Edge corrections for parallel-plate capacitors

As a result of field fringing, the capacitance of a parallel-plate capacitor differs from that predicted by the textbook formula. Using singular perturbations and conformal mapping techniques, we calculate the leading-order correction to the capacitance in the limit of large aspect ratio. We additionally obtain a comparable approximation for the electrostatic attraction between the plates.


Introduction
The calculation of the capacitance of a parallel-plate condenser appears in any electrostatics textbook. Assuming that the field lines are (i) confined between the plates and (ii) perpendicular to them, one readily finds from Gauss law that the capacitance per unit area is given by the ratio of the vacuum permittivity ε 0 to the distance between the plates. In the convenient two-dimensional idealisation, involving two plates of length 2l and separation 2h, the capacitance per unit length is given by ε 0 λ, λ = l/h being the aspect ratio.
Of course, for any finite value of λ, however, large, the field lines are not confined between the plates; in the vicinity of the edges, moreover, they are clearly not perpendicular to the plates. There is accordingly an interest in correcting the 'idealised-capacitor' formula. Assuming plates of zero thickness, Thomson [30] suggested the following approximation to the capacitance (per unit length), normalised by ε 0 , λ + ln(πλ) + 1 π + · · · . It therefore appears desirable to present a systematic derivation of the Bromwich-Love approximation (1.2), which applies in the simplest problem of zero-thickness plates. The scope of such a goal is more modest than that achieved by Shaw [28], which has actually gone to higher asymptotic orders. However, Shaw's work is very general and not easy to read. The restricted derivation which follows may accordingly serve as a convenient introduction for non-experts, supplementing both fundamental analyses of capacitance calculations [15,16,32] as well as less mathematical discussions of the fringing effect [6,10,20,24,26]. This educational goal will be accomplished in the present paper using matched asymptotic expansions [12], where the electric potential is solved in different subdomains (defined via appropriate limit processes). The capacitance may then be obtained by carefully adding the contributions from the respective regions, with the result being independent of the exact manner by which these regions are delineated. The asymptotic solution of the electric potential will also be used to obtain an approximation for the attractive force between the plates; since the mathematical analog of that force has no physical meaning in the problem of parallel-plate viscometer, this quantity has not been sought by Shaw [28].
As will become evident, the major task in the present work involves the calculation of the electric potential near the 'edge' of the capacitor. This task is accomplished using conformal mapping techniques. Consistently with the expository nature of this paper, two alternative mappings will be described. Similarly, the electrostatic attraction is determined using two different approaches, one direct, using the Maxwell stress concept, and one indirect, using the principle of virtual work.

Formulation
Consider a two-dimensional capacitor, made out of two thin conducting plates of length 2l. The plates are parallel to each other and separated at a distance 2h apart (see Figure 1). The plates are set at electric potentials ±V relative to the potential at infinity. Our goal is the capacitance, defined by in which, Q is the charge on the positive plate. Since Q is linear in V , the capacitance is independent of it. Normalising length variables by l, we employ Cartesian xyz coordinates with the x-axis running parallel to the plates, which extend between x = −1 and x = 1, and the y-axis running perpendicular to them. We employ a dimensionless notation where electric potentials are normalised by V and charges (per unit length in the z-direction) are normalised by ε 0 V , ε 0 being the permittivity of vacuum; the capacitance (per unit length in the z-direction) is consistently normalised by ε 0 . In what follows, we find it preferable to employ = h/l instead of the aspect ratio λ.
The electric potential ϕ is governed by (i) Laplace's equation, (ii) the Dirichlet condition ϕ = ±1 at the two plates (respectively, positioned at y = ± ); and (iii) far-field decay, the latter condition following from the absence of net charge in the two-plate system. It is evident from the problem formulation that ϕ is an even function of x and an odd function of y. We hereafter consider only y > 0, adding the symmetry condition (see Figure 2) Since the dimensionless potential difference between the plates is 2, the dimensionless capacitance is given by c = q/2, wherein q is dimensionless charge on the upper plate (cf. (2.1)). Making use of the two-dimensional variant of Gauss law, we find that q = − ∇ϕ ·n ds, (2.5) where the line integral may be carried out over any simple closed contour which encircles (only) the upper plate,n denotes an outward-pointing unit vector normal to that contour and ds is a differential length element. Upon deforming the contour and making use of the far-field decay (2.3), we represent c as an integral over the x-axis, namely The dimensionless capacitance c is a function of , the single parameter in the problem. Our goal is to derive an approximation in the small-gap limit, 1. When the fringing effect is ignored and one consider only the region between the plates, it is evident that, in that region, How is this 'idealised-capacitor' value affected by the finite aspect ratio?

Asymptotic regions
As → 0, it is convenient to distinguish between three 'asymptotic regions'. The 'outer region' corresponds to The different regions then correspond to the manner by which the limit → 0 is attained. In the outer region, x and y are held fixed as → 0; in the gap region, x and Y are held fixed (with x ∈ (−1, 1)); in the edge region, X and Y are held fixed.

