Electromagnetic radiation and the self torque of an oscillating magnetic dipole
11 Electromagnetic radiation and the self torque of an oscillating magnetic dipole
Masud Mansuripur † and Per K. Jakobsen ‡ † College of Optical Sciences, The University of Arizona, Tucson, Arizona, USA ‡ Department of Mathematics and Statistics, UIT The Arctic University of Norway, Tromsø, Norway [Published in the
Proceedings of SPIE , Plasmonics: Design, Materials, Fabrication, Characterization, and Applications XVIII, (2020); doi: 10.1117/12.2569137]
Abstract . A uniformly-charged spherical shell of radius 𝑅𝑅 , mass 𝓂𝓂 , and total electrical charge 𝑞𝑞 , having an oscillatory angular velocity 𝜴𝜴 ( 𝑡𝑡 ) around a fixed axis, is a model for a magnetic dipole that radiates an electromagnetic field into its surrounding free space at a fixed oscillation frequency 𝜔𝜔 . An exact solution of the Maxwell-Lorentz equations of classical electrodynamics yields the self-torque of radiation resistance acting on the spherical shell as a function of 𝑅𝑅 , 𝑞𝑞 , and 𝜔𝜔 . Invoking the Newtonian equation of motion for the shell, we relate its angular velocity 𝜴𝜴 ( 𝑡𝑡 ) to an externally applied torque, and proceed to examine the response of the magnetic dipole to an impulsive torque applied at a given instant of time, say, 𝑡𝑡 = 0 . The impulse response of the dipole is found to be causal down to extremely small values of 𝑅𝑅 (i.e., as 𝑅𝑅 → ) so long as the exact expression of the self-torque is used in the dynamical equation of motion of the spherical shell.
1. Introduction . In a recent paper, we examined the electromagnetic (EM) radiation by a small spherical electric dipole and pointed out the significance of the role played by the exact radiation reaction function in the context of the causal behavior of the dipole’s response to external excitations. The present paper extends the results of [1] to an oscillating magnetic dipole modelled as a rotating electrically-charged spherical shell. While the general conclusions of the present paper parallel those of the previous one, we believe the magnetic dipole’s inherent advantages over an electric dipole make it worthy of its own separate analysis. Whereas an electric dipole’s positive and negative charges physically separate from each other during each oscillation period, the magnetic dipole’s rotary motion does not entail a similar separation of charges. This feature of the magnetic dipole not only removes restrictions on its oscillation amplitude, but also eliminates the restoring force that the opposite charges of an electric dipole, continually and unavoidably, exert upon each other. Such simplifications and reductions in the number of physical constraints on the system under investigation enable one to focus attention on the salient features of the system that impact its causal or acausal behavior. Another difference between the two dipoles is that the inertial masses of the charged particles constituting an electric dipole remain fixed as the dipole’s radius is made to approach zero, whereas in the case of a magnetic dipole, the moment of inertia of the rotating particle diminishes along with its shrinking radius. All in all, an oscillating spherical shell imitating a magnetic dipole provides a simpler model for studying the causal or acausal behavior of small electric charges in the limit when their dimensions are made to approach zero. The organization of the paper is as follows. In Sec.2, we describe the spinning spherical shell model of a classical magnetic dipole, and derive exact expressions for the single-frequency vector potential in the free space regions inside and outside the sphere. The calculated vector potential is used in Sec.3 to arrive at rigorous expressions for the EM fields surrounding the shell, and also to compute the rate of EM energy radiation as a function of the oscillation frequency 𝜔𝜔 for a given radius 𝑅𝑅 and electric charge 𝑞𝑞 of the spherical dipole. The self 𝐸𝐸 -field of the dipole is then used in Sec.4, in conjunction with Newton’s second law of motion, to relate the angular velocity 𝜴𝜴 ( 𝑡𝑡 ) = 𝛺𝛺 𝒛𝒛�𝑒𝑒 −i𝜔𝜔𝜔𝜔 of the spherical shell to an externally applied torque 𝑻𝑻 ( 𝑡𝑡 ) = 𝑇𝑇 𝒛𝒛�𝑒𝑒 −i𝜔𝜔𝜔𝜔 that drives the oscillations of the charged sphere at the desired frequency 𝜔𝜔 . The end result of this section is an expression for the transfer function 𝛺𝛺 𝑇𝑇 ⁄ of the dipole as a function of its excitation frequency 𝜔𝜔 for arbitrary values of the radius 𝑅𝑅 , the overall charge 𝑞𝑞 , and the total mass 𝓂𝓂 of the spinning spherical shell. Section 5 is devoted to a discussion of the role played by the poles of the transfer function 𝛺𝛺 𝑇𝑇 ⁄ in the causal response of our magnetic dipole to an impulsive excitation. Specifically, we argue that the presence of any number of poles in the upper-half of the complex 𝜔𝜔 -plane provides a clear indication that the impulse-response of the dipole is acausal. Here, we also show numerical results that confirm that, while the small-radius approximation to the self-torque (i.e., radiation resistance) leads to the prediction of acausal behavior, the exact self-torque places all the poles of 𝛺𝛺 𝑇𝑇 ⁄ in the lower-half plane, thus ensuring the dipole’s causal response. In Sec.6, we argue that the radiation reaction function Γ ( 𝜔𝜔 ) can be split into two parts: (i) a part that is in-phase with the oscillating electric current around the spherical shell and, therefore, accounts for the radiated EM energy; and (ii) a part that is out-of-phase with the electric current and can be associated with the underlying mechanism that drives the internal exchange between a “material component” and an “EM component” of the mass of the dipole. Considering that the dipole’s inertial mass 𝓂𝓂 is already taken into account through its contribution to the moment of inertia of the sphere, it is tempting to remove the out-of-phase component of the radiation reaction function from the equation of motion — ostensibly because its effect has already been accounted for through the use of a fixed moment of inertia for the particle. However, it will be shown in Sec.6 that removing even a small fraction of the out-of-phase component of Γ ( 𝜔𝜔 ) brings about acausal behavior by putting an infinite number of poles into the upper-half plane of the argument 𝜔𝜔 of the transfer function. A brief discussion of this curious behavior of the transfer function 𝛺𝛺 𝑇𝑇 ⁄ is relegated to the final section of the paper.
