Exact Expression for Radiation of an Accelerated Charge in Classical Electrodynamics

The present expression of radiation of an accelerated point charge is only approximately valid. The exact expression of radiation of an accelerated point charge is derived based on special relativity, and using the Larmor formulation for the radiation of an charged particle being accelerated, but instantaneously at rest. The totaled radiation power obtained by the exact expression is the same as Liénard’s generalization of the Larmor formula.     

Radiation from a Uniformly Accelerated Charge

The emission of radiation by a uniformly accelerated charge is analyzed. According to the standard approach, a radiation is observed whenever there is a relative acceleration between the charge and the observer. Analyzing difficulties that arose in the standard approach, we propose that a radiation is created whenever a relative acceleration between the charge and its own electric field exists. The electric field induced by a charge accelerated by an external (nongravitational) force is not accelerated with the charge. Hence the electric field is curved in the instantaneous rest frame of the accelerated charge. This curvature gives rise to a stress force, and the work done to overcome the stress force is the source of the energy carried by the radiation. In this way, the energy balance paradox finds its solution.     

Origin of Radiation Reaction

The emission of radiation from an accelerated charge is analyzed. It is foundthat at zero velocity, the radiation emitted from the charge imparts no countermomentum to the emitting charge, and no radiation reaction force is created bythe radiation. A reaction force is created by the stress force that exists in thecurved electric field of the charge, and the work done in overcoming this forceis the source of the energy carried by the radiation.     

Characteristic Length in a Linear Acceleration

The role of the characteristic length that characterizes linear acceleration is studied, in order to find how does this length determine the characteristic wavelength of the radiation created by the accelerated charge. Unruh equation for the temperature observed by a detector accelerated relative to the vacuum is used to determine the wavelength distribution of the radiation emitted by a linearly accelerated charge, and it is found that this distribution is peaked close to the characteristic length that characterizes the linear acceleration, which is the radius of curvature of the curved electric field created by the acelerated charge. PACS numbers: 11.10; 41.60.m.     

The Equivalence Principle and an Electric Charge in a Gravitational Field II. A Uniformly Accelerated Charge Does Not Radiate

The electromagnetic field of a charge supported in a uniform gravitational field is examined from the viewpoint of an observer falling freely in the gravitational field. It is argued that such a charge, which from the principle of equivalence is moving with a uniform acceleration with respect to the (inertial) observer, could not be undergoing radiation losses at a rate implied by Larmor's formula. It is explicitly shown that the total energy in electromagnetic fields, including both velocity and acceleration fields, of a uniformly accelerated charge, at any given instant of the inertial observer's time, is just equal to the self-energy of a non-accelerated charge moving with a velocity equal to the instantaneous present velocity of the accelerated charge. At any given instant of time, and as seen with respect to the present position of the uniformly accelerated charge, although during the acceleration phase there is a radially outward component of the Poynting vector, there is throughout a radially inward Poynting flux component during the deceleration phase, and a null Poynting vector at the instant of the turn around. From Poynting's theorem, defined for any region of space strictly in terms of fixed instants of time, it is shown that a uniformly accelerated charge does not emit electromagnetic radiation, in contrast to what is generally believed. Contrary to some earlier suggestions in the literature, there is no continuous passing of electromagnetic radiation from a uniformly accelerated charge into the region inaccessible to a co-accelerating observer.     

Radiation from a uniformly accelerated charge

The electromagnetic field associated with a uniformly accelerated charge is studied in some detail. The equivalence principle paradox that the co-accelerating observer measures no radiation while a freely falling observer measures the standard radiation of an accelerated charge is resolved by noting that all the radiation goes into the region of space time in-accessible to the co-accelerating observer.     

Electrodynamics of hyperbolically accelerated charges V. The field of a charge in the Rindler space and the Milne space

We describe the electromagnetic field of a uniformly accelerated charge in its co-moving Rindler frame. It is shown that the electrical field lines coincide with the trajectories of photons. The self force of a charged particle at rest in Rindler space, and the increase of its weight due to its charge, is calculated. The general case of an accelerated charge in Rindler space is also considered. It is shown that the electrical field inside a uniformly charged spherical shell can be used as a measure of it 4-acceleration. A result that has earlier been deduced in a different way by Fugmann and Kretzschmar is confirmed, namely that the intensity of radiation from a point charge instantaneously at rest in an accelerated frame is proportional to the square of the relative acceleration of the charge and the observer. In particular it is shown that a freely falling charge in Rindler space radiates in accordance with Larmor’s formula. In this case the radiation energy is taken from the Schott energy. The energy of the electromagnetic field is analysed from the point of view of the Hirayama-separation, which generalizes the Teitelboim-separation to non-inertial frames, of the field in a bound part and an unbound part. A detailed account, with reference to the Rindler frame, of the field energy and particle energy is given for the case of a charge entering and leaving a region with hyperbolic motion. We also consider the electromagnetic field of a uniformly accelerated charge with reference to the Milne frame, which covers a different part of spacetime than the Rindler frame. The radiating part of the electromagnetic field is found in the Milne sector of spacetime.     

