Electromagnetic radiation and the self field of a spherical dipole oscillator
11 Electromagnetic radiation and the self field of a spherical dipole oscillator
Masud Mansuripur † and Per K. Jakobsen ‡ † College of Optical Sciences, The University of Arizona, Tucson ‡ Department of Mathematics and Statistics, UIT The Arctic University of Norway, Tromsø, Norway [Published in the
American Journal of Physics , 693 (2020); doi: 10.1119/10.0001348] Abstract . For an oscillating electric dipole in the shape of a small, solid, uniformly-polarized, spherical particle, we compute the self-field as well as the radiated electromagnetic field in the surrounding free space. The assumed geometry enables us to obtain the exact solution of Maxwell’s equations as a function of the dipole moment, the sphere radius, and the oscillation frequency. The self field, which is responsible for the radiation resistance, does not introduce acausal or otherwise anomalous behavior into the dynamics of the bound electrical charges that comprise the dipole. Departure from causality, a well-known feature of the dynamical response of a charged particle to an externally applied force, is shown to arise when the charge is examined in isolation, namely in the absence of the restraining force of an equal but opposite charge that is inevitably present in a dipole radiator. Even in this case, the acausal behavior of the (free) charged particle appears to be rooted in the approximations used to arrive at an estimate of the self-force. When the exact expression of the self-force is used, our numerical analysis indicates that the impulse-response of the particle should remain causal.
1. Introduction . Classical electrodynamics, an elegant theory based on Maxwell’s equations and the Lorentz force law, is a wide-ranging and self-consistent physical theory that is also consistent with special relativity and with the principles of conservation of energy as well as linear and angular momenta.
The theory, however, runs into trouble when attempting to explain the action of small, point-like charged particles upon themselves.
An accelerated charged particle radiates an electromagnetic (EM) field, the action of which on the particle itself could, according to the classical theory, elicit an acausal response from the particle. The particle may thus be required to behave as if it were reacting or responding to an external excitation before the onset of that excitation. In the early years of the twentieth century, Max Abraham and Hendrik Lorentz studied the self-action of an accelerated, electrically-charged particle in the shape of a small sphere, and pointed out the possibility of its acausal behavior. Some thirty years later, Paul Dirac analyzed the relativistic version of the same problem, found a clever way to eliminate the troublesome infinities that had previously hampered the investigations of point particles, and derived an exact solution for the self-force of an accelerated point charge. Unfortunately (for the classical theory), Dirac’s exact solution exhibits the same anomaly of runaway solutions and causality violation that the approximate Abraham-Lorentz theory had previously encountered. With the advent of quantum mechanics and the recognition that Heisenberg’s uncertainty principle forbids the simultaneous specification of the position and momentum of small atomic and sub-atomic particles, it was hoped that the aforementioned foundational problems of the classical theory could be resolved, and that quantum electrodynamics would provide a satisfactory answer to such vexing problems as the violation of causality by an accelerated point charge. Although substantial progress has been made since the formulation of quantum electro-dynamics and a number of classical puzzles have been resolved, the problems associated with the self-force of an accelerated point-charge continue to attract the attention of theoretical physicists to this day.
Much has been written about these problems in textbooks, review papers, monographs, and research articles (to cite only a few), thus making it unnecessary here to discuss the history and the current state of affairs in any great detail. The interested reader can find a good overview of the subject in J. D. Jackson’s
Classical Electrodynamics (Ref. [2], Chapter 16). The monograph by A. D. Yaghjian is an invaluable resource for in-depth understanding of the electrodynamics of charged spheres. Steane describes the pathological behavior exhibited by certain equations of motion in the presence of self-force, and examines a class of formulations that do not show such pathologies. For an overview of attempts to mitigate or eliminate the runaway solutions and/or the predicted acausal behavior of charged particles in the presence of radiation reaction (either according to the Abraham-Lorentz theory, or due to the self-force in Dirac’s fully relativistic formulation), the reader is referred to Rohrlich. Our rather narrow goal in the present paper is to examine a special case of the Abraham-Lorentz problem for which an exact analytical expression for the self-force can be obtained. When the approximate form of this self-force is used to derive the response of the particle to an externally applied impulsive excitation, we find that the emerging acausal behavior is the same as that predicted by the Abraham-Lorentz theory. However, a numerical analysis of the same problem that takes into account our exact expression of the self-force, indicates that the particle’s impulse-response is causal — even when the radius of the particle assumes extremely small values. The conclusion, in agreement with the current understanding of the pathologies associated with small charged particles, is that the acausal behavior predicted by the Abraham-Lorentz theory for a charged particle of finite size may not herald a failure of the Maxwell-Lorentz electrodynamics, but rather be an artifact of the approximations made to arrive at an estimate of the self-force. Our results, however, do not contradict the existence of runaway solutions or the acausal behavior of point charges as predicted by Dirac’s theory, since Dirac’s self-force pertains to a particle of zero-size (i.e., a true point-particle), whereas our analysis applies to particles of finite (albeit very small) dimensions. The organization of the paper is as follows. After a brief synopsis in Sec.2, we solve Maxwell’s equations in Sec.3 for a uniformly-polarized solid sphere of radius 𝑅𝑅 , whose polarization 𝑷𝑷 ( 𝑡𝑡 ) oscillates at a constant frequency 𝜔𝜔 . Here we derive the electric and magnetic fields both inside and outside the sphere and, among other things, find an exact formula for the electric field that is responsible for the radiation resistance. In Sec.4, the damping effect of this self-field is incorporated into the Lorentz oscillator model of the spherical dipole, where we show that well-known classical results such as broadening and shifting of the resonant line-shape and the Thomson scattering cross-section of the dipole emerge when an approximate form of the radiation reaction function is used in the model. We switch gears in Sec.5 and use the exact frequency-dependent transfer function of the Lorentz oscillator obtained in Sec.4 to examine the response of the dipole to an impulsive excitation. This requires a numerical analysis of the poles of the transfer function in the complex 𝜔𝜔 -plane. The exact radiation reaction function will be seen to endow the transfer function with an infinite number of poles, none of which appears to reside in the upper half of the 𝜔𝜔 -plane. We track the evolution of these poles by following their 𝜔𝜔 -plane trajectories as functions of the dipole radius 𝑅𝑅 , and find them to remain in the lower half-plane — even when 𝑅𝑅 becomes exceedingly small (i.e., below the femtometer scale of the classical electron radius † ). The conclusion is that the impulse-response of the dipole is going to be causal, whether the dipole is large (i.e., 𝑅𝑅 ~1 nm ), or has the typical dimensions of a hydrogen atom ( 𝑅𝑅 ~ ) , or becomes as small as a nuclear particle ( 𝑅𝑅 ~1 fm) , or when its radius assumes even smaller values. Finally, in Sec.6, we relax the constraints of the Lorentz oscillator model on the ball of negative charge by eliminating the restraining force of the dipole’s positive charge as well as that † Let a spherical shell of radius 𝑟𝑟 𝑐𝑐 and charge 𝑞𝑞 , where the charge is uniformly distributed over the sphere’s surface, be a model for a stationary electron. Upon integration over the entire space, the 𝐸𝐸 -field energy-density ½ 𝜀𝜀 𝐸𝐸 ( 𝑟𝑟 ) = 𝑞𝑞 (32 𝜋𝜋 𝜀𝜀 𝑟𝑟 ) ⁄ outside the shell yields the total EM energy of the electron as ℰ = 𝑞𝑞 (8 𝜋𝜋𝜀𝜀 𝑟𝑟 𝑐𝑐 ) ⁄ . Equating ℰ to the mass-energy 𝑚𝑚 𝑐𝑐 , one obtains the classical diameter of the electron as 𝑟𝑟 𝑐𝑐 = 𝜇𝜇 𝑞𝑞 (4 𝜋𝜋𝑚𝑚 ) ⁄ ≅ . of the phenomenological spring that is built into the Lorentz oscillator model. The free ball of charge now responds to an impulsive excitation in a way that parallels the behavior predicted by the Abraham-Lorentz theory when we use an approximate form of the radiation reaction force. (Specifically, the free charged particle responds in an acausal manner to the impulsive force.) However, when the model incorporates the exact radiation reaction force, we find that the poles of the transfer function do not leave the lower half of the 𝜔𝜔 -plane, indicating that the impulse-response is causal no matter how small a value is assumed for the radius 𝑅𝑅 of the charged particle. Once again, we find that the predicted acausal behavior reported in the older literature is likely a consequence of the approximations used to estimate the self-force, and that the exact solution of Maxwell’s equations yields a causal impulse response — down to extremely small radii of the charged particle that may even go below the classical electron radius.
2. Synopsis . This paper presents the special case of a spherical dipole for which an exact expression for the self-field (i.e., the electric field responsible for radiation resistance) can be found. This is done by a straightforward (albeit tedious) solution of the equations of classical electrodynamics presented in Appendices A and B. To our knowledge, the particular expression for the self-field of an oscillating dipole appearing in Sec.3, Eq.(13), has not been reported in the extant literature. In contrast to the well-known Abraham-Lorentz-Dirac (ALD) equation that is the usual point of departure in traditional discussions of charged-particle dynamics, our self-field does not yield an equation of motion that is readily recognizable as a finite-order differential equation. Incorporating this exact expression of the self-field into the Lorentz oscillator model enables us to view the classical Abraham-Lorentz problem from a somewhat different perspective, one in which the response of a solid, uniformly-charged, non-deformable spherical ball to an impulsive excitation can be examined with and without the small-radius approximations. Rohrlich has discussed the case involving a similarly exact solution of the Maxwell-Lorentz equations for a spherical shell of uniform surface charge, stating that “the case of a volume charge is considerably more complicated and adds nothing to the understanding of the problem.” Be it as it may, we believe it is worthwhile to bring to the community’s attention the existence of an exact expression for the self-force of a solid, uniformly-charged sphere under the conditions reported in the following sections — if for no other reason than to ensure that the neglect of “nonlinear terms” that is inherent to the otherwise “exact” solution discussed by Rohrlich does not adversely affect his conclusions. In another departure from the conventional approach, we do not renormalize the mass of the charged particle, keeping the full expression of the self-force throughout our analysis. Traditionally, renormalization is done by subtracting a term proportional to 𝑅𝑅⁄ from the observed inertial mass 𝑚𝑚 of the particle, yielding what is known as the bare mass. In our case, this would require subtracting 𝜇𝜇 𝑞𝑞 /5 𝜋𝜋𝑅𝑅 from 𝑚𝑚 ; see the coefficient of 𝜔𝜔 in the denominator of the approximate transfer function given in Eq.(21). Our main reason for setting mass renormalization aside is that we are not convinced that the coefficient of 𝜔𝜔 should carry the entire burden of accounting for the contribution of electrodynamic mass to the particle inertia and, consequently, would like to postpone any discussion of mass renormalization until such time as we have a better grasp of the role played by 𝑚𝑚 in the charged-particle dynamics. In the meantime, the absence of mass renormalization from our analysis should have very little effect as long as the radius 𝑅𝑅 is somewhat greater than the classical radius 𝑟𝑟 𝑐𝑐 of the particle. As 𝑅𝑅 approaches 𝑟𝑟 𝑐𝑐 from above and then goes below this critical radius, mass renormalization (in one form or another) should no longer be ignored, but at this point we are already deep inside the non-classical territory. According to Rohrlich, “ the classical equations of motion have their validity limits where quantum mechanics becomes important: they can no longer be trusted at distances of the order of (or below) the Compton wavelength .” (The Compton wavelength of the electron is 𝜆𝜆 = ℎ 𝑚𝑚 𝑐𝑐⁄ ≅ .) Thus, in the regime where 𝑅𝑅 ≲ 𝑟𝑟 𝑐𝑐 , our analysis of the charged-particle’s impulse-response should only be taken at face value since, deep inside the non-classical regime and absent a reliable understanding of the role played by the inertial mass 𝑚𝑚 , it is undeniable that any purely mathematical result is devoid of physical content. ‡ In a nutshell, the existing literature pertaining to the Abraham-Lorentz problem and relying solely on the ALD equation contends that a small ball of charge exhibits runaway solutions and that, in response to external excitations, it could behave in acausal fashion. Such pathologies in the predicted behavior of the particle disappear when more accurate estimates of the self-force are used in its equation of motion, with the caveat of taking the bare mass to be non-negative. (For an electron, this constraint on the bare mass translates into 𝑅𝑅 ≳ 𝑟𝑟 𝑐𝑐 .) The results reported in the present paper similarly indicate that, (i) when small-radius approximations are invoked, the predicted acausal behavior parallels those reported in the literature in accordance with the ALD equation, and (ii) our exact solution, examined numerically and down to very small particle radii shows no such acausal behavior. Thus, our exact results for uniformly-charged solid spheres agree with those in the literature for uniformly-charged spherical shells. One potentially important difference is that our exact expression of the self-force does not suffer from the neglect of the aforementioned nonlinear terms. Another difference is that our predicted causal behavior (for the bound charge within the dipole of Sec.5 as well as the free particle of Sec.6) extends to small solid spheres well below the classical electron diameter of ~2.818 fm — although this finding may not survive if and when we find a proper mass-renormalization scheme and, in any case, such classical speculations deep inside a non-classical regime are bereft of physical value.