2. Model of magnetic dipole as an oscillating electrical current around a spherical shell . Figure 1 shows a uniformly-charged spherical shell of radius 𝑅𝑅 and total electrical charge 𝑞𝑞 , spinning with a time-dependent angular velocity around the 𝑧𝑧 -axis. Let the total mass 𝓂𝓂 of the shell be uniformly distributed over its surface area. The moment of inertia 𝐼𝐼 of the shell is readily computed in the spherical ( 𝑟𝑟 , 𝜃𝜃 , 𝜑𝜑 ) coordinate system, as follows: 𝐼𝐼 = � � 𝓂𝓂4𝜋𝜋𝑅𝑅 � ( 𝑅𝑅 sin 𝜃𝜃 ) (2 𝜋𝜋𝑅𝑅 sin 𝜃𝜃 )d 𝜃𝜃 𝜋𝜋𝜃𝜃=0 = ⅔𝓂𝓂𝑅𝑅 . (1) Defining 𝛺𝛺 = | 𝛺𝛺 | 𝑒𝑒 i𝜙𝜙 as the complex amplitude of the sinusoidal oscillations around the 𝑧𝑧 -axis, we write the angular velocity of the spherical shell as follows: 𝜑𝜑̇ ( 𝑡𝑡 ) 𝒛𝒛� = | 𝛺𝛺 | cos( 𝜔𝜔𝑡𝑡 − 𝜙𝜙 ) 𝒛𝒛� = Re( 𝛺𝛺 𝑒𝑒 −i𝜔𝜔𝜔𝜔 ) 𝒛𝒛� = Re[ 𝜴𝜴 ( 𝑡𝑡 )] . (2) Fig.1 . A thin spherical shell of radius 𝑅𝑅 , mass 𝓂𝓂 , and total electric charge 𝑞𝑞 rotates around the 𝑧𝑧 -axis with angular velocity 𝜴𝜴 ( 𝑡𝑡 ) . The current density 𝓙𝓙 ( 𝒓𝒓 , 𝑡𝑡 ) is maximum at the equator and drops to zero (in proportion to sin 𝜃𝜃 ) as the polar angle 𝜃𝜃 approaches zero at the north pole, and 𝜋𝜋 at the south pole. 𝑦𝑦 𝜴𝜴 ( 𝑡𝑡 ) 𝑥𝑥 𝑧𝑧 𝓙𝓙 ( 𝒓𝒓 , 𝑡𝑡 ) Introducing 𝒥𝒥 𝑠𝑠0 = 𝑞𝑞𝛺𝛺 (4 𝜋𝜋𝑅𝑅 ) ⁄ as the (complex) amplitude of the 𝜑𝜑 -directed surface current density at the equator, the complete expression of the surface-current-density will be 𝓙𝓙 𝑠𝑠 ( 𝑟𝑟 = 𝑅𝑅 , 𝜃𝜃 , 𝜑𝜑 , 𝑡𝑡 ) = � 𝑞𝑞4𝜋𝜋𝑅𝑅 � ( 𝑅𝑅 sin 𝜃𝜃 ) 𝛺𝛺 ( 𝑡𝑡 ) 𝝋𝝋� = 𝒥𝒥 𝑠𝑠0 sin 𝜃𝜃 𝑒𝑒 −i𝜔𝜔𝜔𝜔 𝝋𝝋� = � 𝑞𝑞4𝜋𝜋𝑅𝑅 � 𝜴𝜴 ( 𝑡𝑡 ) × 𝒓𝒓� . (3) The magnetic dipole moment of the spinning sphere is thus given by 𝒎𝒎 ( 𝑡𝑡 ) = 𝜇𝜇 � ( 𝜋𝜋𝑅𝑅 sin 𝜃𝜃 ) � 𝑞𝑞4𝜋𝜋𝑅𝑅 � ( 𝑅𝑅 sin 𝜃𝜃 𝜴𝜴 ) 𝑅𝑅 d 𝜃𝜃 𝜋𝜋𝜃𝜃=0 = ⅓𝜇𝜇 𝑞𝑞𝑅𝑅 𝜴𝜴 ( 𝑡𝑡 ) , (4) where 𝜇𝜇 = 4 𝜋𝜋 × 10 −7 henry/meter is the permeability of free space in the SI system of units. Writing 𝒎𝒎 ( 𝑡𝑡 ) = 𝑚𝑚 𝑒𝑒 −i𝜔𝜔𝜔𝜔 𝒛𝒛� , the complex dipole moment amplitude is 𝑚𝑚 = 𝜇𝜇 (4 𝜋𝜋𝑅𝑅 ⁄ ) 𝒥𝒥 𝑠𝑠0 . We now write the current density of the spinning sphere as follows: 𝓙𝓙 ( 𝒓𝒓 , 𝑡𝑡 ) = 𝒥𝒥 𝑠𝑠0 𝛿𝛿 ( 𝑟𝑟 − 𝑅𝑅 ) sin 𝜃𝜃 𝑒𝑒 −i𝜔𝜔𝜔𝜔 𝝋𝝋� = 𝒥𝒥 𝑠𝑠0 𝛿𝛿 ( 𝑟𝑟 − 𝑅𝑅 ) 𝑒𝑒 −i𝜔𝜔𝜔𝜔 𝒛𝒛� × 𝒓𝒓� . (5) Appendix A shows that the vector potential produced inside and outside the shell in accordance with the standard (i.e., Maxwellian) theory of electrodynamics are given by 𝑨𝑨 in ( 𝒓𝒓 , 𝑡𝑡 ) = 𝜇𝜇 𝒥𝒥 𝑠𝑠0 sin 𝜃𝜃 ( − i𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) [ sin ( 𝑟𝑟𝜔𝜔 𝑐𝑐⁄ ) − ( 𝑟𝑟𝜔𝜔 𝑐𝑐⁄ ) cos ( 𝑟𝑟𝜔𝜔 𝑐𝑐⁄ )] 𝑟𝑟 ( 𝜔𝜔 𝑐𝑐⁄ ) 𝑒𝑒 −i𝜔𝜔 ( 𝜔𝜔−𝑅𝑅 𝑐𝑐⁄ ) 𝝋𝝋� . (6) 𝑨𝑨 out ( 𝒓𝒓 , 𝑡𝑡 ) = 𝜇𝜇 𝒥𝒥 𝑠𝑠0 sin 𝜃𝜃 ( − i𝑟𝑟𝜔𝜔 𝑐𝑐⁄ ) [ sin ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) − ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) cos ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ )] 𝑟𝑟 ( 𝜔𝜔 𝑐𝑐⁄ ) 𝑒𝑒 −i𝜔𝜔 ( 𝜔𝜔−𝑟𝑟 𝑐𝑐⁄ ) 𝝋𝝋� . (7) Here, 𝑐𝑐 = 1 �𝜇𝜇 𝜀𝜀 ⁄ is the speed of light in vacuum, with 𝜇𝜇 and 𝜀𝜀 being the permeability and permittivity of free space. Note the continuity of the vector potential at the sphere’s surface, as well as its compliance with the Lorenz gauge, which, in the absence of a scalar potential, requires that 𝜵𝜵 ∙ 𝑨𝑨 be zero.
