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Old 24th January 2012, 01:45 PM   #6134
Tim Thompson
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Lightbulb Magnetic Field Lines and Flux Tubes Defined


In anticipation of the eventual re-awakening of this thread (it seems that "Mozina vs The World" is its sole reason for existence) I would like to address one of the key problems I see: There is a great deal of discussion that hinges on semantics and the sloppy use of language, rather than a properly precise discussion of the scientific aspects. Science requires a precise use of language. We have to understand that colloquial language has to give way to precise scientific language when the circumstances require it. So, here I want to present proper, authoritative scientific definitions for key elements of the discussion. In this case I want to make sure that, before we proceed, we have proper definitions in hand for "field lines" and "flux tubes", the key elements of the magnetic fields involved in plasma physics.

I have highlighted what I think are the key points, but I have tried to cite enough language to provide proper context, as well as to avoid insufficiently brief descriptions. Any other emphasis besides the highlight is carried over from the original text. I encourage the reader to pay attention to the full text of the quotes, and to refer directly to the original sources, which I have linked to as best I can, wherever possible.

Lines of Force Defined

From 26 January 2010
Originally Posted by Tim Thompson View Post
What is a "field line"?

"In electric and magnetic phenomena, the magnitude and direction of the resultant force at any point is the main subject of investigation. Suppose that the direction of the force at any point is known, then, if we draw a line so that in every part of its course it coincides with the direction of the force at that point, this line may be called a line of force, since it indicates the direction of the force in every part of its course. By drawing a sufficient number of lines of force, we may indicate the direction of the force in every part of the space in which it acts. Thus if we strew iron filings on paper near a magnet, each filing will be magnetized by induction, and the consecutive filings will unite by their opposite poles, so as to form fibres, and those fibres will indicate the direction of the lines of force. The beautiful illustration of the presence of magnetic force afforded by this experiment, naturally tends to make us think of the lines of force as something real, and as indicating something more than the mere resultant of two forces, whose seat of action is at a distance, and which do not exist there at all until a magnet is placed in that part of the field. We are dissatisfied with the explanation founded on the hypothesis of attractive and repellant forces directed towards the magnetic poles, even though we may have satisfied ourselves that the phenomenon is in strict accordance with that hypothesis, and we cannot help thinking that in every place where we find these lines of force, some physical state or action must exist in sufficient energy to produce the actual phenomena."

From the paper "On Physical Lines of Force" by James Clerk Maxwell, originally published in The Philosophical Magazine, vol. XXI (1861)
See The Scientific Papers of James Clark Maxwell, volume I; Dover publications, 2003, pages 451-452 (a republication of the 1965 Dover reprint of the original, published in 1890 by Cambridge University Press). Emphasis in the quote is from the original.
The concept for "lines of magnetic force" comes from Michael Faraday and was adopted by Maxwell (see On Faraday's Lines of Force; page 155 in the same volume I of Maxwell's collected papers; read Dec 10, 1855 and Feb 11, 1856). He later generalized the concept and used it to literally invent the theory of electromagnetic fields, and really the general topic of field theory (see A Dynamical Theory of the Electromagnetic Field; page 526 in the same volume I of Maxwell's collected papers; read Dec 8, 1864; and see the subsection On Lines of Magnetic Force, page 551 and On Magnetic Equipotential Surfaces, page 553). In the passage above, where Maxwell begins "We are dissatisfied ...", he is expressing his dissatisfaction with the notion of "action at a distance" (an idea Newton did not like either), and his belief that some "physical state or action" must permeate the space around the magnet. That "physical state or action" of 1861 became Maxwell's electromagnetic field of 1864.

{ ... }

PDF copies of Maxwell's papers On Physical Lines of Force and A Dynamical Theory of the Electromagnetic Field are available on the Wikipedia page A Dynamical Theory of the Electromagnetic Field.

From 20 November 2011
Originally Posted by Tim Thompson View Post
Just in case Maxwell is not good enough, here is another, more recent definition for a field line:

"If we join end-to-end infinitesimal vectors representing E, we get a curve in space - called a line of force - that is everywhere normal to the equipotential surfaces. The vector E is everywhere tangent to a line of forces."
From Electromagnetic Fields and Waves, Lorrain & Corson, W.H. Freeman & Co., 1970 (2nd edition), page 46 [this was my undergrad textbook]. This definition for an electric field line works just fine for a magnetic field, just replace E with B and you have the definition for a magnetic field line; the definition is general, so just substitute your favorite letter for your favorite field and you've got it.

