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

[John Denker <jsd@av8n.com>]

Let's take a broader look at the question of how to understand the
charge distribution on the wires leading up to capacitor plates.

Here is a more constructive answer.  I did the calculation, by brute
force finite-element analysis, using relaxation to find the potential,
which then gives us the charge distribution.

Here is the wiring layout:
  https://www.av8n.com/physics/img48/rwire_split_wiring.png

The wires are one unit wide (which you can see) and they extend
one unit in the Z direction (which you can't see).

The capacitor plates are 7 units wide (which you can see) and
extend 7 units in the Z direction (which you can't see).


Here is the charge density:
  https://www.av8n.com/physics/img48/rwire_split_charge.png

Charge density is charge per unit volume.  Note that because the
wire is thin, a high charge per unit volume is not necessarily
a high charge per unit length.

As expected, there is a high charge density where the red wire
runs close to the edge of the blue capacitor plate.

As expected, there is no charge in the interior of the metal
parts;  all charge sits on the surfaces.

As expected, there is zero charge density in the interior corners.
They act almost like little Faraday cages.  Conversely there is a
high charge density on the exterior corners.  This is why corona
points are pointy.

The capacitor gap is rather large, so the fringing fields are
quite significant.  If the gap were smaller and/or the lead
wires were farther away, the charge distribution would be
simpler, more like the ideal "parallel plate" picture you see
in textbooks.

Note that contrary to what it says in certain "modern PER-based"
textbooks, the charge density is not proportional to the local
voltage.  Each of the two wires is an equipotential, but that
does not mean its charge density is uniform, or even monotonically
varying.

Here is the potential:
  https://www.av8n.com/physics/img48/rwire_split_potl.png

Alas my software is not smart enough to draw the field lines.
You can sorta imagine what they look like.  They are everywhere
perpendicular to the equipotential contours, and they are
strongest where the potential gradient is steepest.

==================
Pedagogical and philosophical remarks:

I have no patience with the PER literature that says we should
not teach such things on the grounds that students cannot construct
the diagrams on their own, by hand.

I expect students to learn to /interpret/ such a diagram when
they see it, even though they cannot construct it.

For crying out loud, *I* cannot construct precise charge
distribution diagrams by hand.  That's what computers are for.
Similarly I don't do FFTs and SVDs by hand.  That's what
computers are for.  I cannot draw a detailed map of Africa
freehand, but I can /interpret/ such a map when I see it.

On the other hand, having looked at a bazillion charge distribution
diagrams, and having some clue about the Maxwell equations, I know
qualitatively what the distribution must look like.  In particular,
if there were a gross bug in the program I would probably notice.