3. Electric and magnetic fields . Having found the vector potential, the 𝐻𝐻 -field is derived from the standard relation 𝜇𝜇 𝑯𝑯 ( 𝒓𝒓 , 𝑡𝑡 ) = 𝜵𝜵 × 𝑨𝑨 ( 𝒓𝒓 , 𝑡𝑡 ) and, in the absence of a scalar potential, the 𝐸𝐸 -field is obtained from 𝑬𝑬 ( 𝒓𝒓 , 𝑡𝑡 ) = −𝜕𝜕 𝜔𝜔 𝑨𝑨 ( 𝒓𝒓 , 𝑡𝑡 ) . Denoting the impedance of free space by 𝑍𝑍 , where 𝑍𝑍 = �𝜇𝜇 𝜀𝜀 ⁄ , we will have 𝑬𝑬 in ( 𝒓𝒓 , 𝑡𝑡 ) = 𝑍𝑍 𝒥𝒥 𝑠𝑠0 sin 𝜃𝜃 [( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) + i ] [ sin ( 𝑟𝑟𝜔𝜔 𝑐𝑐⁄ ) − ( 𝑟𝑟𝜔𝜔 𝑐𝑐⁄ ) cos ( 𝑟𝑟𝜔𝜔 𝑐𝑐⁄ )]( 𝑟𝑟𝜔𝜔 𝑐𝑐⁄ ) 𝑒𝑒 −i𝜔𝜔 ( 𝜔𝜔−𝑅𝑅 𝑐𝑐⁄ ) 𝝋𝝋� . (8) 𝑬𝑬 out ( 𝒓𝒓 , 𝑡𝑡 ) = 𝑍𝑍 𝒥𝒥 𝑠𝑠0 sin 𝜃𝜃 [( 𝑟𝑟𝜔𝜔 𝑐𝑐⁄ ) + i ] [ sin ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) − ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) cos ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ )]( 𝑟𝑟𝜔𝜔 𝑐𝑐⁄ ) 𝑒𝑒 −i𝜔𝜔 ( 𝜔𝜔−𝑟𝑟 𝑐𝑐⁄ ) 𝝋𝝋� . (9) 𝑯𝑯 in ( 𝒓𝒓 , 𝑡𝑡 ) = 𝒥𝒥 𝑠𝑠0 (1 − i 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) × � [ sin ( 𝑟𝑟𝜔𝜔 𝑐𝑐⁄ ) − ( 𝑟𝑟𝜔𝜔 𝑐𝑐⁄ ) cos ( 𝑟𝑟𝜔𝜔 𝑐𝑐⁄ )]( + sin 𝜃𝜃𝜽𝜽� )( 𝑟𝑟𝜔𝜔 𝑐𝑐⁄ ) – sin ( 𝑟𝑟𝜔𝜔 𝑐𝑐⁄ ) sin 𝜃𝜃𝜽𝜽�𝑟𝑟𝜔𝜔 𝑐𝑐⁄ � 𝑒𝑒 −i𝜔𝜔 ( 𝜔𝜔−𝑅𝑅 𝑐𝑐⁄ ) . (10) 𝑯𝑯 out ( 𝒓𝒓 , 𝑡𝑡 ) = 𝒥𝒥 𝑠𝑠0 [sin( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) − ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) cos(
𝑅𝑅𝜔𝜔 𝑐𝑐⁄ )] × � ( − i𝑟𝑟𝜔𝜔 𝑐𝑐⁄ )( + sin 𝜃𝜃𝜽𝜽� )( 𝑟𝑟𝜔𝜔 𝑐𝑐⁄ ) − sin 𝜃𝜃𝜽𝜽�𝑟𝑟𝜔𝜔 𝑐𝑐⁄ � 𝑒𝑒 −i𝜔𝜔 ( 𝜔𝜔−𝑟𝑟 𝑐𝑐⁄ ) . (11) Note that, while 𝐸𝐸 𝜑𝜑 and 𝐻𝐻 𝑟𝑟 are continuous at the shell surface, the discontinuity of the tangential 𝑯𝑯 at 𝑟𝑟 = 𝑅𝑅 is precisely matched by the surface current density. The time-averaged rate of energy flow outside the sphere is readily found to be 〈𝑺𝑺 out ( 𝒓𝒓 , 𝑡𝑡 ) 〉 = ½Re( 𝑬𝑬 out × 𝑯𝑯 out ∗ ) = 𝑍𝑍 | 𝒥𝒥 𝑠𝑠0 | [ sin ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) − ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) cos ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ )] sin 𝜃𝜃𝒓𝒓�2 ( 𝑟𝑟𝜔𝜔 𝑐𝑐⁄ ) . (12) Integrating Eq.(12) over a spherical surface of arbitrary radius 𝑟𝑟 yields Emitted power = ∫ 𝜋𝜋𝑟𝑟 sin 𝜃𝜃 〈𝑆𝑆 out ( 𝒓𝒓 , 𝑡𝑡 ) 〉 d 𝜃𝜃 𝜋𝜋𝜃𝜃=0 = | 𝒥𝒥 𝑠𝑠0 | [ sin ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) − ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) cos ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ )] ( 𝜔𝜔 𝑐𝑐⁄ ) . (13) It is easy to show that the emitted power equals the negative of the time-averaged work done by the self-field on the surface current of the shell; that is, � ½Re �𝑬𝑬 self ( 𝑟𝑟 = 𝑅𝑅 , 𝜃𝜃 , 𝜑𝜑 , 𝑡𝑡 ) ∙ 𝒥𝒥 𝑠𝑠0 ∗ sin 𝜃𝜃 𝑒𝑒 i𝜔𝜔𝜔𝜔 𝝋𝝋� � 𝜋𝜋𝑅𝑅 sin 𝜃𝜃 d 𝜃𝜃 𝜋𝜋𝜃𝜃=0 = − | 𝒥𝒥 𝑠𝑠0 | [ sin ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) − ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) cos ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ )] ( 𝜔𝜔 𝑐𝑐⁄ ) . (14) For sufficiently small values of 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ , the Taylor series expansions sin 𝑥𝑥 = 𝑥𝑥 − 𝑥𝑥 ⁄ + ⋯ and cos 𝑥𝑥 = 1 − 𝑥𝑥 ⁄ + ⋯ lead to the approximate expression | 𝑚𝑚 | 𝜔𝜔 (12 𝜋𝜋𝑍𝑍 𝑐𝑐 ) ⁄ for the time-averaged EM power emitted by the dipole moment of amplitude 𝑚𝑚 = 𝜇𝜇 (4 𝜋𝜋𝑅𝑅 ⁄ ) 𝒥𝒥 𝑠𝑠0 oscillating at the frequency 𝜔𝜔 .
4. Response of the spherical magnetic dipole to an externally applied torque . Let the external torque 𝑻𝑻 ( 𝑡𝑡 ) = 𝑇𝑇 𝑒𝑒 −i𝜔𝜔𝜔𝜔 𝒛𝒛� act on our magnetic dipole. † The dipole responds by acquiring an angular velocity 𝜴𝜴 ( 𝑡𝑡 ) , which oscillates with the frequency 𝜔𝜔 of the applied torque. Using Eq.(8) or Eq.(9), we compute the self torque of radiation resistance (produced by the radiated 𝐸𝐸 -field) acting on the uniformly-charged spherical shell, as follows: 𝑻𝑻 self ( 𝑡𝑡 ) = 𝒛𝒛� � � 𝑞𝑞4𝜋𝜋𝑅𝑅 � 𝐸𝐸 self ( 𝑟𝑟 = 𝑅𝑅 , 𝜃𝜃 , 𝜑𝜑 , 𝑡𝑡 )( 𝑅𝑅 sin 𝜃𝜃 )(2 𝜋𝜋𝑅𝑅 sin 𝜃𝜃 )d 𝜃𝜃 𝜋𝜋𝜃𝜃=0 = � � 𝑍𝑍 𝒥𝒥 𝑠𝑠0 [( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) + i ] [ sin ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) − ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) cos ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ )] (
𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) 𝑒𝑒 −i𝜔𝜔 ( 𝜔𝜔−𝑅𝑅 𝑐𝑐⁄ ) 𝒛𝒛� = � 𝑍𝑍 𝑞𝑞 𝛺𝛺 � [ sin ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) − ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) cos ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ )] × [(
𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) +i ] 𝑒𝑒 i𝑅𝑅𝑅𝑅 𝑐𝑐⁄ ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) 𝑒𝑒 −i𝜔𝜔𝜔𝜔 𝒛𝒛� . (15) Newton’s second law may now be invoked to write the dynamic equation of motion for the shell. In the presence of a dynamic friction torque (friction coefficient = 𝛽𝛽 ), we will have 𝑻𝑻 ( 𝑡𝑡 ) + 𝑻𝑻 self ( 𝑡𝑡 ) − 𝛽𝛽𝜴𝜴 ( 𝑡𝑡 ) = 𝐼𝐼𝜴𝜴̇ ( 𝑡𝑡 ) . (16) † Suppose the driving agent is a spatially uniform magnetic field 𝑯𝑯 ( 𝑡𝑡 ) = 𝐻𝐻 𝒛𝒛� cos( 𝜔𝜔𝑡𝑡 ) . For a ring of the shell at the polar coordinate 𝜃𝜃 , Maxwell’s equation 𝜵𝜵 × 𝑬𝑬 = −𝜕𝜕 𝜔𝜔 𝑩𝑩 yields 𝜋𝜋𝑅𝑅 sin 𝜃𝜃 𝐸𝐸 𝜑𝜑 = 𝜋𝜋 ( 𝑅𝑅 sin 𝜃𝜃 ) 𝜇𝜇 𝐻𝐻 𝜔𝜔 sin( 𝜔𝜔𝑡𝑡 ) , or 𝐸𝐸 𝜑𝜑 ( 𝜃𝜃 , 𝑡𝑡 ) = ½ 𝜇𝜇 𝑅𝑅𝐻𝐻 𝜔𝜔 sin 𝜃𝜃 sin( 𝜔𝜔𝑡𝑡 ) . The torque acting on the uniformly charged spherical shell will then be 𝑇𝑇 𝑧𝑧 ( 𝑡𝑡 ) = � ( 𝑞𝑞 𝜋𝜋𝑅𝑅 ⁄ ) 𝐸𝐸 𝜑𝜑 ( 𝜃𝜃 , 𝑡𝑡 )( 𝑅𝑅 sin 𝜃𝜃 )(2 𝜋𝜋𝑅𝑅 sin 𝜃𝜃 )d 𝜃𝜃 𝜋𝜋𝜃𝜃=0 = ⅓𝜇𝜇 𝑞𝑞𝑅𝑅 𝐻𝐻 𝜔𝜔 sin( 𝜔𝜔𝑡𝑡 ) . This, of course, is an approximation, as the other relevant Maxwell equation, 𝜵𝜵 × 𝑯𝑯 = 𝜀𝜀 𝜕𝜕 𝜔𝜔 𝑬𝑬 , has not been considered in this derivation. Nevertheless, it demonstrates the feasibility of generating a contactless EM torque. Substitution from the preceding equations into Eq.