My big mistake was to treat the enclosure as an
ordinary object#4. How did I dare to think that
so many authors, including J. Maxwell himself,
were wrong in saying that the Bij coefficients
CAN BE calculated from Cij coefficients by solving
the n equations with n unknowns? Why did it take
so long to realize that I was thinking about V’s in
the traditional sense while John was thinking about
them in very unusual (for me) way. John kept saying
this to me but I was blind for several days. In any
way, let me summarize how the problem of n floating
conductors can be solved in a traditional way.
1) Keep in mind that each V is a difference of
potentials with respect to a reference. This term
reference is used for a very special object. What
makes it special? IT IS NOT HOW FAR IT IS. It is
the fact that its potential remains constant NO
MATTER HOW much charge is received or given away.
Our planet is big enough to nearly satisfy this
requirement in most cases.
The answer to a funny capacitor problem (either
theoretical or experimental) will depend on the
distance to the enclosure, and on the shape of
that enclosure, unless it is very far away. A
common silent assumption in electrostatic is that
the reference is a very distant body but this
is not essential.
2) I begin by writing down the n equations with
n unknowns, one for each object (excluding the
reference where V=0). In my case n=3 and:
3) Next I must find the Cij coefficients. To
accomplish this I impose V1=1, while V2=V3=0,
and start cranking. Dr. Laplace tells me what
Q1, Q2 and Q3 are under such conditions. These
are the values of C11, C21 and C31. Then I
impose V2=1, while V1=V3=0 . Dr. Laplace gives
me different values for Q1, Q2 And Q3. This
time I identify them as C12, C22 and C32. And
finally the C13, C23 and C33 determined under
the assumption that V3=1 while V1=V2=0. It
took three runs, of 5000 iterations each,
produced the following set of Cij.
4) This matrix can be used to solve any Q(V) problem,
that is to Calculate Qs when Vs are given (using
equations 1). For example, with V1=-12, V2=9 and
V3=-3 one gets Q1=-48.82, Q2= +50.21 and Q3=-7.578.
But our goal is to solve the V(Q) problem, that is
to calculate Qs for the desired Vs when conductors
are electrically floating. To accomplish this
the equations 1 are written as
where numbers on the left side of the equality symbol
are Cij coefficients. The numbers on the right side
are charges on objects 1, 2 and 3, calculated with
the Laplace equation at V1=V2=V3=1. The above set of
equations was solved to obtain the formulas
expressing floating potentials in terms of net charges.
For example, V1, V2 and V3 are –1200, +900 and –300
when Q1, Q2 and Q3 are -4882, +5021 and -757.8,
respectively. In other words, the general solution
of the V(Q) problem is:
where Bij are voltage coefficients for a given set of
capacitance coefficients, Cij. Any set of n conductors
can be analyzed in the same way, that is by solving
the n equations in n unknown. A Laplace code provides
the numbers needed to compose the equations 1. We
invert equations 1 and calculate floating potentials
with the resulting equations 2. Tedious and error-prone?
Yes. Difficult? Not at all. The entire Chapter III in
Maxwell's book is devoted to this fundamental
electrostatic problem.
So why was I confused? Because I accepted John's
potentials. They are not physical concepts defined
in terms of work per unit charge. His potentials,
and potential vectors, are set of numbers which
can be converted to traditional potentials. The
unsolvable set of equations was created by treating
the reference object in the same way as any other
object in which a change of Q leads to a change
in V. Any conductor, no matter how small, can be
used as a gauge in John's model of reality. A
traditional model, on the other hand, does not
allow small objects to be references. (A traditional
reference maust be very very large to keep its
potential constant when its net charge is changing.
John was able to find a unique solution by a trick
of dropping a row and a column from the "full Cij
matrix". In my opinion it is not a good method of
modeling electrostatic Q(V) and V(Q) problems. What
do we gain by turning a problem into something that
seems to be unsolvable and which can only be solved
by a trick? But who am I to impose that opinion on
others? I learned a lot and would again like to
thank John for the guidance. Was I the only
one to benefit from it?
Ludwik Kowalski