Electric Charge Radiation 3.pdf

Electric Charge Radiation Didier François Viel Keywords : Paradox about radiation of an accelerated electron, Feynman’s formula, Larmor’s formula DOI: 10.5281/zenodo.10563107 email : arcopodias@gmail.com 1 Abstract “How wonderful that we have met with a Paradox. Now we have some hope of making progress.” Niels Bohr Niels Bohr satisfaction was concerning the statement he had made some years before, that a gyrating electron around a proton nuclei cannot emit radiation. In fact, if it were, the electron will fall down on the proton, which is not what is observed! This statement was in total contradiction to what Joseph Larmor had demonstrated mathematically in 1897. And when general relativity theory was adopted, Larmor’s formula induced the "Paradox of radiation of charged particles in a gravitational field". With this Paradox, Bohr was hoping that resolving it will theoretically justify his statement. The derivation of the radiation of an electrostatic field emitted by an electric charge moving with constant velocity, done by Oliver Heaviside, was puzzling since it suggested that the field propagates faster than light. The resolution of this issue shows that the wave propagates at the speed of light and that there is no aberration phenomenon when a charge is moving. This conclusion about the aberration of a moving charge can be extended, surprisingly, to any pushing gravity theory, like the one of Nicolas Fatio or Le Sage. But, more than that, the creation of a dynamic electric field leads us to pose the issue of the existence of an absolute universe, which has been denied by Mach and Einstein, for inertia. Also, the radiation by electrons circulating in a circular ring at high constant velocity, like in a synchrotron facility, which radiation is observed everyday, is not so clear a situation to state if an uniformly accelerated charge does radiate or not. By addressing these issues, we will demonstrate, following Feynman’s proposal, that a charged particle can create a photonic radiation (and not a wave as too often said) only when the charged particle is subject to a variable acceleration. Therefore, since there is no radiation when an electron is uniformly accelerated, it resolves both the Paradox and the case of the Niels Bohr hydrogen atom model. We will also show that the electrostatic field is an emergent phenomenon induced in the vacuum by the sole presence of electric charges (electron and positron only, no quarks! [1]). 2 2.1 Electric Charge moving at a constant velocity Heaviside Derivation The electric field “radiated” from a moving electric charge at a constant velocity has been derived for the first time by Oliver Heaviside in 1888 and then was confirmed afterwards by many scientists with 1 different methods. The result has puzzled the scientists since seemingly, with this formula, the electric field was going faster than light speed! But hopefully, we will see that’s not the case. The verb “radiated” is used as if we knew the field was emanating from the electric charge, but we don’t really know where it comes from. In “quantum field theory” of quantum mechanics the electric and magnetic fields produced by electric charges are carried through space by quantum particles, called bosons. Nevertheless, to simplify, we will continue to hypothesize that the fields are “emanating” from the charge. The equations for the electric field and the magnetic fields, radiated by an electric charge q moving with a velocity v, and received by an observer, established by Heaviside, are given by: 2 − → E = (1 − vc2 ) q − → r0 ∗ 4πϵ0 r03 [1 − vc22 sin θ2 ] 23 (1) with ε0 the vacuum permittivity, r0 the distance observer-charge at the instant of observation, θ the angle → → between − r0 and − v , c the speed of light. Since the electric field is varying, due to the electric charge velocity, there is a magnetic field given by: − → → v ×E − → − (2) B = c2 → As it can be seen, the direction of the electric field vector is in the direction “observer-charge”, − r0 , at the time of the observation, whereas the electric field should propagate at a limited speed! This is quite strange, but a close examination of the derivation of the formula can show that the electric field is truly propagating at the limited speed c, with a direction of the vector electric field such that it is in the proper direction, "observer-charge", at the future time of observation. It was this result which astonished Feynman, he said [2] "Nature seems to be attempting to guess what the field at the present time is going to be, by taking the rate of change and multiplying by the time that is delayed". The formula for the electric field was so important for the development of the theory of a moving electron, that many scientists provided alternative