proof that magnetic field lines can begin/end at a neutral point
According to Wikipedia's
current article on magnetic fields:
Wikipedia said:
magnetic B field lines neither start nor end
As I'll prove below, that isn't quite true. (To be fair, I've taken that excerpt a little out of context. In the Wikipedia article, those words are immediately qualified by a parenthetical statement that states the complete truth in a mathematically correct way.)
Magnetic
B field lines can begin or end at the same neutral points where magnetic reconnection can occur. In what follows, I'll provide a mathematically rigorous proof of that fact, using a simple counterexample to Wikipedia's informal statement above. That counterexample involves the same X-shaped magnetic field that's often used to illustrate magnetic reconnection. First, however, we need a mathematically precise definition of magnetic field lines.
[size=+1]
Magnetic Field Lines[/size]
The magnetic field lines of a magnetic field
B are the curves that run parallel to
B. By a curve in three-dimensional space, we mean a one-parameter function c(γ) that maps values of its real parameter γ to points in 3-space. By parallel to
B, we mean that the derivative of c(γ) with respect to γ is
B(c(γ)) at every point on the curve (in other words, for every real number γ). Formally:
[latex]
\[
\begin{align*}
\frac{\partial c}{\partial \gamma} &= \hbox{{\bf B}}(c(\gamma))
\end{align*}
\]
[/latex]
In other words, a magnetic field line is a curve that satisfies that differential equation. To compute the magnetic field line through some particular point p
0=<x
0, y
0, z
0>, we can set c(0)=p
0=<x
0, y
0, z
0> and use numerical integration to trace the locus of points along
[latex]
\[
\begin{align*}
c (\gamma) &=
\langle x_0, y_0, z_0 \rangle +
\int_0^\gamma \hbox{{\bf B}}(c(\gamma^\prime)) d \gamma^\prime
\end{align*}
\]
[/latex]
Using a
standard library for Runge-Kutta integration, the magnetic field line through <x
0, y
0, z
0> can be computed in only a few lines of code:
Code:
(define (integrate-field-line B x0 y0 z0 stepsize)
(integrate-system
(lambda (p)
(let ((x (xpart p))
(y (ypart p))
(z (zpart p)))
(B x y z)))
(vector x0 y0 z0)
stepsize))
That's how I computed the magnetic field lines shown in my
simple derivation of magnetic reconnection. Approximating magnetic field lines via numerical integration provides great insight, and is good enough for almost all applications in science and engineering, but a rigorous mathematical counterexample must use exact calculations. To make those exact calculations feasible, we'll use the magnetic field
B4 I
defined in one of my previous posts.
[size=+1]
An X-shaped Magnetic Field[/size]
B4 is the magnetic field generated by four rods that run parallel to the z axis and are positioned exactly 1 meter east, west, north, and south of the origin in the xy plane at z=0. A current of magnitude I flows through each of the east and west rods in the z direction. The north and south rods each carry current I in the opposite (minus z) direction.
If we zoom in on the unit square (so all four of the conducting rods are out of the picture),
B4 looks like this:
The false colors in that graph indicate the intensity of the magnetic field. The field is weakest within the black region near the origin, and is actually zero at the origin. Within the area shown in that graph, the intensity of the field increases with distance from the origin (from which you can infer the color scale I'm using).
There are four magnetic lines on the diagonals. Naming these four magnetic lines after the compass direction for their quadrant:
- Magnetic line NE (with x = y > 0) points toward the origin and fades into nothingness at the origin.
- Magnetic line SW (with x = y < 0) also points toward the origin and fades into nothingness at the origin.
- Magnetic line NW (with x = -y < 0) points away from the origin and grows out of nothing at the origin.
- Magnetic line SE (with x = -y > 0) also points away from the origin and grows out of nothing at the origin.
[size=+1]
The Counterexample[/size]
The equations for that magnetic field (
B4) are shown in
part 3 of my simple derivation of magnetic reconnection. (If you don't trust me, you can re-derive the equations yourself. It's a straightforward exercise in first-year physics.) Let B
x and B
y be the x and y components of
B4. (The z component is always zero.)
If x = y (as for magnetic field lines NE and SW), then the equations for
B4 simplify to
[latex]
\[
\begin{align*}
B_x &= B_y = \frac{\mu_0 I}{2 \pi}
\left(
\frac{1}{2 x^2 + 2x + 1} -
\frac{1}{2 x^2 - 2x + 1}
\right)
\end{align*}
\]
[/latex]
Since B
x=B
y, we know that magnetic field lines NE and SW never leave the line x=y. First-year calculus tells us that
[latex]
\[
\begin{align*}
\lim_{\delta \rightarrow 0} B_x(\delta) &= \lim_{\delta \rightarrow 0} B_y(\delta) = 0
\end{align*}
\]
[/latex]
and both components actually become zero at the origin. That proves that the magnetic field lines NE and SW either begin or end at the origin.
Finally, a bit of high school algebra shows that
- Bx is negative when x is positive.
- Bx is positive when x is negative.
That tells us that magnetic field lines NE and SW are both directed toward the origin (as was already evident from the graph), so both of those magnetic field lines end at the origin.
Repeating the above calculations for x = - y (as for magnetic field lines NW and SE) shows that magnetic field lines NW and SE remain on the diagonal and are directed away from the origin, so both of them begin at the origin.
Lines NW and SE are the only magnetic field lines of
B4 that begin anywhere within the xy plane. Lines NE and SW are the only magnetic field lines of
B4 that end anywhere within the xy plane.
That's true for every xy plane, regardless of the value of z.
B4 therefore has an uncountable infinity of magnetic field lines that begin at a definite point in space, and another uncountable infinity of magnetic field lines that end at a definite point in space.
Both of those sets have measure zero, and a set of measure zero isn't likely to affect any scientific measurements or engineering calculations. That's why the popular myth that's repeated within Wikipedia's article on magnetic fields is only slightly incorrect. The only people who are likely to notice that it's incorrect at all are called mathematicians.
I'm so happy that I am not taking a side!