I need time to digest John's explanation. But
some general comments are perhaps worth sharing
and discussing. I think it is wise, from the
pedagogical point of view, to be consistent
with the use of V4. In other problems we make
V4=0, either at infinity or on one object called
"grounded", usually anything connected to earth,
our big conductor.
1) Telling students that mother nature has many
solutions [(of the Q(V) or V(Q) problems] is not
very helpful; it can confuse beginners. So let
us agree that "V4=0 at infinity", as we do in
less demanding problems. Let us use one gauge
and focus on essentials. It is true that one
may sometimes take advantage of another choice
but in my opinion this is not worth emphasizing.
BTW, what are numerical values of V1, V2, V3
for our example when V4=0? I keep calling the
fourth object "infinity" because textbooks do
so. But I agree, any object of any shape is an
acceptable reference. The only requirement is
that the total flux through a closed surface
containing the four objects is practically zero.
2) Yes, I was impressed that Dr. Laplace took
care of the charge conservation, and of the
Cij=Cji automatically, I did not have to ask
for it, it was the consequence of solving the
equation.
3) Most students are familiar with the law of
charge conservation. Thus the fact that sums
of Cij are zeros, in each column, does not need
to be explained. When a net charge Q=Q1+Q2+Q3
is supplied to plates of the funny capacitor,
then, it is natural, that -Q must be supplied
to the "only remaining" object #4. Objects
which are neither sources nor sinks of E lines
can be totally ignored.
4) But how can one justify Cij=Cji? John
justified it be referring to Dr. Maxwell. Can
this be done on the basis of what is explained
in a typical introductory physics textbook? The
question is "why does the coefficient of influence
of object i on object j is the same as that of
object j on object i? That may be obvious for
two identical spherical conductors but not for
two conductors whose sizes and shapes are very
different.
Ludwik Kowalski