derivations, but in turn they all got the same result, which confirmed this weird result for the electric field. One of the confirmations of the derivation was done in 1906 by H. Poincaré, using Lorentz transformation of coordinates. This is also confirmed by A. Hofmann, in § 2.4 of [3], by reducing this situation to the classical electrostatic case in the electron reference frame and the use of the Lorentz transformation to the observer reference frame. The reader must be aware that using the Lorentz transformation to find the emission in the laboratory rest frame is only a mathematical operation which has nothing to do with special relativity transformation, as we will show with Oleg D. Jefimenko’s derivation: the nature of the phenomenon is explained by simple physics and not by mathematics! 2.2 Jefimenko Derivation Indeed, a very interesting derivation has been obtained by Oleg D. Jefimenko [4], which leads to the Heaviside formulas, by only taking into account the deformation of the source, viewed by the static observer, due to the constant velocity of the propagating fields. If we suppose that the electric field is “emitted” from the moving charge, with non null dimensions, it can be shown that the geometry of the charge is altered by the velocity of the charge as viewed from the static observer. There are two retardation effects resulting from the motion of the charge and the finite propagation velocity of the field. The first effect is the change of apparent length of the moving charge. If we take the example of a train moving toward a human observer, the light received from the rear end took place at an earlier time than the light received at the same instant from the front. In consequence the train 2 appears longer. The second effect is the apparent change of the shape of the front and the rear end which are slanted by the motion of the charge for the same reason and the rear is more slanted than the front. In conclusion to this demonstration, it is only the geometric deformation of the moving charge viewed from a static observer which leads to Heaviside’s mathematic formula. Also, from this formula we can see that the electric field is always in the actual direction of observation, making the dynamic electrostatic force a central force. This natural derivation by Jefimenko is important since it demonstrates that there is no aberration phenomenon when electric charges are in relative motion. This conclusion about aberration is valid for every theory in which a force is due to the propagation of information from a source, in all directions with a constant velocity to a receptor, for example like in Nicolas Fatio’s or Le Sage’s theory of gravitation. Since the electric charge does not know the presence of another possible electric charge and its relative velocity, it is reasonable to think that the dynamic electrostatic field of an electric charge is created in the vacuum, considered as an "absolute" static reference, by its sole presence, whatever the presence of another charge. The existence of an "absolute" static universe has been considered in the past for the phenomenon of inertia of a mass body or for the propagation of electromagnetic waves. For inertia, Newton shows from his bucket experiment that the motion of the fluid, when the bucket is turning around an axis, can only come from the exterior environment of the turning bucket and not from a consideration of relative velocity or relativity with respect to the observer or the bucket’s wall [5]. For Maxwell and its predecessors, it was the medium responsible of the electromagnetic wave propagation. Nowadays, its reality is still at discussion, even if Einstein GR theory is supposed to have the right answer! Tony Rothman [6] said in the American Scientist: "A century after Ernst Mach and Albert Einstein cast doubt on absolute space, we still don’t know how a gyroscope stays pointed in a fixed direction". This unexplained question, about the gyroscope behavior or Newton’s bucket experiment and then the existence of a static universe, could it be clarified by the phenomenon of the creation of dynamic electric fields? We can’t respond to this question, but it seems reasonable to think that the electrostatic field doesn’t come from the charge itself but is created in the (static) vacuum, which possesses the necessary energy. Also, in the phenomenon of inertia for mass, there are attempts to consider a static universe, simply being the vacuum ether which embedded all matter atoms, which would be responsible for this phenomenon of mass inertia. In this context, Giorgio Toro’s experiment, using a pendulum and a lead disk moving with different orientations with respect to the direction of motion of the pendulum, shows a possible effect of the vacuum ether on the period’s pendulum and then on the inertia of the discs [7]. Nevertheless, this experimental effect has to be confirmed and interpreted before stating that it validate the concept of an absolute static universe. Since the electric field is varying, there is also the creation of a magnetic field given by (2). Then the question is: is it really an electrostatic field or an electromagnetic wave? The electric field in the formula above is decreasing as 1/r2 meaning that it is not an electromagnetic radiation but solely a varying dynamic electrostatic field. 