3. The electromagnetic field inside and outside an oscillating spherical dipole.
Consider an electric dipole in the form of a small sphere of radius 𝑅𝑅 and uniform polarization 𝑃𝑃 𝒛𝒛� cos( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) , sitting at the origin of coordinates and oscillating at the source frequency 𝜔𝜔 𝑠𝑠 . Working in the spherical coordinate system 𝒓𝒓 = ( 𝑟𝑟 , 𝜃𝜃 , 𝜑𝜑 ) , and defining the function sphere( 𝑟𝑟 ) to be zero when 𝑟𝑟 > 1 and when 𝑟𝑟 ≤ , the polarization distribution may be written as 𝑷𝑷 ( 𝒓𝒓 , 𝑡𝑡 ) = 𝑃𝑃 𝒛𝒛� cos( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) sphere( 𝑟𝑟 𝑅𝑅⁄ ) . § (1) The particle’s dipole moment is readily seen to be 𝒑𝒑 ( 𝑡𝑡 ) = (4 𝜋𝜋𝑅𝑅 ⁄ ) 𝑃𝑃 cos( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) 𝒛𝒛� . The first step in evaluating the EM field that emanates from the oscillating dipole is to compute the four-dimensional Fourier transform 𝑷𝑷 ( 𝒌𝒌 , 𝜔𝜔 ) of 𝑷𝑷 ( 𝒓𝒓 , 𝑡𝑡 ) . Appendix A shows that this Fourier transformation yields 𝑷𝑷 ( 𝒌𝒌 , 𝜔𝜔 ) = 3 𝜋𝜋𝑝𝑝 𝒛𝒛� [ 𝛿𝛿 ( 𝜔𝜔 + 𝜔𝜔 𝑠𝑠 ) + 𝛿𝛿 ( 𝜔𝜔 − 𝜔𝜔 𝑠𝑠 )] [sin( 𝑘𝑘𝑅𝑅 ) − 𝑘𝑘𝑅𝑅 cos( 𝑘𝑘𝑅𝑅 )] ( 𝑘𝑘𝑅𝑅 ) ⁄ . (2) In classical electrodynamics, the bound electric charge and current densities are given by 𝜌𝜌 bound ( 𝑒𝑒 ) ( 𝒓𝒓 , 𝑡𝑡 ) = −𝜵𝜵 ∙ 𝑷𝑷 ( 𝒓𝒓 , 𝑡𝑡 ) → 𝜌𝜌 bound ( 𝑒𝑒 ) ( 𝒌𝒌 , 𝜔𝜔 ) = − i 𝒌𝒌 ∙ 𝑷𝑷 ( 𝒌𝒌 , 𝜔𝜔 ) . (3) 𝑱𝑱 bound ( 𝑒𝑒 ) ( 𝒓𝒓 , 𝑡𝑡 ) = 𝜕𝜕𝑷𝑷 ( 𝒓𝒓 , 𝑡𝑡 ) 𝜕𝜕𝑡𝑡⁄ → 𝑱𝑱 bound ( 𝑒𝑒 ) ( 𝒌𝒌 , 𝜔𝜔 ) = − i 𝜔𝜔𝑷𝑷 ( 𝒌𝒌 , 𝜔𝜔 ) . (4) ‡ As a minor solace, one might argue that, if the goal is to show that the classical physics of a charged particle remains causal when the particle radius shrinks to extremely small (but nonzero) values, then it is perhaps advisable to stay away from the conventional — and arguably non-classical — stratagem of mass renormalization. § In the literature, sphere( 𝑟𝑟 𝑅𝑅⁄ ) is sometimes written as the Heaviside step function Θ ( 𝑅𝑅 − 𝑟𝑟 ) . In the absence of free charges, free currents, and magnetization, the bound charge and current densities of Eqs.(3) and (4) will be the total charge and current densities, which are directly related to the scalar potential 𝜓𝜓 ( 𝒓𝒓 , 𝑡𝑡 ) and vector potential 𝑨𝑨 ( 𝒓𝒓 , 𝑡𝑡 ) , as follows: 𝑨𝑨 ( 𝒓𝒓 , 𝑡𝑡 ) = 𝜇𝜇 ( ) � 𝑱𝑱 ( 𝒌𝒌 , 𝜔𝜔 ) 𝑘𝑘 − ( 𝜔𝜔 𝑐𝑐⁄ ) exp[i( 𝒌𝒌 ∙ 𝒓𝒓 − 𝜔𝜔𝑡𝑡 )] d 𝒌𝒌 d 𝜔𝜔 ∞−∞ . (5) 𝜓𝜓 ( 𝒓𝒓 , 𝑡𝑡 ) = ( ) 𝜀𝜀 � 𝜌𝜌 ( 𝒌𝒌 , 𝜔𝜔 ) 𝑘𝑘 − ( 𝜔𝜔 𝑐𝑐⁄ ) exp[i( 𝒌𝒌 ∙ 𝒓𝒓 − 𝜔𝜔𝑡𝑡 )]d 𝒌𝒌 d 𝜔𝜔 ∞−∞ . (6) A detailed step-by-step calculation of these potentials is relegated to Appendix B, where the final results for the regions inside and outside the spherical dipole are given in Eqs.(B17) - (B20). Subsequently, the scalar and vector potentials are used to determine the electric and magnetic fields inside and outside the dipole using the standard formulas 𝑬𝑬 ( 𝒓𝒓 , 𝑡𝑡 ) = −𝜵𝜵𝜓𝜓 ( 𝒓𝒓 , 𝑡𝑡 ) − 𝜕𝜕𝑨𝑨 ( 𝒓𝒓 , 𝑡𝑡 ) 𝜕𝜕𝑡𝑡⁄ , (7) 𝑩𝑩 ( 𝒓𝒓 , 𝑡𝑡 ) = 𝜵𝜵 × 𝑨𝑨 ( 𝒓𝒓 , 𝑡𝑡 ) . (8) In the region outside the sphere of radius 𝑅𝑅 , the 𝑬𝑬 and 𝑩𝑩 fields are found to be 𝑬𝑬 out ( 𝒓𝒓 , 𝑡𝑡 ) = − � 𝑟𝑟 � � sin ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) − ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) cos ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) � � ( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) cos[ 𝜔𝜔 𝑠𝑠 ( 𝑡𝑡 − 𝑟𝑟 𝑐𝑐⁄ )] sin 𝜃𝜃 𝜽𝜽� +{( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) sin[ 𝜔𝜔 𝑠𝑠 ( 𝑡𝑡 − 𝑟𝑟 𝑐𝑐⁄ )] − cos[ 𝜔𝜔 𝑠𝑠 ( 𝑡𝑡 − 𝑟𝑟 𝑐𝑐⁄ )]}(2 cos 𝜃𝜃 𝒓𝒓� + sin 𝜃𝜃 𝜽𝜽� ) � . (9) 𝑩𝑩 out ( 𝒓𝒓 , 𝑡𝑡 ) = − � 𝜔𝜔 𝑠𝑠 𝑝𝑝 sin 𝜃𝜃𝝋𝝋�4𝜋𝜋𝑟𝑟 � � sin ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) − ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) cos ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) � × {( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) cos[ 𝜔𝜔 𝑠𝑠 ( 𝑡𝑡 − 𝑟𝑟 𝑐𝑐⁄ )] + sin[ 𝜔𝜔 𝑠𝑠 ( 𝑡𝑡 − 𝑟𝑟 𝑐𝑐⁄ )]} . (10) The EM fields inside the particle are given by 𝑬𝑬 in ( 𝒓𝒓 , 𝑡𝑡 ) = − � 𝒛𝒛�4𝜋𝜋𝜀𝜀 𝑅𝑅 � cos( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) + � 𝑅𝑅 � � sin ( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) − ( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) cos ( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) (2 cos 𝜃𝜃 𝒓𝒓� + sin 𝜃𝜃 𝜽𝜽� ) − sin ( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) sin 𝜃𝜃 𝜽𝜽�� × {cos[ 𝜔𝜔 𝑠𝑠 ( 𝑡𝑡 − 𝑅𝑅 𝑐𝑐⁄ )] − ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) sin[ 𝜔𝜔 𝑠𝑠 ( 𝑡𝑡 − 𝑅𝑅 𝑐𝑐⁄ )]} . (11) 𝑩𝑩 in ( 𝒓𝒓 , 𝑡𝑡 ) = − � 𝜔𝜔 𝑠𝑠 𝑝𝑝 sin 𝜃𝜃 𝝋𝝋�4𝜋𝜋𝑟𝑟 � � [ cos ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) + ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) sin ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] sin ( 𝜔𝜔 𝑠𝑠 𝑡𝑡 )( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) − [ sin ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) − ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) cos ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] cos ( 𝜔𝜔 𝑠𝑠 𝑡𝑡 )( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) � [sin( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) − ( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) cos( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] . (12) The spatially averaged 𝐸𝐸 -field inside the dipole may now be computed, as follows: 〈𝑬𝑬 in ( 𝒓𝒓 , 𝑡𝑡 ) 〉 = � � � ∫ 𝑬𝑬 in ( 𝒓𝒓 , 𝑡𝑡 )2 𝜋𝜋𝑟𝑟 sin 𝜃𝜃 d 𝑟𝑟 d 𝜃𝜃 𝜋𝜋𝜃𝜃=0𝑅𝑅𝑟𝑟=0 = � 𝒛𝒛�4𝜋𝜋𝜀𝜀 𝑅𝑅 � � [ sin ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) − ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) cos ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] × { cos [ 𝜔𝜔 𝑠𝑠 ( 𝑡𝑡−𝑅𝑅 𝑐𝑐⁄ )] − ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) sin [ 𝜔𝜔 𝑠𝑠 ( 𝑡𝑡−𝑅𝑅 𝑐𝑐⁄ )]} ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) − cos( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) � = � 𝑃𝑃 𝒛𝒛�𝜀𝜀 � Re �� [ sin ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) − ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) cos ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] × [ ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] 𝑒𝑒 i𝑅𝑅𝜔𝜔𝑠𝑠 𝑐𝑐⁄ ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) − � exp( − i 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) � . (13) In deriving Eq.(13), we used the symmetry of Eq.(11), which dictates that the components of 𝑬𝑬 in ( 𝒓𝒓 , 𝑡𝑡 ) parallel to the 𝑥𝑥𝑥𝑥 -plane average out to zero. When integrated over 𝜃𝜃 , the projection along 𝒛𝒛� of (2 cos 𝜃𝜃 𝒓𝒓� + sin 𝜃𝜃 𝜽𝜽� ) yields ∫ (2 cos 𝜃𝜃 − sin 𝜃𝜃 ) sin 𝜃𝜃 d 𝜃𝜃 𝜋𝜋0 = 0 , whereas that of sin 𝜃𝜃 𝜽𝜽� yields − ∫ sin 𝜃𝜃 d 𝜃𝜃 𝜋𝜋0 = − ⁄ . We also used the fact that ∫ 𝑥𝑥 sin 𝑥𝑥 d 𝑥𝑥 𝑎𝑎0 = sin 𝑎𝑎 − 𝑎𝑎 cos 𝑎𝑎 . Note that the Lorentz force exerted on the dipole’s moving charge by the magnetic field 𝑩𝑩 in ( 𝒓𝒓 , 𝑡𝑡 ) is everywhere parallel to the 𝑥𝑥𝑥𝑥 -plane and radially symmetric with respect to the 𝑧𝑧 -axis. Consequently, there cannot be any radiation reaction forces due to this (internal) magnetic field. In the last line of Eq.(13), the term inside the square brackets can be expanded in powers of 𝜁𝜁 = 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ to yield ( sin 𝜁𝜁−𝜁𝜁 cos 𝜁𝜁 )( ) 𝑒𝑒 i𝜁𝜁 𝜁𝜁 − − + 𝜁𝜁 + 𝜁𝜁 − 𝜁𝜁 + ⋯ . (14) For typical atomic dipoles, the condition
𝑅𝑅 ≪ 𝜆𝜆 𝑠𝑠 = 2 𝜋𝜋𝑐𝑐 𝜔𝜔 𝑠𝑠 ⁄ implies that 𝜁𝜁 = 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ≪ . Therefore, the dominant contribution to the average internal 𝐸𝐸 -field of Eq.(13) comes from the static term −𝑃𝑃 𝒛𝒛� 𝜀𝜀 ⁄ . The third term on the right-hand side of Eq.(14) is primarily responsible for radiation damping (or radiation resistance). Outside the dipole, the time-averaged Poynting vector 〈𝑺𝑺 ( 𝒓𝒓 , 𝑡𝑡 ) 〉 is computed as follows: 〈𝑺𝑺 ( 𝒓𝒓 , 𝑡𝑡 ) 〉 = 〈𝑬𝑬 out × 𝑯𝑯 out 〉 = � � � 𝜔𝜔 𝑠𝑠 sin 𝜃𝜃 ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) 𝑟𝑟 � [sin( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) − ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) cos( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] 𝒓𝒓� . (15) The total radiated power is obtained by integrating the above 〈𝑺𝑺 ( 𝒓𝒓 , 𝑡𝑡 ) 〉 over a spherical surface of radius 𝑟𝑟 . We find emitted power = � � 𝑠𝑠 ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) [sin( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) − ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) cos( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] ≅ 𝜇𝜇 𝑝𝑝 𝜔𝜔 𝑠𝑠4 . (16) The approximate form in Eq.(16) is the result of small-angle approximations, sin 𝑥𝑥 ≅ 𝑥𝑥 −𝑥𝑥 ⁄ and cos 𝑥𝑥 ≅ − 𝑥𝑥 ⁄ , which yield sin( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) − ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) cos( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) ≅ ⅓ ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) . The total radiated power is thus seen to be proportional to the square of the dipole moment 𝑝𝑝 and, in its approximate form, proportional to the fourth power of the oscillation frequency 𝜔𝜔 𝑠𝑠 .
