes the vector potential (but which is absent any B field by the Meissner effect and of course is absent any E field because it is a superconductor). Feynman gives a readable account of this in his lecture series. (9) It seems totally amazing that the vector potential can exist in a region where neither of its derivative fields (E and B) can exist, although this has been tested only under d.c. conditions. We don’t know whether there may exist materials or machines which will shield, absorb, reflect, augment, minimize, or otherwise interact with the vector potential. To the extent that the E waves provide a gravitational interaction between neutral masses, it is not obvious that these transient effects can be taken independently, i.e. that the effects are simply additive, as is needed for the effect to respond directly as the product of the masses. The very short periods of time when the electric fields attain their peak value helps persuade one that the effects may be simply additive. Newton’s law of universal gravitation indicates that the only factor of importance is mass, and this can be accommodated in this theory only by a fortuitous concordance between the electrical effects of the spectrum of transient electric fields created by atoms of different elements. A more disturbing problem is the implicit suggestion that there is a temperature effect in gravity. This will be difficult to measure. Temperatures of the sort needed to produce an effect are such as to lead to ionization, and so we are immediately talking about

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plasmas. Or, black holes. On the other hand, cosmology is concerned with the gravitational interaction of plasmas. And, black holes. The spectrum of these E waves is very complex, even from a body consisting of a few atoms. The E wave spectrum from a planet sized body is hopelessly complex, at least with our present and foreseeable technology. On the other hand, black holes exhibit gravity which suggests that the spectrum of the E waves may be rich with information about the interior. Information which cannot be had from electromagnetic radiation. It might appear from [22] that large masses (protons) are less effective than are electrons in sending the 1/r electric field. This is not true. Consider [25], where the electric field at the “surface” is diminished by distance as b/r. The field F at the surface is F = (1/4πε0)q/b2 [27] and so the far−field E is proportional to 1/rb (i.e. is proportional to mass/r). This means that the larger electric field at the surface of a proton (compared with an electron) leads to a larger far−field electric field. The contribution of neutrons to gravity by this mechanism is less clear. Neutrons exist in nuclei in close proximity to protons, and the electric field at the “surface” of a proton is of the order of 1024 v/m. Polarization may occur within a neutron subject to this kind of stress, and the acceleration of the positive and negative portions of the neutron occur in opposite directions. The vector potentials created by a positive charge accelerated in one direction and by a negative charge accelerated in the opposite direction have the same sign, and their contributions to the vector potential are additive.

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The equivalence of inertial and gravitational mass has intrigued physicists for a long time. Experimentally, they are the same. But why? The answer from this study is simply that each is a measure of the encapsulated energy. The inertial force, −d(mv)/dt, can be identified as a Coulombic force upon recognizing that the momentum can equally be written as qA. (6) That is, one can write the momentum of an elementary charged particle as the product of its charge times the internal vector potential in which a particle moves, qA. It is then clear that −d(mv)/dt = − d(qA)/dt, = −(q)d(A)/dt = qE. It is necessary that one accept that mass is wholly electrostatic, if one is to account fully for the inertial force by a Coulombic force. In fact, it was the identification of the inertial force as Coulombic which suggested the present paper. The gravitational mass also derives from the assumption that the mass of an elementary charged particle is wholly electrostatic, as indicated in the discussion at [27]. The most disturbing result of this line of inquiry is that gravity may not turn out to be universal. That is, big “G” may depend on the elements involved and even on temperature. Experiments to measure G are inherently difficult because the forces available are tiny, and so one normally uses a dense metal such as lead for the spheres of a Cavendish experiment. Extension of these studies to include other substances is highly desirable. There have been several recent studies along this line, with materials other than lead, in an attempt to find a new “fifth force”. No success, so far. Conversely, the thrust of this paper is the elimination of gravity as a fundamental force. Preliminary papers dealing with the E spectrum (10) and its possible application to gravity (11) have been posted on the aps e−print server.

CONCLUSIONS

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The wave−particle argument about the proper description of electromagnetic radiation has been long and bitter, and there has until now been no resolution between the two ideologies. This study suggests that “classical” TEM waves are fiction. The electromagnetic spectrum consists of photons, compact assemblies of electromagnetic fields which assume an independent status when they leave the source. These photons also have a wave nature such that they exhibit interference and diffraction and coherence, despite their frozen and flattened appearance to an observer. Perhaps this wave nature of the photon is expressed in the periodic changes of field amplitudes transverse to the propagation? The correct solution for Hertz’ oscillating dipole is E waves, classical electric waves which carry away no energy and remain tied to their source. The region surrounding a charge under acceleration is a busy place, even when no energy is being emitted. The long range E fields interact with other materials within their range. To the extent that they do, small exchanges of energy occur via the Stark effect. This lowers the energy of the combination, replicating the effects of gravity. The effect is weak, always attractive, non−saturating, and, acting independently, additive. This exchange of energy as a source of the gravitational force is in harmony with other “exchange” forces of physics, and suggests that gravity should no longer be considered a “fundamental” force.

REFERENCES 1. J.D. Jackson, CLASSICAL ELECTRODYNAMICS, (John Wiley & Sons, NY, 1975), 780. 2. D. Halliday and R. Resnick, PHYSICS, PARTS I AND II, (John Wiley & Sons, New York, 1978), A18. 3. J.R. Reitz and F.J. Milford, FOUNDATIONS OF 14

ELECTROMAGNETIC THEORY, (Addison−Wesley, Reading, MA, 1960), 344−347. 4. Y. Aharanov and D. Bohm, Phys. Rev. 115, 485−491 (1959). 5. F. London, SUPERFLUIDS VOL. I (John Wiley & Sons, NY 1950), 152. Also see: B.S. Deaver and W.M. Fairbank, Phys. Rev. Letters 7, 43 (1961) and, independently, R. Doll and M. Nabauer, Phys. Rev. Letters 7, 51 (1961). 6. R.L. Collins, eprint aps1997feb28_006. 7. J. Stark, Berl. Akad. Wiss., 40, 932 (1913) and also Ann. d. Phys., 43, 965 (1919). Also see H.E. White, INTRODUCTION TO ATOMIC SPECTRA (McGraw−Hill, NY, 1934), 401. 8. R.L. Collins, eprint aps1997feb11_002. 9. R.P. Feynman, R.B. Leighton, and M. Sands, THE FEYNMAN LECTURES ON PHYSICS, VOL. III (Addison−Wesley, NY, 1965), 21−1. 10. R.L. Collins, eprint aps1997may27_003. 11. R.L. Collins, eprint aps1997apr15_001.

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