Whereas nonrelativistic mechanics always connects the total momentum of a system to the motion of the center of mass, relativistic systems, such as interacting electromagnetic charges, can have internal linear momentum in the absence of motion of the system's center of energy. This internal linear momentum of a system is related to the controversial concept of “hidden momentum.” We suggest that the term “hidden momentum” be abandoned. Here, we use the relativistic conservation law for the center of energy to give an unambiguous definition of the “internal momentum of a system,” and then we exhibit this internal momentum for the system of a magnet (modeled as a circular ring of moving charges) and a distant static point charge. The calculations provide clear illustrations of this system for three cases: (a) the moving charges of the magnet are assumed to continue in their unperturbed motion; (b) the moving charges of the magnet are free to accelerate but have no mutual interactions; and (c) the moving charges of the magnet are free to accelerate and also interact with each other. When the current-carrying charges of the magnet are allowed to interact, the magnet itself will contain internal electromagnetic linear momentum, something that has not been described clearly in the research and teaching literature.

1.
D. J.
Griffiths
, “
Resource Letter EM-1: Electromagnetic Momentum
,”
Am. J. Phys.
80
,
7
18
(
2012
).
2.

Reference 1 contains a section listing 23 articles on hidden momentum.

3.
See, for example,
A.
Zangwill
,
Modern Electrodynamics
(
Cambridge U.P.
,
Cambridge, UK
,
2013
), pp.
519
520
, 854–855, and 857.
4.
See, for example,
J. D.
Jackson
,
Classical Electrodynamics
, 3rd ed. (
Wiley
,
New York
,
1999
), pp.
609
610
and Problem 12.19 on p. 622.
5.
W. H.
Furry
, “
Examples of momentum distributions in the electromagnetic field and in matter
,”
Am. J. Phys.
37
,
621
636
(
1969
).
6.
T. H.
Boyer
, “
Illustrations of the relativistic conservation law for the center of energy
,”
Am. J. Phys.
73
,
953
961
(
2005
);
T. H.
Boyer
, “
Illustrating some implications of the conservation laws in relativistic mechanics
,”
Am. J. Phys.
77
,
562
569
(
2009
);
T. H.
Boyer
Examples and comments related to relativity controversies
,”
Am. J. Phys.
80
,
962
971
(
2012
).
7.
C. G.
Darwin
, “
The dynamical motions of charged particles
,”
Philos. Mag.
39
,
537
551
(
1920
). The Darwin Lagrangian appears in a number of advanced textbooks. See, for example, J. D. Jackson, Ref. 3, pp. 596–598,
or
J.
Schwinger
,
L. L.
,Jr.
,
K. A.
Milton
, and
W.
Tsai
,
Classical Electrodynamics
(
Perseus Books
,
,
1998
), p.
365
,
or
L. D.
Landau
and
E. M.
Lifshitz
,
The Classical Theory of Fields
, 4th ed. (
Pergamon
,
New York
,
1975
), pp.
165
168
.
8.
L.
Page
and
N. I.
, “
Action and reaction between moving charges
,”
Am. J. Phys.
13
,
141
147
(
1945
).
9.
T. H.
Boyer
, “
Faraday induction and the current carriers in a circuit
,”
Am. J. Phys.
(to be published), e-print arXiv:1410.1197.
10.
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
).
11.
T. H.
Boyer
, “
Lorentz-transformation properties of energy and momentum in electromagnetic systems
,”
Am. J. Phys.
53
,
167
171
(
1985
).
12.
See, for example,
D. J.
Griffiths
,
Introduction to Electrodynamics
, 4th ed. (
Pearson
,
New York
,
2013
), Problem 8.15 on p.
379
, Problem 8.20 on p. 380, and Problem 8.22 on p. 381.
13.
See, for example,
J. D.
Jackson
, Ref. 4, Problems 6.5 and 6.6 on p. 286.
14.
W.
Shockley
and
R. P.
James
, “
‘Try simplest cases' discovery of ‘hidden momentum’ forces on ‘magnetic currents’
,”
Phys. Rev. Lett.
18
,
876
879
(
1967
).
15.
S.
Coleman
and
J. H.
Van Vleck
, “
Origin of ‘hidden momentum forces’ on magnets
,”
Phys. Rev.
171
,
1370
1375
(
1968
).
16.

We will use Gaussian units throughout our discussion. When treating relativistic aspects of electromagnetic theory, Gaussian units are far more natural than S.I.

17.

In the past, the author has referred to Eq. (5) as the “fourth conservation law.” However, it has been objected that the relativistic conservation law is actually Eq. (3), while Eq. (5) holds when this quantity is not conserved due to the presence of external forces. Although the analogous relationship between “energy conservation” and the “work-energy theorem” appears in textbooks, there does not appear to be any comparable standard terminology involving the unfamiliar relativistic conservation law and its modified form when external forces are present.

18.

See, for example, Ref. 3, p. 857.

19.

See, for example, Ref. 3, p. 520.

20.

The calculation of $1/(4πc)∫d3r Ei×Bj$ for two point charges is essentially the same as the calculation of the vector potential in the Coulomb gauge given on page 597 of Jackson's text in Ref. 4.

21.
See
T. H.
Boyer
, “
Classical Interaction of a Magnet and a Point Charge: The Classical Electromagnetic Forces Responsible for the Aharonov-Bohm Phase Shift
,” e-print arXiv:1408.3745. The final sections of this manuscript are still undergoing revision.
22.

See Jackson's text in Ref. 4, Problem 6.6 on pp. 286–287 and Problem 12.8 on p. 618–619.

23.

See Griffiths' text in Ref. 12, Example 12.13 on pp. 547–549.