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The flow of electromagnetic energy in the decay of an electric dipole
Abstract
In a typical dipole decay, the induction magnetic field and the radiation magnetic field are in opposing directions. The former is dominant close to the decay, and the latter is dominant far away. Thus, there exists a surface on which these two components cancel each other, so that the magnetic field reverses direction. Because the magnetic field changes direction, so also will the Poynting vector field, provided the electric field is reasonably well behaved. This defines a "causal surface," within which electromagnetic energy will collapse, and outside which the electromagnetic energy will radiate away. I present a simple example, the decay of a point electric dipole, to argue that the radiated energy comes, not necessarily from the accelerating charges themselves, but from the energy stored in the far field.
... We make special note that the Larmor formula describes radiation in the 'far zone', and does not tell the story closer to the radiation charge. For example, if a (classical) electric dipole starts to collapse at a certain time, the Larmor formula describes the radiation in the far zone; however, close to the dipole the energy flows from that initially stored in the electromagnetic field onto the collapsing dipole; see, e.g., [16]. (7). ...
... Considering the close association between accelerating mirrors and charges, we postulate that the electron spectrum has the same form as Equation (16). See Figure 6 for an illustration of the symmetry between the modes, and see Figure 7 for a plot of the spectrum after integration over ω , (17)), which is number of particles at any given frequency ω, demonstrating the usual infrared divergence that signals a divergent total particle count N. ...
... The |β ωω | 2 spectrum of Equation(16) in a contour plot. The symmetry demonstrates that the summing of either ω quanta (Equation(23)) or ω quanta (Equation(24)) results in the same energy being radiated. ...
We examine the extreme situation of radiation from an electron that is asymptotically accelerated to the speed of light, resulting in finite emission energy. The analytic solution explicitly demonstrates the difference between radiation power loss and kinetic power loss (null).
... In Refs. [4,23,24], the time-varying electric field was derived and written for the electric dipole moment taking into account the retardation effect. According to those references, we have ...
... Thus, the temporal electromotive force and the radiated power are subsequently the same as what we derived before. Regarding magnetic dipole moments, the magnetic flux density is expressed as [4,23,24] ...
Radiation from magnetic and electric dipole moments is a key subject in theory of electrodynamics. Although people treat the problem thoroughly in the context of frequency domain, the problem is still not well understood in the context of time domain, especially if dipole moments arbitrarily vary in time under action of external forces. Here, we scrutinize the instantaneous power radiated by magnetic and electric dipole moments, and report findings that are different from the conventional understanding of their instantaneous radiation found in textbooks. In contrast to the traditional far-field approach based on the Poynting vector, our analysis employs a near-field method based on the induced electromotive force, leading to corrective terms that are found to be consistent with time-domain numerical simulations, unlike previously reported expressions. Beyond its theoretical value, this work may also have significant impact in the field of time-varying metamaterials, especially in the study of radiation from subwavelength meta-atoms, scatterers and emitters that are temporally modulated.
... Just before and after r = r s , the Poynting flow is nonzero (although it changes sign when crossing the critical sphere). This zeroing out of Poynting flow appears to have been first observed by Schantz in [1, 59,60], though from a different scope and perspective of the present work. ...
We provide a conceptual and theoretical analysis of nonsinusoidal antennas with emphasis on how electromagnetics and communication theories can be integrated to propose ideas for near-field (NF) communications systems utilizing future antennas. It is shown through rigorous analysis that in nonsinusoidal antennas it is possible to derive and solve ordinary differential equations giving specialized time-domain excitation signals that lead to exact cancellation of the near field at specific radiation spheres. This opens the door to building NF communications systems with far-field-like communication receiver infrastructures utilized if the receive antenna is placed at the special sphere where the NF component is made to vanish.
... Just before and after r = r s , the Poynting flow is nonzero (although it changes sign when crossing the critical sphere). This zeroing out of Poynting flow appears to have been first observed by Schantz in [1, 59,60], though from a different scope and perspective of the present work. ...
We provide a conceptual and theoretical analysis of nonsinusoidal antennas with emphasis on how electromagnetics and communication theories can be integrated to propose ideas for near-field (NF) communications systems utilizing future antennas. It is shown through rigorous analysis that in nonsinusoidal antennas it is possible to derive and solve ordinary differential equations giving specialized time-domain excitation signals that lead to exact cancellation of the near field at specific radiation spheres. This opens the door to building NF communications systems with far-field-like communication receiver infrastructures utilized if the receive antenna is placed at the special sphere where the NF component is made to vanish.
... Just before and after r = r s , the Poynting flow is nonzero (although it changes sign when crossing the critical sphere). ‡ ‡ ‡ ‡ This zeroing out of Poynting flow appears to have been first observed by Schantz in [1,58,59], though from a different scope and perspective of the present work. ...
