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Charged sphere vs. modern 2-plate capacitors
1998 Bill Beaty

PHYS-L mailing list archives
From: William Beaty 
Date: Thu, 19 Feb 1998 22:57:25 -0800 (PST)
Subject: what IS a capacitor anyhow?

On Thu, 19 Feb 1998, Prof. John P. Ertel (wizard) wrote:
> How could anyone ever think that most students don't know or understand
> the CONTINUITY EQUATION.  They know and understand the concept very well
> --- the simply don't recognize it in the mathematical form that we
> commonly show them.
> I usually find that if I take time to teach them only a little extra math
> (that they may not already know) I don't have to make up "virtual
> currents" to make Kirchhoff's Current Law work.

What you say makes sense, but I would make an important addition. 
Capacitors are not simply pairs of adjacent conductive objects.  If they
were, then whenever a charge flow was directed into one capacitor plate,
charge would simply build up there.  End of story.  But capacitors do much
more than this, and it's these extra phenomena which require "displacement
current" or some similar concept.

To all:

Let me discuss this in detail.  The following thought experiments cleared
up quite a number of capacitor concepts for me.  I apologize to any who
are already familiar with these ideas. 

First I will propose a capacitor which has a unique shape.  It takes the
form of two solid metal hemispheres, A and B below, positioned face to
face, with a narrow gap between them.

Note: the gap is so narrow that the capacitance BETWEEN the metal chunks
is many orders of magnitude larger than the capacitance between each chunk
and ground (or infinity.)  My device not like two adjacent spheres,
instead the capacitance between the plates is >> than the capacitance of a
single sphere of similar size.  Picture my capacitor as a bisected solid
sphere with a thin plastic film separating its two halves. In this way I
define "capacitor."  In my view, two adjacent metal objects might have
capacitance, but they are not a "capacitor."  A "capacitor" behaves
differently than a pair of adjacent metal spheres.  This is analogous to a
situation with coils: a laminated iron-core power transformer behaves
entirely differently than a pair of air-core coils laying near each other 
on a tabletop.

                   ___---  ---___
                  |      ||      | 
                 |       ||       |
                |        ||        |     "capacitor" in the form of
                |        ||        |     two solid hemispheres
                 |       ||       |
              A   |___   ||   ___|   B
                      ---  ---

Now, what happens if I direct a current into just one of the plates?  More
specifically, what will happen if I place a dollop of positively charged
particles upon hemisphere B?  How will the charges arrange themselves?

                   ___---  ---___                  +++
                  |      ||      |               ++++++  12 pos
                 |       ||       |     <---     ++++++  charges
                |        ||        |               +++
                |        ||        |     
           A     |       ||       |    B
                  |___   ||   ___|
                      ---  ---

The result is interesting.  Half of the positive charges dive into the
narrow slot, where they induce an equal negative charge on the opposing
metal face.  The induced negative charges on the A face "create" an equal
complement of positive charges which arrange themselves on the outside of
the A hemisphere. 

                     +    +
                +___---  ---___ +
                /     -||+     \       Half of the (+) particles on B
              +|      -||+      |+     dive into the gap, but they induce
              |       -||+       |     charges of identical polarity
             +|       -||+       |     to appear on the OUTER surface of
               |      -||+      |+     hemisphere A
               +\___  -||+  ___/
                    ---  --- +
             A       +   +      B

We put no charges on hemisphere A, yet as far as the world outside the
narrow gap is concerned, A is now charged.  The narrow gap makes A and B
interact strongly. It SEEMS as though there has been a current between A
and B.  This "current" caused half of our 12 positive charges to travel to
the OUTER SURFACE of A.  They didn't jump the gap.  This wasn't an
electric current, but the end result is almost exactly the same as if
charges had crossed the gap.  To a first approximation, charge has
flowed THROUGH the capacitor.

So, see why "Displacement Current" is not easily removed from capacitor

Also, look at this below.  Suppose I ignore the closely-spaced charges
and short-range e-fields in the gap.  This isn't forbidden.  After all, to
an outside instrument the gap is tiny and its fields are vanishingly small
in comparison to the fields associated with those charges on the outer
surface.  I could adjust the gap size to subatomic dimensions, which would
raise the capacitance to kilofarads or higher, and essentially produce
this result:
                       +    +
                  +___--------___ +
                  /              \
                +|                |+
                |                  |
               +|                  |
                 |                |+
                 +\___        ___/
                      -------- +
                       +   +      

By attempting to charge one plate, we have essentially charged our entire 
capacitor.  This has implications for real-world capacitors.  Suppose we 
take a large-value capacitor and try to place a microcouloumb upon just 
one plate.  The result?  The capacitor ends up (ahem!) "electrified," (not 
"charged!") with half a microcoulomb on each face within the gap.  
Perhaps a few volts appears across the capacitor's leads.  But the full 
microcoulomb also appears on the surface of the capacitor's outer case.  
A typical capacitor has *picofarads* from its plates to distant ground.  
Therefor a FEW HUNDRED THOUSAND VOLTS appears upon the entire capacitor 
with respect to ground.  Clearly we cannot put charge on just one plate of 
a typical real-world capacitor.

Our capacitor acts very much like a wire.  If I tried to put a 1 uC charge
on a short wire, I would expect hundreds of kilovolts to appear on the
wire with respect to distant ground.  Or, if I place charge upon one end
of the wire, the charge immediately should distribute itself along the
entire wire.  Just like it did with my hemispheres-capacitor. 

Capacitors are not wires.  However, their actions are very close to those
of wires.  Imagine a capacitor like this:

  Metal wire:

     |                             |                              |

                               Incredibly narrow slot

If I place charges on one end, as far as an outside observer is concerned
they will redistribute themselves along the entire surface.  If I hook the
above "wire" into a circuit, and if I cause a loop of current to appear in
that circuit, charge will flow through the "wire."  THROUGH it? 

Now at the end of my speil I will add the traditional capacitor effects
back in. If I cause a charge to flow through the above wire, opposite
charges will be trapped within the gap, and a potential difference will
grow between the two halves.  Current in one side of the "wire" forces an
equal current to appear in the other side of the "wire" via electrostatic
induction.  If the current is constant with time, then the potential
difference across the halves will increase linearly with time.

Capacitors do not act like two separate metal objects.  As far as currents
are concerned, they almost act like a "short circuit."  They act like
conductors, but they have an interesting "back pressure effect" which
grows with time. 

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