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messy electrostatics



Any two separated metallic pieces form a capacitor. How to find C?
Experimentally (assuming there are no other objects nearby) this can
done, at least in principle, by connecting the pieces to a power supply
of known d.o.p. and by measuring Q on one piece. Then C=Q/V.
It is true that measuring small Q can be complicated but that is not an
issue here.

I want to know how to calculate C without doing an experiment, for
example, to predict the value of Q. The problem was approximately
solved for many simple configurations, such as parallel plates, coaxial
cables and two spheres. But how to find C when a configuration is
not simple? For example, an aluminum foil on top of a balloon and
a suspended cylinder nearby. In other words, how to calculate C for
an arbitrary, but well defined, geometric configuration of two
metallic pieces?

The analytical approach is hopeless while a numerical integration
code should be possible. I do not know start writing it. Suppose we
subdivided the surface of each piece into 1,000,000 small elements;
and view C as a parallel connection of 1,000,000 capacitors. Then
C=dC1+dC2+dC3+ etc. How do we calculate individual dCi between
arbitrary oriented elementary planes whose sizes are different? And
how do we know which two pieces “go together” (are linked by the
same field lines)? A messy problem. Can it be solved?

Some say “what can be measured it can also be calculated”. Hmm.