3 Electric Charge uniformly accelerated This other situation, the radiation of an electric field by an electric charge uniformly accelerated, has been treated by Joseph Larmor in 1897. But, since then, there has been no definite consensus by the scientific community about the real existence of a radiation or not, created by the accelerated charge. All the more since it generates a paradox in the context of general relativity. Maybe scientists think that it is not an issue since there is electromagnetic radiation in a synchrotron facility where the electrons are supposed to be uniformly accelerated. But, we will see that it is not so clear. To say it shortly, theoretically according to Larmor and others it should radiate and according to Richard Feynman it should not! [8] [9]. 3 According to Larmor, the value of the possible radiated electric field in a direction θ with respect to the accelerated direction, is given by: q 1 E= a sin θ (3) 4πε0 rc2 with a the constant acceleration. And the radiated power: q 2 a2 (4) P = 6πc3 ε0 Since the decay of the electric field is in 1/r, it was concluded by Larmor that it radiates an electromagnetic wave. However, the mathematical description of electromagnetic phenomena was explored long before the discovery of the duality of the electromagnetic radiation as a particle and a wave (photon). In consequence, the mathematical description of these physical phenomena was done in the context of the wave theory. This is the issue of the evolution of the hypothesis with the progress of science which implies that we have to be careful about previous conclusions which can become totally obsolete. Richard Feynman disagreed and said [9] “we have inherited a prejudice that an accelerating charge should radiate”. He says “the work per unit time done against the radiation reaction force for a particle moving along the x-axis is given by: 2 e2 dx d3 x dW = − 3 ( )( 3 ) (5) dt 3 c dt dt meaning that the reaction force (and therefore the radiated power) is proportional to the third derivative of position”. According to Richard Feynman’s formula, a uniformly accelerated charge doesn’t radiate. This can be seen from the power emitted, according to Feynman, and given by: − →→ q 2 ȧ − v P = (6) 6πε0 c3 − → and since ȧ = 0, then no energy is radiated. In the Feynman approach it is the work done on the electron to accelerate it which will be transmitted to something . . . maybe a particle called a photon! Who, between Larmor or Feynman, has the correct explanation? As said before, there is no consensus in the scientific community. But according to A.Y.Shiekh and C.C. Vuille, 2022 [10], the Larmor approach is not convincing since they showed that it needs an infinite force required for accelerating a charged particle initially at rest! They also proposed some experiments in order to definitely decide between these two approaches. There are situations wherein we could state in favor of one or the other theory: a bounded electron in the hydrogen atom and the synchrotron radiation. In the hydrogen atom model, a bounded electron moving around a proton, is an example of non radiation often cited in the scientific literature. This electron is constantly accelerated toward the proton, due to the dynamic electrostatic central force, and if the electron were radiating according to the Larmor formula, it will loose energy and will fall down on the proton in a fraction of a second and we know that this is not the case. This issue was a nightmare for the works on the atomic model and since nobody found a clear reason for the non emission of a bounded electron, Niels Bohr postulated that orbiting electrons did not radiate. Fot the synchrotron radiation, at first sight, we could say that it is Larmor who is right, since electrons gyrating in the facility are radiating photons. But as we will see in the following this is not true. The point we will develop in the chapter dedicated to the synchrotron radiation is that the acceleration on the electrons is not constant when it radiates! But still, in the literature most scientists are using Larmor’s formula. Before that, we just have a look at the Paradox of radiation of charged particles in a gravitational field. 