4. The Lorentz oscillator model . With reference to Fig.1, we imagine the dipole as consisting of two overlapping balls of radius 𝑅𝑅 and uniformly distributed positive and negative charges ± 𝑞𝑞 , with the positive ball being massive and, therefore, immobile, while the negative ball of mass 𝑚𝑚 is free to move (ever so slightly) up and down along the 𝑧𝑧 -axis. A short spring, having spring constant 𝛼𝛼 , connects the centers of the two balls and exerts a restoring force on the negatively-charged ball along the 𝑧𝑧 direction. A dynamic friction coefficient 𝛽𝛽 accounts for the internal losses due to a force that is assumed to be proportional to the instantaneous velocity of the oscillating ball. The only other forces acting on the negatively-charged ball are due to an externally applied uniform electric field 𝑬𝑬 ext ( 𝑡𝑡 ) = 𝐸𝐸 cos( 𝜔𝜔𝑡𝑡 ) 𝒛𝒛� , and the (spatially-averaged) internal dipolar 𝐸𝐸 -field 〈𝑬𝑬 in ( 𝒓𝒓 , 𝑡𝑡 ) 〉 given by Eq.(13). The spatial averaging of the internal 𝐸𝐸 -field is justified by our earlier assumption that both spheres are rigidly and uniformly charged. Fig.1 . A pair of solid spherical balls of identical radius 𝑅𝑅 overlap each other in 3-dimensional space. The spheres are uniformly filled with electric charge, one containing a total charge of + 𝑞𝑞 , the other a total charge of −𝑞𝑞 . The positively-charged ball is massive and immobile, whereas the negative ball, having a finite inertial mass 𝑚𝑚 , can oscillate along the 𝑧𝑧 -axis, its minute displacement from the equilibrium position being denoted by 𝑧𝑧 ( 𝑡𝑡 ) . The electric dipole moment thus produced is 𝒑𝒑 ( 𝑡𝑡 ) = −𝑞𝑞𝑧𝑧 ( 𝑡𝑡 ) 𝒛𝒛� . Sinusoidal motion of the negative ball gives rise to the time-dependent dipole moment 𝑝𝑝 𝒛𝒛� cos( 𝜔𝜔𝑡𝑡 ) , which radiates an EM field of frequency 𝜔𝜔 into the surrounding free space. Also created is an oscillatory EM field inside the negatively-charged sphere that is responsible for the radiation resistance. 𝑧𝑧 𝑅𝑅 − − − − − − + + + + + + − + −𝑞𝑞 + 𝑞𝑞 The critical underlying assumptions in our model of the dipole are that the two inter-penetrating balls of charge depicted in Fig.1 are solid (i.e., each occupies the entire volume of its corresponding sphere), are non-deformable and that, within their respective volumes, they have a uniform charge-density distribution. These are essential assumptions that allow one to find the self-field of radiation resistance as the spatial average of the internal 𝐸𝐸 -field given by Eq.(13). It is true, of course, that no material object can exhibit infinite rigidity in view of the constraints of special relativity; nevertheless, our assumption of particle non-deformability is in keeping with the underlying hypotheses of the Abraham-Lorentz theory, at least to the extent that it pertains to the structure of sub-atomic particles. Denoting by 𝑧𝑧 ( 𝑡𝑡 ) the displacement of the negatively-charged ball along the 𝑧𝑧 -axis, we may now write the Newtonian law of motion ** for this ball as follows: 𝑚𝑚 d 𝑧𝑧 ( 𝑡𝑡 ) d𝑡𝑡 = −𝑞𝑞𝐸𝐸 𝑒𝑒 −i𝜔𝜔𝑡𝑡 − 𝑞𝑞〈𝐸𝐸 in ( 𝒓𝒓 , 𝑡𝑡 ) 〉 − 𝛼𝛼𝑧𝑧 ( 𝑡𝑡 ) − 𝛽𝛽 d𝑧𝑧 ( 𝑡𝑡 ) d𝑡𝑡 . (17) The induced dipole is thus given by 𝒑𝒑 ( 𝑡𝑡 ) = −𝑞𝑞𝑧𝑧 ( 𝑡𝑡 ) 𝒛𝒛� = 𝑝𝑝 𝒛𝒛�𝑒𝑒 −i𝜔𝜔𝑡𝑡 . Defining the normalized parameters 𝜔𝜔 = �𝛼𝛼 𝑚𝑚 ⁄ (resonance frequency) and 𝛾𝛾 = 𝛽𝛽 𝑚𝑚 ⁄ (damping coefficient), while denoting the charged ball’s volume by 𝑣𝑣 = 4 𝜋𝜋𝑅𝑅 ⁄ , and the bracketed term in the last line of Eq.(13) by Γ ( 𝜔𝜔 ) 𝑒𝑒 −i𝜔𝜔𝑡𝑡 , the streamlined version of Eq.(17) becomes d d𝑡𝑡 �𝑝𝑝 𝑒𝑒 −i𝜔𝜔𝑡𝑡 � = � 𝑞𝑞 𝑚𝑚 � 𝐸𝐸 𝑒𝑒 −i𝜔𝜔𝑡𝑡 + � 𝑞𝑞 𝑚𝑚 � � 𝑝𝑝 𝜀𝜀 𝑣𝑣 � Γ ( 𝜔𝜔 ) 𝑒𝑒 −i𝜔𝜔𝑡𝑡 − 𝜔𝜔 𝑝𝑝 𝑒𝑒 −i𝜔𝜔𝑡𝑡 − 𝛾𝛾 dd𝑡𝑡 �𝑝𝑝 𝑒𝑒 −i𝜔𝜔𝑡𝑡 � . (18) The so-called “radiation reaction” function Γ ( 𝜔𝜔 ) appearing in the preceding equation is given by Γ ( 𝜔𝜔 ) = [ sin ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) − ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) cos ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ )] × [ − i ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ )] 𝑒𝑒 i𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) − . (19) Recall that the first few terms in the Taylor series expansion of Γ ( 𝜔𝜔 ) in terms of 𝜁𝜁 = 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ are listed in Eq.(14). Introducing the plasma frequency 𝜔𝜔 𝑝𝑝 = 𝑞𝑞 �𝜀𝜀 𝑚𝑚 𝑣𝑣⁄ , and proceeding to simplify Eq.(18), we finally arrive at the transfer function relating the induced dipole amplitude 𝑝𝑝 to the applied 𝐸𝐸 -field amplitude 𝐸𝐸 , namely, 𝑝𝑝 = 𝑞𝑞 𝑚𝑚 ⁄𝜔𝜔 − 𝜔𝜔 − 𝜔𝜔 𝑝𝑝2 Γ ( 𝜔𝜔 ) − i𝛾𝛾𝜔𝜔 𝐸𝐸 . (20) Figure 2 shows computed profiles of (a) the absolute value, (b) the real part, and (c) the imaginary part of the transfer function 𝑝𝑝 𝐸𝐸 ⁄ of Eq.(20) versus the excitation frequency 𝜔𝜔 for a particle of radius 𝑅𝑅 = 1 Å , charge 𝑞𝑞 = 1.6 × 10 −19 coulomb , and mass 𝑚𝑚 = 9.11 × 10 −31 kg , corresponding to a spherical ball of charge roughly equivalent to a single electron in the 1 s -state of the hydrogen atom. We have also set 𝜔𝜔 = 3 × 10 rad s ⁄ to obtain a reasonable resonance frequency in the uv range of the optical spectrum, and 𝛾𝛾 = 10 rad s ⁄ for a relatively small contribution by non-radiative damping. (The constants of nature are 𝜀𝜀 ≅ −12 farad m ⁄ , 𝜇𝜇 = 4 𝜋𝜋 × 10 −7 henry m ⁄ , and 𝑐𝑐 = ( 𝜇𝜇 𝜀𝜀 ) − ½ ≅ m s ⁄ .) In the case of sinusoidal excitation of the dipole, it suffices to use the first three terms in the Taylor series expansion of Γ ( 𝜔𝜔 ) to arrive at the following approximate transfer function: ** Since the displacement 𝑧𝑧 ( 𝑡𝑡 ) of the negatively-charged sphere must be much less than the radius 𝑅𝑅 of the spherical particle, its velocity 𝑉𝑉 should be well below 𝑅𝑅𝜔𝜔 and, therefore,
𝑉𝑉 𝑐𝑐⁄ ≪ 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ . One can, therefore, push
𝑅𝑅𝜔𝜔 𝑐𝑐⁄ to values as high as 10 before having to worry about the validity of Newton’s (non-relativistic) equation of motion. 𝑝𝑝 ≅ 𝑞𝑞 𝑚𝑚 ⁄ ( 𝜔𝜔 +⅓𝜔𝜔 𝑝𝑝2 ) − ( 𝑞𝑞 𝑅𝑅⁄ ) 𝜔𝜔 − i ( 𝜇𝜇 𝑞𝑞 𝑐𝑐⁄ ) 𝜔𝜔 − i𝛾𝛾𝜔𝜔 𝐸𝐸 . (21) However, to compute the dipole’s impulse-response, it is necessary to use the exact form of Γ ( 𝜔𝜔 ) , given in Eq.(19), in order to locate all the poles of the transfer function of Eq.(20) in the complex 𝜔𝜔 -plane. We defer a discussion of the dipole’s impulse-response to Sec.5, continuing for the time being with an examination of the approximate transfer function of Eq.(21). Fig. 2 . Normalized profiles of (a) the absolute value, (b) the real part, and (c) the imaginary part of the transfer function 𝑝𝑝 𝐸𝐸 ⁄ of Eq.(20) versus the frequency 𝜔𝜔 for a spherical particle of radius 𝑅𝑅 = 1 Å . The parameter values are chosen so that the negative ball of charge is roughly equivalent to a single electron in the 1 s -state of the hydrogen atom. We have also set 𝜔𝜔 = 3 × 10 rad s ⁄ to obtain a reasonable resonance frequency in the ultraviolet range of the optical spectrum, and 𝛾𝛾 = 10 rad s ⁄ for a relatively small contribution from non-radiative damping. Recalling the Clausius-Mossotti correction of the Lorentz oscillator model, one could argue that the term ⅓𝜔𝜔 𝑝𝑝 added to 𝜔𝜔 in the denominator of Eq.(21) has already been taken into account when defining the spring constant 𝛼𝛼 . Leaving this argument aside, the effective resonance frequency is going to be 𝜔𝜔� = ( 𝜔𝜔 + ⅓𝜔𝜔 𝑝𝑝2 ) ½ . Also, for typical atomic radii, the correction 𝜇𝜇 𝑞𝑞 (5 𝜋𝜋𝑅𝑅 ) ⁄ to the mass 𝑚𝑚 appearing in the coefficient of 𝜔𝜔 in Eq.(21) is quite small and, for all practical purposes, negligible. In the absence of internal loss mechanisms (e.g., 𝜔𝜔 ) (a) 𝜔𝜔 (b) 𝜔𝜔 (c) ) ) − − absorption followed by non-radiative decay), one would set 𝛾𝛾 to zero and rely solely on the radiation resistance term for damping. Under these circumstances, when the dipole is excited by a monochromatic plane wave of amplitude 𝐸𝐸 and frequency 𝜔𝜔 , we will have | 𝑝𝑝 | ≅ ( 𝑞𝑞 𝑚𝑚 ⁄ ) 𝐸𝐸 ( 𝜔𝜔� − 𝜔𝜔 ) + ( 𝜇𝜇 𝑞𝑞 𝑐𝑐⁄ ) 𝜔𝜔 . (22) Defining the parameter 𝜏𝜏 = 𝜇𝜇 𝑞𝑞 (6 𝜋𝜋𝑚𝑚 𝑐𝑐 ) ⁄ , it is not difficult to show that the resonance line-width is ∆𝜔𝜔 ~ 𝜏𝜏𝜔𝜔� , and that the resonance peak occurs at 𝜔𝜔 peak ≅ 𝜔𝜔� − ¾ 𝜏𝜏 𝜔𝜔� . This estimate of the peak shift away from the resonance frequency 𝜔𝜔� , which is solely due to radiation reaction, is much smaller than typical atomic line-widths ∆𝜔𝜔 . (One must resort to quantum mechanical treatments of atomic and molecular radiation to arrive at accurate and realistic estimates of both the resonance line-width and the corresponding peak shift. ) In the example depicted in Fig.2, using 𝑣𝑣 = 4 𝜋𝜋𝑅𝑅 ⁄ for the particle volume, at 𝑅𝑅 = we find 𝜔𝜔 𝑝𝑝 = 2.753 × 10 rad s ⁄ and 𝜏𝜏 = 6.245 × 10 −24 𝑠𝑠 , yielding an effective resonance frequency 𝜔𝜔� = 1.617 × 10 rad s ⁄ and a line-width ∆𝜔𝜔 ~1.63 × 10 rad s ⁄ . Using Eqs.(16) and (22), one can also compute the radiation emission rate per unit time. Considering that the rate of incident optical energy (per unit area per unit time) is 𝐸𝐸 𝑍𝑍 ⁄ , where 𝑍𝑍 = ( 𝜇𝜇 𝜀𝜀 ⁄ ) ½ ≅ Ω is the impedance of free space, the dipole’s scattering cross-section is straightforwardly found to be 𝑠𝑠 ≅ ( 𝜇𝜇 𝑞𝑞 𝑚𝑚 ⁄ ) 𝜔𝜔 [( 𝜔𝜔� −𝜔𝜔 ) + ( 𝜇𝜇 𝑞𝑞 𝑐𝑐⁄ ) 𝜔𝜔 ] . (23) In the above equation, the radiation resistance term is essentially negligible everywhere except in the vicinity of the resonance frequency 𝜔𝜔 ≅ 𝜔𝜔� , where it governs the peak value 𝑠𝑠 max of the scattering cross-section and the width ∆𝜔𝜔 of the resonance peak, as follows: 𝑠𝑠 max ≅ 𝜋𝜋 ( 𝑐𝑐 𝜔𝜔� ⁄ ) = 3 𝜆𝜆 𝜋𝜋⁄ , (24) ∆𝜔𝜔 ≅ 𝜇𝜇 𝑞𝑞 𝜔𝜔� (6 𝜋𝜋𝑚𝑚 𝑐𝑐 ) ⁄ . (25) In accordance with Eq.(23), at frequencies well below resonance, we will have 𝑠𝑠 ≅ ( 𝜇𝜇 𝑞𝑞 𝑚𝑚 ⁄ ) ( 𝜔𝜔 𝜔𝜔� ⁄ ) 𝜋𝜋⁄ , (26) and at frequencies well above resonance (but not extremely high), we will have 𝑠𝑠 ≅ ( 𝜇𝜇 𝑞𝑞 𝑚𝑚 ⁄ ) 𝜋𝜋⁄ . (27) All the results obtained thus far by approximating the exact radiation reaction function Γ ( 𝜔𝜔 ) of Eq.(19) are in complete accord with the classical results. In the remaining part of the paper, we argue that any acausal behavior predicted by the classical theories is likely due to the approximate nature of those theories — the exception being Dirac’s demonstration of runaway behavior based on his formula for the self-force of an accelerated point-charge, which is an exact relativistic solution of the Maxwell-Lorentz equations of classical electrodynamics.
Specifically, upon examining the impulse-response of the dipole in the next section using the exact radiation reaction function of Eq.(19), we do not find any indication of departure from causal behavior — even in the limit when the dipole radius 𝑅𝑅 becomes exceedingly small, going far below the classical radius 𝑟𝑟 𝑐𝑐 of the electron. Similarly, in Sec.6, where we eliminate the restoring force of the positive charge as well as that of the (phenomenological) spring, and proceed to examine the impulse-response of a free-standing, negatively-charged sphere, we find acausal behavior only when the analysis relies on an approximate form of Γ ( 𝜔𝜔 ) . In other words, 0 the exact radiation reaction function of Eq.(19) acting on a small, electrically-charged sphere driven by an impulsive force, does not appear to give rise to acausal behavior.