(16) yields the following transfer function for the system: 𝛺𝛺 𝑇𝑇 = i𝐼𝐼𝜔𝜔 + Γ ( 𝜔𝜔 ) + i𝛽𝛽 , (17) where Γ ( 𝜔𝜔 ) = � 𝑍𝑍 𝑞𝑞 � [ sin ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) − ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) cos ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ )] × ( − i𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) 𝑒𝑒 i𝑅𝑅𝑅𝑅 𝑐𝑐⁄ ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) . (18) For sufficiently small values of 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ , the Taylor series expansions sin 𝑥𝑥 = 𝑥𝑥 − 𝑥𝑥 ⁄ + ⋯ and cos 𝑥𝑥 = 1 − 𝑥𝑥 ⁄ + ⋯ lead to the following approximate expression for the radiation reaction function: Γ ( 𝜔𝜔 ) ≅ � 𝑍𝑍 𝑞𝑞 � [( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) + ⅖ ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) + ⅓ i( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) ] . (19) In this approximation, it is clear that radiation reaction contributes additive terms to both the moment of inertia 𝐼𝐼 , and the friction coefficient 𝛽𝛽 as they appear in Eq.(17). ‡ Depending on the various parameter values, it is conceivable for the transfer function of Eq.(17) with the approximate Γ ( 𝜔𝜔 ) of Eq.(19) to have one or more poles in the upper-half of the 𝜔𝜔 -plane, thereby rendering the system acausal. This, in fact, turns out to be the case for the broad range of parameter values examined in the next section. An important question for the numerical analysis taken up in Sec.5 is whether Eq.(17), in conjunction with the exact radiation reaction function of Eq.(18), could exhibit causal behavior. In the following analysis the total charge and total mass of the spherical shell will be assumed to be those of a free electron, namely 𝑞𝑞 = − −19 C and 𝓂𝓂 = 9.11 × 10 −31 kg . Our equation of motion will not change if we imagine a system consisting of two oppositely charged spherical shells (one immediately inside and essentially in contact with the other) that have equal and opposite charges ± 𝑞𝑞 ⁄ , equal masses 𝓂𝓂 ⁄ , equal friction coefficients 𝛽𝛽 ⁄ , and equal but opposite angular velocities ± 𝜴𝜴 ( 𝑡𝑡 ) . The particle now becomes charge-neutral but, since its opposite (internal) charges rotate in opposite directions, it will have the same overall magnetic dipole moment 𝒎𝒎 ( 𝑡𝑡 ) = ⅓𝜇𝜇 𝑞𝑞𝑅𝑅 𝜴𝜴 ( 𝑡𝑡 ) as before. Thus, with a judicious choice of the parameter values 𝑞𝑞 , 𝓂𝓂 , and 𝑅𝑅 , Eqs.(17) -(19) can be applied not only to a charged particle, but also to a neutral particle such as neutron ( 𝓂𝓂 neutron = 1.675 × 10 −27 kg, 𝑅𝑅 neutron ≅ .
5. Causality and the absence of poles in the upper-half plane . The transfer function of Eq.(17) is the Fourier transform of the impulse-response of the spherical magnetic dipole presently under consideration. As we have argued in [1], if one or more poles of this transfer function happen to be in the upper-half of the complex 𝜔𝜔 -plane, the impulse-response will have nonzero values before the arrival (in time) of the externally-applied torque 𝑻𝑻 ( 𝑡𝑡 ) = 𝑇𝑇 𝛿𝛿 ( 𝑡𝑡 ) 𝒛𝒛� . This, of course, is a clear indication that the response of the dipole to (externally applied) driving torques is acausal. ‡ The first term of the approximate Γ ( 𝜔𝜔 ) in Eq.(19) increments the mechanical moment of inertia 𝐼𝐼 = ⅔𝓂𝓂𝑅𝑅 by an EM contribution equal to 𝜇𝜇 𝑞𝑞 𝑅𝑅 (18 𝜋𝜋 ) ⁄ . Thus, the non-EM contribution to the overall mass 𝓂𝓂 of the particle must be negative if 𝑅𝑅 shrinks below 𝜇𝜇 𝑞𝑞 (12 𝜋𝜋𝓂𝓂 ) ⁄ . This critical radius is of the same order of magnitude as the classical radius of a charged particle [1,2]. (A similar argument also applies to a charge-neutral particle consisting of two identical spherical shells of equal and opposite charge rotating in opposite directions, with one shell immediately inside the other.) It is customary to resort to a mass-renormalization scheme by reducing the mass 𝓂𝓂 of the particle in order to compensate for the electrodynamic contribution to the inertial mass. In the context of the present paper, mass-renormalization would entail subtracting 𝜇𝜇 𝑞𝑞 (12 𝜋𝜋𝑅𝑅 ) ⁄ from the mass 𝓂𝓂 . However, since we are not convinced that this is the best way to handle the electrodynamic contribution to the inertial mass, we eschew this approach to mass-renormalization in favor of the alternative scheme that is discussed in Sec.6. In the complex 𝜔𝜔 -plane depicted in Fig.2(a), red dots mark the locations of the four poles of the transfer function of Eq.(17), when the small-radius approximation to Γ ( 𝜔𝜔 ) of Eq.(19) is used along with the parameter values 𝑞𝑞 = − −19 C , 𝓂𝓂 = 9.11 × 10 −31 kg (corresponding to a single electron), 𝑅𝑅 = 1.0 nm , and 𝛽𝛽 = 0 . Two of the poles are seen to be in the upper-half of the 𝜔𝜔 -plane, thus revealing the acausal nature of the impulse-response. Shrinking the radius 𝑅𝑅 down to , or increasing the mass 𝓂𝓂 (e.g., using 𝓂𝓂 neutron in place of 𝓂𝓂 electron ), or raising 𝛽𝛽 by as much as −10 kg ∙ m s ⁄ did not make the upper poles move into the lower half plane. In contrast, Fig.2(b) shows that, for the exact radiation reaction function Γ ( 𝜔𝜔 ) of Eq.