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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 explanations?

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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