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Re: [Phys-L] Field Lines and charges



On 07/03/2012 09:06 AM, LaMontagne, Bob wrote:
Here is a pair of field lines that do not follow the usual maxim of
starting and stopping on a charge.

I don't buy it, for a reason nobody has mentioned yet: in addition to the
lines flowing out from the origin in the plane of symmetry, there are also
lines flowing *into* the origin along the axis of symmetry.

Just because we didn't mention them doesn't mean they are not there. I'm
kicking myself for not mentioning them earlier. The inflowing lines have
just as much significance (i.e. not much) as the outflowing lines. These
lines have virtually no significance because the density of lines is zero
at the origin.

It is mildly peculiar to have lines making a sharp 90 degree turn, but that
is not nearly so peculiar as having lines start or stop in a charge-free
region. The sharp-turn behavior is easy to understand as the limiting case,
considering lines that pass very near the origin.

On 07/03/2012 01:27 PM, treborsci@verizon.net wrote:
An unspoken implication in this metaphorical construction is the
expectation that one, and only one, field line goes through a given
space point. It is this expectation of the metaphorical field lines
that is violated in the cited situation. At a given x=0 space point,
several field lines coexist. The usual computerized field line
construction algorithm will balk in this situation.

I don't understand that. The more I think about it, the less I
understand it.

As I see it, the electrostatic potential exists and is continuously
differentiable at the origin and everywhere else, except right on
top of the two charges.

As a corollary, the electric field is well defined and continuously
differentiable at the origin and everywhere else, except right on
top of the two charges.

If you try to represent the field using a polar coordinate basis or
some sort of point/slope representation, it goes haywire at the origin
... but let's be clear, the thing that goes haywire is the representation,
not the field itself. If you choose a better representation, such as a
Cartesian basis, everything is fine.

If I were writing the field-line tracing program, it would instantly
find the two stable fixed points and their basins of attraction,
namely the upper half-space and the lower half-space. It would be
/slightly/ more work to find the third fixed point, namely the origin,
because it is an unstable fixed point and its basin of attraction is
a set of measure zero. However, given two basins of attraction, it
would be theoretical-physics malpractice to not ask what happens on
the boundary between them.

Field lines cannot cross, and I would expect any decent field-line tracing
program to know that. As a related matter, I would expect any decent
program to refuse to draw field lines in places where the field is zero.
If you're arguing based on intuition or based on symmetry that the program
"should" draw field lines in the plane of symmetry, I'm not buying it.
Such lines may be symmetrical but they are not typical. They are highly
atypical. They represent a set of measure zero. They are not a faithful
representation of the /density/ of field lines, which is what the physics
cares about. I like symmetry as much as the next guy, but it would be
better to make the lines symmetrical on each side of the plane of symmetry,
not /in/ the plane of symmetry. The picture I cited previously gets it
right:
http://www.ece.drexel.edu/courses/ece-e304/e3042/conduc4.jpg

The basin-label is discontinuous across the boundary, obviously, but
that is just a discontinuity, not a singularity. It is trivial to
represent such a discontinuity in a computer. Indeed, everything a
computer does is digital i.e. discontinuous ... representing a
continuous quantity is what's hard. Furthermore, the basin-label
has no significance to the laws of motion. It's just a label.
By analogy, if you throw a ball into the air, it makes a seemingly
discontinuous change from "so-called upward" motion to "so-called
downward" motion, but the dynamical equations don't care what you call
it; the dynamics is perfectly well-behaved and continuous.