a simple derivation of magnetic reconnection, part 3
To review the derivation so far:
- part 1 and its erratum used one of Maxwell's equations to derive the magnetic field B around a current-carrying rod.
- part 2 expressed that magnetic field in both cylindrical and Cartesian coordinates.
Both of those results are often found within introductory textbooks.
In this part of the derivation, we go beyond introductory textbooks by showing that the magnetic field around four current-carrying rods reproduces
We will show that this particular magnetic field is a counterexample to three myths that are often repeated by people who don't understand magnetic fields and their associated mathematics.
In part 4, I will describe a simple variation of
the experiment I've been suggesting to Michael Mozina and prove that the topology of the magnetic field changes during that experiment. As Yamada et al explain in their appendix, that change in the topology of the magnetic field is what we mean by magnetic reconnection.
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The physical experiment[/size]
We may as well assume
the experiment will be run in a perfectly shielded room that contains a perfect vacuum. (We needn't worry about the expense, because
Michael Mozina isn't going to run this experiment anyway.)
We position four vertical conducting rods in parallel, each exactly one meter away from the origin of our coordinate system. One rod will be exactly one meter to the west of the origin, another will be exactly east of the origin, one will one meter north, and the other rod will be one meter south.
Sparing no expense, we will run 1000 amperes of well-regulated direct current through each rod. That current will run upward through the west and east rods, but will run downward through the north and south rods. By symmetry, the magnetic field will look the same in any plane that intersects the four rods at a right angle. We will view that magnetic field from above.
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Magnetic field around a single rod at the origin[/size]
In part 2 of this derivation, we calculated the magnetic field around a single rod positioned at the origin. As everyone who has passed a freshman-level course in electromagnetism should know, that magnetic field looks like this:
The colors in that graph reveal the intensity of the magnetic field. The current-carrying rod is at the center of the white disk, with the current travelling toward you out of the xy plane you're viewing. The disk is white because I'm using white to indicate the most intense parts of a magnetic field. The intensity of the magnetic field decreases as you travel away from the rod, so you can infer the color scale I'm using from the graph above. I'm using black to indicate the least intense parts of a magnetic field.
The gray lines show a few of the uncountably infinite number of magnetic field lines. The arrows show the direction of the magnetic field along each line. By using colors to show intensity and magnetic field lines to display direction, my graphs convey more information about the magnetic field than can be deduced from black-and-white graphs of magnetic field lines alone.
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Magnetic field around a single rod away from the origin[/size]
We now modify our equation for the magnetic field around a rod at the origin to obtain an equation for the magnetic field around a single rod that's perpendicular to the xy plane and intersects that plane at a point p=<x
0,y
0>:
[latex]
\[
\begin{align*}
\hbox{{\bf B}}^{(p)} &= \hbox{{\bf B}}^{(p)} (x, y, z, t) = \hbox{{\bf B}}^{(p)} (t) (x, y, z) \\
&= \frac{\mu_0}{2 \pi} \frac{I^{(p)}(t)}{(x-x_0)^2+(y-y_0)^2}
\left( - (y-y_0) \, \hbox{{\bf e}}_x + (x-x_0) \, \hbox{{\bf e}}_y \right)
\end{align*}
\]
[/latex]
where
ex and
ey are the unit vectors in the x and y directions.
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Magnetic fields around multiple rods[/size]
Applying that equation to our four rods, we define
[latex]
\[
\begin{align*}
p_E &= \langle 1, 0 \rangle \\
p_W &= \langle -1, 0 \rangle \\
p_N &= \langle 0, 1 \rangle \\
p_S &= \langle 0, -1 \rangle \\
I_E(t) &= I^{p_E}(t) = \hbox{{1000 amperes}} \\
I_W(t) &= I^{p_W}(t) = I_E(t) \\
I_N(t) &= I_S(t) = - I_E(t) = - I_W(t) \\
\hbox{{B}}_E &= \hbox{{B}}^{(p_E)} \\
\hbox{{B}}_W &= \hbox{{B}}^{(p_W)} \\
\hbox{{B}}_N &= \hbox{{B}}^{(p_N)} \\
\hbox{{B}}_S &= \hbox{{B}}^{(p_S)} \\
\hbox{{B}}_2 &= \hbox{{B}}_E + \hbox{{B}}_W \\
\hbox{{B}}_4 &= \hbox{{B}}_E + \hbox{{B}}_W + \hbox{{B}}_N + \hbox{{B}}_S
\end{align*}
\]
[/latex]
By superposition (which was the last of our five equations in part 1 and its erratrum),
B2 is the magnetic field around the east and west rods when no current is flowing through the north and south rods.