Leading-order electric potential
We now derive the leading-order approximations to the electric potentials in the three respective regions, noting that the far-field condition (2.3) applies only in the outer region. In the leadingorder gap region problem, the ends are removed to infinity. Laplace's equation (4.10) becomes A solution that satisfies the boundary conditions (see Figure 3) is this, of course, is the dimensionless version of (2.8). It may appear that (4.2) is only a leadingorder solution; recalling, however, the form of the eigenfunctions of Laplace's equation in the gap geometry, it is evident that the asymptotic correction to (4.2) is exponentially small [9].
In the outer region, at leading order, one has the discontinuous Dirichlet condition at y = 0 (see Figure 4): Recalling that a polar angle is a solution of Laplace's equation (4.10), we claim that wherein the polar angles θ ± are indicated in Figure 4. Indeed, it is readily seen that (4.4) satisfies both (2.3) and (4.3). Expressing the angles in terms of the xy coordinates, we alternatively have where it is hereafter understood that the arctangent is between 0 and π .

Edge region
The only region where the calculation of ϕ requires some effort is the edge. Defining Z = X + iY , it is here convenient to employ a conformal mapping Z = f (ζ ) from the complex ζ -plane to the upper-half complex Z-plane. In particular, the degenerate polygon geometry of the edge suggests the use of a Schwarz-Christoffel mapping [5] from the upper-half ζ -plane (see Figure 5). Upon setting the critical points of the transformation in the manner described in Figure 5, the mapping f must satisfy Integration thus yields where the principal branch of the logarithm is taken.
Mapping to the edge region.
The coefficient M is determined by the requirement that the semi-width of the capacitor in the XY -plane is unity. In terms of f , this requirement reads Making use of (4.7), we then obtain M = 1/π . To determine N, we impose the condition f (−1) = i (see Figure 5), which gives N = 1/π . We conclude that In what follows, we write ζ = ξ + iη. Since Laplace's equation is conformally invariant [5], we need to solve (4.10) As Dirichlet conditions are also conformally invariant [5], the following applies at η = 0: The solution of (4.10)-(4.11) is simply is the polar angle in the ξη-plane (see Figure 5). Note that the real and imaginary parts of (4.7), respectively, give Edge corrections for parallel-plate capacitors . Alternative mapping to the edge region.
When reaching out to the outer region, ξ and η become large (see Figure 5). Relations (4.14) then give where the relative error is algebraically small (i.e. smaller than some positive power of |ζ | −1 ). At leading order, we then obtain from (4.13) φ ∼ arctan(Y /X ). Reverting to the unscaled coordinates using (3.2b) and substituting into (4.12) give Thus, asymptotic matching with the outer potential (4.5) is trivially satisfied. Similarly, when approaching the gap region, ξ and η are small (see Figure 5), so relations (4.14) become with an algebraically small relative error. From (4.12) and (4.17b), we find that ϕ ∼ Y , so that asymptotic matching with the gap solution (4.2) is, too, trivially satisfied.

Evaluating the capacitance integral
Consider now the integral (2.7). Roughly speaking, the gap solution (4.2) applies for x < 1, while the outer solution (4.5) applies for x > 1. A naive attempt to evaluate (2.7) using these solutions would result in a logarithmically divergent outer contribution. A similar obstacle was encountered by both Landau and Lifshitz [21] and Binns and Lawrenson [3]. The systematic evaluation of (2.7) makes use of two ingredients. The first is the edge solution (4.12), which applies in the vicinity of x = 1. The second is the use of a cutoff-like parameter δ, which naturally disappears in the final evaluation of (2.7). Since that parameter is otherwise arbitrary, we choose it at will. Thus, following Hinch [12], we choose it to satisfy the asymptotic condition δ 1. (5.1) For example, it may be taken as α for any 0 < α < 1.

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Using that parameter, we split the capacitance integral (2.7) as follows: The three integrals appearing in (5.2), associated with the contributions of the gap, edge and outer regions, are, respectively, denoted by c g , c e and c o . Upon using the natural coordinates in each asymptotic region, we find that the gap contribution is the edge contribution is (5.4) and the outer contribution is In general, each of these contribution depends both upon and δ; their sum, however, cannot depend upon the arbitrary parameter δ, which does not appear in the problem formulation. The evaluation of the gap and outer contributions is straightforward. Thus, substitution of the gap potential (4.2) into (5.3) gives with an exponentially small error. In the outer region, we find from (4.5) that Substitution into (5.5) gives where the error is algebraically small (i.e. smaller than some positive power of ). The latter expression clearly exhibits the logarithmic divergence of the outer contribution. In calculating the edge contribution (5.4), we note that both (4.9) and (4.18) map the ξ -axis to the X -axis. Thus, regardless of which one is employ, we have Moreover, making use of the manner by which normal derivatives transform under conformal mappings [5], we note that