5. The impulse-response . When excited by an external impulse, an initially dormant dipole exhibits a damped oscillatory response, with a time dependence that is precisely the inverse Fourier transform of the transfer function 𝑝𝑝 𝐸𝐸 ⁄ of Eq.(20). This is because the impulsive excitation consists of a uniform superposition of all sinusoidal frequencies 𝜔𝜔 from −∞ to ∞ . Consequently, the dipole’s impulse-response is obtained by the inverse Fourier integral of the transfer function 𝑝𝑝 / 𝐸𝐸 over the entire real-axis 𝜔𝜔 . Now, according to Cauchy’s theorem of complex analysis, when 𝑡𝑡 < 0 , the contour of integration can be closed with a large semi-circle in the upper half of the complex 𝜔𝜔 -plane. The absence of poles in the upper-half plane would then imply that the inverse Fourier integral (i.e., the impulse-response) is zero when 𝑡𝑡 < 0 . This is the requirement for causal behavior that will be used throughout the following analysis. (We mention in passing that an impulsive excitation takes the dormant dipole instantaneously to an excited initial state, whence it decays to the ground state following the dynamical equation of motion of the dipole. Physically, this is tantamount to bringing an isolated atom to an excited state, then observing its decay to the ground state via spontaneous emission; see Appendix C for a discussion of the radiated EM energy when an initially dormant dipole is excited by an impulsive force.) Fig.3 . (a) Complex plane diagram showing, within the 4 th quadrant of the 𝜔𝜔 -plane, the zero contours of the real part (blue) and imaginary part (black) of the denominator of Eq.(20). A mirror image of these contours also resides in the 3 rd quadrant. Here 𝑅𝑅 = 1.0 nm , and the remaining parameters are the same as those used in Fig.2. The marked crossing points are the poles of the transfer function 𝑝𝑝 𝐸𝐸 ⁄ . The inset is a magnified view of the region surrounding the dominant pole. (b) Trajectory of the dominant pole within the 4 th quadrant of the 𝜔𝜔 -plane. All the parameters are fixed except for the radius 𝑅𝑅 of the particle, which starts at on the upper left-hand corner of the graph and goes down to at its lower right-hand corner. Causality, an important property of the impulse-response of the dipole, is governed by the location of the poles of its transfer function in the complex 𝜔𝜔 -plane. Causal behavior is ensured if all the poles reside in the lower half of the 𝜔𝜔 -plane. We undertook a detailed numerical study of the poles of the transfer function of Eq.(20) with the exact Γ ( 𝜔𝜔 ) of Eq.(19). Here, we describe the trajectories of these poles as the radius 𝑅𝑅 of the spherical dipole approaches zero, and confirm that, for all nonzero values of 𝑅𝑅 , the poles remain in the lower half of the 𝜔𝜔 -plane. Stated differently, we have found no indication that the dipole’s impulse-response violates causality as the radius of the particle shrinks to extremely small values. (b) (a) (× 10 ) (× 10 )
0 1 2 3 4 5 6 − − − − ( × ) 𝜔𝜔 ′ 𝜔𝜔 ″ −
0 2 4 6 − ( × ) 𝜔𝜔 ′ (× 10 ) − − − − 𝜔𝜔 ″ ( × ) 𝜔𝜔 = 𝜔𝜔 ′ + i 𝜔𝜔 ″ 𝐷𝐷 ( 𝜔𝜔 ) of the transfer function 𝑝𝑝 𝐸𝐸 ⁄ within the 4 th quadrant of the complex 𝜔𝜔 -plane. (A mirror image of these contours also resides in the 3 rd quadrant.) The crossing points marked with red dots identify the poles of the transfer function. For the chosen set of parameters ( 𝑅𝑅 = 1.0 nm , 𝜔𝜔 = 3 × 10 rad s ⁄ , 𝛾𝛾 = 10 rad s ⁄ , 𝜀𝜀 , 𝑐𝑐 , 𝑞𝑞 , and 𝑚𝑚 the same as those in Fig.2), the dominant pole, shown in the inset of Fig.3(a), is at 𝜔𝜔 ≅ − i 4 × 10 rad s ⁄ , whereas all the remaining poles are far below the real-axis. Figure 3(b) shows the trajectory of the dominant pole as the radius 𝑅𝑅 of the particle declines from at the upper left-hand corner of the graph down to at the lower right-hand corner. At 𝑅𝑅 = 1.0 pm , the dominant pole is located at 𝜔𝜔 = 1.5 × 10 − i 0.8 × 10 rad s ⁄ . Fig. 4 . 𝜔𝜔 -plane trajectories of (a) the dominant pole, and (b, c) the st and nd distant poles depicted in Fig.3(a). In (a) and (b), 𝑅𝑅 is in the range of [0.5, 0.1] fm ; in (c) the range of 𝑅𝑅 is [1.0 , 0.05] fm . Figure 4 shows an extended trajectory of the dominant pole as well as those of the first and second distant poles depicted in Fig.3(a) — i.e., poles identified with red dots, counting from left to right. The dominant pole is seen to continually move downward and to the right as 𝑅𝑅 drops from to . The first distant pole initially moves up toward the real axis (but never crosses it), then goes down and to the right. The second distant pole shows two initial humps (again, never crossing the real axis), before it drops down again. We also set 𝜔𝜔 = 0 and 𝛾𝛾 = 0 to rule out the possibility that these phenomenological features of the Lorentz oscillator model might be responsible for the causal behavior of the impulse-response. †† Under no circumstances did we observe any pole of the transfer function to move into the upper half of the 𝜔𝜔 -plane. Application of Cauchy’s argument principle with the aid of numerical integration (spanning the broad frequency range of | 𝜔𝜔 | ≤ rad s ⁄ ) further affirmed these findings. We conclude that the dipole’s response to an impulsive excitation must be causal, even when its radius 𝑅𝑅 assumes exceedingly small values, far below the classical electron radius. †† Ideally, when the radius 𝑅𝑅 becomes comparable to or smaller than the classical radius 𝑟𝑟 𝑐𝑐 of the electron, one should also consider renormalizing the mass 𝑚𝑚 of the oscillating ball of charge. However, for the reasons mentioned in Sec.2, we believe that this subject is best left for future studies. (a) 𝜔𝜔 ″ − − − − − 𝜔𝜔 ′ (× 10 ) ( × ) (c) 𝜔𝜔 ′
1 2 3 4 5 (× 10 ) 𝜔𝜔 ″ − − − − − ( × ) (b) 𝜔𝜔 ″ − − − − 𝜔𝜔 ′ (× 10 ) ( × )
6. Radiation reaction on a small spherical charge . Returning to Eq.(17), if the driving force acting on the negatively-charged ball is 𝒇𝒇 ext ( 𝑡𝑡 ) = −𝑞𝑞𝐸𝐸 exp( − i 𝜔𝜔𝑡𝑡 ) 𝒛𝒛� , and the small deviation from the ball’s equilibrium position is written as 𝑧𝑧 ( 𝑡𝑡 ) = 𝑧𝑧 ( 𝜔𝜔 ) exp( − i 𝜔𝜔𝑡𝑡 ) , we will have 𝑧𝑧 ( 𝜔𝜔 ) = 𝑓𝑓 ext ( 𝜔𝜔 ) 𝛼𝛼 − 𝑚𝑚 𝜔𝜔 − ( 𝑞𝑞 𝜀𝜀 𝑣𝑣⁄ ) Γ ( 𝜔𝜔 ) − i𝛽𝛽𝜔𝜔 . (28) Now, the restoring force of the (stationary) positive ball on the (vibrating) negative ball is built into Γ ( 𝜔𝜔 ) as 𝑙𝑙𝑙𝑙𝑚𝑚 𝜔𝜔→0 Γ ( 𝜔𝜔 ) = −⅓ . Thus, by setting 𝛼𝛼 = −𝑞𝑞 (3 𝜀𝜀 𝑣𝑣 ) ⁄ , we eliminate the restoring force of the positive charge, leaving the negatively-charged ball as the only source of radiation reaction on itself. Also, setting 𝛽𝛽 = 0 would eliminate the effects of non-radiative damping, but it is preferable to retain this friction coefficient for the time being. All in all, the oscillation amplitude 𝑧𝑧 ( 𝜔𝜔 ) of a small solid sphere of radius 𝑅𝑅 , charge −𝑞𝑞 , and mass 𝑚𝑚 , in response to an externally-applied force 𝒇𝒇 ext ( 𝑡𝑡 ) = 𝑓𝑓 exp( − i 𝜔𝜔𝑡𝑡 ) 𝒛𝒛� — taking into account the radiation reaction force as well as a non-radiative damping force — will be 𝑧𝑧 ( 𝜔𝜔 ) = − 𝑓𝑓 ( 𝜔𝜔 ) 𝑚𝑚 𝜔𝜔 + 𝑞𝑞 [ ( 𝜔𝜔 ) +1 ] ( 𝑅𝑅 ) ⁄ + i𝛽𝛽𝜔𝜔 . (29) Invoking the Taylor series expansion (up to the 3 rd order) of Γ ( 𝜔𝜔 ) + 1 ≅ ⅘ ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) + ⅔ i( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) given in Eq.(14), one may approximate Eq.(29) as follows: 𝑧𝑧 ( 𝜔𝜔 ) ≅ − 𝑓𝑓 ( 𝜔𝜔 )[ 𝑚𝑚 + ( 𝜇𝜇 𝑞𝑞 )] 𝜔𝜔 + i ( 𝜇𝜇 𝑞𝑞 ) 𝜔𝜔 + i𝛽𝛽𝜔𝜔 . (30) Note in the above equation that the contribution of radiation reaction to the particle’s mass is 𝜇𝜇 𝑞𝑞 (5 𝜋𝜋𝑅𝑅 ) ⁄ , which, in the case of a single electron, approaches its inertial mass 𝑚𝑚 if 𝑅𝑅 happens to be in the vicinity of the classical electron radius (i.e., ~10 −15 meter). Let us denote by 𝑚𝑚 = 𝑚𝑚 + ( 𝜇𝜇 𝑞𝑞 𝜋𝜋𝑅𝑅⁄ ) the effective mass of the negatively charged ball, then normalize the remaining parameters by 𝑚𝑚 , so that 𝜏𝜏 = 𝜇𝜇 𝑞𝑞 (6 𝜋𝜋𝑚𝑚𝑐𝑐 ) ⁄ and 𝛾𝛾 = 𝛽𝛽 𝑚𝑚⁄ . The transfer function appearing in Eq.(30) has three poles at 𝜔𝜔 = 0 and 𝜔𝜔 = ½i (1 ± � 𝛾𝛾𝜏𝜏 ) 𝜏𝜏⁄ . The response 𝑧𝑧 ( 𝑡𝑡 ) of the negatively-charged sphere to the impulsive force 𝒇𝒇 ext ( 𝑡𝑡 ) = 𝐹𝐹 𝛿𝛿 ( 𝑡𝑡 ) 𝒛𝒛� , computed with the aid of Cauchy’s theorem and complex-plane integration, is readily found to be 𝑧𝑧 ( 𝑡𝑡 ) = � 𝐹𝐹 𝑚𝑚 � ⎩⎨⎧− + ( ) 𝑡𝑡 2𝜏𝜏⁄ �1+4𝛾𝛾𝜏𝜏 + �1+4𝛾𝛾𝜏𝜏 ; 𝑡𝑡 ≤ − ( ) 𝑡𝑡 2𝜏𝜏⁄ �1+4𝛾𝛾𝜏𝜏 − �1+4𝛾𝛾𝜏𝜏 ; 𝑡𝑡 ≥ (31) Both 𝑧𝑧 ( 𝑡𝑡 ) and its derivative 𝑧𝑧̇ ( 𝑡𝑡 ) are seen to be continuous at 𝑡𝑡 = 0 . Since Eq.(30) does not specify an initial condition (due to the presence of a pole at 𝜔𝜔 = 0 ), the solution 𝑧𝑧 ( 𝑡𝑡 ) can be augmented by an arbitrary constant 𝑧𝑧 . Adding 𝑧𝑧 = 𝐹𝐹 (2 𝑚𝑚𝛾𝛾 ) ⁄ to Eq.(31) ensures that 𝑧𝑧 ( −∞ ) = 0 . We now let the damping coefficient 𝛾𝛾 shrink to sufficiently small values to ensure that 𝛾𝛾𝜏𝜏 ≪ , in which case Eq.(31) becomes 𝑧𝑧 ( 𝑡𝑡 ) ≅ � 𝐹𝐹 𝑚𝑚 � �𝜏𝜏 exp( 𝑡𝑡 𝜏𝜏⁄ ) ; 𝑡𝑡 ≤ − exp ( −𝛾𝛾𝑡𝑡 ) 𝛾𝛾 ; 𝑡𝑡 > 0. (32) The acausal behavior during 𝑡𝑡 < 0 , which is manifest in this nonzero response of the charged particle to an impulsive excitation at 𝑡𝑡 = 0 , has been deemed indicative of the failure of 3 classical electrodynamics when applied to point particles. Historically, this has been the lesson of the original Abraham-Lorentz theory pertaining to the effect of self-force on an accelerated point charge. However, as pointed out in Sec. 2, more accurate calculations of the self-force for small charged particles have revealed that the acausal behavior disappears so long as the particle’s bare mass remains positive — a condition that is equivalent to the particle radius being greater than the classical radius associated with its charge 𝑞𝑞 and inertial mass 𝑚𝑚 . When we used the exact Γ ( 𝜔𝜔 ) of Eq.(19) in Eq.(29), and conducted a numerical search for the poles of the transfer function in the 𝜔𝜔 -plane, we found all the poles to reside in the lower half-plane. As was done in our numerical investigation of the impulse-response of a dipole in Sec.5, we allowed the radius 𝑅𝑅 of the solid sphere in the present case to shrink to exceedingly small values, and also examined situations in which 𝛾𝛾 was set to zero. Under no circumstances did we find a pole in the upper half-plane. (These findings were further affirmed for values of 𝑅𝑅 as small as −25 fm by applying Cauchy’s argument principle with the aid of numerical integration spanning the broad frequency range of | 𝜔𝜔 | ≤ rad s ⁄ .) The conclusion is that the acausal behavior exemplified by Eq.(32) is an artifact of the approximate nature of the radiation reaction force used in Eq.(30). In other words, when the transfer function incorporates the exact radiation reaction function of Eq.(19), the predicted behavior under all examined circumstances (even for particle radii far below their classical radius ‡‡ ) remains causal.
7. Concluding remarks . We have examined a special case of EM radiation by an accelerated, uniformly-charged, non-deformable, solid sphere and, based on a numerical evaluation of the location of the poles of its transfer function, concluded that the response of the particle to an impulsive excitation must be causal. Our numerical investigation covered a broad range of particle radii from atomic-scale ( 𝑅𝑅 ~1 nm ) down to sub-nuclear dimensions ( 𝑅𝑅 ≪ ). We examined cases where the particle was confined within a dipole and, therefore, restrained by the attractive coulomb force of an equal but opposite charge, as well as cases where the restraining force of its opposite-charge partner was removed, thus releasing the particle from confinement. While these are special cases of accelerated charged particles, which do not merit blind generalization to other situations for which the Abraham-Lorentz type of analysis is pertinent —and definitely not cases to which Dirac’s exact relativistic treatment applies — it is nonetheless important to recognize that, if not in all cases, then at least in some situations, acausal behavior is not inherent to the classical Maxwell-Lorentz theory, but rather is an artifact of the approximations used to estimate the self-force of an accelerated particle. We did not automatically renormalize the inertial mass 𝑚𝑚 of our charged particle, not so much because its necessity is in doubt, but rather because we are not convinced that a simple reduction of 𝑚𝑚 by 𝜇𝜇 𝑞𝑞 𝜋𝜋𝑅𝑅⁄ is the best way to account for the electrodynamic contributions to the particle inertia. We have, therefore, opted to postpone the issue of mass renormalization until such time as we have attained a better understanding for the role of 𝑚𝑚 in our equations of motion. In the meantime, for particle radii that are more or less greater than the critical radius 𝑟𝑟 𝑐𝑐 , our main conclusions are not significantly affected, given that electrodynamic contributions to the inertial mass in this regime are fairly small. For 𝑅𝑅 ≲ 𝑟𝑟 𝑐𝑐 , however, the conclusions could change, but here, as pointed out by Rohrlich, we are already deep inside the non-classical regime where such mathematical results are devoid of physical meaning. Be it as it may, the analytical as well as numerical methods that we have introduced here to investigate the transfer ‡‡ Again, for
𝑅𝑅 ≲ 𝑟𝑟 𝑐𝑐 , proper accounting for mass-renormalization could change this conclusion. It was pointed out in Sec.1, that Dirac “found a clever way to eliminate the troublesome infinities that had previously hampered the investigations of point particles.” Having carefully examined Dirac’s argument, we find his “clever way” to not be a trick, nor an ad hoc attempt at extracting a meaningful answer from an ill-defined problem in the classical electrodynamics of an accelerated point-charge. As explained in some detail in a two-part paper by one of the authors, Dirac’s courageous and unconventional approach — using one-half each of the retarded and advanced potentials — is inevitable when one attempts to solve Maxwell’s equations for a true point-particle. What is more, Dirac’s result can be obtained directly based on the conservation laws of energy and linear momentum, without invoking his unconventional assumption. We contend, therefore, that Dirac’s relativistic solution to the Abraham-Lorentz problem is indeed the exact solution of Maxwell’s equations for an accelerated point-charge. Dirac’s solution for a zero-size particle removes the infinite contribution of the self-force to the particle’s mass, making his solution fundamentally different from any exact or approximate solution that may be found for a particle of finite radius 𝑅𝑅 . This is why we have refrained from using the notation 𝑅𝑅 → in this paper, lest it imply that our conclusions remain valid when the particle size shrinks to zero. The various appearances throughout the paper of the qualifiers “small,” “extremely small,” and “exceedingly small” for the particle size are intended to forestall a potential misunderstanding that our results should remain applicable when the particle under consideration reduces to a true (i.e., zero-dimensional) point-charge. Acknowledgement . The authors express their gratitude to Vladimir Hnizdo for commenting on an early draft of this paper, and for generously sharing with them his extensive knowledge of the electrodynamics of charged particles. This work has been supported in part by the AFOSR grant FA9550-19-1-0032.