B4 is the magnetic field when 1000 amperes is flowing through each rod, with the north and south rods carrying their current in the opposite direction from the east and west rods.
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Dungey's figure 2[/size]
B2 looks like this:
That reproduces the "figure eight" field of the second figure in [Dungey 1958]:
The magnetic field's intensity is zero at the neutral point N, which is the point at which magnetic field lines merge or separate. For a closer look at two magnetic field lines that are almost but not quite touching at the neutral point, we can zoom in:
That magnetic field gives us a counterexample to two myths about magnetic field lines.
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Myth 1: magnetic field lines have constant intensity[/size]
It's easy to see that all of the magnetic field lines shown in the graph above travel through points of different colors. Since different colors indicate different intensities of the magnetic field, it is obvious that the intensity of the magnetic field varies as we follow those magnetic field lines.
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Myth 2: magnetic field lines are isobars (contour lines)[/size]
Since the intensity of the magnetic field can vary as we follow a magnetic field line, magnetic field lines are not always isobars. The isobar myth comes from looking only at the simplest magnetic fields, and from trying to explain what's going on in terms that might be comprehensible to a humanities major. Contour lines are a metaphor, but that metaphor is not exact or precise.
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Dungey's figure 1[/size]
Dungey's figure 1 shows a classical example of magnetic reconnection at the neutral point of an X-type magnetic field:
Because our four conducting rods are positioned symmetrically, running current through all four rods (with positive current in the east and west rods, and negative current in the north and south) gives us a more symmetrical magnetic field:
That's a graph of
B4. Zooming in improves its resemblance to Dungey's figure:
Rotating the above graph by 90 degrees gives us a corrected version of figure 3a from [Yamada et al 2010]:
Note, however, that there's a mistake in that figure 3a: The arrows are reversed on the line that runs from the northwest to the southeast. Rotating my graph by 90 degrees shows the correct direction of those arrows.
We now have a clear counterexample to yet another myth.
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Myth 3: Magnetic field lines can neither begin nor end[/size]
In my two colored graphs of the magnetic field around four conducting rods, there are two magnetic field lines that begin at the neutral point and two other magnetic field lines that end at the neutral point.
The magnetic field line that runs from the northeast toward the neutral point is decreasing in intensity as it approaches the neutral point, and ends completely at that neutral point. The same is true of the magnetic field line that runs from the southwest toward the neutral point.
The magnetic field line that runs from the neutral point toward the northwest begins at the neutral point. The same is true of the magnetic field line that runs from the neutral point toward the southeast.
Do these magnetic field lines violate Gauss's law for magnetism? Not at all!
I have already sketched a
geometric proof that Gauss's law holds for the magnetic field generated by a single current-carrying rod. That result is confirmed by starting from its equation and calculating the divergence. By linearity, Gauss's law must therefore hold for the magnetic field around four such rods. That result can also be confirmed by starting from the equation for
B4 and calculating its divergence.
The idea that Gauss's law prevents magnetic field lines from beginning or ending is a fairy tale. Gauss's law doesn't really talk about magnetic field lines at all. Gauss's law is really about certain integrals and limits.
Gauss's law for magnetism says the magnetic flux through any smooth closed surface is zero. That's true at the neutral point: Intuitively speaking, the two lines that end there are offset by the two lines that begin there.
(Those four lines have measure zero anyway, but those who know what that means will be able to confirm Gauss's law for themselves. Throughout this derivation, I've been trying to limit myself to math that sophomore physics majors should understand.)
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Conclusion to part 3[/size]
J W Dungey was one of the pioneers who suggested that solar phenomena might be explained by magnetic reconnection. The two-dimensional field shown in figure 1 of [Dungey 1958] can be reproduced and understood in vacuo, with no currents or electric field at all within the region of space shown in that figure.
That magnetic field also gives us a simple counterexample to three common myths about magnetic field lines.
Dungey's X-shaped magnetic field is of fundamental importance for understanding magnetic reconnection. In part 4 of this derivation, I will prove that magnetic reconnection must occur during a simple variation of
the experiment I suggested to Michael Mozina almost a year ago, on 28 December 2010.
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References[/size]
[Dungey 1958] J W Dungey.
The neutral point discharge theory of solar flares. A reply to Cowling's criticism. Proceedings of
Electromagnetic Phenomena in Cosmical Physics, edited by Bo Lehnert. International Astronomical Union number 6, Cambridge University Press, page 135.
[Yamada et al 2010] Masaaki Yamada, Russell Kulsrud, and Hantao Ji. Magnetic reconnection.
Reviews of Modern Physics volume 82, January-March 2010, pages 603-664.
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