Re: funny capacitor

["John S. Denker" <jsd@MONMOUTH.COM>]

At 01:12 AM 3/12/01 -0500, Ludwik Kowalski wrote:

> We agreed on the Cij coefficients for the funny capacitor:
>
>    +2.86, -0.46,  -1.15,  -1.25
>    -0.46, +2.86,  -1.15,  -1.25
>    -1.15, -1.15,  +3.26,  -0.96
>    -1.25, -1.25,  -0.96,  +3.46

OK.


> John wrote "we have four independent voltages. We
> can set voltages on anything we want, and turn the
> crank on Laplace's equation." We did this to find the
> Cij coefficients.

OK.

> At the same time we found that an
> arbitrarily chosen set of potentials (V1=-100, V2=+100
> and V3=+20) produced the well defined charges
> (Q1=-198, Q2=143 and Q3=+65.2). In other words
> we found the "one and only one" solution of the Q(V)
> problem.

That is, we found the unique Q that corresponds to this V.
In particular, we found that Q(V) is a _function_
which means it associates one ordinate "Q" with each
abscissa "V" in the domain.

We did _not_ prove that this function was a one-to-one relationship.

> We think that no other combination of Vs
> can produce the same set of Qs.

Actually we can, if we include V4 as part of the V vector.
In particular,
        V1=k-100, V2=k+100, V3=k+20, V4=k
all produce the same Q values, independent of the gauge k.

> As often emphasized by John, V1, V2 and V3 are really
> potential differences with respect to object #4; we call them
> potentials but this is only a verbal shortcut.

We are getting into murky water here.

It would clarify the discussion to speak in terms of delta (!) V.  That is,
delta_V1 etc. are measured according to the expression
        delta_V1 := V1 - V4
etc.

> The gauge lead
> of a voltmeter would  always go to the object #4. In other
> words, the object #4 is associated with the gauge.

You _may_ choose that but others _may_ choose differently.

> John
> wrote: "the V(Q) problem does have four independent
> variables, namely three charges and one gauge."

And I distinguished the V(Q) problem from the delta_V(Q) problem.

> The entire universe, in the context of this problem, consists
> of four objects which either receive or give away charges.
> The first three objects are plates of our funny capacitor. Let
> the fourth object be an enclosure. Why not? By knowing Q1,
> Q2 and Q3 we can always calculate Q4 as -(Q1+Q2+Q3).

OK.

> The value of V4 remains zero no matter how large is Q4.

I would rather say that delta_V4 remains zero.  The reason for this
distinction is that the 4x4 capacitance matrix given above operates on the
V vector, not the delta_V vector.  If you want to explain C44, you can't
explain it in terms of the delta_V vector.

> A difference of potentials "with respect to itself" is always zero.

OK.

> A non-gauge object i whose potential does not change when
> its net Q changes would have Cii=infinity.

But the definition of C44 involved setting V4=1 and all the other
Vk=0.  This is inconsistent with the assumption that we are measuring
voltages relative to node 4.

Again we see that thinking in terms of delta_V is not the same as thinking
in terms of V.

> So what does it mean that C44=3.46? Doesn't this imply that V4 must change
> when Q4 changes?

Yes, it implies exactly that.  If V1=V2=V3 are held at zero, then V4
changes when Q4 changes.  This says nothing about delta_V4.

This choice, choosing to treat all four objects on the same footing, was
highly convenient for the number-crunching phase.

If you dare to !assume! gauge invariance, then the step of setting V4=1 and
all the others to zero must be equivalent to leaving V4=0 and setting all
the others to -1.  However, it was much more fun to !not! assume gauge
invariance, and let Dr. Laplace demonstrate it to us by giving us a Cij
matrix with the appropriate symmetries and the appropriate values in the
4th column.