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The four-dimensional Fourier transform of 𝑷𝑷 ( 𝒓𝒓 , 𝑡𝑡 ) given by Eq.(1) is calculated as follows: 𝑷𝑷 ( 𝒌𝒌 , 𝜔𝜔 ) = ∫ 𝑷𝑷 ( 𝒓𝒓 , 𝑡𝑡 ) exp[ − i( 𝒌𝒌 ∙ 𝒓𝒓 − 𝜔𝜔𝑡𝑡 )] d 𝒓𝒓 d 𝑡𝑡 ∞−∞ = 𝑃𝑃 𝒛𝒛��∫ sphere( 𝑟𝑟 𝑅𝑅⁄ ) 𝑒𝑒 −i𝒌𝒌∙𝒓𝒓 d 𝒓𝒓 ∞−∞ � × � ½ ∫ ( 𝑒𝑒 i𝜔𝜔 𝑠𝑠 𝑡𝑡 + 𝑒𝑒 −i𝜔𝜔 𝑠𝑠 𝑡𝑡 ) 𝑒𝑒 i𝜔𝜔𝑡𝑡 d 𝑡𝑡 ∞−∞ � = 𝜋𝜋𝑃𝑃 𝒛𝒛� [ 𝛿𝛿 ( 𝜔𝜔 + 𝜔𝜔 𝑠𝑠 ) + 𝛿𝛿 ( 𝜔𝜔 − 𝜔𝜔 𝑠𝑠 )] ∫ ∫ 𝜋𝜋𝑟𝑟 sin 𝜃𝜃 exp( − i 𝑘𝑘𝑟𝑟 cos 𝜃𝜃 ) d 𝑟𝑟 d 𝜃𝜃 𝜋𝜋𝜃𝜃=0𝑅𝑅𝑟𝑟=0 = 2 𝜋𝜋 𝑃𝑃 𝒛𝒛� [ 𝛿𝛿 ( 𝜔𝜔 + 𝜔𝜔 𝑠𝑠 ) + 𝛿𝛿 ( 𝜔𝜔 − 𝜔𝜔 𝑠𝑠 )]( − i 𝑘𝑘⁄ ) ∫ 𝑟𝑟 exp( − i 𝑘𝑘𝑟𝑟 cos 𝜃𝜃 )| 𝜃𝜃=0𝜋𝜋 d 𝑟𝑟 𝑅𝑅0 = 2 𝜋𝜋 𝑃𝑃 𝒛𝒛� [ 𝛿𝛿 ( 𝜔𝜔 + 𝜔𝜔 𝑠𝑠 ) + 𝛿𝛿 ( 𝜔𝜔 − 𝜔𝜔 𝑠𝑠 )]( − i 𝑘𝑘⁄ ) ∫ i2 𝑟𝑟 sin( 𝑘𝑘𝑟𝑟 ) d 𝑟𝑟 𝑅𝑅0 = 4 𝜋𝜋 ( 𝑃𝑃 𝑘𝑘 ⁄ ) 𝒛𝒛� [ 𝛿𝛿 ( 𝜔𝜔 + 𝜔𝜔 𝑠𝑠 ) + 𝛿𝛿 ( 𝜔𝜔 − 𝜔𝜔 𝑠𝑠 )] ∫ 𝑥𝑥 sin 𝑥𝑥 d 𝑥𝑥 𝑘𝑘𝑅𝑅0 = 4 𝜋𝜋 ( 𝑃𝑃 𝑘𝑘 ⁄ ) 𝒛𝒛� [ 𝛿𝛿 ( 𝜔𝜔 + 𝜔𝜔 𝑠𝑠 ) + 𝛿𝛿 ( 𝜔𝜔 − 𝜔𝜔 𝑠𝑠 )] �−𝑥𝑥 cos 𝑥𝑥 | 𝑥𝑥=0𝑘𝑘𝑅𝑅 + ∫ cos 𝑥𝑥 d 𝑥𝑥 𝑘𝑘𝑅𝑅0 � = 4 𝜋𝜋 𝑅𝑅 𝑃𝑃 𝒛𝒛� [ 𝛿𝛿 ( 𝜔𝜔 + 𝜔𝜔 𝑠𝑠 ) + 𝛿𝛿 ( 𝜔𝜔 − 𝜔𝜔 𝑠𝑠 )] [sin( 𝑘𝑘𝑅𝑅 ) − 𝑘𝑘𝑅𝑅 cos( 𝑘𝑘𝑅𝑅 )] ( 𝑘𝑘𝑅𝑅 ) ⁄ = 3 𝜋𝜋𝑝𝑝 𝒛𝒛� [ 𝛿𝛿 ( 𝜔𝜔 + 𝜔𝜔 𝑠𝑠 ) + 𝛿𝛿 ( 𝜔𝜔 − 𝜔𝜔 𝑠𝑠 )] [sin( 𝑘𝑘𝑅𝑅 ) − 𝑘𝑘𝑅𝑅 cos( 𝑘𝑘𝑅𝑅 )] ( 𝑘𝑘𝑅𝑅 ) ⁄ . (A1) In the limit when 𝑘𝑘𝑅𝑅 → , we find that [sin( 𝑘𝑘𝑅𝑅 ) − 𝑘𝑘𝑅𝑅 cos( 𝑘𝑘𝑅𝑅 )] ( 𝑘𝑘𝑅𝑅 ) ⁄ → ⅓ . The function 𝑷𝑷 ( 𝒌𝒌 , 𝜔𝜔 ) is thus well-behaved for all values of 𝑅𝑅 throughout the ( 𝒌𝒌 , 𝜔𝜔 ) space. Appendix B
The scalar potential 𝜓𝜓 ( 𝒓𝒓 , 𝑡𝑡 ) and the vector potential 𝑨𝑨 ( 𝒓𝒓 , 𝑡𝑡 ) are directly computed from the charge- and current-densities 𝜌𝜌 ( 𝒓𝒓 , 𝑡𝑡 ) and 𝑱𝑱 ( 𝒓𝒓 , 𝑡𝑡 ) , given by Eqs.(3) and (4). In the step-by-step derivations that follow, we seek analytic expressions for the potentials in the regions both inside ( 𝑟𝑟 < 𝑅𝑅 ) and outside ( 𝑟𝑟 > 𝑅𝑅 ) the spherical dipole. B1. The vector potential . In the ( 𝒓𝒓 , 𝑡𝑡 ) spacetime domain The vector potential is obtained by an inverse Fourier integral over the total electric current density distribution, as follows: 𝑨𝑨 ( 𝒓𝒓 , 𝑡𝑡 ) = 𝜇𝜇 ( ) � 𝑱𝑱 ( 𝒌𝒌 , 𝜔𝜔 ) 𝑘𝑘 − ( 𝜔𝜔 𝑐𝑐⁄ ) exp[i( 𝒌𝒌 ∙ 𝒓𝒓 − 𝜔𝜔𝑡𝑡 )] d 𝒌𝒌 d 𝜔𝜔 ∞−∞ = − i3𝜋𝜋𝜇𝜇 𝑝𝑝 𝒛𝒛� ( ) � sin ( 𝑘𝑘𝑅𝑅 ) −𝑘𝑘𝑅𝑅 cos ( 𝑘𝑘𝑅𝑅 )( 𝑘𝑘𝑅𝑅 ) 𝑒𝑒 i𝒌𝒌 ∙ 𝒓𝒓∞−∞ ∫ 𝜔𝜔 [ 𝛿𝛿 ( 𝜔𝜔+𝜔𝜔 𝑠𝑠 ) +𝛿𝛿 ( 𝜔𝜔−𝜔𝜔 𝑠𝑠 )] 𝑘𝑘 − ( 𝜔𝜔 𝑐𝑐⁄ ) 𝑒𝑒 −i𝜔𝜔𝑡𝑡 d 𝜔𝜔 d 𝒌𝒌 ∞−∞ = − 𝑝𝑝 𝜔𝜔 𝑠𝑠 sin ( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) 𝒛𝒛�8𝜋𝜋 𝑅𝑅 � sin ( 𝑘𝑘𝑅𝑅 ) −𝑘𝑘𝑅𝑅 cos ( 𝑘𝑘𝑅𝑅 ) 𝑘𝑘 [ 𝑘𝑘 − ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) ] 𝑒𝑒 i𝒌𝒌 ∙ 𝒓𝒓 d 𝒌𝒌 ∞−∞ = − 𝑝𝑝 𝜔𝜔 𝑠𝑠 sin ( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) 𝒛𝒛�8𝜋𝜋 𝑅𝑅 � ∫ sin ( 𝑘𝑘𝑅𝑅 ) −𝑘𝑘𝑅𝑅 cos ( 𝑘𝑘𝑅𝑅 ) 𝑘𝑘 [ 𝑘𝑘 − ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) ] 𝑒𝑒 i𝑘𝑘𝑟𝑟 cos 𝜑𝜑 𝜋𝜋𝑘𝑘 sin 𝜑𝜑 d 𝑘𝑘 d 𝜑𝜑 𝜋𝜋𝜑𝜑=0∞𝑘𝑘=0 = − 𝑝𝑝 𝜔𝜔 𝑠𝑠 sin ( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) 𝒛𝒛�4𝜋𝜋 𝑅𝑅 � sin ( 𝑘𝑘𝑅𝑅 ) −𝑘𝑘𝑅𝑅 cos ( 𝑘𝑘𝑅𝑅 ) 𝑘𝑘 [ 𝑘𝑘 − ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) ] ∫ 𝑘𝑘 sin 𝜑𝜑 𝑒𝑒 i𝑘𝑘𝑟𝑟 cos 𝜑𝜑 d 𝜑𝜑 d 𝑘𝑘 𝜋𝜋𝜑𝜑=0 ∞𝑘𝑘=0 = − i3𝜇𝜇 𝑝𝑝 𝜔𝜔 𝑠𝑠 sin ( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) 𝒛𝒛�4𝜋𝜋 𝑅𝑅 𝑟𝑟 � sin ( 𝑘𝑘𝑅𝑅 ) −𝑘𝑘𝑅𝑅 cos ( 𝑘𝑘𝑅𝑅 ) 𝑘𝑘 [ 𝑘𝑘 − ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) ] 𝑒𝑒 i𝑘𝑘𝑟𝑟 cos 𝜑𝜑 � 𝜑𝜑=0𝜋𝜋 d 𝑘𝑘 ∞𝑘𝑘=0 = − 𝑝𝑝 𝜔𝜔 𝑠𝑠 sin ( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) 𝒛𝒛�4𝜋𝜋 𝑅𝑅 𝑟𝑟 � [ sin ( 𝑘𝑘𝑅𝑅 ) −𝑘𝑘𝑅𝑅 cos ( 𝑘𝑘𝑅𝑅 )] sin ( 𝑘𝑘𝑟𝑟 ) 𝑘𝑘 [ 𝑘𝑘 − ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) ] d 𝑘𝑘 ∞−∞ = − i3𝜇𝜇 𝑝𝑝 𝜔𝜔 𝑠𝑠 sin ( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) 𝒛𝒛�16𝜋𝜋 𝑅𝑅 𝑟𝑟 � � ( 𝑘𝑘𝑅𝑅+i ) 𝑒𝑒 i𝑘𝑘𝑅𝑅 + ( 𝑘𝑘𝑅𝑅−i ) 𝑒𝑒 −i𝑘𝑘𝑅𝑅 � ( 𝑒𝑒 i𝑘𝑘𝑟𝑟 −𝑒𝑒 −i𝑘𝑘𝑟𝑟 ) 𝑘𝑘 [ 𝑘𝑘 − ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) ] d 𝑘𝑘 ∞−∞ = − i3𝜇𝜇 𝑝𝑝 𝜔𝜔 𝑠𝑠 sin ( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) 𝒛𝒛�16𝜋𝜋 𝑅𝑅 𝑟𝑟 � ( 𝑘𝑘𝑅𝑅+i ) 𝑒𝑒 i ( 𝑟𝑟+𝑅𝑅 ) 𝑘𝑘 − ( 𝑘𝑘𝑅𝑅−i ) 𝑒𝑒 −i ( 𝑟𝑟+𝑅𝑅 ) 𝑘𝑘 + ( 𝑘𝑘𝑅𝑅−i ) 𝑒𝑒 i ( 𝑟𝑟−𝑅𝑅 ) 𝑘𝑘 − ( 𝑘𝑘𝑅𝑅+i ) 𝑒𝑒 −i ( 𝑟𝑟−𝑅𝑅 ) 𝑘𝑘 𝑘𝑘 [ 𝑘𝑘 − ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) ] d 𝑘𝑘 ∞−∞ = 𝑝𝑝 𝜔𝜔 𝑠𝑠 sin ( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) 𝒛𝒛�8𝜋𝜋 𝑅𝑅 𝑟𝑟 Im � ( 𝑘𝑘𝑅𝑅+i ) 𝑒𝑒 i ( 𝑟𝑟+𝑅𝑅 ) 𝑘𝑘 + ( 𝑘𝑘𝑅𝑅−i ) 𝑒𝑒 i ( 𝑟𝑟−𝑅𝑅 ) 𝑘𝑘 𝑘𝑘 [ 𝑘𝑘− ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] [ 𝑘𝑘+ ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] d 𝑘𝑘 ∞−∞ . (B1) The preceding integral can be evaluated in the complex plane using Cauchy’s theorem. Outside the spherical particle, where 𝑟𝑟 > 𝑅𝑅 , both exponential functions appearing in the integrand approach zero on a large semi-circular contour in the upper half-plane. The half-residues at the first-order poles 𝑘𝑘 = ± 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ are easy to evaluate. As for the second-order pole at 𝑘𝑘 = 0 , the product of 𝑘𝑘 and the integrand turns out to have a Taylor series expansion around 𝑘𝑘 = 0 that has no constant term and no first-order term. As such, the pole at 𝑘𝑘 = 0 makes no contribution to the integral. We will have 𝑨𝑨 out ( 𝒓𝒓 , 𝑡𝑡 ) = 𝑝𝑝 𝜔𝜔 𝑠𝑠 sin ( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) 𝒛𝒛�8𝜋𝜋 𝑅𝑅 𝑟𝑟 Im � i 𝜋𝜋 [( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) +i ] 𝑒𝑒 i ( 𝑟𝑟+𝑅𝑅 ) 𝜔𝜔𝑠𝑠 𝑐𝑐⁄ + [( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) −i ] 𝑒𝑒 i ( 𝑟𝑟−𝑅𝑅 ) 𝜔𝜔𝑠𝑠 𝑐𝑐⁄ ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) +i 𝜋𝜋 [( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) −i ] 𝑒𝑒 −i ( 𝑟𝑟+𝑅𝑅 ) 𝜔𝜔𝑠𝑠 𝑐𝑐⁄ + [( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) +i ] 𝑒𝑒 −i ( 𝑟𝑟−𝑅𝑅 ) 𝜔𝜔𝑠𝑠 𝑐𝑐⁄ ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) + i 𝜋𝜋 dd𝑘𝑘 ( 𝑘𝑘𝑅𝑅+i ) 𝑒𝑒 i ( 𝑟𝑟+𝑅𝑅 ) 𝑘𝑘 + ( 𝑘𝑘𝑅𝑅−i ) 𝑒𝑒 i ( 𝑟𝑟−𝑅𝑅 ) 𝑘𝑘 𝑘𝑘 − ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) � 𝑘𝑘=0 � = 𝑝𝑝 𝜔𝜔 𝑠𝑠 sin ( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) 𝒛𝒛�8𝜋𝜋𝑅𝑅 𝑟𝑟 � ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) cos [( 𝑟𝑟+𝑅𝑅 ) 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ] −sin [( 𝑟𝑟+𝑅𝑅 ) 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ]( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) + ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) cos [( 𝑟𝑟−𝑅𝑅 ) 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ] +sin [( 𝑟𝑟−𝑅𝑅 ) 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ]( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) � = − � 𝑝𝑝 𝜔𝜔 𝑠𝑠 𝒛𝒛�4𝜋𝜋𝑟𝑟 � [ sin ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) − ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) cos ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] cos ( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) sin ( 𝜔𝜔 𝑠𝑠 𝑡𝑡 )( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) . (B2) A similar method applies to Eq.(B1) inside the spherical particle ( 𝑟𝑟 < 𝑅𝑅 ) , except that the second term of the integrand now vanishes on a large semi-circular contour in the lower half of the complex plane. Also, this time the 2 nd order pole at 𝑘𝑘 = 0 does make a contribution. We have 𝑨𝑨 in ( 𝒓𝒓 , 𝑡𝑡 ) = 𝑝𝑝 𝜔𝜔 𝑠𝑠 sin ( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) 𝒛𝒛�8𝜋𝜋 𝑅𝑅 𝑟𝑟 Im � i 𝜋𝜋 [( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) +i ] 𝑒𝑒 i ( 𝑟𝑟+𝑅𝑅 ) 𝜔𝜔𝑠𝑠 𝑐𝑐⁄ − [( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) −i ] 𝑒𝑒 i ( 𝑟𝑟−𝑅𝑅 ) 𝜔𝜔𝑠𝑠 𝑐𝑐⁄ ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) +i 𝜋𝜋 [( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) −i ] 𝑒𝑒 −i ( 𝑟𝑟+𝑅𝑅 ) 𝜔𝜔𝑠𝑠 𝑐𝑐⁄ − [( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) +i ] 𝑒𝑒 −i ( 𝑟𝑟−𝑅𝑅 ) 𝜔𝜔𝑠𝑠 𝑐𝑐⁄ ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) +i 𝜋𝜋 dd𝑘𝑘 ( 𝑘𝑘𝑅𝑅+i ) 𝑒𝑒 i ( 𝑟𝑟+𝑅𝑅 ) 𝑘𝑘 − ( 𝑘𝑘𝑅𝑅−i ) 𝑒𝑒 i ( 𝑟𝑟−𝑅𝑅 ) 𝑘𝑘 𝑘𝑘 − ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) � 𝑘𝑘=0 � = 𝑝𝑝 𝜔𝜔 𝑠𝑠 sin ( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) 𝒛𝒛�8𝜋𝜋𝑅𝑅 𝑟𝑟 � ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) cos [( 𝑟𝑟+𝑅𝑅 ) 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ] −sin [( 𝑟𝑟+𝑅𝑅 ) 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ]( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) Note that the first term in the Taylor series expansion around 𝑘𝑘 = 0 is zero. − ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) cos [( 𝑟𝑟−𝑅𝑅 ) 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ] +sin [( 𝑟𝑟−𝑅𝑅 ) 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ]( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) + ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) � = 𝑝𝑝 𝜔𝜔 𝑠𝑠 𝒛𝒛�4𝜋𝜋 ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) 𝑟𝑟 {( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) − [cos( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) + ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) sin( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] sin( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )} sin( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) . (B3) Now, in the far field, the vector potential 𝑨𝑨 out ( 𝒓𝒓 , 𝑡𝑡 ) of Eq.