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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Let a spherical shell of radius and charge , 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 -field energy density outside the shell yields the total EM energy of the electron as . Equating to the mass energy , one obtains the classical diameter of the electron as .
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.
In the literature, is sometimes written as the Heaviside step function .
Since the displacement of the negatively charged sphere must be much less than the radius of the spherical particle, its velocity should be well below and, therefore, . One can, therefore, push to values as high as before having to worry about the validity of Newton's (non-relativistic) equation of motion.
Ideally, when the radius becomes comparable to or smaller than the classical radius of the electron, one should also consider renormalizing the mass 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.
Again, for , proper accounting for mass renormalization could change this conclusion.