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Re: [Phys-L] Field Lines and charges

On 07/03/2012 01:27 PM, treborsci@verizon.net wrote:
Gauss' law is a physical fact. The field line is a metaphorical
construct to help us visualize the electric field distribution.

That's true as stated ... but the field lines are actually
quite a good construct, quite a good representation, so if/when
it looks like they're not working, it behooves us to understand
why.

In that spirit, I would remind people that it is the /density/
of field lines that matters. If you quantize them too coarsely,
you will get into trouble, which is part of what is happening
here.

If you understand the physics, you see that the "suspicious"
field lines are only causing trouble on a set of measure zero,
which is no trouble at all. For the given charge distribution,
draw field lines more and more densely. You will find that
infinitely many lines terminate on the actual charges, and at
most one of them might not. This is a zero-sized problem.

To say the same thing another way, you know it is not in general
safe to interchange the order of limits. Here there are two
limiting processes:
a) the limit as the field lines become more finely spaced, and
b) the limit as we look closer to the plane of symmetry.

If you pass to limit (b) without due regard for limit (a), you
are breaking the rules, and whatever trouble you get is your
own fault.

This is a more-verbose explanation of my previous suggestion
to put a charge of infinitesimal magnitude at the end of the
offending field line(s).

==============

What follows isn't relevant to the question that was asked, but
to head off future trouble, I hasten to add that it makes much
more sense to treat the field as a bivector in 3+1 dimensional
spacetime (as opposed to treating it as vectors or "lines" in
3 dimensional Euclidean space).
http://www.av8n.com/physics/magnet-relativity.htm

In static situations, you can sometimes kinda sorta get away
with looking at the lines, which represent one component of
the actual bivector.