(B2) contains both retarded terms in the form of sin[ 𝜔𝜔 𝑠𝑠 ( 𝑡𝑡 − 𝑟𝑟 𝑐𝑐⁄ )] and advanced terms in the form of sin[ 𝜔𝜔 𝑠𝑠 ( 𝑡𝑡 + 𝑟𝑟 𝑐𝑐⁄ )] . To eliminate the advanced terms, the heretofore neglected contributions of source-free (i.e., vacuum) terms must be added to both 𝑨𝑨 out of Eq.(B2) and 𝑨𝑨 in of Eq.(B3). We will return to this task later, after computing the scalar potential in the next section. B2. The scalar potential . The scalar potential in the spacetime domain is obtained similarly, via an inverse Fourier transformation, as follows: 𝜓𝜓 ( 𝒓𝒓 , 𝑡𝑡 ) = (2 𝜋𝜋 ) −4 � 𝜌𝜌 ( 𝒌𝒌 , 𝜔𝜔 ) 𝜀𝜀 [ 𝑘𝑘 − ( 𝜔𝜔 𝑐𝑐⁄ ) ] exp[i( 𝒌𝒌 ∙ 𝒓𝒓 − 𝜔𝜔𝑡𝑡 )]d 𝒌𝒌 d 𝜔𝜔 ∞−∞ = − i3𝜋𝜋𝑝𝑝 𝒛𝒛� ( ) 𝜀𝜀 ∙ � 𝒌𝒌 [ sin ( 𝑘𝑘𝑅𝑅 ) −𝑘𝑘𝑅𝑅 cos ( 𝑘𝑘𝑅𝑅 )] exp ( i𝒌𝒌 ∙ 𝒓𝒓 )( 𝑘𝑘𝑅𝑅 ) ∞−∞ � [ 𝛿𝛿 ( 𝜔𝜔+𝜔𝜔 𝑠𝑠 ) +𝛿𝛿 ( 𝜔𝜔−𝜔𝜔 𝑠𝑠 )] exp ( −i𝜔𝜔𝑡𝑡 ) 𝑘𝑘 − ( 𝜔𝜔 𝑐𝑐⁄ ) d 𝜔𝜔 ∞−∞ d 𝒌𝒌 = i3𝑝𝑝 cos ( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) 𝒛𝒛�8𝜋𝜋 𝜀𝜀 ∙ � 𝒌𝒌 [ 𝑘𝑘𝑅𝑅 cos ( 𝑘𝑘𝑅𝑅 ) −sin ( 𝑘𝑘𝑅𝑅 )] exp ( i𝒌𝒌 ∙ 𝒓𝒓 )( 𝑘𝑘𝑅𝑅 ) [ 𝑘𝑘 − ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) ] d 𝒌𝒌 ∞−∞ = i3𝑝𝑝 cos ( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) 𝒛𝒛�8𝜋𝜋 𝜀𝜀 ∙ � ∫ ( 𝑘𝑘𝒓𝒓� cos 𝜑𝜑 )[ 𝑘𝑘𝑅𝑅 cos ( 𝑘𝑘𝑅𝑅 ) −sin ( 𝑘𝑘𝑅𝑅 )] exp ( i𝑘𝑘𝑟𝑟 cos 𝜑𝜑 )( 𝑘𝑘𝑅𝑅 ) [ 𝑘𝑘 − ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) ] 𝜋𝜋𝑘𝑘 sin 𝜑𝜑 d 𝑘𝑘 d 𝜑𝜑 𝜋𝜋𝜑𝜑=0 ∞𝑘𝑘=0 = i3𝑝𝑝 cos ( 𝜔𝜔 𝑠𝑠 𝑡𝑡 )( 𝒛𝒛� ∙ 𝒓𝒓� ) 𝜀𝜀 � 𝑘𝑘𝑅𝑅 cos ( 𝑘𝑘𝑅𝑅 ) −sin ( 𝑘𝑘𝑅𝑅 ) 𝑅𝑅 [ 𝑘𝑘 − ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) ] ∫ sin 𝜑𝜑 cos 𝜑𝜑 exp(i 𝑘𝑘𝑟𝑟 cos 𝜑𝜑 ) d 𝜑𝜑 d 𝑘𝑘 𝜋𝜋𝜑𝜑=0 ∞𝑘𝑘=0 = cos ( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) cos 𝜃𝜃 𝜀𝜀 � 𝑘𝑘𝑅𝑅 cos ( 𝑘𝑘𝑅𝑅 ) −sin ( 𝑘𝑘𝑅𝑅 ) 𝑅𝑅 [ 𝑘𝑘 − ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) ] × 𝑘𝑘𝑟𝑟 cos ( 𝑘𝑘𝑟𝑟 ) −sin ( 𝑘𝑘𝑟𝑟 )( 𝑘𝑘𝑟𝑟 ) d 𝑘𝑘 ∞0 = cos ( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) cos 𝜃𝜃 𝜀𝜀 𝑅𝑅 𝑟𝑟 � � ( 𝑘𝑘𝑅𝑅+i ) 𝑒𝑒 i𝑘𝑘𝑅𝑅 + ( 𝑘𝑘𝑅𝑅−i ) 𝑒𝑒 −i𝑘𝑘𝑅𝑅 � × � ( 𝑘𝑘𝑟𝑟+i ) 𝑒𝑒 i𝑘𝑘𝑟𝑟 + ( 𝑘𝑘𝑟𝑟−i ) 𝑒𝑒 −i𝑘𝑘𝑟𝑟 �𝑘𝑘 [ 𝑘𝑘 − ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) ] d 𝑘𝑘 ∞0 = cos ( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) cos 𝜃𝜃 𝜀𝜀 𝑅𝑅 𝑟𝑟 �� [ 𝑅𝑅𝑟𝑟𝑘𝑘 −1+i ( 𝑟𝑟+𝑅𝑅 ) 𝑘𝑘 ] 𝑒𝑒 i ( 𝑟𝑟+𝑅𝑅 ) 𝑘𝑘 + [ 𝑅𝑅𝑟𝑟𝑘𝑘 −1−i ( 𝑟𝑟+𝑅𝑅 ) 𝑘𝑘 ] 𝑒𝑒 −i ( 𝑟𝑟+𝑅𝑅 ) 𝑘𝑘 𝑘𝑘 [ 𝑘𝑘 − ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) ] d 𝑘𝑘 ∞−∞ + � [ 𝑅𝑅𝑟𝑟𝑘𝑘 +1−i ( 𝑟𝑟−𝑅𝑅 ) 𝑘𝑘 ] 𝑒𝑒 i ( 𝑟𝑟−𝑅𝑅 ) 𝑘𝑘 + [ 𝑅𝑅𝑟𝑟𝑘𝑘 +1+i ( 𝑟𝑟−𝑅𝑅 ) 𝑘𝑘 ] 𝑒𝑒 −i ( 𝑟𝑟−𝑅𝑅 ) 𝑘𝑘 𝑘𝑘 [ 𝑘𝑘 − ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) ] d 𝑘𝑘 ∞−∞ � = cos ( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) cos 𝜃𝜃 𝜀𝜀 𝑅𝑅 𝑟𝑟 Re � [ 𝑅𝑅𝑟𝑟𝑘𝑘 −1+i ( 𝑟𝑟+𝑅𝑅 ) 𝑘𝑘 ] 𝑒𝑒 i ( 𝑟𝑟+𝑅𝑅 ) 𝑘𝑘 + [ 𝑅𝑅𝑟𝑟𝑘𝑘 +1−i ( 𝑟𝑟−𝑅𝑅 ) 𝑘𝑘 ] 𝑒𝑒 i ( 𝑟𝑟−𝑅𝑅 ) 𝑘𝑘 𝑘𝑘 [ 𝑘𝑘− ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] [ 𝑘𝑘+ ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] d 𝑘𝑘 ∞−∞ . (B4) Once again, the preceding integral can be evaluated in the complex plane using Cauchy’s theorem. Outside the spherical particle, where 𝑟𝑟 > 𝑅𝑅 , both exponential functions appearing in the integrand approach zero on a large semi-circular contour in the upper half-plane. The half-residues at the first-order poles 𝑘𝑘 = ± 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ are easy to evaluate. As for the second-order pole at 𝑘𝑘 = 0 , the product of 𝑘𝑘 and the integrand turns out to have a Taylor series expansion around sin( 𝑎𝑎 ± 𝑏𝑏 ) = sin 𝑎𝑎 cos 𝑏𝑏 ± cos 𝑎𝑎 sin 𝑏𝑏 cos( 𝑎𝑎 ± 𝑏𝑏 ) = cos 𝑎𝑎 cos 𝑏𝑏 ∓ sin 𝑎𝑎 sin 𝑏𝑏 𝑘𝑘 = 0 that has no constant term and no first-order term. As such, the pole at 𝑘𝑘 = 0 makes no contribution to the integral. We will have 𝜓𝜓 out ( 𝒓𝒓 , 𝑡𝑡 ) = cos ( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) cos 𝜃𝜃 𝜀𝜀 𝑅𝑅 𝑟𝑟 × Re � i 𝜋𝜋 [ 𝑅𝑅𝑟𝑟 ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) −1+i ( 𝑟𝑟+𝑅𝑅 )( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] 𝑒𝑒 i ( 𝑟𝑟+𝑅𝑅 ) 𝜔𝜔𝑠𝑠 𝑐𝑐⁄ + [ 𝑅𝑅𝑟𝑟 ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) +1−i ( 𝑟𝑟−𝑅𝑅 )( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] 𝑒𝑒 i ( 𝑟𝑟−𝑅𝑅 ) 𝜔𝜔𝑠𝑠 𝑐𝑐⁄ ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) − i 𝜋𝜋 [ 𝑅𝑅𝑟𝑟 ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) −1−i ( 𝑟𝑟+𝑅𝑅 )( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] 𝑒𝑒 −i ( 𝑟𝑟+𝑅𝑅 ) 𝜔𝜔𝑠𝑠 𝑐𝑐⁄ + [ 𝑅𝑅𝑟𝑟 ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) +1+i ( 𝑟𝑟−𝑅𝑅 )( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] 𝑒𝑒 −i ( 𝑟𝑟−𝑅𝑅 ) 𝜔𝜔𝑠𝑠 𝑐𝑐⁄ ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) +i 𝜋𝜋 dd𝑘𝑘 [ 𝑅𝑅𝑟𝑟𝑘𝑘 −1+i ( 𝑟𝑟+𝑅𝑅 ) 𝑘𝑘 ] 𝑒𝑒 i ( 𝑟𝑟+𝑅𝑅 ) 𝑘𝑘 + [ 𝑅𝑅𝑟𝑟𝑘𝑘 +1−i ( 𝑟𝑟−𝑅𝑅 ) 𝑘𝑘 ] 𝑒𝑒 i ( 𝑟𝑟−𝑅𝑅 ) 𝑘𝑘 𝑘𝑘 − ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) � 𝑘𝑘=0 � = − cos ( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) cos 𝜃𝜃 𝑅𝑅 𝑟𝑟 � [ 𝑅𝑅𝑟𝑟 ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) −1 ] sin [( 𝑟𝑟+𝑅𝑅 ) 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ] + ( 𝑟𝑟+𝑅𝑅 )( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) cos [( 𝑟𝑟+𝑅𝑅 ) 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ]( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) + [ 𝑅𝑅𝑟𝑟 ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) +1 ] sin [( 𝑟𝑟−𝑅𝑅 ) 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ] − ( 𝑟𝑟−𝑅𝑅 )( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) cos [( 𝑟𝑟−𝑅𝑅 ) 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ]( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) � = � cos 𝜃𝜃4𝜋𝜋𝜀𝜀 𝑟𝑟 � [ sin ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) − ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) cos ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] × [ cos ( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) + ( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) sin ( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] cos ( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) . (B5) A similar method applies to Eq.(B4) inside the spherical particle ( 𝑟𝑟 < 𝑅𝑅 ) , except that the second term of the integrand now vanishes on a large semi-circular contour in the lower half of the complex plane. Again, the 2 nd order pole at 𝑘𝑘 = 0 fails to make a contribution. We will have 𝜓𝜓 in ( 𝒓𝒓 , 𝑡𝑡 ) = cos ( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) cos 𝜃𝜃 𝜀𝜀 𝑅𝑅 𝑟𝑟 × Re � i 𝜋𝜋 [ 𝑅𝑅𝑟𝑟 ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) −1+i ( 𝑟𝑟+𝑅𝑅 )( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] 𝑒𝑒 i ( 𝑟𝑟+𝑅𝑅 ) 𝜔𝜔𝑠𝑠 𝑐𝑐⁄ − [ 𝑅𝑅𝑟𝑟 ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) +1−i ( 𝑟𝑟−𝑅𝑅 )( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] 𝑒𝑒 i ( 𝑟𝑟−𝑅𝑅 ) 𝜔𝜔𝑠𝑠 𝑐𝑐⁄ ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) − i 𝜋𝜋 [ 𝑅𝑅𝑟𝑟 ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) −1−i ( 𝑟𝑟+𝑅𝑅 )( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] 𝑒𝑒 −i ( 𝑟𝑟+𝑅𝑅 ) 𝜔𝜔𝑠𝑠 𝑐𝑐⁄ − [ 𝑅𝑅𝑟𝑟 ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) +1+i ( 𝑟𝑟−𝑅𝑅 )( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] 𝑒𝑒 −i ( 𝑟𝑟−𝑅𝑅 ) 𝜔𝜔𝑠𝑠 𝑐𝑐⁄ ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) +i 𝜋𝜋 dd𝑘𝑘 [ 𝑅𝑅𝑟𝑟𝑘𝑘 −1+i ( 𝑟𝑟+𝑅𝑅 ) 𝑘𝑘 ] 𝑒𝑒 i ( 𝑟𝑟+𝑅𝑅 ) 𝑘𝑘 − [ 𝑅𝑅𝑟𝑟𝑘𝑘 +1−i ( 𝑟𝑟−𝑅𝑅 ) 𝑘𝑘 ] 𝑒𝑒 i ( 𝑟𝑟−𝑅𝑅 ) 𝑘𝑘 𝑘𝑘 − ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) � 𝑘𝑘=0 � = − cos ( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) cos 𝜃𝜃 𝑅𝑅 𝑟𝑟 � [ 𝑅𝑅𝑟𝑟 ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) −1 ] sin [( 𝑟𝑟+𝑅𝑅 ) 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ] + ( 𝑟𝑟+𝑅𝑅 )( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) cos [( 𝑟𝑟+𝑅𝑅 ) 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ]( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) − [ 𝑅𝑅𝑟𝑟 ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) +1 ] sin [( 𝑟𝑟−𝑅𝑅 ) 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ] − ( 𝑟𝑟−𝑅𝑅 )( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) cos [( 𝑟𝑟−𝑅𝑅 ) 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ]( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) � = � 𝑟𝑟 cos 𝜃𝜃4𝜋𝜋𝜀𝜀 𝑅𝑅 � [ cos ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) + ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) sin ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] × [ sin ( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) − ( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) cos ( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] cos ( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) ( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) . (B6) As before, the scalar potential 𝜓𝜓 out ( 𝒓𝒓 , 𝑡𝑡 ) of Eq.