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

On Mon, 07 Apr 1997 John Mallinckrodt wrote:

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.
___________________________________________________________________________
I am not going to use this analysis in my undergraduate course. But in
order to appreciate it better I want to know how the electrodes of the
battery must be treated. Are they charged conductors with constnat q3 and
q4? And what about that corrosive substance between them? Is it another
conductor with some q5? Or is it treated as an isolator? And where does
the energy for charging the capacitor come from? Something must separate
mutually attracting charges and deliver them to electrodes.

I do not see how the formalism presented can be used to deal with an
elctrochemical cell, unless you treat it as a "black box" which mentains
a constant difference of potentials and which, BY DEFINITION remains neutral.
This is not a way for EXPLAINING the neutrality of a battery. Talking about
something and explaining something are two different things, in my opinion.
___________________________________________________________________________
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|.
----------------------------------------------------------------------
John, I was much more satisfied (intellectually) when the worm problem
solution was explained to me. But I do appreciate your effort. And I
know that my own limitations are usually responsible for uncomfortable
intellectual situations.
Ludwik Kowalski