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