4 4 Paradox of radiation of charged particles in a gravitational field The simple situation of an electric charge uniformly accelerated in a gravitational field is an issue that give rise to a Paradox in the context of general relativity (Wikipedia,[11]). The paradox comes when you imagine an electric charge at rest in an earth laboratory. In order to maintain the particle at rest, it must be maintained by something which compensates for the downward earth gravitational force of 1g. In the general relativity theory, due to the equivalence principle between gravitation and inertia, it is assumed that this situation is equivalent as being in outer space and constantly accelerated upward at 1g, which we know by resolving Maxwell’s equations should radiate electromagnetic energy given by Larmor’s formula (3). But the fact is that an electron at rest in an earth laboratory has never been reported to radiate energy. Then, either the equivalence principle in the context of general relativity theory is untrue, or the Larmor formula is untrue. There have been some discussions to show how to solve this paradox and we let the reader have a view to Anwar Shiekh and Chris Vuille, July 2022 [10]. In Anwar Shiekh and Chris Vuille paper, the Feynmann’s version of the power emitted, in the context of quantum mechanics, alternative to the Larmor’s formula, shows that a uniformly accelerated charge does not radiate, which seems to solve the Paradox and is coherent with the equivalence principle. In this Feynmann’s formula appears the time derivative of the acceleration, as we have seen in the preceding chapter, which is zero in the case of constant acceleration. Then there is no Paradox in the context of general relativity if we believe Feynman is right. Nevertheless, it is still a controversy in the scientific community. 5 Synchrotron radiation The synchrotron radiation is an example of the emission of a photonic radiation by a charge moving with a supposed uniform (centripetal) acceleration in a circular trajectory around a central point. This type of radiation is created by man in a facility called a synchrotron (Nicolas Carmignani, [12]) and created naturally in stars and galaxies. A synchrotron facility is able to accelerate a bunch of electrons to reach a speed close to the speed of light until they are stored in a “quasi-circular” storage-ring. Due to their high speed they are called relativistic electrons, but this doesn’t automatically mean that special relativity is responsible for the radiation emitted. In such a facility the electrons stored in the “quasi-circular” ring emit photonic radiation and we could conclude that it contradicts what was developed in chapter III. In reality we have to go into the details of the facility to see where the devil is hiding! The fact is that the storage ring is not at all circular. It is constituted of straight sections, laid end-to-end, connected by bending magnets which produce a strong acceleration at this location to keep the bunch of electrons in the new direction of the following straight section and moving at a constant velocity. So it is entirely false to say that, in a synchrotron facility, the electrons are moving with a circular motion. The radiation occurs only at the location of the bending magnets. There, the acceleration is not continuous but varying very strongly from zero to whatever value and then decreases again to zero. It should be this rapid variation of the acceleration which is responsible for the synchrotron radiation, according to Feynman’ equation. This looks like the phenomenon of radiation from an atom bound electron, leaving its orbit to another orbit with a different velocity, then subject to a variable acceleration. To sum up the situation: 5 1. In the storage ring, an electron is moving with a constant velocity in straight sections, near the speed of light. As shown in the preceding chapters, this constant motion of the electron produces a dynamic electrostatic field. This electrostatic field is of no importance in the experiment, especially since it decreases as 1/r2 , 2. At a certain location there is a bending magnet which is giving a very short variable acceleration to the electron towards its guiding center, which produces the photonic radiation at a certain angle. Traditionally the emission in the reference frame of the electron is derived using the Larmor formula in this frame of reference. But we have seen it is probably not correct. To get the opening angle of the synchrotron radiation in the laboratory rest frame it is done by the use of the Lorentz transformation. Again, using the Lorentz transformation to find the emission opening angle in the laboratory rest frame is only a mathematical operation which has nothing to do with the special relativity transformation. The angle of the radiation in the facility rest frame is in a small