(B5) contains both advanced and retarded terms in the form of cos[ 𝜔𝜔 𝑠𝑠 ( 𝑡𝑡 ± 𝑟𝑟 𝑐𝑐⁄ )] and sin[ 𝜔𝜔 𝑠𝑠 ( 𝑡𝑡 ± 𝑟𝑟 𝑐𝑐⁄ )] . To eliminate the advanced terms, the heretofore neglected contributions of source-free (i.e., vacuum) terms must be added to both 𝜓𝜓 out of Eq.(B5) and 𝜓𝜓 in of Eq.(B6). The vacuum potentials are computed in the following section, after which we will return to derive the complete expressions for the scalar and vector potentials of our spherical dipole. sin( 𝑎𝑎 ± 𝑏𝑏 ) = sin 𝑎𝑎 cos 𝑏𝑏 ± cos 𝑎𝑎 sin 𝑏𝑏 cos( 𝑎𝑎 ± 𝑏𝑏 ) = cos 𝑎𝑎 cos 𝑏𝑏 ∓ sin 𝑎𝑎 sin 𝑏𝑏 sin( 𝑎𝑎 ± 𝑏𝑏 ) = sin 𝑎𝑎 cos 𝑏𝑏 ± cos 𝑎𝑎 sin 𝑏𝑏 cos( 𝑎𝑎 ± 𝑏𝑏 ) = cos 𝑎𝑎 cos 𝑏𝑏 ∓ sin 𝑎𝑎 sin 𝑏𝑏 B3. Computing the vacuum potentials . The vacuum vector potential 𝑨𝑨 vac ( 𝒓𝒓 , 𝑡𝑡 ) is a super-position of plane-waves propagating in free space, having frequencies ± 𝜔𝜔 𝑠𝑠 , 𝑘𝑘 -vectors 𝒌𝒌 =( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) 𝒌𝒌� , and the overall Fourier domain distribution 𝑨𝑨 vac ( 𝒌𝒌 , 𝜔𝜔 ) = 𝐴𝐴 𝒛𝒛�𝛿𝛿 [ 𝑘𝑘 − ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )][ 𝛿𝛿 ( 𝜔𝜔 − 𝜔𝜔 𝑠𝑠 ) + 𝛿𝛿 ( 𝜔𝜔 + 𝜔𝜔 𝑠𝑠 )] . (B7) The constant coefficient 𝐴𝐴 will be determined further below. The spacetime profile of the above distribution is found straightforwardly via the following inverse Fourier transformation: 𝑨𝑨 vac ( 𝒓𝒓 , 𝑡𝑡 ) = (2 𝜋𝜋 ) −4 ∫ 𝐴𝐴 𝒛𝒛�𝛿𝛿 [ 𝑘𝑘 − ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )][ 𝛿𝛿 ( 𝜔𝜔 − 𝜔𝜔 𝑠𝑠 ) + 𝛿𝛿 ( 𝜔𝜔 + 𝜔𝜔 𝑠𝑠 )] exp[i( 𝒌𝒌 ∙ 𝒓𝒓 − 𝜔𝜔𝑡𝑡 )] d 𝒌𝒌 d 𝜔𝜔 ∞−∞ = (2 𝜋𝜋 ) −4 𝐴𝐴 𝒛𝒛� ∫ 𝛿𝛿 [ 𝑘𝑘 − ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] 𝑒𝑒 i𝒌𝒌 ∙ 𝒓𝒓 d 𝒌𝒌 ∫ [ 𝛿𝛿 ( 𝜔𝜔 − 𝜔𝜔 𝑠𝑠 ) + 𝛿𝛿 ( 𝜔𝜔 + 𝜔𝜔 𝑠𝑠 )] 𝑒𝑒 −i𝜔𝜔𝑡𝑡 d 𝜔𝜔 ∞−∞∞−∞ = 2(2 𝜋𝜋 ) −4 𝐴𝐴 𝒛𝒛� cos( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) ∫ 𝛿𝛿 [ 𝑘𝑘 − ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] 𝑒𝑒 i𝒌𝒌 ∙ 𝒓𝒓 d 𝒌𝒌 ∞−∞ = 2(2 𝜋𝜋 ) −4 𝐴𝐴 𝒛𝒛� cos( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) � ∫ 𝛿𝛿 [ 𝑘𝑘 − ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] 𝑒𝑒 i𝑘𝑘𝑟𝑟 cos 𝜑𝜑 𝜋𝜋𝑘𝑘 sin 𝜑𝜑 d 𝑘𝑘 d 𝜑𝜑 𝜋𝜋𝜑𝜑=0 ∞𝑘𝑘=0 = 2(2 𝜋𝜋 ) −3 𝐴𝐴 𝒛𝒛� cos( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) � 𝛿𝛿 [ 𝑘𝑘 − ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] 𝑘𝑘 ∫ sin 𝜑𝜑 𝑒𝑒 i𝑘𝑘𝑟𝑟 cos 𝜑𝜑 d 𝜑𝜑 d 𝑘𝑘 𝜋𝜋𝜑𝜑=0 ∞𝑘𝑘=0 = 2(2 𝜋𝜋 ) −3 𝐴𝐴 𝒛𝒛� cos( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) ∫ 𝛿𝛿 [ 𝑘𝑘 − ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] 𝑘𝑘 [2 sin( 𝑘𝑘𝑟𝑟 ) ( 𝑘𝑘𝑟𝑟 ) ⁄ ]d 𝑘𝑘 ∞0 = � 𝜔𝜔 𝑠𝑠 𝐴𝐴 𝒛𝒛�2𝜋𝜋 𝑐𝑐𝑟𝑟 � sin( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) cos( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) . (B8) Comparison with Eq.(B2) reveals that, in order to eliminate the advanced term in the overall vector potential, the coefficient 𝐴𝐴 of the vacuum potential must be set to 𝐴𝐴 = � 𝑐𝑐𝜇𝜇 𝑝𝑝 � sin ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) − ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) cos ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) . (B9) We thus arrive at the desired form of the vacuum vector potential, namely, 𝑨𝑨 vac ( 𝒓𝒓 , 𝑡𝑡 ) = � 𝜔𝜔 𝑠𝑠 𝑝𝑝 𝒛𝒛�4𝜋𝜋𝑟𝑟 � � sin ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) − ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) cos ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) � sin( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) cos( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) . (B10) The corresponding vacuum scalar potential 𝜓𝜓 vac ( 𝒓𝒓 , 𝑡𝑡 ) is a superposition of plane-waves in free space, having frequencies ± 𝜔𝜔 𝑠𝑠 , 𝑘𝑘 -vectors 𝒌𝒌 = ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) 𝒌𝒌� , and the Fourier domain distribution 𝜓𝜓 vac ( 𝒌𝒌 , 𝜔𝜔 ) = 𝜓𝜓 𝛿𝛿 [ 𝑘𝑘 − ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )][ 𝛿𝛿 ( 𝜔𝜔 − 𝜔𝜔 𝑠𝑠 ) − 𝛿𝛿 ( 𝜔𝜔 + 𝜔𝜔 𝑠𝑠 )] 𝒌𝒌� ∙ 𝒛𝒛� . (B11) The constant coefficient 𝜓𝜓 will be determined further below. The spacetime profile of the above distribution is found straightforwardly via inverse Fourier transformation, as follows: 𝜓𝜓 vac ( 𝒓𝒓 , 𝑡𝑡 ) = (2 𝜋𝜋 ) −4 � 𝜓𝜓 𝛿𝛿 [ 𝑘𝑘 − ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )][ 𝛿𝛿 ( 𝜔𝜔 − 𝜔𝜔 𝑠𝑠 ) − 𝛿𝛿 ( 𝜔𝜔 + 𝜔𝜔 𝑠𝑠 )] 𝒌𝒌� ∙ 𝒛𝒛� exp[i( 𝒌𝒌 ∙ 𝒓𝒓 − 𝜔𝜔𝑡𝑡 )] d 𝒌𝒌 d 𝜔𝜔 ∞−∞ = (2 𝜋𝜋 ) −4 𝜓𝜓 𝒛𝒛� ∙ ∫ 𝛿𝛿 [ 𝑘𝑘 − ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] 𝒌𝒌�𝑒𝑒 i𝒌𝒌 ∙ 𝒓𝒓 d 𝒌𝒌 ∫ [ 𝛿𝛿 ( 𝜔𝜔 − 𝜔𝜔 𝑠𝑠 ) − 𝛿𝛿 ( 𝜔𝜔 + 𝜔𝜔 𝑠𝑠 )] 𝑒𝑒 −i𝜔𝜔𝑡𝑡 d 𝜔𝜔 ∞−∞∞−∞ = − 𝜋𝜋 ) −4 𝜓𝜓 sin( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) 𝒛𝒛� ∙ ∫ 𝛿𝛿 [ 𝑘𝑘 − ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] 𝒌𝒌�𝑒𝑒 i𝒌𝒌 ∙ 𝒓𝒓 d 𝒌𝒌 ∞−∞ = − 𝜋𝜋 ) −4 𝜓𝜓 sin( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) 𝒛𝒛� ∙ � ∫ 𝛿𝛿 [ 𝑘𝑘 − ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] cos 𝜑𝜑 𝒓𝒓� 𝑒𝑒 i𝑘𝑘𝑟𝑟 cos 𝜑𝜑 𝜋𝜋𝑘𝑘 sin 𝜑𝜑 d 𝑘𝑘 d 𝜑𝜑 𝜋𝜋𝜑𝜑=0∞𝑘𝑘=0 = − 𝜋𝜋 ) −3 𝜓𝜓 sin( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) 𝒛𝒛� ∙ 𝒓𝒓� � 𝛿𝛿 [ 𝑘𝑘 − ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] 𝑘𝑘 ∫ sin 𝜑𝜑 cos 𝜑𝜑 𝑒𝑒 i𝑘𝑘𝑟𝑟 cos 𝜑𝜑 d 𝜑𝜑 d 𝑘𝑘 𝜋𝜋𝜑𝜑=0∞𝑘𝑘=0 = 4(2 𝜋𝜋 ) −3 𝜓𝜓 sin( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) cos 𝜃𝜃 � 𝛿𝛿 [ 𝑘𝑘 − ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] 𝑘𝑘 � sin ( 𝑘𝑘𝑟𝑟 ) − 𝑘𝑘𝑟𝑟 cos ( 𝑘𝑘𝑟𝑟 )( 𝑘𝑘𝑟𝑟 ) � d 𝑘𝑘 ∞0 = 𝜓𝜓 cos 𝜃𝜃2𝜋𝜋 𝑟𝑟 [sin( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) − ( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) cos( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] sin( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) . (B12) Comparison with Eq.(B5) reveals that, in order to eliminate the advanced terms in the overall scalar potential, the coefficient 𝜓𝜓 must be set to 𝜓𝜓 = � 𝑝𝑝 � sin ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) − ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) cos ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) . (B13) We thus arrive at the desired vacuum scalar potential, namely, 𝜓𝜓 vac ( 𝒓𝒓 , 𝑡𝑡 ) = � cos 𝜃𝜃4𝜋𝜋𝜀𝜀 𝑟𝑟 � [ sin ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) − ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) cos ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )][ sin ( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) − ( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) cos ( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] sin ( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) . (B14) Upon adding the vacuum potentials of Eqs.(B10) and (B14) to the dipolar potentials found previously in Sections B1 and B2, the terms corresponding to advanced potentials are eliminated, leaving the exterior fields of the dipole with retarded terms only. Digression : The Lorenz gauge, 𝜵𝜵 ∙ 𝑨𝑨 ( 𝒓𝒓 , 𝑡𝑡 ) + (1 𝑐𝑐 ⁄ ) 𝜕𝜕𝜓𝜓 ( 𝒓𝒓 , 𝑡𝑡 ) 𝜕𝜕𝑡𝑡⁄ = 0 , must be satisfied by the vacuum potentials of Eqs.(B10) and (B14). To confirm this, note that the potentials in the Fourier domain are given by 𝑨𝑨 vac ( 𝒌𝒌 , 𝜔𝜔 ) = � 𝑐𝑐𝜇𝜇 𝑝𝑝 � � sin ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) − ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) cos ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) � 𝛿𝛿 [ 𝑘𝑘 − ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )][ 𝛿𝛿 ( 𝜔𝜔 − 𝜔𝜔 𝑠𝑠 ) + 𝛿𝛿 ( 𝜔𝜔 + 𝜔𝜔 𝑠𝑠 )] 𝒛𝒛� , (B15) 𝜓𝜓 vac ( 𝒌𝒌 , 𝜔𝜔 ) = � 𝑝𝑝 � � sin ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) − ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) cos ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) � 𝛿𝛿 [ 𝑘𝑘 − ( 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )][ 𝛿𝛿 ( 𝜔𝜔 − 𝜔𝜔 𝑠𝑠 ) − 𝛿𝛿 ( 𝜔𝜔 + 𝜔𝜔 𝑠𝑠 )] 𝒌𝒌� ∙ 𝒛𝒛� . (B16) These expressions clearly satisfy the Fourier version of the Lorenz gauge, namely 𝒌𝒌 ∙ 𝑨𝑨 ( 𝒌𝒌 , 𝜔𝜔 ) = ( 𝜔𝜔 𝑐𝑐 ⁄ ) 𝜓𝜓 ( 𝒌𝒌 , 𝜔𝜔 ) . B4. Complete expressions of the vector and scalar potentials . Adding the vacuum vector potential of Eq.(B10) to Eqs.