Re: [Phys-L] charge distribution leading up to a capacitor

[John Denker <jsd@av8n.com>]

On 07/30/2016 08:31 AM, Robert Cohen wrote:
> Thank you John.

:-)

> I was particularly interested to see that the high charge density is
> not just on the plates (or the side of the wire that is close to the
> opposite plate/wire) but also along the side of the wires far from
> the opposite plate (i.e., not just "leakage" off the plates).

To understand what the bottom blue wire segment is doing, it
may help to switch back and forth between the charge-density
view:
  https://www.av8n.com/physics/img48/rwire_split_charge.png

and the potential view:
  https://www.av8n.com/physics/img48/rwire_split_potl.png

The wire is an equipotential.  Far away from the wire, the
potential must fall off to zero (in accordance with the usual
choice of gauge).  If we were talking about a flat plate
with thickness t and width w, the potential would fall off
only slowly, on a length-scale comparable to w, independent
of t.  However, for a wire of radius r, the potential starts
falling off immediately, on a length-scale comparable to r.
So there will be a lot of field lines spewing out of the
wire in all directions, including out towards the edge of
the diagram.

To say the same thing another way, an ideal parallel-plate
capacitor keeps all the fields inside, with minimal stray
fringing fields.  In contrast, two parallel wires make a
horribly non-ideal capacitor, with enormous fringing fields.

The bottom blue wire segment can be considered paired with 
the bottom red wire segment;  they care more about each other
than about the far-away capacitor plates.

The diagram shows the XY plane.  If you were to look instead
at the YZ plane, you would see (to a first approximation) 
those two wire segments producing a dipole field pattern in
two dimensions, like the flow-field of pair of vortex lines.

It is tempting but not quite correct to talk about the self
capacitance of a long straight wire.  Strictly speaking, its
self capacitance is zero.  However each pair of wires has
a well-defined mutual capacitance, and a single wire has a
well-defined mutual capacitance to the nearby "chassis" or
"ground".

Kirchhoff's so-called «laws» embody the assumption that all
wires have zero capacitance ... but physics says otherwise.

=====

Also keep in mind that my charge-density diagrams show charge
per unit /volume/.  So a thin wire of radius r, even if it
lights up prominently in the diagram, will have a smallish
charge per unit length, proportional to r.  To say the same
thing another way, the diagram shows a slice through the
middle of the wire at Z=0, and you have to ask how many other
slices, at other Z-values, would show a similar result.  The
answer is proportional to r.

If you wanted to show the charge per unit length of wire,
you could perhaps multiply by a factor of r.  This would be
in some ways more informative but in other ways more confusing.
It would be a disaster in non-ideal geometries such as corners.

It might be possible to improve the presentation using 3D
graphics, but that's more work than I feel like doing at the
moment.