capacitor with non constant V REVIEWED
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Short Word note from Phil's transmission lines files, dated 9.14.14 and reviewed 10.8.14, in which he concedes he got nowhere. He relates the question to an averaged voltage-over-charge idea used in Appendix S. Plan A splits the dielectric between two cylinders along E field lines into sliver capacitors dq = C(θ)dθ V(θ) and shows q is not Cav times Vav. Plan B uses differential patch pairs with C(x1,x2)dA1dA2.
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Capacitor with non constant V ? 9.14.14
I got nowhere with this. I was trying to physically model the notion of an "average capacitance" if the surfaces are not really at constant potential. In Appendix S I never had any need for an average C. But I did use this notion:
= (S.3)
and then
< >C1,C2 = < V(x1,x2) >C1,C2 / q(z)
and the double average on the right makes complete sense and then defines what is on the left. It was I think a valid question to ask! Reviewed 10.8.14.
Plan A. Suppose we took our two cylinders and following the E field lines, partitioned the dielectric space into a set of small capacitors, each of which has a particular voltage across it:
Perhaps we parameterize things with an angle θ around the left conductor. Then for a particular one of these small capacitors,
dq(θ) = [C(θ)dθ] V(θ)
A tiny sliver capacitor has capacitance C(θ)dθ which we assume we could calculate from the geometry.
We regard V(θ) as a specified function. Then the total charge on the left conductor is
q = ∫dq(θ) = ∫dθ C(θ) V(θ)
We might also talk about a mean voltage V as
Vav = (1/2πa)∫dθ V(θ)
The average capacitance would have to be
Cav = (1/2πa) ∫dθ C(θ)
Notice that we can NOT then say that
q = Cav Vav
Plan B. Use this different picture
A differential capacitor consists of two tiny metal patches of area dA1 and dA2 (patches are shown in red). The left patch is at x1 and the right patch is at x2 . The voltage between these patches is V(x1, x2) . We could then write
dq = [C(x1,x2)dA1dA2] V(x1,x2) dC = C(x1,x2)dA1dA2