Not-so-parallel Plate Capacitor
المؤلف:
Sidney B. Cahn Boris E. Nadgorny
المصدر:
A GUIDE TO PHYSICS PROBLEMS
الجزء والصفحة:
part 1 , p 51
11-8-2016
1630
Not-so-parallel Plate Capacitor
a) A capacitor is formed by two rectangular conducting plates having edges L1 and L2. The plates are not parallel. One pair of edges of length L1 is separated by a distance d1 everywhere, and the other pair of edges of length L1 is separated by d2 everywhere; d2 > d1
Figure 1.1
(see Figure 1.1). Neglecting edge effects, when a voltage difference V is placed across the two conductors, find the potential everywhere between the plates.
b) Determine the capacitance.
SOLUTION
a) Consider this problem in cylindrical coordinates, so that the plates are along the radii (see Figure 1.2). The Laplace equation, 
then
Figure 1.2
becomes
(1)
Writing ϕ = R(ρ)χ (φ), we may separate (1) into two differential equations
where R is the radial part of the potential, χ is the azimuthal part, and n is some constant. In the small-angle approximation (which can be assumed since we are allowed to disregard the edge effects), we can say that R is independent of ρ, and then
for which we have the solution χ = Aφ + B. From the boundary condition χ(0) = 0, we have B = 0, and using the other condition χ(φ0) = V, we find that
where
(4)
b) Using the result of (a) and the expressions for the of the field energy U contained in a capacitor, we may find the capacitance. We have
(5)
Here,
so (5) becomes
Hence, the capacitance
(6)
In the limit of d2 → d1 = d (the case of a parallel plate capacitor), (6) reduces to
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