This article treats Faraday induction from an untraditional, particle-based point of view. The electromagnetic fields of Faraday induction can be calculated explicitly from approximate point-charge fields derived from the Liénard–Wiechert expressions, or from the Darwin Lagrangian. Thus the electric fields of electrostatics, the magnetic fields of magnetostatics, and the electric fields of Faraday induction can all be regarded as arising from charged particles. Some aspects of electromagnetic induction are explored for a hypothetical circuit consisting of point charges that move frictionlessly in a circular orbit. For a small number of particles in the circuit (or for non-interacting particles), the induced electromagnetic fields depend upon the mass and charge of the current carriers while energy is transferred to the kinetic energy of the particles. However, for an interacting multiparticle circuit, the mutual electromagnetic interactions between the particles dominate the behavior so that the induced electric field cancels the inducing force per unit charge, the mass and charge of the individual current carriers become irrelevant, and energy goes into magnetic energy.

## References

See, for example, Ref. 2, p. 664. When treating relativistic aspects of electromagnetism, Gaussian units are far more natural than S.I. In S.I. units, one would have a factor of $1/(4\pi \epsilon o)$ multiplying the right-hand side of Eq. (1) and a factor of $1/c$ multiplying the right-hand side of Eq. (2).

See, for example, Ref. 1, pp. 458–459.

See, for example, Ref. 1, p. 303.

See, for example, Ref. 2, problem 14.24 on pp. 705–706. There is no radiation emitted, and the fields of the circuit are those of electrostatics and magnetostatics.

Essén in his appendix discusses the necessary approximations to connect the self-inductance of this hypothetical circuit back to that of a wire of nonzero thickness that is bent in a circle. See the appendix of Ref. 4. We have followed Essén's basic analysis in the appendix of the present article.

See, for example, Ref. 1, pp. 271–272.

A continuous circular wire of nonzero cross section provides the analogue within traditional electromagnetism texts of our hypothetical circuit. Our hypothetical circuit is not continuous but rather involves a finite number *N* of charges with spaces between the charges. It seems interesting that for our discrete circular charge arrangement, the typical multiparticle behavior requires at least four charges. The summation in Eq. (29) is steadily increasing with increasing particle number *N*. However, the summation starts out negative $(\u22120.5)$ for $N=2$, and is still negative $(\u22120.289)$ for *N* = 3. Only for *N* = 4 does the summation first become positive $(+0.207)$. It should be emphasized that even for small numbers of charges, provided that we observe the restriction that $mR\u226be2/c2$, the total inertia is positive. For a few charges, the mass of the current carriers is crucial; for large *N*, the mass of the current carriers is unimportant. The negative value of *L* for a small number of charges corresponds to total energy in the combined magnetic fields that is smaller than that of the individual magnetic fields of the charges when located far from each other. The Darwin–Lagrangian approximation contains some of the run-away aspects that are seen elsewhere in classical electromagnetic theory.

See, for example, Ref. 2, p. 234.

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