Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

Re: Neutrality of a battery; was Capacitor or condenser?



Because of the fact that the earth itself deviates substantially from
electrical neutrality (with typical fields near the surface of 100 V/m
implying a total charge on the order of half a million C), no object--
capacitor, battery, or flying squirrel--is likely to be neutral either.
Fortunately, as William Beaty points out, this need not overly concern us
when we talk about "the charge on a capacitor" or any other circuit
property related to the potential *differences* which we are interested
in.

To show this more quantitatively we can make use of the "coefficients of
voltage"--a formalism that I only recently discovered as a result of
recent phys-l discussion on this same general topic. It can be (and has
been!) shown that, for any specified configuration of charged conductors,
one can write a linear system of equations relating their absolute
potentials to the charges they carry. That is,

V1 = k11*q1 + k12*q2 + k13*q3 + ...
V2 = k12*q1 + k22*q2 + k23*q3 + ...
V3 = k13*q1 + k23*q2 + k33*q3 + ...
....

where I have used the fact (not proven here) that the coefficient of
voltage matrix is symmetric, i.e. kij = kji. The coefficients themselves
can be calculated strictly on the basis of geometry.

Now, let conductors 1 and 2 be the "plates" of our capacitor. We can
solve the equations above for "the capacitor voltage," DV = V2 - V1, and
get

DV = kdiff*(q2-q1) + ksum*(q2+q1) + DVbgnd(q3, q4, ...)

where kdiff = (k11+k22-2*k12)/2, ksum = (k22-k11)/2, and DVbgnd is a
linear function of all charges except q1 and q2. Note that kdiff and
ksum depend *only* upon the geometric configuration of the two plates of
the capacitor.

Now suppose that we start with an "uncharged capacitor" by which we
really mean

DVo = 0 = kdiff*(q2o-q1o) + ksum*(q2o+q1o) + DVbgnd(q3o, q4o, ...)

Notice, that the phrase "uncharged capacitor" does *not* imply that the
individual charges on the plates of the capcitor are zero; it only implies
a specific relationship between all of the charges on all of the
conductors.

Now let us charge the capacitor by taking Dq from q10 and adding it to
q2o leaving all other charges the same. That is let

q1 = q1o - Dq
q2 = q20 + Dq

Thus, we find

DV = 2*kdiff*Dq

or

DV = Dq/C where C = 1/(2*kdiff)

which is to say that "the capacitor voltage" depends *linearly* and
*only* upon the amount of charge transferred from one plate of the
capacitor to the other as long as we don't alter the charges on the other
conductors or the positions and orientations of any of the conductors.
In this event we need not concern ourselves *at all* with the possibility
(and even likelihood) that |q1| does not equal |q2|.

One could argue, correctly, that we regularly violate the assumption that
none of the other conductors move or have their charges altered.
However, I strongly suspect that typical capacitor geometries make the
effect of such alterations all but negligible for any reasonable values
of the charges q3, q4, ...

John
----------------------------------------------------------------
A. John Mallinckrodt email: mallinckrodt@csupomona.edu
Professor of Physics voice: 909-869-4054
Cal Poly Pomona fax: 909-869-5090
Pomona, CA 91768 office: Building 8, Room 223