Maxwell's equations and magnetic reconnection
In recent days,
Michael Mozina has cited Maxwell's equations in alleged support of two extremely silly claims he has made concerning magnetic reconnection:
Extremely silly claim #1: Maxwell's equations can be used to "convert all the B's to E's" in any paper on magnetic reconnection.
Extremely silly claim #2: Magnetic reconnection is forbidden by Maxwell's equations, hence pseudoscience.
The first of those extremely silly claims is incorrect in a straightforwardly trivial sense, but provides insight into the "thinking" that led to Mozina's second and more profoundly silly claim. In this post I will use freshman-level calculus and physics to explain why both of the above claims are extremely silly.
Maxwell's equations are one of the more important topics in freshman physics. They are usually covered late in the freshman year, after students have had time to encounter vector derivatives in calculus. Here, for future reference, are Maxwell's equations (in vacuum) from a typical freshman-level textbook, using nabla/del notations for the
div and
curl operators:
[latex]
\begin{align}
\nabla \times \hbox{{\bf E}} & = - \frac{1}{c} \frac{\partial\hbox{{\bf B}}}{\partial t} \\
\nabla \times \hbox{{\bf B}} & = \frac{1}{c} \frac{\partial\hbox{{\bf E}}}{\partial t} + \frac{4\pi}{c}\hbox{{\bf J}} \\
\nabla \cdot \hbox{{\bf E}} & = 4\pi\rho \\
\nabla \cdot \hbox{{\bf B}} & = 0
\end{align}
[/latex]
Extremely silly claim #1 was part of Mozina's claim that the details missing from his pet "theory" of circuit reconnection would be provided by a mathematical transformation of the equations that appear in any paper on magnetic reconnection:
Why wait for me? Just take any paper on MR theory, convert all the B's to E's and you'll have it. I'm not your math mommy.
When I pointed out that changing all the B's to E's would contradict Maxwell's equations, Mozina pretended to have been talking about a more sophisticated transformation based on Maxwell's equations:
First off I suggested you start with a paper on MR theory, not Gauss's Law. By the term "convert" I *ASSUMED* that an intelligent individual like yourself would immediately understand that you would need to USE MAXWELL"S EQUATIONS to convert from one orientation to the other (B->E).
I then suggested that Mozina demonstrate his alleged transformation on a simple example taken from one of the standard papers on MR theory. He ignored that challenge, even after he gave me cause to repeat it.
A casual reader of this thread might think my challenge was intended to dramatize the fact that Mozina lacks the mathematical chops to perform his alleged transformation on any real-world example. I must admit that I knew Mozina would be unable to demonstrate his alleged transformation, but that didn't have anything to do with Mozina's incompetence. I knew Mozina wouldn't be able to do it because it can't be done:
Theorem. In general, the electric field
E cannot be reconstructed from the magnetic field
B.
Sketch of proof: We can describe two distinct physical situations that have exactly the same magnetic field
B but wildly differing electric fields
E. (Once an example of that has been described, its generality becomes obvious.)
Anyone who has taken freshman physics should be able to fill in the details of that proof; doing so is a typical homework exercise or test question in freshman physics.
In short,
extremely silly claim #1 is extremely silly because the alleged transformation is impossible.
Had Mozina's claim referred to the current density field
J instead of the electric field
E, the transformation would have been possible provided the partial derivative of the electric field
E with respect to time is known independently. That transformation is expressed by the second of Maxwell's equations (as stated above), which is known as Ampere's law with Maxwell's correction.
Several of our more knowledgeable posters gave Mozina a free ride by assuming he had meant
J instead of
E, and proceeded to explain that (because of Maxwell's correction) even that transformation is impossible without additional knowledge. I did not give Mozina that free ride, because I have seen no evidence that Mozina even understands the difference between the electric field
E and the current density field
J.
The distinction between
E and
J is pretty basic. Ignorance of that distinction implies vast ignorance of freshman physics. Now that I have said this, of course, Mozina may claim to have understood that difference all along; if so, then why did his 10 to 20 subsequent posts on this subject never reveal any hint of a clue about that vital technical distinction?
