Magnetic reconnection refers to the merging and separation of magnetic field lines that can occur at neutral points of magnetic fields as those fields change over time. Alternatively, magnetic reconnection refers to changes over time in the topology of magnetic field lines.

Although magnetic reconnection is a simple consequence of Maxwell's equations, it is usually discussed in connection with plasma physics. Magnetic reconnection has been observed in space plasma and in laboratory experiments. Magnetic reconnection is now known to be responsible for rapid movements and bursts of light in the aurora borealis, and is believed to be responsible for similar phenomena seen in solar flares. Magnetic reconnection may also play some role in heating the solar corona.

The reality of magnetic reconnection is denied by a motley group of folklorists and amateur physicists who promote an Electric Universe based on Immanuel Velikovsky's speculations combined with long-refuted scientific conjectures put forth by notable scientists such as Kristian Birkeland and Hannes Alfvén. By misinterpreting some unfortunate remarks by Alfvén, these pseudoscientists denounce magnetic reconnection as pseudoscience.

In particular, the Electric Universe folk often declare that magnetic reconnection violates Maxwell's equations, specifically Gauss's law for magnetism. Their argument is based upon the following over-simplification of Gauss's law for magnetism, which is widely repeated even by people who should know better:

Gauss's law for magnetism is equivalent to the statement that the field lines have neither a beginning nor an end: Each one either forms a closed loop, winds around forever without ever quite joining back up to itself exactly, or extends to infinity.

-- Wikipedia's article on Gauss's law for magnetism, retrieved 12 November 2011.

Although that's a useful white lie when we try to explain magnetic field lines to humanities or engineering majors, it isn't quite true: Gauss's law for magnetism allows magnetic field lines to begin or to end at neutral points of a magnetic field.

Because magnetic reconnection is most often relevant to plasma physics, it's hard to find a tutorial on magnetic reconnection that doesn't start with plasma. That complicates the discussion immensely, because plasma can react to changes in the magnetic field by forming electric fields and currents that may cause further changes in the magnetic field. Magnetic reconnection is far easier to understand in vacuo.

Because I am not a physicist, and was trained in mathematics, it was easier for me to derive magnetic reconnection directly from Maxwell's equations than to understand published derivations that start with plasma. Because my derivation avoids the complications of plasma, and uses only freshman-level physics and vector calculus, it might help a wider audience to understand magnetic reconnection.

I may improve my derivation later. (If no one can show me a published derivation of magnetic reconnection in vacuo that's freely available and at least as simple, I may write this up as a tutorial paper.) In the meantime, my derivation is online at the JREF subforum on Science, Mathematics, Medicine, and Technology:

- a simple derivation of magnetic reconnection, part 1: the fundamental equations as they appear in the third edition of Classical Electrodynamics by John David Jackson
- an erratum for part 1 above
- a simple derivation of magnetic reconnection, part 2: derivation of equation for the magnetic field around a current-carrying rod
- a simple derivation of magnetic reconnection, part 3: the X-shaped magnetic field with neutral point produced by a particular configuration of four conducting rods
- proof that magnetic field lines can begin/end at a neutral point
- a simple derivation of magnetic reconnection, part 4: demonstration and mathematical proof of magnetic reconnection
- a simple derivation of magnetic reconnection, part 5: using the results of parts 1 through 4 to replicate a non-trivial result about plasma physics and magnetic reconnection that dates back to 1958

In what follows, we assume a laboratory that's so well-shielded from external electromagnetic fields that we can calculate the magnetic fields generated in our experiments directly from Maxwell's equations. The graphs shown below simply display those calculated fields and field lines.

Consider a long metal rod that's perpendicular to the xy plane. If we run an electrical current through the rod in the direction of positive z, that current will generate a magnetic field that looks like this (when viewed from above):

The false colors in that figure show the magnitude of the magnetic field. The most intense part of the magnetic field is the white disk centered on the conducting rod. As the intensity of the magnetic field falls off away from the rod, the color goes from white to blue to purple to red to black.

That figure also shows three magnetic field lines, with arrows to show the direction of the magnetic field.

All of the graphs below show magnetic fields that are generated by four long conducting rods parallel to the z axis of a Cartesian coordinate system. Each of the four rods is exactly one meter from the origin of the xy plane at z=0.

If we run 1000 amperes through the east and west rods in the z direction, and run 1000 amperes through the north and south rods in the opposite (negative z) direction, we get the following magnetic field:

In the center of that figure, the magnetic fields generated by the four rods cancel each other perfectly, so the magnetic field is zero at that point. We refer to such points as neutral points. This particular neutral point happens to be at the origin of our xy coordinate system, just because it was mathematically convenient to arrange the rods symmetrically around the origin.

It may look as though two magnetic field lines cross at the neutral point. Magnetic field lines can't cross each other, however, so look again, and pay attention to the arrows. In reality, there are four distinct magnetic field lines that come infinitesimally close to touching at the neutral point. Two of those magnetic field lines begin at the neutral point; their arrows point away from the neutral point. The other two fade into nothingness at the neutral point.

(You may have heard that magnetic field lines never begin or end.
That's mostly true, but it's an over-simplification.
Magnetic field lines can begin or end at neutral points,
as shown in the figure above.
There is no need to worry about Gauss's law for magnetism.
It's trivial to confirm that the mathematically correct
statement of Gauss's law for magnetism (**div B** = 0)
holds for the magnetic field generated by a single rod.
By linearity, Gauss's law for magnetism also holds for the
linear superposition of four such rods that we are examining
here.)

The graphs above show the magnetic field when each of the four rods is carrying 1000 amperes of current (although the north and south rods are carrying their current in the opposite direction from the east and west rods).

If we vary the current in the north and south rods between 999.9 and 1000.1 amperes, we see magnetic reconnection at the neutral point:

The animation above includes one frame that shows magnetic field lines when the current in the north and south rods is exactly 1000 amperes. At that brief instant, the two magnetic field lines that are almost touching divide into four field lines that begin or end at the neutral point. This division of magnetic field lines, like the subsequent merger of magnetic field lines, is possible only at neutral points (where the magnetic field is zero).

In the animation below, we start with 1000 amperes of current in the north and south rods and slowly reduce the current in those two rods to zero, without changing the current running through the east and west rods.

The animation above begins with a magnetic field line topology in which

- Every field line either
- begins at the neutral point,
- ends at the neutral point, or
- loops back on itself.

- Given any 5 field lines that loop back on themselves, at least one of the 5 field lines lies within the loop formed by another of the 5 field lines.

Those are topological properties of the magnetic field lines, which means those properties are preserved by any homeomorphism of the field lines.

Reducing the current within the north and south rods to zero yields a magnetic field line topology in which

- Every field line either
- begins at the neutral point,
- ends at the neutral point, or
- loops back on itself.

- Given any 3 field lines that loop back on themselves, at least one of the 3 field lines lies within the loop formed by another of the 3 field lines.

The original magnetic field did not have that last property. That means the topology of the magnetic field lines must have changed as the current in the north and south rods was reduced to zero. The topology changes again as the current is increased back to 1000 amperes.

Those changes in topology are what we mean by magnetic reconnection.

Last updated 24 November 2011.

Added 2 April 2013:

Tim Thompson's web page links to many of his informative posts concerning magnetic reconnection and other topics. Sadly, the LaTeX embedded in our posts is no longer being rendered properly by the JREF Forum's software.