Re: [Phys-l] Electric Field

[Hugh Haskell <hhaskell@mindspring.com>]

At 22:03  -0500 2/20/06, Alvin Bachman wrote:
> 1. For the case of two identical charges located on the X axis at +1m and -1m,
> consider the field lines that go along the X axis from each charge 
> toward the origin.
>  They simply disappear at the origin!
>     There are an infinite number of lines that can be drawn in the 
> central Y-Z plane
>  from near the origin, outward, but no way to connect any one of 
> them to one of the
> lines on the X-axis.

I'm not sure why this issue has come up. It also was raised on 
another list I belong to. Even in this case the electric field is 
continuous in every region that does not contain one charge or the 
other. At the origin of the coordinate system described the magnitude 
of the field is zero, as one would expect, but it is not 
discontinuous there. If one constgructs the field by superposition of 
the fields from each charge, one finds that the field in the Y-Z 
plane is radial outward from the center. It increases in magnitude 
from zero at the origin to a maximum at some radius from the origin 
and thereafter it decreases, with the decrease approaching 1/r^2 
asymptotically as r goes to infinity. Why this should be considered 
remarkable is beyond me. Maybe I am missing something.

I'm not sure why anyone would expect to be able to connect a field 
line in the Y-Z plane to a line on the X-axis. Because of the 
symmetry of the charges, the field vector everywhere in the Y-Z plane 
is confined to the Y-Z plane, with no component in the X-direction. 
Of course, the field magnitude goes to zero at the origin, so at that 
point it is true that there is no vector, but the variation in the 
field is still continuous.

> 2. Just beyond the charges, the field lines that move outward along 
> the X-axis give
>  no indication of what they should do "at" infinity. Of course, 
> there are no negative
>  charges in this space for any lines to "close" on. But since the 
> field is going to be
>  zero "at" infinity, we really can't say what their behavior will be.

But neither do any of the field lines that radiate from a single 
isolated charge. I think that is it usually taken that there is an 
equal negative charge smeared over the infinite sphere centered on 
the central charge, upon which all of the field lines can terminate. 
Since this arrangement of charges behaves asymptotically as a single 
charge for very large distances, it seems to me that the same 
structure should apply.

> 3. In short, field lines are a way to visualize the geometry of the 
> electric field in a
> limited region of space.
I'm not sure how this is particularly relevant to the preceding two 
items, or why the qualification "limited region" has to be applied. 
It seems to me that the field lines are a visualization of the field 
in all space (with the exception of the singularities at the 
locations of the charges).

I will admit to an inability to use the "density" of field lines to 
illustrate the magnitude of the field in the vicinity of the origin, 
but it can be depicted by putting arrows of appropriate length and 
orientation at a number of points on the Y-Z plane. A simple analysis 
shows that, for the configuration given, the maximum field in the Y-Z 
plane occurs on a circle of radius about .51 m from the origin.

Is there more to it that I have missed?

Hugh
--

Hugh Haskell
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<mailto:hhaskell@mindspring.com>

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