(18), the transfer function 𝛺𝛺 𝑇𝑇 ⁄ of Eq.(17) possesses an infinite number of poles, all residing in the lower-half of the 𝜔𝜔 -plane. The exact location of the poles, of course, varies with the system parameters ( 𝑅𝑅 , 𝑞𝑞 , 𝓂𝓂 , 𝛽𝛽 ) , but our numerical studies indicate that the impulse-response in this case remains causal over a broad range of the parameter values. (As was done in Ref.[1], we also used Cauchy’s argument principle to confirm that the poles of the transfer function remain in the lower-half of the 𝜔𝜔 -plane.) We mention in passing that, in the case of 𝛽𝛽 = 0 , the half-residue of the first-order pole at 𝜔𝜔 = 0 adds a constant term to the impulse-response during the time-interval 𝑡𝑡 < 0 . This, however, is not indicative of acausal behavior, but rather a reminder that the initial condition of the dipole can be adjusted to ensure that 𝛺𝛺 ( 𝑡𝑡 ) = 0 for 𝑡𝑡 < 0 . The conclusion is that the predicted acausal behavior based on the approximate Γ ( 𝜔𝜔 ) of Eq.(19) is not a reliable indicator of the actual response of our magnetic dipole to an impulsive excitation. When the exact form of the radiation reaction function given by Eq.(18) is used in the calculations, the dipole is found to respond in a causal way. Fig.2 . Complex-plane diagrams showing the zero contours of the real part (dashed blue) and imaginary part (solid black) of the denominator of Eq.(17); the marked crossing points are the poles of the transfer function 𝛺𝛺 𝑇𝑇 ⁄ . (a) In the case of the approximate Γ ( 𝜔𝜔 ) of Eq.(19), aside from the trivial pole at 𝜔𝜔 = 0 , there reside one purely imaginary pole in the lower half plane, and two (symmetrically-positioned with respect to the imaginary axis) poles in the upper half plane. Locations of the nonzero poles vary with 𝑅𝑅 , but as 𝑅𝑅 → , there always remain two symmetrically-positioned poles in the upper-half plane and one imaginary pole in the lower-half plane. When 𝓂𝓂 neutron is substituted for 𝓂𝓂 electron , the general pattern of the pole locations remains the same, although the nonzero poles move further apart. (b) In the case of the exact Γ ( 𝜔𝜔 ) of Eq.(18), an infinite number of poles are symmetrically-distributed (again, with respect to the imaginary axis) in the lower half plane. Aside from the trivial pole at 𝜔𝜔 = 0 , none of these poles coincide with those of the approximate transfer function depicted in (a). Locations of the nonzero poles vary with 𝑅𝑅 , but as 𝑅𝑅 → , the poles remain in the lower-half plane and retain their symmetry with respect to the imaginary axis. 𝜔𝜔 ′ 𝜔𝜔 ″ 𝜔𝜔 = 𝜔𝜔 ′ + i 𝜔𝜔 ″ ( × ) (× 10 ) − − −
3 0 3 6 9 − − (a) 𝜔𝜔 ′ 𝜔𝜔 ″ ( × ) (× 10 ) − − −
1 0 1 2 3 − − (b) The standard way to infer the causality of the impulse-response from the absence of poles in the upper-half of the 𝜔𝜔 -plane is to begin by noting that 𝛺𝛺 𝑇𝑇 ⁄ → when 𝜔𝜔 → ∞ in the upper-half plane. For 𝑡𝑡 < 0 , the inverse Fourier transform integral of ( 𝛺𝛺 𝑇𝑇 ⁄ ) 𝑒𝑒 −i𝜔𝜔 ′ 𝜔𝜔 over the real axis 𝜔𝜔 ′ is then equated with the integral over a large upper-half semi-circle plus the sum of the residues at the upper-half poles. Considering that the integral over the (infinitely large) semi-circle vanishes, the absence of poles in the upper-half plane heralds the vanishing of the impulse-response over the interval 𝑡𝑡 < 0 . A less formal, but perhaps more intuitive, way to arrive at the same conclusion is to begin by supposing that the dipole’s impulse-response is, in fact, causal. If the transfer function happens to have a first-order pole at the origin, (e.g., when the friction coefficient 𝛽𝛽 in Eq.(17) is set to zero), one should eliminate this pole by multiplying 𝛺𝛺 𝑇𝑇 ⁄ with − i 𝜔𝜔 , which is tantamount to replacing the impulse-response with its own time-derivative, thus avoiding situations in which the impulse-response may have a constant nonzero value during the time interval 𝑡𝑡 < 0 . Stated differently, our starting assumption here is that the impulse-response (or its time-derivative) is precisely zero for 𝑡𝑡 < 0 , and is sufficiently well-behaved during 𝑡𝑡 ≥ to have the Fourier transform function 𝛺𝛺 𝑇𝑇 ⁄ (or − i 𝜔𝜔𝛺𝛺 𝑇𝑇 ⁄ if 𝜔𝜔 = 0 happens to be a pole). Now, if we multiply this well-behaved impulse-response (or its time-derivative) by exp( −𝛼𝛼𝑡𝑡 ) , where 𝛼𝛼 is some positive real number, the Fourier transform of the product function must also be well-behaved. However, the Fourier transform of the product is just our transfer function 𝛺𝛺 𝑇𝑇 ⁄ (or − i 𝜔𝜔𝛺𝛺 𝑇𝑇 ⁄ if 𝜔𝜔 = 0 happens to be a pole) evaluated at 𝜔𝜔 ′ + i 𝛼𝛼 , that is, on a straight line parallel to the real axis 𝜔𝜔 ′ in the upper-half 𝜔𝜔 -plane. Since our starting assumption was that the product function is well-behaved and that the positive number 𝛼𝛼 is arbitrary, the transfer function in the upper-half-plane cannot go to infinity. The conclusion is that the presence of even one pole in the upper-half-plane is proof that the impulse-response is acausal.
