>Note that the sum of three charges is not zero, unless V3=0.
That is neither a necessary nor sufficient condition for Q=0.
A sufficient condition is V1 = -V2, and V3=V4=0, under the stated
symmetry.
>I think that this is OK because some negative charge was sent
>to infinity to satisfy the imposed potentials.
Right. Keep that in mind.
>Here is my dilemma. I want to calculate the Cij coefficients in
>
> Q1=C12*V1 + C12*V2 + C13*V3
> Q2=C21*V1 + C22*V2 + C23*V3 Equations (2)
> Q3=C31*V1 + C32*V2 + C33*V3
>
>from the known values of potentials and charges. Let me rewrite
>these equations by taking under account the "gauge invariance"
>(Cij=Cji) and by using single lower case letters instead of indexed
>coefficients C. In this case it simplifies notation, I think.
>
> Q1=a*V1 + b*V2 + c*V3
> Q2=b*V1 + d*V2 + e*V3 equations (2)
> Q3=c*V1 + e*V2 + f*V3
>
>The "conservation of charge" requires that:
>
> a + b + c = 0
> b + d + e = 0
> c + e + f = 0
No, that's not a correct way to express conservation of charge.
What happened to the charge at "infinity"? This has to be part of the
charge-conservation equation.
The rule that each column of the C matrix must sum to zero assumes the
sum runs over all (!) relevant objects. In this case, you need a 4x4
matrix, including the object at infinity. Even if that object is just a
cloud of space charge, it is still must be included in the sum.
Actually, since V4 is (for the moment at least) assumed to be zero, you can
get by with a 4x3 matrix! That's 4 rows of 3 columns. But charge
conservation requires summing over all rows in each column, and each sum
necessarily includes 4 terms. Of course I recommend the 4x4 matrix, and
just multiplying the fourth column by V4=0 when necessary.
>What is the content of equations 2? They tell us us a net charge
>received by a conductor (from its battery) depends not only on
>its own potential but also on potentials imposed on other objects.
Including object 4.
>I would expect C12 and C13, for example, to become smaller
>than C11 when objects 2 and 3 are moved further away from
>object 1.
Yes.
>So I should be able to calculate the Cij coefficients for
>any set of distances. .... What is the accepted method of calculaing
>the Cij coefficients from known charges and potentials?