## Abstract

In this paper, we write a system of integral equations for a capacitor composed of two discs of different radii, generalizing Love’s equation for equal discs. We compute the complete asymptotic form of the capacitance matrix for both large and small distances obtaining a generalization of Kirchhoff’s formula for the latter case.

## 1. Introduction

The analytic calculation of the capacitance and potential coefficients of a couple of conductors of various definite shapes has a very long and distinguished history. Several eminent scientists have tackled the issue [1,2,3], and many researchers have produced specialized studies in the past decades using more and more frequently the newly available numerical resources [4,5]. Owing to the availability of approximate solutions when the conductors are considered very far from or very close to each other, much attention has been devoted to the asymptotic behaviour of the parameters in the far and in the near limit as a suitable check of elaborate numerical procedures. Moreover, even an approximate closed form is very useful in order to ascertain or conjecture on relevant and general properties of the electrostatic parameters.

It is worth noting that the long history is punctuated by definite periods of revival of the issue, often in conjunction with a particular field of technical application. In the 1990s considerable effort was applied to the calculation of the capacitance of planar systems in conjunction with the analysis of microstrip transmission lines [4].

Today, the extensive use of MEMS devices such as electrostatically driven actuators or detectors sensitive to the static charge carried by movable parts (e.g. capacitive sensors) encouraged an accurate knowledge of their electrical properties for selected geometries such as those occurring with parallel conducting plates or spherical–spherical and spherical–plane electrodes [5].

The potential or capacitance coefficients are the primary parameters involved in the description of the charge (voltages) variations following a deformation of the shape of conductors or a change in their separation or in the dielectric constant of the gap material. Those coefficients are also important in the investigation of the interelectrode forces arising when small charged conducting particles approach each other or one of them interacts with a close conducting surface [6].

In this paper, we present a generalization of Love’s integral equation [7] treating the case of discs of different diameters and giving the exact result in the near limit. To the best of our knowledge, these results are new. Moreover, we put the specific problem explored in this paper in a more general framework: the analysis of divergences of the coefficients of capacitance when the mutual distance between conductors vanishes.

The paper is organized as follows. In §2, we put the problem in the more general framework of small distances behaviour of capacitance matrix elements. In §3, we give a brief review of the results for identical discs. In §4, we present a summary of our results, particularly the expression of the capacity coefficients. We perform comparisons with large distances results and the arguments expressed in §2. Section 5 provides a formal proof for the basic system of integral equations which fix the solution and, in §6, we perform the asymptotic small distance expansion, both for capacity and for solutions of the integral equations.

## 2. Physical constraints on the coefficients *C*_{ij}

Let us consider a unique conductor, ideally composed of many approached parts, of capacitance *C*_{F}, which is the charge *Q*_{F} acquired if the body is maintained at unitary potential. The charge is equal to the flux of the electric field across the surface of the body or, more intuitively, to the number of lines of force leaving the surface and ending at infinity. Now let us perform a small displacement, ℓ, between two parts of the body holding fixed the potential: no force line joining the detached parts appears, if the potential of each part is held fixed, and the complex of lines acquires a small distortion. In more formal language, the new total charge, *Q*(ℓ) is close to the original one, i.e. the total charge is a continuous function of ℓ. In particular, the charge *Q*_{F} is finite for ℓ→0, i.e. in the limit reproducing the original situation. Using the same kind of reasoning, one can conclude that the charge of each part remains finite in that limit. A more algebraic proof is obtained by observing that the sum of the individual charges *Q*_{i}(ℓ) gives the total charge *Q*(ℓ) and that *Q*_{i}(ℓ) is positive, being the sum of coefficients *C*_{ij} for fixed *i* ([1], §89). Then if *Q*(ℓ) is finite, the single *Q*_{i}(ℓ) cannot diverge for ℓ→0.

In the particular case of a system of two conductors, the above considerations amount to say
*a*and
*b*Let us stress that (2.1b) implies a complete cancellation between possible divergences in each conductor separately, while (2.1a) implies a limit value for a combination of the *finite* part of fringe effects.

These constraints are particularly effective in the case of parallel plates, the limit of two plates being again a plate, and even more effective for identical parallel plates, where by symmetry
*A*: we expect two divergent terms: a ‘bulk’ divergence which goes like *A*/ℓ, and a ‘fringe’ divergence growing like *C*_{12}. The sum of the two terms must cancel out when *C*_{12} is combined with two *different* self-capacitance coefficients, *C*_{11},*C*_{22}, in accordance with the constraints of equations (2.1). We have thought that it would be instructive to verify how this could be achieved in a system disregarded in the literature, and this has been the original motivation for the study of a capacitor composed by two different coaxial discs.

