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Re: [Phys-L] The status of Kirchhoff''s laws



On 3/2/2013 9:25 PM, Bruce Sherwood wrote:
Nice that the Wikipedia article (now, maybe not before?) specifically talks
about the violation of the Kirchhoff Current Law (KCL) for a single plate
of a capacitor (conventional current flows onto the positive plate and
doesn't flow off that plate, and the plate becomes more and more positively
charged).

It is interesting to see the mechanism for how and why the same current
flows off the negative plate as flows onto the positive plate. An
underappreciated aspect of capacitors in circuits is the role of the fringe
field. From the superposition of the fields contributed by two charged
disks that are close together it is easy to show that the field just
outside a disk, at the center of the disk, is 0.5(s/R) times the field in
the gap, where s is the gap length and R is the radius of the disks. This
is typically a very small field (s << R), but its role is crucial. Note
that this center-point fringe field points away from the positive plate and
toward the negative plate (the direction of the fringe field is determined
by the closer plate).

For years I was puzzled as to how charge could leave a discharging
capacitor, since the charge on a plate is strongly attracted to the
opposite charge on the nearby other plate. Charges in the neighboring wires
move due to the field in the wires. The fringe field just outside the
positive plate points away from the positive plate, driving positive charge
away from the positive plate (so conventional current leaves the positive
plate). The fringe field just outside the negative plate points toward the
negative plate, driving positive charge onto the negative plate (so
conventional current flows onto the negative plate).

Note that for a field at some location to have an effect on matter, there
has to be matter at that location! From this fundamental view of charges
and field, it's the fringe field that matters, not the field in the gap
(whereas when analyzing in terms of potential one focuses on the potential
difference across the gap, which is Es if s << R, where E is the nearly
uniform field in the gap of length s).

Now for the punch line. Suppose in a circuit that more conventional current
is flowing onto the positive plate of a capacitor than is flowing off the
negative plate. In that case the field in the neighboring wires contributed
by the capacitor, just outside the capacitor, is no longer the tiny fringe
field of a device with zero net charge. Now field in the wires can be
large, because the capacitor has a nonzero net charge, in this case a
positive net charge. What will this large field do? It will slow down the
conventional current approaching the positive plate, and it will increase
the conventional current leaving the negative plate. There is a built-in
feedback mechanism that will bring the two currents back to being equal.

A footnote: Note the value of considering a capacitor to have finite rather
than infinite plates. What matters for approximate analyses is that the
size of the capacitor be large compared to the gap distance, but don't go
to the infinite limit too soon. An infinite-plate capacitor has no fringe
field and cannot affect neighboring wires. As an astute student once
pointed out to Ruth and me, "You could never discharge an infinite-plate
capacitor". There are many situations in physics, including intro physics,
where for a sense of mechanism it's important to stick with not quite
"ideal" objects.

Bruce
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I was hoping to read about the role of a dielectric in providing displacement current?

Brian W