In a recent TPT article,1 I addressed a common miscommunication about Faraday's law, namely, that introductory texts often say the law expresses a causal relationship between the magnetic fields time variation and the electric fields circulation. In that article, I demonstrated that these field behaviors share a common cause in a time-varying current density. From that, many readers may have rightly guessed at a symmetric conclusion: while the Ampère-Maxwell law is commonly said to express a causal relation between the electric fields time variation and the magnetic fields circulation, these field behaviors share a distinct, common cause. Together, Faraday's law and the Ampère-Maxwell law constitute half of Maxwell's laws that form a foundation for almost all of electricity and magnetism. By misrepresenting these two laws, introductory texts not only present students with unnecessary conceptual hurdles early in their physics educations but also leave them with enduring misunderstandings about the very foundation of electricity and magnetism. Fortunately, compared to what is commonly taught, the actual cause of these field variations is conceptually simpler and more consistent with what the students will have already learned in the introductory texts' own earlier chapters.

1.
S. Eric
Hill
, “
Rephrasing Faraday's law
,”
Phys. Teach.
48
,
410
412
(
Sept. 2010
).
2.
D.
Halliday
,
R.
Resnick
, and
J.
Walker
,
Fundamentals of Physics
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Wiley
,
New York
,
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), pp.
758
759
.
3.
H.
Young
and
R.
Freedman
,
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Pearson Addison-Wesley
,
San Francisco
,
2007
), p.
1010
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4.
N. H.
Frank
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Introduction to Electricity and Optics
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McGraw-Hill
,
New York
,
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157
;
D.
Halliday
and
R.
Resnick
,
Fundamentals of Physics
(
Wiley
,
New York
,
1970
), p.
633
;
H.
Ohanian
,
Physics
, 2nd ed. (
Norton
,
New York
,
1989
), p.
849
;
R.
Wolfson
and
J.
Pasachoff
,
Physics for Scientists and Engineers
, 3rd ed. (
Addison-Wesley
,
San Francisco
,
1999
), p.
887
;
E.
Hecht
,
Physics: Calculus
, 2nd ed. (
Brooks/Cole
,
Pacific Grove, CA
,
2000
), p.
885
;
P.
Tipler
and
G.
Mosca
,
Physics for Scientists and Engineers
, 5th ed. (
Freeman
,
New York
,
2004
), p.
973
;
R.
Serway
and
J.
Jewett
Jr.
,
Physics for Scientists and Engineers
, 6th ed. (
Brooks/Cole-Thomas Learning
,
Belmont, CA
,
2004
), p.
944
;
D.
Giancoli
,
Physics for Scientists & Engineers with Modern Physics
, 4th ed. (
Pearson Prentice Hall
,
Upper Saddle River, NJ
,
2008
), p.
813
;
R.
Knight
,
Physics for Scientists and Engineers: A Strategic Approach
, 2nd ed. (
Pearson Addison-Wesley
,
San Francisco
,
2008
), p.
1094
.
5.
E.
Purcell
,
Electricity and Magnetism Berkeley Physics Course
, Volume
2
, 2nd ed. (
McGraw-Hill
,
New York
,
1985
), p.
326
.
6.
D.
Griffiths
,
Introduction to Electrodynamics
, 3rd ed. (
Prentice Hall
,
Upper Saddle River, NJ
,
1999
), p.
323
.
7.

The verb “induce” may be as pedagogically problematic and inescapable as is the noun “heat.”

8.
J. D.
Jackson
,
Classical Electrodynamics
, 2nd ed. (
Wiley
,
New York
,
1975
), p.
218
.
9.
Ref. 6, pp.
422
425
.
F.
Rohrlich
, “
Causality, the Coulomb field, and Newton's law of gravitation
,”
Am. J. Phys.
70
(
4
),
411
414
(
2002
).
Be aware that a series of Comments and Replies appears in 70 (9), 71 (7), and 72 (3) of the same journal. Equivalent to requiring causality, one can require that finite energy is initially invested in the fields; see Ref. 11.
10.
O. D.
Jefimenko
, “
Presenting electromagnetic theory in accordance with the principle of causality
,”
Eur. J. Phys.
25
(
2
),
287
296
(
2004
).
11.
R.
Jones
, “
Resource Letter CD-1: Causality and determinism in physics
,”
Am. J. Phys.
64
(
3
),
208
215
(
1996
).
12.
J. L.
Anderson
, “
Why we use retarded potentials
,”
Am. J. Phys.
60
(
5
),
465
467
(
1992
).
13.
O.
Jefimenko
,
Electricity and Magnetism
, 2nd ed. (
Electret Scientific Company
,
Star City, WV
,
1966
), pp.
515
516
;
An equivalent set of equations were independently produced by Panofsky and Phillips.
K. T.
McDonald
, “
The relation between expressions for time-dependent electromagnetic fields given by Jefimenko and by Panofsky and Phillips
,”
Am. J. Phys.
65
(
11
),
1074
1076
(
1997
).
14.
Ref. 6, p.
302
.
15.
D. J.
Griffiths
and
M. A.
Heald
, “
Time-dependent generalizations of the Biot-Savart and Coulomb laws
,”
Am. J. Phys.
59
,
111
117
(
1991
).
16.
O.
Jefimenko
, “
Solution of Maxwell's equations for electric and magnetic fields in arbitrary media
,”
Am. J. Phys.
60
,
899
902
(
1992
).
17.
J. A.
Heras
, “
The exact relation between the displacement current and the conduction current: Comment on ‘Time-dependent generalizations of the Biot-Savart and Coulomb laws' by D. J. Griffiths and M. A. Heald, [Am. J. Phys. 59 (2), 111–117 (1991)],
Am. J. Phys.
76
(
6
),
592
595
(
2008
).
18.
Ibid.
19.
For example:
B.
Batell
and
A.
Ferstl
, “
Electrically induced magnetic fields; a consistent approach
,”
Am. J. Phys.
71
,
925
929
(
2003
);
H.
Young
and
R.
Freedman
, p.
1016
;
R.
Wolfson
and
J.
Pasachoff
, pp.
887
888
;
R.
Knight
, pp.
1097
1098
;
The Mechanical Universe…and Beyond
,” Part II (39/40), video produced by the California Institute of Technology and the Southern California Consortium (distributed by Intellimation, P.O. Box 1922, Santa Barbara, CA 93116-1922; released 1985),
VHS
, color, 60 min.
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