cone, which length value is 1/γ, with γ = p 1 2 , v the constant electron velocity and c the speed of light [3] 1− vc2 [12]. The power radiated by the electron is: 1. with the Larmor formula: 1 e2 c 1 4 γ 6 πε0 ρ2 (7) − →→ 1 e2 c ȧ − v 4 γ 6 πε0 c4 (8) P = with ρ the radius of the pseudo circular ring. 2. with the Feynman formula: P = Since the acceleration in the bending magnet short section goes from zero to a, and the velocity of the electrons is v, then we can approximate the acceleration by: a = v 2 /ρ, and supposing v = c, we can rewrite Larmor’s version (7) as: P = 1 e2 c a2 4 γ 6 πε0 c4 (9) − →→ We can see the difference between the two versions by comparing ȧ − v to a2 . This difference may or may not be significant. In the literature you don’t see any comparison of the power radiated by a synchrotron with the theoretical value obtained by Larmor’s formula. Thus, it is not possible to conclude which approach, Larmor’s or Feynman’s, is the correct one, but it should be possible to do. Nevertheless, there is a high probability that Feynman’s version is the true one, since a bounded electron, which is uniformly accelerated, doesn’t radiate when it is in its orbit and radiates only when it leave its orbit. This conclusion is validating Bohr’s premonition. 6 Radio Emission In an antenna wire, the electrons are pushed back and forth with a variable acceleration. The simplest way to see that, is to imagine a particle circulating on a circle and to project the motion of this particle on a straight line. The motion of the particle on the circle being at constant velocity, we can see that the motion projected on the straight line can be described by a point going back and forth as a sinusoid and with a variable acceleration. Then, from what precede, we know that an accelerated electron with a variable acceleration will radiate an electromagnetic wave. 6 Another thing is about the photon existence for a radio emission which is often an interrogation. There is no clear answer to that question, people just saying that they think there are photons or there are not, which is not a scientific answer. The term radio emission covers a large frequency domain from about 100 kHz to 500 MHz. It is also employed in astronomy for radio sources which are emitting at the order of the GHz. Then why this question, since all electromagnetic waves are supposed to be dual, as a particle and a wave? This question comes from the wavelength of a radio emission which can be around 103 m. With such a large wavelength, it is difficult to imagine a particle the size of an electron with an associated wave 1020 times its size. However, it is explained by Planck’s formula, which relates the energy of the photon, transported by the wave, to the wave frequency, E = hν, with ν the frequency of the wave. It probably means that to make the vacuum “vibrate” at a frequency ν you have to provide a corresponding energy given by the Planck’s formula. Then, this question is more a question about the physical nature of the vacuum, which is able to transform an energy given by an electron, to another particle with an associated wave propagating in it via the Planck’s constant. For the nature of the vacuum we have to consider not only the permittivity and the permeability characteristics of the medium but also the Planck’s constant to be part of the process of creation of an electromagnetic radiation. But, to go back to the question of the reality of a particle and a wave in radio emission, I have seen that this question leads to the question of how to detect such a radiation? Like we detect a light photon with a certain size of a detector, for a radio emission we have also to adjust the size (or the impedance) of the detector to the corresponding wavelength. And doing so, we are sure that if there is a particle in the radiation, it will be detected. Then it is reasonable to think that the notion of a particle, like a photon in what we know for light, is real also for a radio emission. Maybe the last question about radio emission is: what the radio particle will do to an electron in the detector? The size of the detector was to get the entirety of the radio wave, and in consequence the particle, but what the particle will do? Radio detection is done by the wave itself, then what could be the role of the particle? In conclusion, all electromagnetic radiation, up to the km wavelength, are constituted of a particle and a wave, with the wave frequency given by the Planck’s formula. In order to understand how we can detect such a point particle, we have to have in mind that the detector must have an appropriate size to the size of the wavelength of the associated wave. 7 Conclusion It is reasonable to think that the dynamic electrostatic field of a moving electric charge is created in the