## 3. Parallel discs capacitor: a short review of Love’s equation

The problem of a capacitor consisting of two parallel coaxial discs of radius *a* and at distance ℓ has a long and rich history in electrostatics research. The first classic result is the Kirchhoff [2] formula (3.11). A crucial step for our purposes has been done by Love [7], who, generalizing some previous results of Nicholson [9], reduced the Laplace problem to a solution of a linear integral equation. The problem has been reformulated in a useful way using cylindrical coordinates by Sneddon [10]. In this section, we review some aspects of the problem to fix the notations and for comparison with subsequent work. We follow the clear presentation of these results given in [11].

In short, the solution of the general case of two equal discs at different potentials *V*_{1},*V*_{2} is reduced to a system of integral equations
*a* is the disc radius and ℓ the distance between discs. Clearly, the solutions depend parametrically on *κ*, so a more correct notation would be *F*_{i}(*t*;*κ*), but we omit the second argument when it is not necessary. The charges on the discs are
*F*_{i} are related to the densities *σ*_{i} on the two discs by an Abel transformation:

Specializing (3.1) to the case *V*_{1}=*V*_{0}, *V*_{2}=0 and with *F*_{i}=*V*_{0}*f*_{i}, we have
*V*_{1}=*V*_{0}=−*V*_{2} and with *f*_{H}=*f*_{L}/2, in such a way that the normalizations in all the integrals are the same. Clearly *f*_{H} satisfies the integral equation (3.8) with

It is quite difficult to extract the asymptotic short-distance behaviour, *κ*→0, from (3.8), mainly because at *κ*→0 the ‘Lorentzian’ kernel tends to a delta function and the equation becomes singular. This difficult task has been performed by Hutson [12], who obtained
*κ*→0 are neglected: in the following, we will use *f*∼*g* if *f*=*g*+*o*(1). The estimate (3.10) is valid away from the edge *t*∼1, but this is sufficient for the computation of capacity, because an interval of order *κ* gives negligible contributions, as it is easily shown that the solution is bounded on the whole interval, and in effect is of order 1 at *t*∼1. Integration of (3.10) gives directly the classical Kirchhoff [2] formula for the relative capacity *C*

For two equal conductors, the relative capacity is related to coefficients *C*_{ij} by
*Q* the relative capacity is defined (see [13] §2-prob.1) by the ratio *Q*/*ΔV* , where *ΔV* is the potential difference between the bodies. It is easily seen that for equal conductors this definition reduces to (3.12). Let us note that the divergent behaviour of the solution for *κ*→0 is specifically due to the fact that the kernel *K* becomes a *δ* in this limit: this rules out evidently any finite solution in the limit *κ*→0. Translating (3.8) in matrices language, the solution is
*u* is a vector filled with 1’s. When the operator *K* has an eigenvalue 1 the solution diverges. The completely opposite behaviour of *C* with respect to the sum of coefficients in (2.1a) clearly reflects the different physical realization of the two limits, for *C* the charges are held fixed for ℓ→0, for the sum (2.1a) the potential in fixed, while the charges change as ℓ varies.

For two equal discs of radius *a*, the limit *κ*→0 produces a single disc with the same radius and with a known capacitance *C*_{F}=2*a*/*π*. The arguments introduced in §2 and the expression equation (2.2), fix in this case the asymptotic behaviour of *C*_{11} and *C*_{12}:
*a*and
*b*

To confirm our statement (3.13), we have to solve (3.5) and find the expected result from (3.6). The equations can be decoupled using the change of variables
*a*and
*b*The last equation is the usual Love equation, (3.8), then *f*(*t*)=*f*_{L}(*t*). Equation (3.14a) is similar to (3.8) but with a *crucial* difference in sign: now the operator (1+*K*) is clearly invertible for *κ*→0 and the solution of (3.14a) in this limit, where *K* behaves like an identity operator, is
*κ*→0

From the physical point of view, it is instructive to expose the mechanism of cancellation of divergences in (3.13) by translating in equations the simple considerations of the foregoing section. The physical procedure outlined there is here realized by considering equations (3.1) with *V*_{1}=*V*_{2}=1 and *F*_{i}=*V*_{i}*f*_{i}=*f*_{i}. The two equations are identical and we arrive at the single relation
*κ*→0, i.e. *C*_{11}+*C*_{12}, i.e. as *κ*→0:
*computation* of the capacity of a single disc, independent in principle from the usual one ([13], §4). A similar observation will be discussed for the general case of large distance expansion in the next section.

