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.

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31.

Let a spherical shell of radius rc and charge q, 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 E-field energy density ½ε0E2r=q2/32π2ε0r4 outside the shell yields the total EM energy of the electron as E=q2/8πε0rc. Equating E to the mass energy m0c2, one obtains the classical diameter of the electron as 2rc=μ0q2/4πm02.818fm.

32.

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 non-zero) values, then it is perhaps advisable to stay away from the conventional—and arguably non-classical—stratagem of mass renormalization.

33.

In the literature, spherer/R is sometimes written as the Heaviside step function ΘRr.

34.

Since the displacement zt of the negatively charged sphere must be much less than the radius R of the spherical particle, its velocity V should be well below Rω and, therefore, V/cRω/c. One can, therefore, push Rω/c to values as high as 10 before having to worry about the validity of Newton's (non-relativistic) equation of motion.

35.

Ideally, when the radius R becomes comparable to or smaller than the classical radius rc of the electron, one should also consider renormalizing the mass m0 of the oscillating ball of charge. However, for the reasons mentioned in Sec. II, we believe that this subject is best left for future studies.

36.

Again, for Rrc, proper accounting for mass renormalization could change this conclusion.

37.
See supplementary material at https://doi.org/10.1119/10.0001348 for a step-by-step calculation of the scalar and vector potentials of the oscillating dipole described in Sec. III.

Supplementary Material

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