vacuum, considered as an "absolute" static reference, by its sole presence whatever the presence of another charge. The existence of an absolute static universe as a reference frame is still at discussion, particularly for the phenomenon of inertia. From the Heaviside equation for a moving electric charge with constant velocity, we can conclude that there is no aberration phenomenon between two electric charges in relative motion. This aberration conclusion can be extended to every theory in which the information is propagated at a finite speed. Then this applies to pushing gravity theories like Nicolas Fatio’s or Le Sage’s gravitational theories. When an electron is subject to a variable (short) acceleration with respect to the vacuum (supposed to be static), it radiates a photonic emission, composed of a particle and a wave (called a photon), in the sense of Bohm quantum mechanics, if we accept Feynman’s approach. The energy of the reaction of the vacuum to the variable acceleration of the electron is transferred to the radiated photon. In the case of the synchrotron radiation, the use of the Larmor formula is erroneous. In reality, synchrotron radiation is due to a strong variable acceleration, located at the bending magnets, instead of a uniformly acceleration around a circular ring, as it is often said. A comparison between the two 7 approaches, Larmor and Feynman, could be made by the use of the synchrotrons’s data and it seems it has never been done. For a uniformly accelerated electric charge, Larmor’s formula is still used in the scientific community. Feynman’s formula should be use instead, and then, in this case there is no radiation and no more Paradox in the general relativity context. It is reasonable to think that these radiations are done by the reaction of the vacuum to the motion of electric charges. By its velocity in the case of electrostatic radiation and by its high variation in acceleration in the case of photonic radiation. The value of the electric field produced is dependent on the velocity (measured in the observer referential) or the varying acceleration (absolute), but once the radiation is emitted in the vacuum, it can be observed in whatever other referential. It can be noted that the direction of the electric field is different in these two cases: along the direction charge-observer for electrostatic field’s radiation, whatever value of the charge speed is, and perpendicular to the direction of radiation in the photonic radiation. It is untrue to say that electromagnetic radiation is decreasing with 1/r since it is not simply a wave, but a photonic radiation, which is not decreasing at all with distance. If it was, we would never see any bright stars in the sky. Nevertheless, it is true that the energy of electromagnetic radiation is decreasing in 1/r2 , since it obeys Newton’s square law for radiation from a point source, whose derivation is totally different from the one considering a propagating wave. This new point of view of the radiation by moving electric charges is of importance for those who want to develop a model for the nature of the vacuum in presence of electric charges. In particular, we have to consider not only the permittivity and the permeability characteristics of the medium but also the Planck’s constant to be part of the process of creation of a photonic radiation. The existence or not of a static universe is also part of the issue. References [1] Proton and Neutron Electric Charges. DOI:10.5281/zenodo.7623966. Didier F. Viel. [2] Dynamic effects in Le Sage Model. Paul Stowe; Edited by Matthew R. Edwards: Pushing Gravity. 2002. [3] Characteristics of synchrotron radiation. A. Hofmann. CERN, Geneva Switzerland. [4] Oleg D. Jefimenko. 1993. Direct calculations of the electric and magnetic fields of an electric point charge moving with constant velocity. American Journal of Physics 62, 79-85 (1994). [5] https://www.youtube.com/watch?v=PIKx-fGH5e0 [6] https://www.americanscientist.org/article/the-forgotten-mystery-of-inertia. American Scientist. The Forgotten Mystery of Inertia. Tony Rothman. [7] Inertia & Shape 2. November 2023. Giorgio Toro. Academia. [8] On the radiation from a uniformly accelerated charge. Ikka Mäkinen. Uniformly accelerated charge Makinen.pdf [9] mathpages.com/home/kmath528/kmath528.htm. [10] A.Y. Shiekh and C.C. Vuille. Does a charge in a gravitational field radiates? (The accelerating charge Paradox). Anwar Shiekh and Chris Vuille, july 2022. DOI:10.13140/RG.2.2.29515.46888/2. Research Gate. 8 [11] Wikipedia. Paradox of radiation of charged particles in a gravitational field. https://en.wikipedia.org/wiki/Paradox of radiation of charged particles in a gravitational field. [12] Nicolas Carmignani. 2021. Principle of synchrotron radiation.The European Synchrotron. ESRF. 9