Mozina's
extremely silly claim #2 involves a more profound error:
That same lack of a conceptual understanding of subatomic physics shows up in your great love of what Alfven himself called "pseudoscience". You can't physically tell me the difference between induction and magnetic reconnection, or ordinary particle collisions in plasma. Instead you simply cling to the concept in spite of the fact that every electrical textbook on the planet points out that magnetic fields form as a complete and full continuum, without beginning and without end. They are physically *INCAPABLE* of "disconnecting" or "reconnecting" to any other magnetic line. Induction is not "magnetic reconnection". Circuit reorientation is not "magnetic reconnection". Alfven rejected that whole concept as pseudoscience his *ENTIRE* career, yet you prattle on about it in paper after paper.
There's never a problem with your math, just a serious problem with the physics. What's the point in discussing the math when your entire physical premise is based upon a non-sequitur?
Mozina claims that "every electrical textbook on the planet" points out his highlighted phrase, but a Google search on the highlighted phrase brings up only this thread. If you leave off the "without beginning and without end" part of the phrase, a Google search will bring up this thread and one of Mozina's previous posts at another forum. In short, Mozina is the only authority for that highlighted phrase.
It appears that Mozina's highlighted phrase represents his limited personal understanding of the implications of Gauss's law for magnetism (the fourth of Maxwell's four equations above), as dumbed down by textbooks that don't use calculus because they're written for poets and other students in non-technical majors.
Although it mentions three technical terms ("magnetic fields", "complete", and "continuum"), Mozina's highlighted prose is actually quite vague. Suppose, for example, that we want to know whether there is any magnetic analogue to electric charge. Does the highlighted prose tell us? Beats me. On the other hand, the five symbols of ∇∙
B=0 tell us the answer is no. Suppose we want to know whether the magnetic field could be written as the curl of a vector potential. Does the highlighted prose tell us? I don't think so. Yet ∇∙
B=0 tells us the answer is yes.
The equation evidently contains a great deal more information than Mozina's highlighted prose, but Mozina relies on his highlighted prose even when he makes the mathematical (and radically incorrect) claim that magnetic fields
are physically *INCAPABLE* of "disconnecting" or "reconnecting" to any other magnetic line.
Instead of asking Mozina's highlighted prose whether that's true, let's ask Gauss's law for magnetism, which appears to have inspired the dumbed-down prose. That law answered our previous questions,
but it doesn't answer this one.
The reason ∇∙
B=0 has nothing to say about magnetic reconnection is that magnetic reconnection involves the changes in
B over time, and the equation ∇∙
B=0 doesn't have anything to do with time. It doesn't even mention time. That means the equation is true at every instant of time, but it says nothing about how
B changes over time.
Maxwell's first equation above (Faraday's law of induction) does involve the change of
B over time, and its mention of the electric field brings in Maxwell's second equation above (Ampere's law with Maxwell's correction). Although those two equations state certain relationships, and suggest that both the magnetic and electric fields should be differentiable with respect to time, those equations
most definitely do not rule out magnetic reconnection.
That's a mathematical fact. We can prove it by constructing solutions of Maxwell's equations that exhibit magnetic reconnection. (The equations I quoted in my challenge to Mozina were part of one such solution.) The existence of solutions to Maxwell's equations that involve magnetic reconnection prove, beyond any shadow of any doubt, that Mozina's
extremely silly claim #2 is simply incorrect.
Epilogue. We know Mozina's claims are incorrect because we can use mathematics and the accepted laws of physics to prove they are incorrect.
From the nature of Mozina's incorrect claims, we can conclude he does not understand mathematics or physics, even at the level of freshman calculus and freshman physics.
It is important to understand that our rejection of Mozina's claims does not depend upon our knowledge of Mozina's ignorance. We do not conclude that his claims are incorrect because he doesn't understand math or physics; that would be a fallacy. We conclude instead that, because his claims are incorrect, he does not understand math or physics.
When Mozina made the extremely silly claims I have dissected in this post, he was blowing smoke. Because he understands neither calculus nor freshman physics, blowing smoke is all he can do.
!