This definition from Lorrain & Corson is quite the same as Maxwell's and is the standard definition for field lines in general; not just electric and magnetic fields, but for any classical field, that is how "field lines" are defined. For example, we find this definition in another, older, standard textbook:

Originally Posted by Smythe, 1950
A most useful method of visualizing an electric field is by drawing the "lines of force" and "equipotentials". A line of force is a directed curve in an electric field such that the forward drawn tangent at any point has the direction of the electric intensity there. It follows that if ds is an element of this curve, ds = λE, where λ is a scalar factor. Writing out the components in rectangular coordinates and equating values of λ, we have the differential equation of the line of force, dx/Ex = dy/Ey = dz/Ez.

This definition comes from Static and Dynamic Electricity, William R. Smythe, McGraw-Hill 1950, 2nd edition (1st 1939), section 1.08 "Lines of Force" on page 7. As before, note that this definition is general despite the specific reference to an electric field. Just replace electric field with magnetic field and the definition is precisely the same, and is in fact so for any classical field.

Flux Tubes defined

Originally Posted by Deiter Biskamp, 1993
A fundamental concept in MHD theory is that of a magnetic flux tube. It is based on the conservation of the magnetic flux

\phi = \int_F \boldsymbol B \cdot d \boldsymbol F

through an arbitrary surface F(t) bounded by a curve which moves with the fluid as illustrated in Fig. 2.1

Taking the surface integral of equation 2.11 one obtains

 \int_F \partial_t \bold B \cdot d \bold F = \oint (\bold v \times \bold B) \cdot d \bold l = -\oint \bold B \cdot (\bold v \times d \bold l)

and hence

 \dfrac {d\phi}{dt} = \int \partial_t \bold B \cdot d \bold F + \oint \bold B \cdot (\bold v \times d \bold l) = 0


\int_{dF} \bold B \cdot d \bold F =  \oint \bold B \cdot (\bold v \times d \bold l) dt .

Sweeping the boundary curve along the field lines defines a magnetic flux tube. Because of flux conservation the picture of field lines frozen to the fluid has a well defined physical meaning as flux tubes of infinitesimal diameter

The definition quote above is from the book Nonlinear Magnetohydrodynamics by Deiter Biskamp (Max Planck Institute for Plasma Physics); Cambridge University Press 1993. His reference to equation 2.11 is just Faraday's Law after replacing E with vxB:

Biskamp Equation 2.11:

 \partial_t \boldsymbol B = \nabla \times (\boldsymbol v \times \boldsymbol B)


All magnetic fields are originally generated by electric currents. However, while the current flows in a confined volume, the consequent magnetic field will fill a vastly larger volume than the current. Hence it is possible to measure an active and time variable magnetic field in a vacuum far removed from the current that generated it. This is a point which seems to be overlooked to me so I want to make sure the point is made explicitly somewhere. Furthermore, magnetic fields and plasmas commonly couple together, so that the plasma will carry the "frozen in" magnetic field with it. So a plasma can be magnetized by a magnetic field that is not generated by that plasma, but by another completely independent plasma far away. As an example, the solar wind carries the solar magnetic field along with it. The magnetic field was originally generated in the sun, but is carried to the outermost reaches of the solar system by the solar wind, which can deform that magnetic field, but has nothing to do with the generation of that magnetic field. Likewise, magnetic fields generated deep inside the sun will pass through the photosphere of the sun and couple with it, despite not being formed in the photosphere by the plasma it is coupled to. So it is important to understand that we can have a magnetic field in a plasma, but not assign the task of generating that magnetic field to that particular plasma.

Also note that, as defined, both field lines and flux tubes are strictly mathematical objects, not physical objects. There are, of course, physical manifestations that go along with the mathematical theory. The mathematics becomes in essence a second language, far more efficient than our own human languages, used to describe the physics. The language of mathematics is far more precise than any human language, and it is truly universal across humanity; everyone has the same understanding of an equation, regardless of the human language they use to discuss that equation. In my short preface I spoke of the "semantics and the sloppy use of language". This thread is replete with attempts to replace mathematical and physical rigor with colloquial English and that is a serious mistake, which we should make an effort to repudiate in the future, for this or any other thread.
The point of philosophy is to start with something so simple as not to seem worth stating, and to end with something so paradoxical that no one will believe it. -- Bertrand Russell
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