## 4. Coaxial discs with different radii

Let us consider now the more general case of two coaxial discs with radii *a*,*c* with *β*=*c*/*a*>1. The distance between the discs is ℓ. In this section, we summarize and discuss the results obtained in this case; the formal developments are given in the next two sections.

### (a) Equations and potential for different discs

The first result is that we can deal with this problem using the previous approach, based on the separability of Laplace equation in cylindrical coordinates. One arrives at a system of equations similar to the previous one:
*κ*=ℓ/*a* and the kernel *K* is always given by (3.2). Charges are given by

The exact form of the potential is
*r*^{2}=*x*^{2}+*y*^{2}. A proof of these three statements is given in §5.

### (b) Matrix of *C*_{ij} in the near limit

A second result is that system (4.1) can be solved, in the asymptotic regime *κ*→0, in a region excluding an interval of order *κ* near the boundary *t*=1. This is sufficient to compute the capacitance coefficients extending the classical Kirchhoff formula to this more general case. The result is
*a*and
*b*Here, *f*_{L}(*t*;2*κ*) is the solution of Love’s equation with scale 2*κ*. It is also convenient to put these relations in the form

### (c) Discussion of the asymptotic solutions

Before starting the mathematical analysis of the results it may be useful to have a look at the solutions and perform some checks.

Let us start from large distances, *κ*≫1. As in the case of equal discs, it is quite easy to make an expansion in powers of 1/*κ*: the solutions *F*_{i} are polynomial in *t*, as can be easily seen by an iterative solution of (4.1). This allows us to perform a check of some historical interest: computing the solutions and integrating we obtain the capacitance matrix
*a*
*b*
*c*Inverting the matrix, we can write the potential coefficients *M*_{ij} and we obtain, in usual units:
*a*and
*b*The coefficient *M*_{11},*M*_{22} up to order 1/ℓ^{10} and *M*_{12} up to order 1/ℓ^{7} have been computed by Maxwell in [14] and coincide with expressions (4.9). The first terms in (4.8), up to order 1/ℓ^{4} included, are in agreement with the general result of [15], expressed in terms of the self-capacitance, the polarizability tensor *α*_{ij} and the quadrupole moment *D*_{ij} of the conductors. In the case at hand for a disc of radius *a*, the relevant parameters are [13]

The second region of interest is *κ*→0. In this limit, the two conductors form a disc of radius *c*=*aβ* (the bigger of the two) so we expect from (2.1a)

Let us note that the definition of short distance in the present case, while clear from a mathematical viewpoint, requires some physical specifications. We have *two* possibly small parameters, ℓ, the distance between the discs, and, in some cases, *c*−*a*, or, in scaled units, *κ* and *β*−1. On the mathematical side, the limit is understood in the sense *κ*→0 with *β*>1 fixed. In the following, it will be clear that the limits *β*→1 and *κ*→0 do not commute so special care must be taken when comparing capacitance with approximate numerical results in this regime.

### (d) Physical and numeric analysis of results

Having passed these preliminary checks, let us now sketch the procedure followed in the analysis of the problem. A clue to the solution of problem (4.1) comes from the physical considerations given in §2. We expect that in the *κ*→0 region the complex of force lines is determined mainly by the smaller disc, then let us first consider the limit case *V*_{1}=1, *V*_{2}=0 and *x*–*y* plane and grounded, i.e. *V*_{2}=0, the smaller disc is placed at *z*=*aκ*=ℓ.

Using the image method for the solution, it is clear that in the half-space *z*>0 the potential is identical to the one obtained by two discs, oppositely charged, at distance 2ℓ, the second disc being the image of the first at coordinate *z*=−ℓ. So we are dealing with a problem of two identical discs, and the charge accumulated on disc 1, i.e. *C*_{11}, will be simply given by the usual expression for two discs with a potential difference *ΔV* =2 at distance ℓ=2*κa*, i.e. (see equation (3.11))

(1) The ‘geometrical’ term in the capacitance is dictated by the smaller disc, as expected from elementary considerations on the flux of the electric field.

(2) The logarithmic (and finite) part of edge corrections is

*different*from (3.13), this signals a crossover region in the limit*κ*→0 when*β*→1, i.e. the two limits do not commute.

