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Re: A funny capacitor.



On Tue, 27 Feb 2001, Ludwik Kowalski wrote:

where C3 is the capacity of each component. The
prediction that C2>C1 would be correct only if
C3 were larger than 2*C1. How can one be sure that
this is true?

Start with the 2-plate capacitor and the extra plate far away. As the
extra plate approaches from a distance, there is attraction by induction
and work is done (the plate accelerates.) The mechanical energy had to
come from the electrostatic energy stored in the system, PE = 1/2 V^2 / C,
so the approaching extra plate caused V to decrease, or C to increase, or
both. Since Q = C * V, and since Q is constant (nothing connected to the
plates), any change in C causes a simultaneous opposite change in V, so if
the stored energy decreased, then V *had* to decrease and C *had* to
increase. (C is the total capacitance as measured between the two
terminals.)

Said another way: The only time that the total C would FALL rather than
rise, is when the extra plate was repelled. If induction can only create
an attractive force on an uncharged conductor, then the addition of an
uncharged conductor must always increase the capacitance of the system.


It is not at all
obvious to me that C3 is larger than 2*C1.

Right. The creation of C3 causes the value of C1 to decrease, but does it
also cause the TOTAL capacitance to increase or decrease? The answer is
found by noting the mechanical work being performed as the extra plate is
added. I'm familiar with this because of a similar, simpler problem:
watch what happens when two parallel oppositely-charged capacitor plates
are allowed to accelerate towards each other. Voltage falls, capacitance
proportionally rises, work is done, and stored electrostatic energy is
lost. These are linked by simple algebra, call it "electrostatic motor
equations", and whenever a change in geometry causes electrostatic energy
to be converted into mechanical work, the other changes necessarily
follow.

Visually, the e-fields act like rubber bands, and allowing them to shorten
themselves causes V to fall and C to rise. Changes in Q act to
create/destroy rubber bands, but that is not happening here. The
approaching extra plate cuts the pre-existing rubber bands in half as
opposite charge is induced on its surface, but then lets the individual
halves of the "bands" contract in length, so mechanical work is being done
and the total C is increasing.




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