E. Yariv
We therefore find that (5.4) becomes If we employ the Schwarz-Christoffel transformation (4.9), we have, using (4.12), The numbers Z = ±δ/ are, respectively, associated with points D and E in Figure 5. Their preimages, f −1 (±δ/ ), are associated with points D and E . These preimages have positive real part and zero imaginary part. Substitution of (5.12) into (5.11) accordingly gives . (5.13) All that remains is to evaluate the preimages f −1 (±δ/ ). Since δ/ 1 (recall (5.1)), it is readily seen from (4.9) that with a relative error which is algebraically small. Similarly, we find that with an algebraically small error. Substitution of (5.14)-(5.15) into (5.13) provides the requisite edge contribution: with an algebraically small error.
As an alternative, we may employ the mapping (4.18). We then note that (5.9)-(5.10) still hold at η = 0. The edge contribution (5.4) is then still given by (5.11). Here, however, substitution of (4.20) yields (cf. (5.13)) The numbers Z = ±δ/ are now, respectively, associated with points N and P in Figure 6. Their preimages, f −1 (±δ/ ), are associated with points N and P . All that remains is to evaluate these preimages. Since δ/ 1 (recall (5.1)), it is readily seen from (4.18) that with a relative error which is algebraically small. In this approximation, the term e πζ in (4.18) is exponentially small. On the other hand, when seeking f −1 (δ/ ), which is presumably large and negative, it is evident that this term dominates both ζ and the constant term, suggesting that Inspection reveals that the leading-order correction to (5.19) is o (1). By substitution of (5.18)-(5.19) into (5.17), we obtain (5.16).

Electric force
Having calculated the electric potential in the three asymptotic regions, we can exploit the results to calculate the attractive electrostatic force (per unit length) between the two plates. Normalised by ε 0 V 2 /l, the force on the upper plate is Our goal is the leading-order correction to (6.4).
Making use of the cutoff parameter δ, we decompose F as The three integrals appearing in the above, respectively, associated with the contributions of the gap, edge and outer regions, are, respectively, denoted by F g , F e and F o . Upon using the natural coordinates in each asymptotic region, we find that the gap contribution is the edge contribution is The evaluation of the gap and outer contributions is straightforward. Thus, substitution of the gap potential (4.2) into (6.6) gives with an exponentially small error. In the outer region, substitution of (5.7) into (6.8) gives where the error is algebraically small. In calculating the edge contribution (6.7), we make use of (5.9)-(5.10) to obtain dξ . (6.11) Substituting (4.9) and (5.12) gives While approximation (5.15) is sufficient for our needs, we need a refinement of (5.14). It is not difficult to see from (4.9) that with a relative error which is algebraically small. Substitution into (6.12) provides the requisite edge contribution: with an algebraically small error. Alternatively, we could use (4.18) and (4.20) instead of the Schwarz-Christoffel transformation (4.9). Substitution into (6.14) then gives, instead of (6.12), Making use of (5.18)-(5.19), we retrieve (6.14).
Adding up the respective contributions to (6.5), namely (6.9), (6.10) and (6.14), we find that the dependence upon δ disappears, giving with an algebraically small error. The ratio of the above approximation to the idealised capacitor formula (6.4) is 1 + /π.
An indirect derivation of (6.16), based upon the principle of virtual work and the capacitance approximation (5.20), is presented in the Appendix.

Concluding remarks
Making use of matched asymptotic expansions, we have analysed the electrostatics of a parallelplate capacitor in the limit of small separation. Our analysis has reconfirmed the result of Bromwich [4] and others, thus showing that the Thomson's approximation for capacitance edge correction misses a term. We have additionally calculated the electrostatic attraction between the plates. Measurement of that force has been used for the determination of ε 0 , the permittivity of free space [11,32]. The refined approximation (6.16) for the dependence of the force upon the separation between the plates may yield a more accurate determination.
There is some resemblance between the present analysis and that of Jeffrey [17], where a two-sphere capacitor was considered. There are two main technical differences between the two problems. The first is obvious: in Jeffrey's analysis, there are two distinct asymptotic regions, namely, the gap and outer regions; in the present problem, one needs to distinguish between three regions.
The second difference has to do with asymptotic corrections. In both Ref. [17] and the present analysis, the parameter appears only in the gap region, due to the non-uniform manner by which the x and y coordinates are stretched (see (4.1)). In Ref. [17], where the gap is analysed using a lubrication approximation, the presence of the small parameter in the differential equation naturally leads to higher order corrections. This is not the case here, where the gap solution (4.2) satisfies the exact equation (4.1), and the error is exponentially small. Since the errors associated with the leading-order solutions in the outer and edge regions are not exponentially small, higher order corrections emerge in these regions as a consequence of higher order asymptotic matching [28]. In the present contribution, where we sought the leading-order correction to the idealised capacitor approximations, no need arises for the calculation of such higher order terms.

Conflicts of interest
None.