Suppose an experiment using a funny capacitor was performed.
The V1 on the left vertical plate was -50 V, V2 on the right
vertical plate was +50 V and V3 on the horizontal plate was
+20 V. All potentials are defined with respect to infinity where
V4, as usual, was assumed to be zero. The corresponding charges
on the plates were found to be Q1=-189, Q2=+144 and Q3=+65
units. Well, I did not try to perform an experiment; the three
charges were calculated by solving the Laplace equation
numerically, as we learned last week.
Note that the sum of three charges is not zero, unless V3=0.
I think that this is OK because some negative charge was sent
to infinity to satisfy the imposed potentials. In my mind V1, V2
and V3 are maintained by three batteries. One terminal of each
battery is connected to its plate while another is connected to
"infinity". In other words, the infinity is a very large object on
which potential is zero. What is wrong with assigning zero
to infinity? Why did my teacher criticize this in his last
message under this thread?
Here is my dilemma. I want to calculate the Cij coefficients in
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.
The determinant of the matrix of coefficients is indeed zero and
I am not able to solve the equations. It shows that the singularity
does occur when one wants to determine the values of Cij. What
am I supposed to do to determine Cij on the basis of experimental
data? The singularity has nothing to do with my values of Q because
these values are not in the matrix of coefficients; they are on the
right side.
The singularity disappears as soon as a single zero (anywhere in the
matrix of coefficients) is replaced by a nonzero number, such as 0.1.
As soon as I replace a zero by a number my program solves the
equations (by inverting the matrix) and verifies that a, b, c, d, e and
f are indeed the correct solutions. What am I doing wrong?
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.
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. So I should be able to calculate the Cij coefficients for
any set of distances. But I am not able to do this. Where did I
goof? What is the accepted method of calculaing the Cij
coefficients from known charges and potentials?
Ludwik Kowalski