Correlation phenomena in the Cherenkov radiation of accelerated multiply charged ions

Correlation phenomena occurring in Cherenkov radiation are considered which are related to fluctuations of the charge states of multiply charged accelerated ions in a medium. The additional correlation contribution to the radiation is determined by the root-mean-square deviation of the ion charge from its equilibrium value and is responsible for the nonzero radiation yield in the event that the threshold condition is not fulfilled. Numerical estimates of the radiation yield of heavy ions in the optical and X-ray frequency ranges are given. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 41–46, January, 2009.     

The charged particle radiation field in the electrodynamics of moving media

We solve Sommerfeld's problem of the accelerated charge radiation in a uniformly moving simple medium.     

Equation of Motion of an Electric Charge

The appearance of the time derivative of the acceleration in the equation of motion (EOM) of an electric charge is studied. It is shown that when an electric charge is accelerated, a stress force exists in the curved electric field of the accelerated charge, and in the case of a constant linear acceleration, this force is proportional to the acceleration. This stress force acts as a reaction force which is responsible for the creation of the radiation (instead of the radiation reaction force that actually does not exist at low velocities). Thus the initial acceleration should be supplied as an initial condition for the solution of the EOM of an electric charge.     

Self-force of a charge in a real current

The analysis of the EM radiation from a single charge shows that the radiated power depends on the retarded acceleration of the charge. Therefore for consistency, an accelerated charge, free from the influence of external forces, should gradually lose its acceleration, until its total energy is radiated. Calculations show that the self force of a charge, which compensates for its radiation, is proportional to the derivative of the acceleration. However, when using this self force in the equation of motion of the charge, one gets a diverging solution, for which the acceleration runs away to infinity. This means that there is an inconsistency in the solution of the single charge problem. However, in the construction of the conserved Maxwell charge density, there is implicitly an integral over the corresponding world line which corresponds to a collection of charged spacetime events. One may therefore consistently think of the “self force” as the force on a charge due to another charge at the retarded position. From this point of view, the energy is evidently conserved and the radiation process appears as an absorbing resistance to the feeding source. The purpose of this work is to learn about the behavior of single charges from the behavior of a real current, corresponding to the set of charges moving on a world line, and to study the analog of the self force of a charge associated with the radiation resistance of a continuum of charges.     

A Coaccelerated Observer

We analyze the situation of an observer coaccelerated relative to a linearly accelerated charge, in order to find whether he can observe the radiation emitted from the accelerated charge. It is found that the seemingly special situation of the coaccelerated observer relative to any other observer, is deduced from a wrong use of the retarded coordinate system, when such a system is inadmissible. It is also found that the coaccelerated observer has no special position other than any other observer, and hence, he can observe any physical events as any other observer.     

Correlation effects of multiple charged accelerated ions in Cherenkov radiation

The correlation effects in Cherenkov radiation associated with fluctuations of multiple charged accelerated ions in the medium are considered. The contribution of additional correlation into the radiation is determined by the ion charge mean-square deviation from its equilibrium value and defines the non-zero radiation yield as the threshold conditions are not fulfilled. The numerical estimations of the yield of Au ion radiation in the optical and X-ray frequency range in carbon are presented.     

Photographic observation of aluminum micropellet ablation in theMT-1M tokamak plasma

Laser beam accelerated aluminum micropellets were injected into the MT-1M tokamak plasma, and the distribution of characteristic line radiation of aluminum atoms and ions of different charge state was detected. The investigations are focused on the visualization of the pellet cloud, on the bending of the pellet path, and on the striation occurring in the radiation of different charge state along the pellet path in the plasma     