In this limit, the second disc (the larger one) has clearly the opposite charge of disc 1, i.e. *C*_{12}=−*C*_{11}, a result expected by simple considerations on flux lines, in the limit

The same conclusions can be drawn more formally from equations (4.1), which for the normalized functions read

This simple result is the one on which is built the solution in §6.

As the following formal analysis can be rather tedious maybe it is of some interest for the reader to have a look at the results in a particular case from a graphical point of view. To illustrate the point, we take the case *β*=1.1, which *a priori* could be problematic as *β* is not so large.

In figure 1, the difference between the numerical solution of system (4.1) and the asymptotic prediction (4.4) is plotted. One can appreciate that the agreement is quite good. In figure 2 is reported the same kind of differences on a linear scale for the three coefficients. The non-monotonic approach to the limit for *C*_{12} is characteristic in the case of *β*∼1 and disappears for large *β*. The details of numerical solutions of system (4.1) will be given elsewhere. Let us note that in the range *κ*∼10^{−4}–10^{−5} the absolute value of capacitance coefficients is of order 10^{4}–10^{5}, so the agreement shown in the figure is, on an absolute scale, of the order of one part in 10^{7}–10^{8}. For the analogous case of identical discs at least four order of magnitudes are simply due to the geometrical capacitance, so the agreement would be good but not so impressive. For *different* discs, this agreement is quite strong numerical evidence for the picture sketched in §2: only the smaller disc dictates the divergences.

Finally, in figure 3 we give a plot of equipotential lines and flux lines computed from (4.3).

## 5. The integral equation

Let us consider two coaxial discs of radii *a* and *c*, with *β*=*c*/*a*>1. The first is placed on the plane *z*=0 and centred at the origin of the reference frame. The second disc is in the plane *z*=ℓ. The potentials are, respectively, *V*_{1} and *V*_{2}. The Laplace equation is clearly separable in cylindrical coordinates and it is easily seen that the general solution vanishing at infinity has the form
*r*^{2}=*x*^{2}+*y*^{2} and *J*_{0} is the Bessel function of order 0. In this section, we freely use a certain number of non-trivial integrals of Bessel functions; the reader is referred to [10,11] for further information on this subject.

The first boundary condition is on the potentials
*a*and
*b*The second condition comes from the continuity of electric field in the region outside the conductors. From (5.1) it appears that a possible discontinuity arises in the *z* derivative of *ϕ*. The *z* derivative of *ϕ* is easily computed and imposing to it the continuity at the planes *z*=0 and *z*=ℓ outside the conductors fixes, respectively, the conditions
*a*and
*b*The solution of the Laplace equation with boundary conditions (5.2) and (5.3) fixes the potential *ϕ*. The discontinuity of the subsequent computed electric field on the discs gives the charge density
*a*and
*b*It is convenient to define adimensional variables in the form
*a*and
*b*and
*a*and
*b*while the charges on the conductors, obtained by integrating *σ*_{1},*σ*_{2} in 2*πr* d*r*, are
*g*_{1},*g*_{2}:
*a*and
*b*This is the point where the problem of two discs differs from the case with equal discs. It turns out that the simple change in the range of *t* is sufficient to take care of the difference in the size of discs. Substitution in (5.7) gives rise to expressions
*t* for Δ*E*^{(1)} and Δ*E*^{(2)} extend to 1 and *β*, respectively, the *θ* function assures that boundary conditions (5.7) are satisfied.

Substitution of (5.9) in (5.6) gives the integral equations (4.1).

The proof below parallels step by step the derivation given in [11]. Let us consider (5.6a), the substitution produces
*F*_{1}, of the general form
*x*
*t* gives

Now we show that the charges of the conductors are directly related to the integral of the solution *F*_{1},*F*_{2} of the system of integral equations, as anticipated in equations (4.2) here repeated:
*Q*_{2}, the charge on the larger disc. Substitution of (5.9) in (5.8) gives
*Q*_{1}, the charge on the smaller disc, finding the first relation of equations (5.18).

Finally, let us give the explicit expression for the potential *ϕ*(*r*,*z*) in terms of the functions *F*_{1},*F*_{2}. Using the integral transformation (5.9) and the definitions (5.5), we can write
*y*]≥0, we have

## 6. Capacitance coefficients in the near limit

Using an argument based on the method of image charges, it has been shown in §4 that for *V*_{1}=1, *V*_{2}=0 is expected to be *f*_{1}(*t*)=*f*_{L}(*t*;2*κ*), where *f*_{L} is the solution of the standard Love equation with a doubled scale. In this approximation

In this section, we confirm this result and compute the corrections for finite *β*, which are shown to be finite. We also compute the asymptotic form of the solutions *f*_{1}(*t*) and *f*_{2}(*t*) for different discs. There are at least three ways to obtain the correction to capacitance: (i) compute directly the corrections to the capacitance matrix, (ii) compute perturbatively the corrections to the basic approximation for *f*_{1} given above in system (4.1), and (iii) transform the system by decoupling the variables.

