Can Magnetic Field Lines Break and Reconnect?
Kirk T. McDonald
Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544
(December 29, 2015; updated June 15, 2017)
1
Problem
In 1956, Sweet [1, 2] 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 [3, 4], which included the figure
below.
Discuss whether this description of field lines “breaking” and “reconnecting” is sensible
by considering a pair of collinear, “point” magnetic dipoles, m = m ẑ located at (x, y, z) =
(0, 0, ±a).
2
2.1
Solution
Collinear Point Dipoles
The magnetic field of the two “point” dipoles can be written (in Gaussian units) as
3((r − a) · z)(r − a)
B(r)
ẑ
3((r + a) · z)(r + a)
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
ẑ
3a(x x̂ + a ẑ)
ẑ
3a(x x̂ − a ẑ)
B(x, 0, 0)
− 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.
(2)
(x2 + a2)5/2
√
The magnetic field is zero on the ring x2 + y 2 = 2a2 of radius 2a in the midplane, z = 0.
All magnetic field lines emanating
from the dipole at z = −a within angle 0 < θ < α to
√
−1
2 ≈ 55◦ , end up on the dipole at z = a. Likewise all field
the z-axis, where α = tan
lines emanating from the dipole at z = −a within angle β < θ < 180◦ to the z-axis, where
1
β has a value greater than 90◦ end up on the dipole at z = a. Only field lines emanating
from the dipole at z = −a within angle α < θ < β return to this dipole. Angles α and β do
not depend on the separation 2a of the two magnetic dipoles, and the pattern of field lines
does not depend on this distance, in contrast to the two left figures above from [3]. Only
the rightmost figure is correct qualitatively.
No field√lines in this example “break” or “reconnect” as the distance 2a changes, although
the radius 2a of the “neutral point/ring” does change.
The figure below was generated by the Wolfram CDF applet at
http://demonstrations.wolfram.com/ElectricFieldLinesDueToACollectionOfPointCharges/
for four electric charges arranged as two dipoles, in which particlar example α ≈ 57◦ and β ≈ 180 − 23◦ .
One can drag the charges around in applet, and the field lines will be recalculated in real time, permitting
one to verify that no lines are broken or reconnected as the geometry of the four charges of the two coplanar
dipoles is varied.
2.2
Coplanar, Perpendicular Dipoles
The notion of “breaking” and “reconnecting” of field lines is better illustrated by an example
featured in Sweet’s second paper [2]. Here, the two dipoles have (fictitious) magnetic charges
that are well separated, and the second dipole is aligned along the midplane of the first, as
sketched below.
As the positive charge of the “horizontal” dipole is moved from left to right, with the
2
other three charged kept fixed, the field lines in the plane of the dipoles evolve as illustrated
below.
In the upper two figures, no field lines go “downwards” from the upper, positive charge
to the lower, negative charge, while in the lower two figures there are field lines that do so.
In the portion of the upper-left figure reproduced below, the four field line segments a, b,
c and d are the separatrices between lines the veer off in opposite directions close to the blue
+ symbol, at which location the magnetic field is zero. It is ambiguous whether segment a
connects to segment b or to segment d, etc.
The argument of Sweet is that as the charge labeled 3 moves to the left, with the other
charges kept fixed, the field line that initially combined segments a and d “breaks” and
“reconnects” to become a combination of segments a and b, while the field line that initially
combined segments b and c “breaks” and “reconnects” to become a combination of segments
c and d.
3
This “breaking” and “reconnecting” occurs at the common point of line segments z, b, c
and c, where the field strength is zero. As such, there would be no field-energy cost to the
process of “breaking/reconnecting,” so this process is consistent with conservation of (field)
energy.1
2.3
Comment
While the notion of “breaking” and “reconnection” of field lines is popular in the astrophysical literature, it is often overdramatized. A possibly better view is that these terms serve
as jargon for interesting phenomena that deserve more complete description.2
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] P.A. Sweet, The Production of High Energy Particles In Solar Flares, Nuovo Cim. Suppl.
8, 188 (1958), http://physics.princeton.edu/~mcdonald/examples/EM/sweet_ncs_8_188_58.pdf
[3] E.N. Parker, Sweet’s Mechanism for Merging Magnetic Fields in Conducting Fluids, J.
Geophys. Res. 62, 509 (1957),
http://physics.princeton.edu/~mcdonald/examples/EM/parker_jgr_62_509_57.pdf
[4] E.N. Parker, The Solar Flare Phenomenon and the Theory of Reconnection and Annihilation of Magnetic Fields, Ap. J. Suppl. 8, 177 (1963),
http://physics.princeton.edu/~mcdonald/examples/EM/parker_apjs_8_177_63.pdf
[5] 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
[6] K.T. McDonald, Can the Field Lines of a Permanent Magnet Be Tied in Knots? (June
12, 2017), http://physics.princeton.edu/~mcdonald/examples/knot.pdf
1
For another discussion by the author of knotted field lines, see [6].
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 [5], 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:
2
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.
4