6. Bypassing the need for mass-renormalization ? One could argue that the relevant self-torque is due only to that part of the self 𝐸𝐸 -field that is in-phase with the surface current; that is, 𝑬𝑬� self ( 𝑟𝑟 = 𝑅𝑅 , 𝜃𝜃 , 𝜑𝜑 , 𝑡𝑡 ) = − 𝑍𝑍 𝒥𝒥 𝑠𝑠0 sin 𝜃𝜃 [ sin ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) − ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) cos ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ )] ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) 𝑒𝑒 −i𝜔𝜔𝜔𝜔 𝝋𝝋� . (20) Consequently, 𝑻𝑻� self ( 𝑡𝑡 ) = − � 𝑍𝑍 𝑞𝑞 𝛺𝛺 � [ sin ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) − ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) cos ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ )] ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) 𝑒𝑒 −i𝜔𝜔𝜔𝜔 𝒛𝒛� . (21) This means that Γ ( 𝜔𝜔 ) of Eq.(18) should be replaced by the following radiation reaction function: Γ� ( 𝜔𝜔 ) = i � 𝑍𝑍 𝑞𝑞 � [ sin ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) − ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) cos ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ )] ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) . (22) In this way, Γ� ( 𝜔𝜔 ) essentially acts as a friction coefficient — albeit one that, unlike 𝛽𝛽 in Eq.(17), is frequency-dependent. In contrast to Γ ( 𝜔𝜔 ) of Eq.(18), the fact that Γ� ( 𝜔𝜔 ) of Eq.(22) does not have a real part clearly indicates that it does not make an undesirable EM contribution to the moment of inertia 𝐼𝐼 in Eq.(17). This is the sense in which mass-renormalization is avoided. The component of the self-torque of Eq.(15) that constitutes the effective self-torque of Eq.(21) is the part that accounts for the rate of outgoing radiation (i.e., the EM energy that leaves the dipole and propagates away, never to return). We contend that this is a reasonable way to account for the radiation resistance torque. One might inquire as to the role of the remaining part of the total self-torque of Eq.(15) — the part that is out-of-phase with the surface current and, therefore, makes no contribution to the radiation. The answer is that, during each oscillation period (frequency = 𝜔𝜔 ), some EM energy goes out into the surrounding 𝑬𝑬 and 𝑩𝑩 fields, but subsequently returns to the dipole, so that the net energy going out (or coming in) during a full period is precisely zero. In a nutshell, the out-of-phase component that is being discounted here is that part of the self-torque whose role is to release some EM energy into the surrounding space during one half of each oscillation cycle, then reclaim that energy during the remaining half of the cycle. Classical electrodynamics contends that a fraction of the inertial mass of a charged particle resides in its surrounding EM field, while the remaining part is in some mysterious “material” stuff, sometimes associated with the Poincaré stresses. Our conjecture here is that the non-radiated EM energy that goes in and out of the dipole during each cycle is intimately tied to its inertial mass. In other words, if the spinning ball of charge is inclined to convert some of its inertial mass to EM energy during one half of each cycle, then bring that energy back in the form of the “mysterious material stuff” during the remaining half of the cycle, then this is just an internal exchange process between one form of mass and another. Therefore, as far as the overall dynamics is concerned, this internal exchange process is irrelevant and may be ignored. Stated differently, since we are already using a fixed value 𝓂𝓂 for the inertial mass, we should not allow the (internal) mass-conversion-related self-torque to enter into the overall dynamics of the particle through the backdoor. The mass-renormalization procedure described in the literature is intended to cancel this double-counting of the self-field contribution to the inertial mass. Thus, by substituting the effective self-torque of Eq.(21) for the total self-torque of Eq.(15), we endeavor to eliminate the need for the conventional mass-renormalization scheme. Figure 3(a) shows the computed 𝜔𝜔 -plane locations of the poles of the transfer function 𝛺𝛺 𝑇𝑇 ⁄ of Eq.(17) in conjunction with the effective radiation reaction function Γ� ( 𝜔𝜔 ) of Eq.(22) and the same set of parameters 𝑅𝑅 , 𝑞𝑞 , 𝓂𝓂 , 𝛽𝛽 as used in Fig.2(b). An infinite number of poles now Fig.3 . (a) Complex-plane diagram showing the zero contours of the real part (dashed blue) and imaginary part (solid black) of the denominator of Eq.(17) with the approximate Γ� ( 𝜔𝜔 ) of Eq.(22). The marked crossing points are the poles of the transfer function 𝛺𝛺 𝑇𝑇 ⁄ . Here, 𝑞𝑞 = − −19 C , 𝓂𝓂 = 9.11 × 10 −31 kg , 𝑅𝑅 = 1.0 nm , and 𝛽𝛽 = 0 . An infinite number of poles, symmetrically-positioned with respect to the imaginary axis, appear in both the upper and lower halves of the 𝜔𝜔 -plane. The exact locations of the poles vary with 𝑅𝑅 , but as 𝑅𝑅 → , they retain their symmetry with respect to the imaginary axis and remain in both the upper and lower-halves of the 𝜔𝜔 -plane. The existence of upper-half-plane poles is evidence that the dipole’s response to an impulsive excitation is acausal. (b) Similar to (a) except that the exact Γ ( 𝜔𝜔 ) of Eq.(18) is only slightly modified here to attenuate its out-of-phase component by one part in . The upper-half-plane poles immediately show up even when a tiny fraction of the out-of-phase component of Γ ( 𝜔𝜔 ) is taken out. 𝜔𝜔 ′ 𝜔𝜔 ″ 𝜔𝜔 ′ 𝜔𝜔 ″ ( × ) 𝜔𝜔 = 𝜔𝜔 ′ + i 𝜔𝜔 ″ ( × ) (× 10 ) − − −
1 0 1 2 3 − − − − −
1 0 1 2 3 (× 10 ) − − (a) (b) appear in the upper-half 𝜔𝜔 -plane, thus making the impulse-response of the dipole acausal. What is more disheartening is that removing any fraction of the out-of-phase component of the self-torque, no matter how small, will have a similar deleterious effect on the dipole’s transfer function. The pole-location plot in Fig.3(b) shows that attenuating the out-of-phase component of Γ ( 𝜔𝜔 ) by as little as one part in causes the transfer function to exhibit an infinite number of poles in the upper-half 𝜔𝜔 -plane. (Appendix B provides a more detailed discussion of the behavior of the poles in the upper-half 𝜔𝜔 -plane.) Causality of the impulse-response is thus seen to be a delicate matter that requires the presence of the radiation reaction function Γ ( 𝜔𝜔 ) in its entirety as given by Eq.(18). This is not to say that efforts at accounting for the EM contribution to the inertial mass 𝓂𝓂 of the particle should be abandoned, but rather that the role of 𝓂𝓂 in the equation of motion of a charged particle is far more nuanced than might appear at a first glance.
7. Concluding remarks . We have examined the rotary motion of a small, uniformly-charged spherical shell (the classical model of a magnetic dipole), and shown that its predicted response to an externally applied torque is causal provided that the exact form of the radiation reaction function is used in its dynamical equation of motion. When small-radius approximations were used to evince the behavior of the dipole in the limit when it approaches a point-particle, the electro-mechanical response of the particle was found to be acausal. The acausal behavior is thus seen to be a consequence of the approximations used to evaluate the radiation reaction function, rather than heralding a failure of the classical (Maxwell-Lorentz) equations of electrodynamics. These findings are fully accordant with the predicted behavior of the electric dipole that was the subject of our recent paper. In an attempt to discount the EM contributions to the inertial mass of the particle, we removed a part of the radiation reaction torque that is not directly involved in the extraction of the radiated EM energy from the dipole. The reduced form of the equation of motion, however, immediately sends the predicted response of the particle to external excitations into acausal territory. This is yet another indication that a better understanding is needed of the role of a charged particle’s inertial mass in its dynamic equations of motion. Nevertheless, we also remain cognizant of the other shortcomings of the models used in our work in that (i) the analysis has relied on the Newtonian equation of motion, not its relativistic counterpart, and (ii) we have totally ignored quantum mechanics and, in particular, the uncertainty principle that forbids the simultaneous knowledge of the position and momentum of the particle under consideration.