We present here only the third method, the most insightful one, having checked that all three methods give the same result.

### (a) Construction of the solutions for *κ*→0

For simplicity and to refer to some known results, we freely extend the solutions for *t*<0 with *f*(−*t*)=*f*(*t*); this is allowed due the form of the kernel *K*(*t*,*s*;*κ*). We will write in such a case

Let us consider system (4.1) for *V*_{1}=1, *V*_{2}=0, i.e. (4.14). Substituting the second equation in the first, we have
*s* in *κ*. The last term vanishes for

Defining
*δf*_{1}(*t*)

The solution of (6.5) gives the correction to the solution of the disc-plane problem when the plane reduces to a disc larger than the first one. It is interesting to calculate that correction in the limit *κ*→0 to find the correct expression of Kirchhoff’s formula for the case of different discs.

In fact, this equation can be easily solved at the leading order in *κ*. In effect due to integration limits *s*≥*β*>1, the kernels *K* in the last two terms are non-singular in *κ* and the leading term in the solution is obtained replacing *f*_{L} by its leading form
*s*≥*β*>1. The equation for *δf*_{1} is then approximated by
*κ*→0 has the form
*κ*, we have
*s*>1, we can neglect the *κ*^{2} term in the denominator of *K*(*t*,*s*;*κ*) and perform the resulting integral
*x*, we have
*C*_{11}

We note that *κ*, then neglecting the second-last term in (6.5) shows to be consistent. Let us observe that for simply computing *δC*_{11} we could have used

To compute *f*_{2}(*t*), it is simpler to consider separately the two intervals 0<*t*<1 and *β*>*t*>1. In the first region, we can apply the general property [16]
*κ*
*κ*, *t*>1, we have, once again, the depression in the kernel and obtain, using (6.7):
*f*_{2} is given by
*κ* near *t*=1, so the divergence in (6.17) is only due to this approximation. The integration in *t* gives

A similar analysis can be done for the second independent system, with *V*_{1}=0, *V*_{2}=1, i.e. for smaller disc at null potential:
*C*_{21} and *C*_{22} and to verify explicitly that *C*_{21}=*C*_{12}. Substituting the second equation in the first, we have
*κ*, with opposite sign, the other terms are depressed in *κ* as *κ*→0. Defining
*f*_{L}−*δg*_{1} by their leading term in *κ*. Then, we can write the solution at once:
*δf*_{1}, with opposite sign, while the first term is, using the Taylor expansion in *κ* for *G*_{0}:
*t*, we can compute *C*_{21}
*g*_{2}, we can use the same procedure adopted for *f*_{2} and we obtain

### (b) Interchange of the limits *κ*→0 and *β*→1

Finally, let us make some comments on the crossover region, i.e. what happens if we interchange the limits *β*→1 and *κ*→0. A look to equation (6.13) brings the suspicion that for *β*∼1+*αk*, where *α* is a numerical factor, the expected coefficient *C*_{ij}. This expectation can be explicitly realized by a rough approximation to the limit *β*→1. Consider again *C*_{11}. Extracting the real part in (6.7) we have, for *x*>1 and *κ*>0,
*κ*=0 in the denominator, we get approximation (6.7). The argument of the square root in *I*(*x*;*k*) is 2(*x*^{2}−1) for *κ*=0 and 2*κ* for *x*=1, then a rough approximation would be
*κ*,
*κ*→0, we find result (6.13), but if instead we perform the limit *β*→1 we have
*I*(*x*;*κ*), but the reader can verify that adding the

## 7. Conclusion

In this paper, we reduce the computation of electrostatic potential for a couple of different circular coaxial discs to a couple of integral equations, generalizing the previous result of Love [7] for equal discs. The full capacitance matrix is evaluated in the short distance limit, obtaining a generalization of classical Kirchhoff [2] result. We think that this result could also have some practical applications when a precise knowledge of fringe effects is needed.

## Authors' contributions

The authors have contributed equally to this work. All the authors gave final approval for publication.

## Competing interests

We declare we have no competing interests.

## Funding

This research received no specific grant funding.

- Received July 18, 2016.
- Accepted September 15, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.