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Old 25th January 2010, 11:29 PM   #104
Tim Thompson
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Lightbulb Fields: Points & Lines

Originally Posted by Zeuzzz View Post
Thanks tim, but if you could answer each of the questions in the OP I think that would resolve any issues far quicker than your long posts summarizing the theory.
I don't think I can do a better job than the answers you already rejected. Rather, I think your reasons for rejecting the answers you already have are not good. Let me concentrate on the issue of field lines.
Originally Posted by Zeuzzz View Post
Every magnetic field is a continuum, i.e., a vector field. Each of the infinite and uncountable points in this continuum has a magnitude and a direction that is associated with it. This continuum is not made of (does not contain) a set of discrete lines. Lines can be drawn on paper to describe the magnetic fields direction and magnitude, however the field itself is not made of these lines.
It is not obvious that this is correct. Consider this:
Originally Posted by sol invictus View Post
One can fully characterize the field by its lines - in the limit of infinite line density, all the information about the field is contained in the line configuration.
It makes no difference. You can't say that lines are wrong and points are right as both tell us exactly the same thing.
And consider this:

"Thus while the coordinates of a particle depend only on time, fields depend continuously on some space variables as well. Therefore, a theory described by fields is usually known as a D+1 dimensional field theory where D represents the number of spatial dimensions on which the field variables depend. For example, the theory describing the displacements of a one dimensional string would constitute a 1+1 dimensional field theory whereas the more familiar Maxwell's equations (in four dimensions) can be regarded as a 3+1 dimensional field theory. In this language, then, it is clear that a theory describing the motion of a particle can be regarded as a special case, namely, we can think of such a theory as a 0+1 dimensional field theory."
From the book Field Theory - A Path Integral Approach by Ashok Das, World Scientific Publishing, 2006 (2nd edition; link is to Google Books 1st edition); Page 1 of the introduction.
The lines constitute a 1+1 dimensional field embedded in a 3+1 dimensional field. It is critical to note from the comments by Ashok Das that this includes moving lines. We assume that the field is a physical entity unto itself (as Maxwell makes explicit in his 1861 paper On Physical Lines of Force). But points and lines are only mathematical methods of describing the behavior of the field. We do not know what the field "really is" nor do we know what the field is "really made of". All we know is that we can describe the physical phenomena associated with the field through some mathematical methodology. The physical reality of the lines is no more relevant than the physical reality of the points; both are in fact mathematical entities and neither is necessarily physical. It is (and indeed it must be) the case that describing changes in the field makes just as much sense using the formalism of moving lines as it does to use any other mathematically & physically correct formalism. Just look at any textbook on electromagnetism and you will find that not only are moving lines perfectly reasonable, but likely the easiest way to characterize the field.
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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