(B2) and (B3) now yields the correct (i.e., retarded) form of the vector potential function both inside and outside the spherical particle, as follows: 𝑨𝑨 out ( 𝒓𝒓 , 𝑡𝑡 ) = − � 𝜔𝜔 𝑠𝑠 𝑝𝑝 𝒛𝒛�4𝜋𝜋𝑟𝑟 � � sin ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) − ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) cos ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) � sin[ 𝜔𝜔 𝑠𝑠 ( 𝑡𝑡 − 𝑟𝑟 𝑐𝑐⁄ )] . (B17) 𝑨𝑨 in ( 𝒓𝒓 , 𝑡𝑡 ) = 𝜔𝜔 𝑠𝑠 𝑝𝑝 𝒛𝒛�4𝜋𝜋 ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) 𝑟𝑟 {[sin( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) − ( 𝑅𝑅 𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) cos( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] sin( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) cos( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) − {[cos( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) + ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) sin( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] sin( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) − ( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )} sin( 𝜔𝜔 𝑠𝑠 𝑡𝑡 )} . (B18) Similarly, adding the vacuum scalar potential of Eq.(B14) to Eqs.(B5) and (B6) yields the retarded form of the scalar potential inside as well as outside the particle; that is, 𝜓𝜓 out ( 𝒓𝒓 , 𝑡𝑡 ) = � cos 𝜃𝜃4𝜋𝜋𝜀𝜀 𝑟𝑟 � � [ sin ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) − ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) cos ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] × [ cos ( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) + ( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) sin ( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] cos ( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) + [ sin ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) − ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) cos ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] × [ sin ( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) − ( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) cos ( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] sin ( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) � = � cos 𝜃𝜃4𝜋𝜋𝜀𝜀 𝑟𝑟 � [ sin ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) − ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) cos ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] × { cos [ 𝜔𝜔 𝑠𝑠 ( 𝑡𝑡−𝑟𝑟 𝑐𝑐⁄ )] − ( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) sin [ 𝜔𝜔 𝑠𝑠 ( 𝑡𝑡−𝑟𝑟 𝑐𝑐⁄ )]} ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) . (B19) 2 𝜓𝜓 in ( 𝒓𝒓 , 𝑡𝑡 ) = � 𝑟𝑟 cos 𝜃𝜃4𝜋𝜋𝜀𝜀 𝑅𝑅 � � [ cos ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) + ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) sin ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] × [ sin ( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) − ( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) cos ( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] cos ( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) ( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) + [ sin ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) − ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) cos ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] × [ sin ( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) − ( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) cos ( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] sin ( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) ( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) � = � 𝑟𝑟 cos 𝜃𝜃 𝑅𝑅 � × � sin ( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) − ( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) cos ( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )( 𝑟𝑟𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) � × {[cos( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) + ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) sin( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] cos( 𝜔𝜔 𝑠𝑠 𝑡𝑡 ) +[sin( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) − ( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ ) cos( 𝑅𝑅𝜔𝜔 𝑠𝑠 𝑐𝑐⁄ )] sin( 𝜔𝜔 𝑠𝑠 𝑡𝑡 )} . (B20) These scalar and vector potentials may now be used to evaluate the electric and magnetic fields inside as well as outside the spherical dipole. Appendix C
In Sec.5, we tracked the poles of the transfer function of a spherical dipole in the complex 𝜔𝜔 -plane and argued that the dipole’s impulse-response must be causal. Here, we demonstrate the conservation of energy by showing that, in the absence of non-radiative damping (i.e., when 𝛾𝛾 = 0 ), the dipole, which instantaneously acquires its energy at 𝑡𝑡 = 0 from the externally applied impulsive 𝐸𝐸 -field, releases all of this energy into the surrounding space via EM radiation. Let the response of the dipole to the impulsive 𝐸𝐸 -field 𝑬𝑬 ( 𝑡𝑡 ) = 𝛿𝛿 ( 𝑡𝑡 ) 𝒛𝒛� be denoted by 𝓅𝓅 ( 𝑡𝑡 ) 𝒛𝒛� , and let 𝓅𝓅 ( 𝜔𝜔 ) = ∫ 𝓅𝓅 ( 𝑡𝑡 ) 𝑒𝑒 i𝜔𝜔𝑡𝑡 d 𝑡𝑡 ∞−∞ be the Fourier transform of 𝓅𝓅 ( 𝑡𝑡 ) . Suppose 𝓅𝓅 ( 𝜔𝜔 ) is sampled at regular intervals of ∆𝜔𝜔 , corresponding to a set of discrete dipoles having amplitudes 𝑝𝑝 =2| 𝓅𝓅 ( 𝜔𝜔 )| ∆𝜔𝜔 𝜋𝜋⁄ , each oscillating at the respective frequency 𝜔𝜔 , which is an integer-multiple of ∆𝜔𝜔 . The radiated power is thus the sum of Eq.(16) over all such single-frequency oscillators across a range of 𝜔𝜔 that extends from to ∞ . The periodic function 𝓅𝓅 ( 𝑡𝑡 ) obtained by a superposition of these discrete frequency dipoles thus has the period 𝑇𝑇 = 2 𝜋𝜋 / ∆𝜔𝜔 . Multiplying the radiation rate of each oscillator with the repetition period 𝑇𝑇 yields the EM energy that the individual dipole oscillator emits during the period 𝑇𝑇 . In the limit when ∆𝜔𝜔 → , we will have 𝑇𝑇𝑝𝑝 → (2 𝜋𝜋⁄ )| 𝓅𝓅 ( 𝜔𝜔 )| d 𝜔𝜔 and, therefore, the total radiated EM energy ℰ in response to the impulsive excitation can be computed by integrating Eq.(16) over all frequencies 𝜔𝜔 ; that is, ℰ = � � | 𝓅𝓅 ( 𝜔𝜔 )| � ( 𝜔𝜔 𝑐𝑐⁄ ) [sin( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) − ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) cos(
𝑅𝑅𝜔𝜔 𝑐𝑐⁄ )] d 𝜔𝜔 ∞0 . (C1) Considering that the Fourier spectrum of 𝑬𝑬 ( 𝑡𝑡 ) = 𝛿𝛿 ( 𝑡𝑡 ) 𝒛𝒛� is unity at all frequencies 𝜔𝜔 ranging from −∞ to ∞ , the transfer function 𝑝𝑝 𝐸𝐸 ⁄ of the dipole is given by Eq.(20), as follows: 𝓅𝓅 ( 𝜔𝜔 ) = 𝑞𝑞 𝑚𝑚 ⁄𝜔𝜔 − 𝜔𝜔 − 𝜔𝜔 𝑝𝑝2 Γ ( 𝜔𝜔 ) − i𝛾𝛾𝜔𝜔 . (C2) Here 𝜔𝜔 𝑝𝑝 = 𝑞𝑞 ( 𝜀𝜀 𝑚𝑚 𝑣𝑣 ) ⁄ , with 𝑣𝑣 = 4 𝜋𝜋𝑅𝑅 ⁄ being the volume of the dipole and, from Eq.(19), Γ ( 𝜔𝜔 ) = [ sin ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) − ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) cos ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ )] × [ − i ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ )] 𝑒𝑒 i𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) − . (C3) The impulse-response 𝓅𝓅 ( 𝑡𝑡 ) is the inverse Fourier transform of the transfer function; that is, 𝓅𝓅 ( 𝑡𝑡 ) = � 𝑞𝑞 𝑚𝑚 ⁄𝜔𝜔 − 𝜔𝜔 − 𝜔𝜔 𝑝𝑝2 Γ ( 𝜔𝜔 ) − i𝛾𝛾𝜔𝜔 𝑒𝑒 −i𝜔𝜔𝑡𝑡 d 𝜔𝜔 ∞−∞ . (C4) 3 Since all the poles of 𝓅𝓅 ( 𝜔𝜔 ) reside in the lower-half of the complex 𝜔𝜔 -plane, 𝓅𝓅 ( 𝑡𝑡 ) = 0 for 𝑡𝑡 ≤ . The vanishing of 𝓅𝓅 (0) can be numerically verified by showing that the total area under the function 𝓅𝓅 ( 𝜔𝜔 ) is zero. In what follows, we shall set 𝛾𝛾 = 0 to eliminate non-radiative losses, thus focusing our attention exclusively on the radiated EM energy. The initial slope 𝓅𝓅̇ ( 𝑡𝑡 )| 𝑡𝑡=0 of the impulse-response function is obtained by differentiating Eq.(C4) with respect to time, then setting 𝑡𝑡 = 0 . We find 𝓅𝓅̇ ( 𝑡𝑡 )| 𝑡𝑡=0 = − i2𝜋𝜋 � ( 𝑞𝑞 𝑚𝑚 ⁄ ) 𝜔𝜔𝜔𝜔 −𝜔𝜔 −𝜔𝜔 𝑝𝑝2 Γ ( 𝜔𝜔 ) d 𝜔𝜔 ∞−∞ = � 𝜔𝜔 𝑝𝑝2 Im [ Γ ( 𝜔𝜔 )] ( 𝑞𝑞 𝑚𝑚 ⁄ ) 𝜔𝜔 | 𝜔𝜔 − 𝜔𝜔 −𝜔𝜔 𝑝𝑝2 Γ ( 𝜔𝜔 )| d 𝜔𝜔 ∞−∞ = � ( 𝑞𝑞 𝑚𝑚 ⁄ ) 𝜔𝜔𝜀𝜀 𝑣𝑣 | 𝜔𝜔 −𝜔𝜔 −𝜔𝜔 𝑝𝑝2 Γ ( 𝜔𝜔 )| × [ sin ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) − ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) cos ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ )] ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) d 𝜔𝜔 ∞−∞ = � | 𝓅𝓅 ( 𝜔𝜔 )| 𝑣𝑣 × ( 𝜔𝜔 𝑐𝑐⁄ ) [sin( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) − ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) cos(
𝑅𝑅𝜔𝜔 𝑐𝑐⁄ )] d 𝜔𝜔 ∞0 . (C5) Considering that the rate of transfer of EM energy from the applied field 𝑬𝑬 ( 𝑡𝑡 ) to the dipole is 𝑬𝑬 ( 𝑡𝑡 ) ∙ 𝓹𝓹̇ ( 𝑡𝑡 ) , and that the overall energy picked up by the dipole is ∫ 𝑬𝑬 ( 𝑡𝑡 ) ∙ 𝓹𝓹̇ ( 𝑡𝑡 )d 𝑡𝑡 ∞−∞ = 𝓅𝓅̇ (0) , a comparison of Eq.(C5) with Eq.(C1) confirms that the energy that is instantaneously picked up by the dipole at 𝑡𝑡 = 0= 0