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.

1.
See, for example, the undergraduate textbook by
D. J.
Griffiths
,
Introduction to Electrodynamics
, 3rd ed. (
Prentice-Hall
,
Upper Saddle River, NJ
,
1999
).
2.
See, for example, the graduate-level textbook by
J. D.
Jackson
,
Classical Electrodynamics
, 3rd ed. (
Wiley
,
New York
,
1999
).
3.
C. G.
Darwin
, “
The inertia of electrons in metals
,”
Proc. R. Soc. London A
154
,
61
66
(
1936
).
4.
One of the rare instances is provided by the work of
H.
Essén
, “
From least action in electrodynamics to magnetomechanical energy—A review
,”
Eur. J. Phys.
30
,
515
539
(
2009
). In an appendix, Essén evaluates the self-inductance of the same hypothetical circuit as used in the present article.
5.
Attention to a current-source point of view is encouraged by
S. E.
Hill
, “
Rephrasing Faraday's law
,”
Phys. Teach.
48
,
410
412
(
2010
), who refers to Jeffimenko's equations when seeking the source of the Faraday induction field.
Hill's reminder is mentioned in a footnote by
D. J.
Griffiths
,
Introduction to Electrodynamics
, 4th ed. (
Pearson
,
New York
,
2013
), p.
313
.
6.

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/(4πεo) multiplying the right-hand side of Eq. (1) and a factor of 1/c multiplying the right-hand side of Eq. (2).

7.
L.
Page
and
N. I.
Adams
,
Electrodynamics
(
D. Van Nostrand
,
New York
,
1940
), p.
175
. This text was reprinted by Dover Publications, New York, in 1965, but now seems to be out of print.
8.
C. G.
Darwin
, “
The dynamical motions of charged particles
,”
Philos. Mag.
39
,
537
551
(
1920
). The Darwin Lagrangian appears in some modern textbooks. See, for example, Ref. 2, pp. 596–598, or L. D. Landau and E. M. Lifshitz, “The classical theory of fields,” 4th ed. (Pergamon, New York, 1975), pp. 165–168.
9.

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

10.
L.
Page
and
N. I.
Adams
, “
Action and reaction between moving charges
,”
Am. J. Phys.
13
,
141
147
(
1945
).
11.
T. H.
Boyer
, “
Lorentz-transformation properties of energy and momentum in electromagnetic systems
,”
Am. J. Phys.
53
,
167
171
(
1985
).
12.
T. H.
Boyer
, “
Example of mass-energy relation: Classical hydrogen atom accelerated or supported in a gravitational field
,”
Am. J. Phys.
66
,
872
876
(
1998
).
13.

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

14.

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.

15.

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.

16.

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

17.
The situation of our one-particle example contains elements of the same physical system as “the Feynman disk paradox.” See
R. P.
Feynman
,
R. B.
Leighton
, and
M.
Sands
,
The Feynman Lectures on Physics
(
Addison-Wesley
,
Reading, MA
,
1964
), Vol.
II
, pp.
17-5
and
27-6
. See also Ref. 1, pp. 359–361. In the present article, we evaluate the induced electromagnetic field arising from the accelerating disk charges by use of the approximate electric field expression (3).
18.

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 (0.5) for N=2, and is still negative (0.289) 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 mRe2/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.

19.
See, for example,
M. D.
Greenberg
,
Advanced Engineering Mathematics
(
Prentice Hall
,
Upper Saddle River, NJ
,
1998
), p.
312
.
20.

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

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