Acknowledgement . The authors express their gratitude to Vladimir Hnizdo for generously sharing with us his extensive knowledge of the electrodynamics of charged particles. This work has been supported in part by the AFOSR grant FA9550-19-1-0032. Appendix A Computing the vector potential of the magnetic dipole
To compute the vector potential 𝑨𝑨 ( 𝒓𝒓 , 𝑡𝑡 ) produced by the electric current distribution 𝓙𝓙 ( 𝒓𝒓 , 𝑡𝑡 ) over the surface of the spherical shell depicted in Fig.1, we begin by Fourier transforming the spatial part of 𝓙𝓙 ( 𝒓𝒓 , 𝑡𝑡 ) given in Eq.(5), as follows: 𝓙𝓙 ( 𝒌𝒌 ) = 𝒛𝒛� × � � 𝒥𝒥 𝑠𝑠0 𝛿𝛿 ( 𝑟𝑟 − 𝑅𝑅 ) cos 𝜗𝜗 𝒌𝒌� exp( − i 𝑘𝑘𝑟𝑟 cos 𝜗𝜗 ) 2 𝜋𝜋𝑟𝑟 sin 𝜗𝜗 d 𝑟𝑟 d 𝜗𝜗 𝜋𝜋𝜗𝜗=0∞𝑟𝑟=0 = − i4𝜋𝜋𝒥𝒥 𝑠𝑠0 [ sin ( 𝑘𝑘𝑅𝑅 ) − 𝑘𝑘𝑅𝑅 cos ( 𝑘𝑘𝑅𝑅 )] 𝑘𝑘 𝒛𝒛� × 𝒌𝒌� . (A1) Consequently, the contribution of the surface current to the spatial part of the vector potential is 𝑨𝑨 ( 𝒓𝒓 ) = (2 𝜋𝜋 ) −3 � 𝜇𝜇 𝓙𝓙 ( 𝒌𝒌 ) 𝑘𝑘 − ( 𝜔𝜔 𝑐𝑐⁄ ) exp(i 𝒌𝒌 ∙ 𝒓𝒓 ) d 𝒌𝒌 ∞−∞ = − i𝜇𝜇 𝒥𝒥 𝑠𝑠0 𝒛𝒛� × � � sin ( 𝑘𝑘𝑅𝑅 ) − 𝑘𝑘𝑅𝑅 cos ( 𝑘𝑘𝑅𝑅 ) 𝑘𝑘 [ 𝑘𝑘 − ( 𝜔𝜔 𝑐𝑐⁄ ) ] cos 𝜗𝜗 𝒓𝒓� exp(i 𝑘𝑘𝑟𝑟 cos 𝜗𝜗 ) 2 𝜋𝜋𝑘𝑘 sin 𝜗𝜗 d 𝑘𝑘 d 𝜗𝜗 𝜋𝜋𝜗𝜗=0∞𝑘𝑘=0 = 𝒥𝒥 𝑠𝑠0 sin 𝜃𝜃 𝝋𝝋�𝜋𝜋𝑟𝑟 � [ sin ( 𝑘𝑘𝑅𝑅 ) − 𝑘𝑘𝑅𝑅 cos ( 𝑘𝑘𝑅𝑅 )] × [ sin ( 𝑘𝑘𝑟𝑟 ) − 𝑘𝑘𝑟𝑟 cos ( 𝑘𝑘𝑟𝑟 )] 𝑘𝑘 [ 𝑘𝑘 − ( 𝜔𝜔 𝑐𝑐⁄ ) ] d 𝑘𝑘 ∞0 = 𝜇𝜇 𝒥𝒥 𝑠𝑠0 sin 𝜃𝜃 𝝋𝝋� � [ cos ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) + ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) sin ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ )] × [ sin ( 𝑟𝑟𝜔𝜔 𝑐𝑐⁄ ) − ( 𝑟𝑟𝜔𝜔 𝑐𝑐⁄ ) cos ( 𝑟𝑟𝜔𝜔 𝑐𝑐⁄ )] 𝑟𝑟 ( 𝜔𝜔 𝑐𝑐⁄ ) ; 𝑟𝑟 ≤ 𝑅𝑅 , [ sin ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) − ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) cos ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ )] × [ cos ( 𝑟𝑟𝜔𝜔 𝑐𝑐⁄ ) + ( 𝑟𝑟𝜔𝜔 𝑐𝑐⁄ ) sin ( 𝑟𝑟𝜔𝜔 𝑐𝑐⁄ )] 𝑟𝑟 ( 𝜔𝜔 𝑐𝑐⁄ ) ; 𝑟𝑟 ≥ 𝑅𝑅 . (A2) To this, we must now add a contribution from the vacuum field to ensure that the overall vector potential outside the sphere acquires the proper (i.e., retarded) spacetime dependence, namely, 𝑒𝑒 −i𝜔𝜔 ( 𝜔𝜔 − 𝑟𝑟 𝑐𝑐⁄ ) . Introducing the vacuum wavenumber 𝑘𝑘 = 𝜔𝜔 𝑐𝑐⁄ and the as-yet-unspecified vacuum field amplitude 𝐴𝐴 , we suggest the following spectral distribution for the vacuum field: 𝑨𝑨 vac ( 𝒌𝒌 , 𝑡𝑡 ) = 𝐴𝐴 𝛿𝛿 ( 𝑘𝑘 − 𝑘𝑘 ) 𝑒𝑒 −i𝜔𝜔𝜔𝜔 ( 𝒛𝒛� × 𝒌𝒌� ) . (A3) The space part of the vacuum potential will thus be 𝑨𝑨 vac ( 𝒓𝒓 ) = (2 𝜋𝜋 ) −3 � � 𝐴𝐴 𝛿𝛿 ( 𝑘𝑘 − 𝑘𝑘 )( 𝒛𝒛� × 𝒌𝒌� ) exp(i 𝑘𝑘𝑟𝑟 cos 𝜗𝜗 ) 2 𝜋𝜋𝑘𝑘 sin 𝜗𝜗 d 𝑘𝑘 d 𝜗𝜗 𝜋𝜋𝜗𝜗=0∞𝑘𝑘=0 = (2 𝜋𝜋 ) −2 𝐴𝐴 𝑘𝑘 ( 𝒛𝒛� × 𝒓𝒓� ) ∫ sin 𝜗𝜗 cos 𝜗𝜗 exp(i 𝑘𝑘 𝑟𝑟 cos 𝜗𝜗 ) d 𝜗𝜗 𝜋𝜋𝜗𝜗=0 = i𝐴𝐴 𝑟𝑟 [sin( 𝑟𝑟𝜔𝜔 𝑐𝑐⁄ ) − ( 𝑟𝑟𝜔𝜔 𝑐𝑐⁄ ) cos( 𝑟𝑟𝜔𝜔 𝑐𝑐⁄ )] sin 𝜃𝜃 𝝋𝝋� . (A4) Comparison with Eq.(A2) for the field outside the sphere ( 𝑟𝑟 ≥ 𝑅𝑅 ) now allows us to fix the (heretofore unknown) coefficient 𝐴𝐴 . We will have 𝑨𝑨 vac ( 𝒓𝒓 ) = i𝜇𝜇 𝒥𝒥 𝑠𝑠0 [ sin ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) − ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) cos ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ )] × [ sin ( 𝑟𝑟𝜔𝜔 𝑐𝑐⁄ ) − ( 𝑟𝑟𝜔𝜔 𝑐𝑐⁄ ) cos ( 𝑟𝑟𝜔𝜔 𝑐𝑐⁄ )] sin 𝜃𝜃𝝋𝝋�𝑟𝑟 ( 𝜔𝜔 𝑐𝑐⁄ ) . (A5) Combining Eqs.(A2) and (A5), we finally arrive at the total vector potential inside as well as outside the spherical shell, as follows: Gradshteyn & Ryzhik -11 residue theorem Gradshteyn & Ryzhik -11 𝑨𝑨 in ( 𝒓𝒓 , 𝑡𝑡 ) = 𝜇𝜇 𝒥𝒥 𝑠𝑠0 sin 𝜃𝜃 ( − i𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) [ sin ( 𝑟𝑟𝜔𝜔 𝑐𝑐⁄ ) − ( 𝑟𝑟𝜔𝜔 𝑐𝑐⁄ ) cos ( 𝑟𝑟𝜔𝜔 𝑐𝑐⁄ )] 𝑟𝑟 ( 𝜔𝜔 𝑐𝑐⁄ ) 𝑒𝑒 −i𝜔𝜔 ( 𝜔𝜔−𝑅𝑅 𝑐𝑐⁄ ) 𝝋𝝋� . (A6) 𝑨𝑨 out ( 𝒓𝒓 , 𝑡𝑡 ) = 𝜇𝜇 𝒥𝒥 𝑠𝑠0 sin 𝜃𝜃 ( − i𝑟𝑟𝜔𝜔 𝑐𝑐⁄ ) [ sin ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) − ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) cos ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ )] 𝑟𝑟 ( 𝜔𝜔 𝑐𝑐⁄ ) 𝑒𝑒 −i𝜔𝜔 ( 𝜔𝜔−𝑟𝑟 𝑐𝑐⁄ ) 𝝋𝝋� . (A7) As expected, the field outside the sphere as given by Eq.(A7) has the retarded spatio-temporal profile. Appendix B Approximate formula for the upper-half-plane poles using the Lambert function 𝑾𝑾 𝒌𝒌 ( 𝒛𝒛 ) If we attenuate the out-of-phase contribution to Γ ( 𝜔𝜔 ) of Eq.(18) by a factor of (1 − 𝜀𝜀 ) while keeping the in-phase contribution intact, we will have Γ ( 𝜔𝜔 ) ≅ � 𝑍𝑍 𝑞𝑞 � [ sin ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) − ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) cos ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ )](
𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) × {(1 − 𝜀𝜀 )[cos( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) + (
𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) sin(
𝑅𝑅𝜔𝜔 𝑐𝑐⁄ )] + i[sin(
𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) − ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) cos(
𝑅𝑅𝜔𝜔 𝑐𝑐⁄ )]} = 𝑍𝑍 𝑞𝑞 ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) {(1 − 𝜀𝜀 )[½ sin(2 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) − ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) cos(2
𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) − ½( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) sin(2 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ )] +i[½ − ½ cos(2 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) − ( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) sin(2
𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) + ½(
𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) + ½( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) cos(2 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ )]} . (B1) When 𝜔𝜔 → ∞ in the upper-half-plane, the dominant terms inside the curly brackets of Eq.(B1) will be ±¼i( 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) 𝑒𝑒 −i2𝑅𝑅𝜔𝜔 𝑐𝑐⁄ (with ± for the cosine and sine terms, respectively). Consequently, Γ ( 𝜔𝜔 ) ≅ i 𝜀𝜀 � 𝑍𝑍 𝑞𝑞 � 𝑒𝑒 −i2𝑅𝑅𝜔𝜔 𝑐𝑐⁄ . (B2) The upper-half-plane poles of Eq.(17) (with 𝛽𝛽 = 0 ) may thus be approximated as follows: 𝐼𝐼𝜔𝜔 + Γ ( 𝜔𝜔 ) ≅ ⅔𝓂𝓂𝑅𝑅 𝜔𝜔 + i 𝜀𝜀 � 𝑍𝑍 𝑞𝑞 � 𝑒𝑒 −i2𝑅𝑅𝜔𝜔 𝑐𝑐⁄ = 0 → (i2 𝑅𝑅𝜔𝜔 𝑐𝑐⁄ ) 𝑒𝑒 i2𝑅𝑅𝜔𝜔 𝑐𝑐⁄ = 𝑍𝑍 𝑞𝑞 𝜀𝜀8𝜋𝜋𝑐𝑐𝓂𝓂𝑅𝑅 → 𝜔𝜔 𝑘𝑘 = − � i𝑐𝑐2𝑅𝑅 � 𝑊𝑊 𝑘𝑘 � 𝑍𝑍 𝑞𝑞 𝜀𝜀8𝜋𝜋𝑐𝑐𝓂𝓂𝑅𝑅 � . (B3) Here, 𝑊𝑊 𝑘𝑘 ( 𝑧𝑧 ) is the 𝑘𝑘 th branch of the Lambert function, defined over the complex 𝑧𝑧 -plane as the inverse of the function 𝑧𝑧 = 𝑤𝑤𝑒𝑒 𝑤𝑤 . The integer 𝑘𝑘 may assume positive, zero, and negative values. Although Eq.(B3) is only asymptotically valid in the limit when 𝜀𝜀 → , numerical evaluations indicate its accuracy over a broad range of the parameters. For 𝑘𝑘 ≠ , the Lambert function has a singularity at the origin, its value approaching −∞ as its argument goes to zero, which shows that the imaginary part of the upper-half-plane poles approaches + ∞ when 𝜀𝜀 → . Thus, any departure from the full radiation reaction function Γ ( 𝜔𝜔 ) by way of attenuating its out-of-phase component will result in the dipole’s transfer function 𝛺𝛺 𝑇𝑇 ⁄ acquiring poles in the upper-half of the 𝜔𝜔 -plane. Equation (B3) also indicates that an increase in the inertial mass 𝓂𝓂 of the particle (for example, switching from 𝓂𝓂 electron to 𝓂𝓂 neutron ) reduces the argument of 𝑊𝑊 𝑘𝑘 ( ∙ ) , thus causing the poles in the upper-half of the 𝜔𝜔 -plane to move further up, a behavior that is confirmed by numerical calculations. 2 References M. Mansuripur and P. K. Jakobsen, “Electromagnetic radiation and the self field of a spherical dipole oscillator,”
American Journal of Physics , 693-703 (2020). 2. J. D. Jackson, Classical Electrodynamics , 3 rd edition, Wiley, New York (1999). 3. R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics , Addison-Wesley, Reading, Massachusetts (1964). 4. L. D. Landau and E. M Lifshitz,
Electrodynamics of Continuous Media , 2 nd edition, Addison-Wesley, Reading, Massachusetts (1984). 5. L. D. Landau and E. M Lifshitz, The Classical Theory of Fields , 4 th revised English edition, translated by M. Hamermesh, Pergamon Press, Oxford, and Addison-Wesley, Reading, Massachusetts (1987). 6. A. Zangwill, Modern Electrodynamics , Cambridge University Press, Cambridge, United Kingdom (2012). 7. M. Mansuripur,
Field, Force, Energy and Momentum in Classical Electrodynamics , revised edition, Bentham Science Publishers, Sharjah, UAE (2017). 8. F. Rohrlich, “The dynamics of a charged sphere and the electron,” Am. J. Phys. , 1051-56 (1997). 9. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products , 7 th edition, Academic Press, Burlington, Massachusetts (2007). 10. R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Advances in Computational Mathematics5