Can Magnetic Field Lines Break and Reconnect?
Kirk T. McDonald
Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544
(December 29, 2015)
1
Problem
In 1956, Sweet [1] argued that in a region where time-dependent magnetic fields “collide,”
field lines near the “neutral point” can be said to “break” and “reconnect.” This notion was
perhaps better illustrated in a subsequent paper by Parker [2], which included the figure
below.
Discuss whether this description of field lines “breaking” and “reconnecting” is sensible
by considering a pair of point magnetic dipoles, m = m ẑ located at (0, 0, ±a).
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Solution
The magnetic field of the two dipoles can be written (in Gaussian units) as
3((r − a) · z)(r − a)
3((r + a) · z)(r + a)
B(r
ẑ
ẑ
=
−
−
,
5
3 +
5
m
|r − a|
|r − a|
|r + a|
|r + a|3
(1)
where a = (0, 0, a). At a point r = (x, 0, 0) on the midplane between the two dipoles, the
magnetic field is
3a(x x̂ + a ẑ)
3a(x x̂ − a ẑ)
B(x, 0, 0)
ẑ
ẑ
− 2
+
− 2
= − 2
2
5/2
2
3/2
2
2
5/2
m
(x + a )
(x + a )
(x + a )
(x + a2)3/2
4a2 − 2x2
=
ẑ.
(x2 + a2)5/2
(2)
The magnetic field is zero on the ring x2 + y 2 = 2a2 in the plane√z = 0. All magnetic field
lines emanating from the dipole at z = −a within angle tan−1 2 to the vertical end up
on the dipole at z = a, while all those outside this angle return to the dipole at z = −a.
That is, the pattern of field lines does not depend on the distance 2a between the dipoles,
in contrast to the figure above from [2].
No field lines “break” or “reconnect” as the distance 2a changes, although the location
of the “neutral point/ring” does change.
1
2.1
Comment
While the notion of “break” and “reconnecting” of field lines is popular in the astrophysical literature, it is not well founded technically. Rather, these terms serve as jargon for
interesting phenomena that deserve more complete description.
There is a tendency among physicists to overuse to compelling language of “field lines,”
following Faraday. A caution against this was published in 1951 by Slepian [3], who took the
view that any field “line” can be thought of as being ”broken” into segments at any point,
with no consequence to the physics:
No observable electromagnetic phenomenon can exist which involves two point in space,
and which depends upon there being a continuous line of force joining these points. Such a
phenomenon would contradict our postulate of the complete sufficiency of the local vector
fields for describing local phenomena.
References
[1] P.A. Sweet, The Neutral Point Theory of Solar Flares, p. 123 of IAU Symposium 6,
Electromagnetic Phenomena in Cosmical Physics, ed. B. Lehnert (Kluwer, 1958),
http://physics.princeton.edu/~mcdonald/examples/EM/sweet_piua_6_123_56.pdf
[2] E.N. Parker, Sweet’s Mechanism for Merging Magnetic Fields in Conducting Fluids, J.
Geophys. Res. 69, 509 (1957),
http://physics.princeton.edu/~mcdonald/examples/EM/parker_jgr_69_509_57.pdf
[3] J. Slepian, Lines of Force in Electric and Magnetic Fields, Am. J. Phys. 19, 87 (1951),
http://physics.princeton.edu/~mcdonald/examples/EM/slepian_ajp_19_87_51.pdf
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