Re: A funny capacitor.

[William Beaty <billb@ESKIMO.COM>]

On Tue, 27 Feb 2001, Ludwik Kowalski wrote:

> Instead of asking "which of the two setup has a larger C?"
> I can now ask "which of the setups should take more Q from
> the battery?"  Why should the presence of the third plate
> require a larger Q?

Things might become slightly clearer if we specifically depict the two new
capacitors formed by the extra plate.

Change this:
>                 B
>
>           *           *
>           *           *
>     *******           *******            C2
>           *           *
>           *           *
>
>             *********



Into this:

>                 B
>
>           *            *
>           *            *
>     *******            *******            C2
>      *    *            *   *
>      *    *            *   *
>      *                     *
>      *                     *
>      *                     *
>      *                     *
>      *******           *****
>            *           *
>            *           *
>        *********   *********
>    C3                          C4
>        *********   *********
>            *           *
>            *           *
>            *           *
>            *************

However, the added plate does not approach the *backs* of the two
capacitor plates, instead it approaches the gap region.  It causes charge
on the original two plates to migrate towards the extra plate, and acts to
"intercept" flux lines connected to that charge, therefore it partially
shields the original two plates from each others' effects.  As far as the
charge is concerned, the two original plates are rotating to face the
extra plate.  As a result, the original capacitor's value would decrease
even as the extra capacitance of C(added) = 1/(1/c3 + 1/c4) causes the
total capacitance to increase.  Arguements using forces and potential
energy show that the drop in value of C2 is more than made up by the extra
C(added).

I don't see other ways to simplify this problem, so perhaps it's in the
same realm as calculating charge distributions on conductors of arbitrary
shape, and is best "solved" with CAD programs (or spreadsheet numerical
simulations.)  It is however very similar to inserting a metal block
between capacitor plates, because even as two new high-value (small-gap)
capacitors are created as the metal block is inserted, the original
wide-gap capacitor is being destroyed.



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