... The time constant, τ = RC, depends on the product of associated resistance (R) and capacitance (C). The fascinating details of the fields and energy flow were originally noted by Mandel [23] and examined elsewhere in more detail [24]. ...
Conventional definitions of ‘near fields’ set bounds that describe where near fields may be found. These definitions tell us nothing about what near fields are, why they exist or how they work. In 1893, Heaviside derived the electromagnetic energy velocity for plane waves. Subsequent work demonstrated that although energy moves in synchronicity with radiated electromagnetic fields at the speed of light, in reactive fields the energy velocity slows down, converging to zero in the case of static fields. Combining Heaviside's energy velocity relation with the field Lagrangian yields a simple parametrization for the reactivity of electromagnetic fields that provides profound insights to the behaviour of electromagnetic systems. Fields guide energy. As waves interfere, they guide energy along paths that may be substantially different from the trajectories of the waves themselves. The results of this paper not only resolve the long-standing paradox of runaway acceleration from radiation reaction, but also make clear that pilot wave theory is the natural and logical consequence of the need for quantum mechanics correspond to the macroscopic results of the classical electromagnetic theory.
This article is part of the theme issue ‘Celebrating 125 years of Oliver Heaviside's ‘Electromagnetic Theory’’.
Radiation from magnetic and electric dipole moments is a key subject in the theory of electrodynamics. Although people treat the problem thoroughly in the context of the frequency domain, the problem is still not well understood in the context of the time domain especially if dipole moments arbitrarily vary in time under the action of external forces. Here, we scrutinize the instantaneous power radiated by magnetic and electric dipole moments and report findings that are different from the conventional understanding of their instantaneous radiation found in textbooks. In contrast to the traditional far-field approach based on the Poynting vector, our analysis employs a near-field method based on the induced electromotive force, leading to corrective terms that are found to be consistent with time-domain numerical simulations, unlike previously reported expressions. Beyond its theoretical value, this paper may also have significant impact in the field of time-varying metamaterials especially in the study of radiation from subwavelength meta-atoms, scatterers, and emitters that are temporally modulated.
Textbooks rarely give time-domain solutions to antenna problems. For the
case of a finite linear antenna along which a fixed current waveform
propagates, we present analytical time-domain solutions for the electric
and magnetic radiation (far) fields. We also give computer solutions for
the total (near and far) fields. The current waveform used as an example
in the computer calculations approximates that of a lightning
return-stroke, a common geophysical example of the type of radiation
source under consideration.
The methods of calculating fields due to quasi-steady currents in closed and unclosed circuits are reviewed. It is emphasized that it is sufficient to apply the Biot-Savart law to all the moving charges and to ignore the vacuum displacement current. Attention is drawn to a basic error in the widespread practice of treating a circuit containing a capacitor as though the fringing fields could be ignored and the displacement current replaced by conduction currents confined to the gap between the plates.
An alternative choice for the energy flow vector of the electromagnetic field is proposed by discarding a term ∇×(phi H) from the Poynting vector. This new choice for the energy flow vector is in harmony with our ordinary intuitions, and yields Larmor's radiation formula for an accelerated charge.
This paper seeks to promote a wider use of the two little-known time-dependent, generalized Coulomb and Biot-Savart laws in the form given by Jefimenko's book. For some problems, these two time-dependent laws giving the electromagnetic field from known distributions of charge and current densities without spatial differentiations provide a more convenient method of solution, but they are not well known, probably due to their difficult derivation. This paper gives an alternative derivation to these formulas using a novel ``light cone transformation'' that may be more appealing to some readers than the derivation given in the book by Jefimenko. The paper then shows that together with the law of conservation of charge these two laws can give back Maxwell's system of four equations when the medium is infinite, homogeneous, linear, and isotropic; consequently, the time-dependent, generalized Coulomb, Biot-Savart, and charge conservation laws can be used for such a case instead of Maxwell's four equations. Three elegant examples, which have been traditionally solved in lengthy ways, or using the postulates of the special relativity theory, are then solved using these formulas: They are the examples of a differential antenna, of an electron in a uniform motion, and in an arbitrary motion (Feynman's formulas). The examples also highlight the necessity of using Lienard-Wiechert's formula in typical applications of these time-dependent laws.
We obtain analytical solutions in the time domain for the electric and
magnetic fields associated with establishing a finite electrostatic
dipole. We assume that a simple source current distribution, a square
pulse of current, produces the dipole, and solve for the fields produced
by that source current distribution using Maxwell's equations. Salient
features of the fields are discussed from a physical point of view. We
outline a technique to determine in the time domain the electric and
magnetic fields produced by any arbitrary time-varying current
propagating along a straight antenna, given the calculated fields due to
a short square pulse of current.