Scott Rebuttal. IV. 'Open' magnetic field lines

Here is another entry in my response to Dr. Scott's 'rebuttal'.

“Open” magnetic field lines is another concept that Dr. Scott condemns (”The Electric Sky”, page 118; “D.E. Scott Rebuts T. Bridgman: Open Magnetic Field Lines'', pg 11), but like so many other of his claims, he is, at best, playing semantic games.  In principle, magnetic and electric field lines can extend to infinity, however, in most cases we wish to examine, we don't want or need to consider the behavior at infinity.  Is Dr. Scott saying that any time you want to visualize something with a magnetic field, you must represent the entire universe? 

In any real analysis, we have to draw the boundary somewhere.  This can leave field lines cut-off.  Particles can still flow along these lines.  In general, they will connect to field lines from another field of a more distant source.  In the case of magnetic dipole fields, these 'open' lines generally occur near the poles.  If Dr. Scott claims these lines don't exist, is he claiming that charged particles cannot travel out from these regions?  Where do the charged particles go?

The recognition that magnetic field lines can never end is acknowledged by many researchers by enclosing the term 'open' in quotes.  I will use that convention here.

Dr. Scott's obsession with 'open' field lines also reveals an hypocrisy on his part.  It's as if he wants to treat them as 'real', in need of being drawn 'complete'.  But Dr. Scott's claim gets even stranger when we begin to explore his justification that 'open' field lines violate Maxwell's equations [Scott 2007].   Specifically, Scott claims that field lines must be closed to satisfy the Maxwell Equation,

But magnetic field lines are not 'real', they are representations of a vector field designed as mere guides to directions of charged particle flow, representing the direction of the vector field at that point.  When we draw a field line as complete between two points, say A and B,  (See Figure 1 at below) we are saying that we expect the particles moving outward from point A will eventually arrive at point B.  If a field line is 'open', such as for point C, we don't expect the particle to ever return to the original system, such as point D, but connect to magnetic fields from other more distant sources.  Dr. Scott's insistence on closed magnetic field lines makes such systems of plasmas permanently CLOSED to any kind of charged particle transfer.

Figure 1: An exploration of closed and 'open' field lines.  The red field lines close back on the source object for the field.  The blue field lines are 'open', connecting to more distant field sources which we don't show in this graphic.

The Maxwell equation,

, is true everywhere in a magnetic field.  This is a consequence of what is called the Divergence Theorem and is related to the observational evidence that there are no (known) magnetic monopoles (source points of magnetic flux) and it means that the net magnetic flux passing through a closed surface is zero: the same amount of magnetic flux passes into the surface as passes out of it.  This condition demands no other constraint!

In fact, the simplest set of 'open' field lines, a constant magnetic field, where the field lines extend from points at 'negative infinite distance' to 'positive infinite distance', trivially satisfies  the divergence condition.  Dr. Scott's claim that 'open' field lines violate the divergence condition for magnetic fields is false by trivial counter example!  An undergraduate physics major taking a year course in electromagnetism should immediately recognize Dr. Scott's statement as a gross error.

The really embarassing part is that Dr. Scott managed to get this fundamentally erroneous claim into the IEEE Transactions on Plasma Science, a journal that is supposed to be peer-reviewed!  This wasn't an obscure paragraph in the paper.  A large section of the paper was devoted to arguing this statement.  The journal allowed a blooper like this through?  What does that say about the quality of the peer-review process for this journal?

But let's cut Dr. Scott some slack.  Maybe Dr. Scott relying on the divergence condition was actually a typo (a really big typo).  Perhaps Dr. Scott really meant to invoke the other Maxwell equation related to magnetic field structure, Ampere's Law:

The only problem with this is that the applicable vector identity is Stokes Theorem, so

This mathematical relationship is true for any arbitrary closed path in the magnetic field, such as path EFGH, or even IJKL, in Figure 1.  It is not just paths representing magnetic field 'lines'!  What we define as magnetic field lines are just lines defined along a path parallel to the local vector field, so that the line segment, dl, is parallel to the magnetic vector, B. 

There is no requirement that the path be closed when defining magnetic field lines, only that if the path is closed, by Stoke's Theorem, it will specify some characteristic of the current and electric field passing through the surface enclosed by the path.

While drawing field lines as closed loops will guarantee that

insistence on 'close' field lines appears to be a human convention not tied to any physical requirement so long as the magnetic field lines never begin or end (Wikipedia: Magnetic Fields). 

As an additional check, I've examined a number of papers written in the past 100 years on the development of electromagnetism and on magnetic fields and field lines, many written by leaders in the field, including works by Alfven, Vasyliunas, Stern, Swann, and Falthammar, and have found no support for Scott's statements that field lines cannot be 'open'.  Many of these researchers use the concept themselves.  If Dr. Scott wishes to continue making this particular claim, he needs to provide more documentation than "Don Scott Says So".  Professionals with a stronger background in electromagnetism than Dr. Scott (or me), disagree with him.

Contrary to Dr. Scott's claims in “The Electric Sky”, 'Open' field lines do not violate Maxwell's Equations!

References
[1] D. E. Scott. Real Properties of Electromagnetic Fields and Plasma in the Cosmos. IEEE Transactions on Plasma Science, 35:822–827, August 2007. doi: 10.1109/TPS.2007.895424.