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Old 1st November 2011, 04:31 PM   #4361
W.D.Clinger
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a simple derivation of magnetic reconnection, part 1

So long as Michael Mozina continues to wallow in the pit of denial he's dug for himself, we might as well finish up the derivation of magnetic reconnection in the experiment he's been running away from for most of the past year. By giving up on Michael Mozina, we free ourselves to use freshman-level math and physics that lie far beyond his knowledge and capability.

Simpler demonstrations of magnetic reconnection have already been presented in this or related threads, includingThe purpose of this derivation is to show how we can get from Maxwell's equations to Dungey's figures, the first half of Yamada et al's figure 3, and Wikipedia's animation using no mathematics beyond freshman calculus.

The derivation proceeds along the outline suggested by the five equations I quoted from Jackson's Classical Electrodynamics. Although that is hardly a freshman-level textbook, equivalents of those particular equations appear within Purcell's Electricity and Magnetism and other introductory textbooks.

Equation 1 (Ampère's law with Maxwell's correction)
<br />
\[<br />
\nabla \times \hbox{{\bf H}} - \frac{\partial \hbox{{\bf D}}}{\partial t} = \hbox{{\bf J}}<br />
\]<br />
That's one of the four Maxwell's equations.

By conducting the experiment in a vacuum and changing the magnetic field slowly, we can make Maxwell's correction as small as desired. (Changing the magnetic field more slowly makes magnetic reconnection happen more slowly, so you might be afraid this demonstration will be like watching grass grow. Never fear: We can compensate by using time-lapse animation to view the reconnection.) We can simplify our math by making Maxwell's correction negligible and dropping it from the equation to obtain Ampère's original law:
<br />
\[<br />
\nabla \times \hbox{{\bf H}} = \hbox{{\bf J}}<br />
\]<br />

Equation 2 (relationship between H and B)

Ampère's law is stated using the H-field. For our derivation, we need to use the B-field. In a vacuum, converting from the H-field to the B-field involves a change of units. The conversion factor µ0 is known as the magnetic constant:
<br />
\[<br />
\hbox{{\bf H}} = \frac{\hbox{{\bf B}}}{\mu_0}<br />
\]<br />
Substituting for H in Ampère's original law, we get
<br />
\[<br />
\nabla \times \hbox{{\bf B}} = \mu_0 \hbox{{\bf J}}<br />
\]<br />

Equation 3 (applying the Kelvin-Stokes theorem)

Let S be any smooth compact 2-dimensional surface, and let C (a 1-dimensional curve) be the boundary of S. Applying the Kelvin-Stokes theorem (which is a corollary of the fundamental theorem of calculus in n dimensions) to the equation above, we get
<br />
\[<br />
\oint_C \hbox{{\bf B}} \cdot d \hbox{{\bf l}} =<br />
\int_S \nabla \times \hbox{{\bf B}} \cdot \hbox{{\bf n}} \; da =<br />
\mu_0 \int_S \hbox{{\bf J}} \cdot \hbox{{\bf n}} \; da<br />
\]<br />

Equation 4 (magnetic field around a current-carrying rod)

For our experiment, we can use long rods and perform all of our measurements of the magnetic field in the vicinity of the rods' centers. Under those conditions, the magnetic fields we measure will be the same (to within experimental error) as the magnetic fields around infinitely long rods.

We need to start by calculating the magnetic field around a single current-carrying rod. By symmetry, the magnetic field will look the same in every plane that intersects the rod at a right angle. Taking S to be a disk of radius R in one of those planes with the rod at its center, we find that the integral of B along the boundary of that disk is equal to the total current flux through the disk. By Ampère's law, the magnetic field is tangent to that boundary at every point (with direction determined by the right hand rule). By symmetry, the magnitude of the field is the same at every point on the circle.

Denoting the current through a single rod at time t by I(t), the current flux through S is I(t). Hence
<br />
\[<br />
2 \pi R | \hbox{{\bf B}} | = \mu_0 I(t)<br />
\]<br />
whence
<br />
\[<br />
| \hbox{{\bf B}} | = \frac{\mu_0}{2 \pi} \frac{I(t)}{R}<br />
\]<br />
which is a simplification of Jackson's equation. (His equation illustrated the Biot-Savart law, which we managed to avoid by appealing to symmetry.)

(We have now answered one of Reality Check's critical questions that Michael Mozina was unable to answer.)


Equation 5 (superposition)

Superposition is so simple that many textbooks don't even bother to state it as an equation. Jackson stated superposition as part of the equation that tells how to convert the B-fields through media of different permeability into the composite H-field:
<br />
\[<br />
H_\alpha &= \sum_{\beta} \mu_{\alpha \beta}^\prime B_\beta<br />
\]<br />
By conducting our experiment in a vacuum, transforming from component notation to vector notation, and translating the left-hand side of that equation into the equivalent B-field, we get the unadorned and uncomplicated equation for superposition of magnetic fields:
<br />
\[<br />
\hbox{{\bf B}} = \sum_{\beta} \hbox{{\bf B}}_\beta<br />
\]<br />
(Yes, the Greek letter that confused Michael Mozina has disappeared altogether. Imagine that.)

To be continued...

Last edited by W.D.Clinger; 1st November 2011 at 05:39 PM. Reason: corrected a minor mistake (while leaving a more serious error untouched)
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