Electrodynamics in accelerated frames revisited

Maxwell's equations are formulated in arbitrary moving frames by means of tetrad fields, which are interpreted as reference frames adapted to observers in space‐time. We assume the existence of a general distribution of charges and currents in an inertial frame. Tetrad fields are used to project the electromagnetic fields and sources on accelerated frames. The purpose is to study several configurations of fields and observers that in the literature are understood as paradoxes. For instance, are the two situations, (i) an accelerated charge in an inertial frame, and (ii) a charge at rest in an inertial frame described from the perspective of an accelerated frame, physically equivalent? Is the electromagnetic radiation the same in both frames? Normally in the analysis of these paradoxes the electromagnetic fields are transformed to (uniformly) accelerated frames by means of a coordinate transformation of the Faraday tensor. In the present approach coordinate and frame transformations are disentangled, and the electromagnetic field in the accelerated frame is obtained through a frame (local Lorentz) transformation. Consequently the fields in the inertial and accelerated frames are described in the same coordinate system. This feature allows the investigation of paradoxes such as the one mentioned above.     

Comment on “Exact Expression for Radiation of an Accelerated Charge in Classical Electrodynamics”

It is shown that a newly derived “exact expression” for radiation of an accelerated charge in the recent literature is simply incorrect, having arisen because of a wrong relativistic transformation of the distance parameter. The ensuing claim that the newly derived expression alone satisfies the energy conservation for the electromagnetic radiation, is based on a wrong reasoning where a proper distinction between the time during which the radiation is received and the time for emission (retarded time of the charge) was not maintained.     

Resolution of the fields of an accelerated charge into their bradyonic and tachyonic parts

A resolution of the fields of an accelerated charge in vacuum into their bradyonic and tachyonic parts is given. It is shown that the bradyonic part contributes only to the induction fields, whereas the tachyodic part gives rise to the radiation emitted by the charge.     

Radiation From an Electric Charge

The conditions in which electromagnetic radiation is formed are discussed. It is found that the main condition for the emission of radiation by an electric charge is the existence of a relative acceleration between the charge and its electric field. Such a situation exists both for a charge accelerated in a free space, and for a charge supported at rest in a gravitational field. Hence, in such situations, the charges radiate. It is also shown that relating radiation to the relative acceleration between a charge and its electric field, solves several difficulties that existed in earlier approaches, like the energy balance paradox, and the relativistic nature of the observation of the emitted radiation.     

On the Energy Radiation from an Accelerated Charge

In the framework of classical electromagnetism, a charge however accelerated with respect to an inertial frame radiates energy, in any circumstance. Regarding the energy as made of photons, the hypothesis is here introduced that the emission of a photon is only possible as a result of a change of the energy of the charge, which requires an energy-work exchange with the accelerating field. On such an hypothesis an elementary impulsive-dissipative model for the photon emission is constructed, in the framework of special relativity, in which no energy radiation is emitted from a charge in a central Coulomb field uniformly describing a circular orbit.     

The symmetry,inferable from Bogoliubov transformation,between processes induced by a mirror in two-dimensional and a charge in four-dimensional space-time

We consider the symmetry between creation of pairs of massless bosons or fermions by an accelerated mirror in (1+1)-dimensional space and emission of single photons or scalar quanta by an electric or scalar charge in (3+1)-dimensional space. The relation of Bogoliubov coefficients describing the processes generated by a mirror to Fourier components of the current or charge density implies that the spin of any disturbances bilinear in the scalar or spinor field coincides with the spin of quanta emitted by the electric or scalar charge. The mass and invariant momentum transfer of these disturbances are essential for the relation of Bogoliubov coefficients to invariant singular solutions and the Green functions of wave equations for both (1+1)-and (3+1)-dimensional spaces, and especially for the integral relations between these solutions. One of these relations leads to the coincidence of the self-action changes and vacuum-vacuum amplitudes for an accelerated mirror in two-dimensional space-time and a charge in four-dimensional space-time. Both invariants of the Lorentz group, spin and mass, play an essential role in the established symmetry. The symmetry embraces not only the processes of real quanta radiation, but also the processes of the mirror and charge interactions with fields carrying spacelike momenta. These fields accompany their sources and determine the Bogoliubov matrix coefficients α _ω′ω ^{B, F} . It is shown that the Lorentz-invariant traces ±trα^B,F describe the vector and scalar interactions of the accelerated mirror with a uniformly moving detector. This interpretation rests essentially on the relation between propagators of the waves with spacelike momenta in two-and four-dimensional spaces. The traces ±trα^{B, F} coincide with the products of the mass shift Δm_{1, 0} of the accelerated electric or scalar charge and the proper time of the shift formation. The symmetry fixes the value of the bare fine structure constant α₀=1/4π.