Bounding Formulae for the Capacitance of a Cylindrical Two-dimensional

Capacitor with Cartesian Orthotropic Dielectric Material

ATTILA BAKSA,ISTVÁN ECSEDI

Institute of Applied Mechanics,

University of Miskolc,

3515 Miskolc-Egyetemváros,

HUNGARY

Abstract: This paper addresses the evaluation of a two-dimensional cylindrical capacitor featuring homogeneous

Cartesian anisotropic dielectric material. The development of a bounding formula forms the crux of the inves-

tigation and is grounded in the principles of the Cauchy-Schwarz inequality, a mathematical concept widely ac-

knowledged for establishing relationships between different mathematical entities. In the course of this study,

a dual-sided bound is systematically derived for the circular cylindrical two-dimensional capacitor through the

application of well-established inequality relations. These bounds play a pivotal role in setting limits on the ca-

pacitance of the system, providing valuable insights into its electrical behavior.

Key-Words: capacitance,orthotropic,hollow circular domain,upper and lower bounds,analytical solution

Received: January 19, 2023. Revised: Octobert 15, 2023. Accepted: November 16, 2023. Published: December 21, 2023.

1 Introduction

This paper delves into the intricacies of capacitance

analysis, specifically focusing on an infinitely long

cylindrical capacitor. The capacitor under consid-

eration is composed of homogeneous Cartesian or-

thotropic dielectric material. The inequalities pre-

sented in this study play a pivotal role in establishing

both upper and lower bounds for the capacitance of

the unit length of the capacitor.

Capacitance, defined as the ability of a capacitor to

store electric charge per unit voltage across its inner

and outer surfaces, is a critical parameter influenced

by the capacitor’s geometry and the permittivity of

the dielectric material between its conductor surfaces.

Exact capacitance values are known only for capac-

itors with simplistic shapes. Therefore, the princi-

ples and methodologies employed to create upper and

lower bounds for the numerical value of capacitance

become crucial, as highlighted in references>1], [2],

[3], [4], and [5].

In prior work[6] an analytical solution is provided

for a spherical capacitor, comparing it with a numer-

ically determined solution achieved through the di-

vision of the spherical surface into numerous subre-

gions. Other contributions’ calculation of potential

distribution for a parallel plate capacitor>7], or an

exploration of capacitance determination for regular

solids[8], and a provision of error bounds for the ca-

pacitance matrix elements in a system of conductions

[9], showcase the diverse approaches in the field.

The paper[10] stands out for calculating improved

upper and lower bounds for the capacitance of a cube,

employing a combination of the Kelvin inversion and

random walk method. A recent work>11] contributes

bounding formulae for the capacitance of a cylindrical

two-dimensional capacitor with a nonhomogeneous

and isotropic dielectric material.

In essence, this paper adds to the body of knowl-

edge by specifically addressing the capacitance of an

infinitely long cylindrical capacitor, providing valu-

able insights into the realm of electrical characteris-

tics influenced by geometric considerations and di-

electric material properties. The establishment of

bounding formulae contributes to the broader under-

standing of capacitance estimation, offering a foun-

dation for practical applications and validation of nu-

merical simulations.

In this paper, lower and upper bounds will be de-

rived for the capacitance of a two-dimensional capac-

itor with homogeneous but Cartesian orthotropic di-

electric.

2 Governing(quations

Figure 1 visually represents a two-dimensional hol-

low plane domain denoted as A. This domain is char-

acterized by an inner boundary curve, ∂A1, and an

outer boundary curve, ∂A2. The Cartesian coordinate

system, labeled as Oxyz, is employed to define the

spatial coordinates within this domain. Notably, the

origin of the coordinate system is situated as an inner

point of the closed curve ∂A1.

The unit vectors of the Oxyz coordinate system

are denoted as ex,ey, and the position vector of any

arbitrary point Pwithin the domain Ais represented

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∂A2

∂A1

P(x,y)

Fig1: Two-dimensional Cartesian orthotropic ca-

pacitor.

by r=xex+yey. Here, xand yare the coordinates

of point Pin the Oxy plane.

It’s essential to note that the domain Ais composed

of the union of the interior region Aand the boundary

∂A, where ∂A =∂A1∪∂A2. This comprehensive

description sets the stage for a detailed understanding

of the geometric configuration and coordinate system

utilized in the subsequent analysis or discussion re-

lated to the depicted hollow plane domain.

To give the concept of a capacitor for a two-

dimensional hollow domain shown in Figure 1 the fol-

lowing boundary value problem is defined

∇ · (ε· ∇U) = 0r∈A, (1)

U(r) = U1r∈∂A1(2)

U(r) = U2r∈∂A2U1=U2.(3)

Equations (1) and (2) introduce the electric poten-

tial, denoted as U=U(r), where εrepresents a two-

dimensional positive definite tensor known as the per-

mittivity tensor for the Cartesian anisotropic dielec-

tric material. The operator ∇corresponds to the two-

dimensional del operator. These expressions are pre-

sented within the framework of a Cartesian coordinate

system

∇=∂

∂x ex+∂

∂y ey.(4)

Equation (1) employs the dot notation to denote

the scalar product. The permittivity tensor εfinds its

matrix representation in equation (5), expressed as

ε=ε10

0ε2(5)

where ε1>0and ε2>0. Let Crepresent the capac-

itance of the two-dimensional capacitor, measured in

units of [F/m]. The electric energy of the capacitor,

[1], [2], [3] and [4], denoted as W, is given by the

expression in equation (6)

W=1

2C(U1−U2)2.(6)

To reformulate formula (6), a new function u=

u(r)is introduced. The relationship between U=

U(r)and u=u(r)is established through equation

(7)

U(r) = (U1−U2)u(r) + U2.(7)

Here, u=u(r)satisfies the Dirichlet’s type

boundary-value problem defined by equations (8) and

(9).

∇ · (ε(r)· ∇u) = 0 r∈A, (8)

u(r) = 1 r∈∂A1, u(r) = 0 r∈∂A2.

(9)

The specific electric energy, [1], [2], [3] and [4],

within the dielectric material is computed using equa-

tion (10)

w=1

2D·E=1

2E·ε·E=1

2∇U·ε· ∇U=

2(U1−U2)2∇u·ε· ∇u. (10)

The overall electric energy of the two-dimensional

capacitor denoted as W, is given by equation (11)

W=

wdA=1

2(U1−U2)2

∇u·ε· ∇udA.

(11)

Comparing equation (6) with equation (11) results

in an explicit formula for capacitance, as given by

equation (12)

C=

∇u·ε· ∇udA. (12)

An alternative expression for capacitance Cis de-

rived, commencing with equation (13)

0 = u∇·(ε·∇u) = ∇·(uε·∇u)−∇u·ε·∇u. (13)

This leads to the derivation of an alternative ex-

pression for capacitance, expressed in equation (14)

C=

∇u·ε· ∇udA=

∂A1

n·ε· ∇uds. (14)

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3 Upper%ound for C

Theorem 1. Suppose the function F=F(r)is con-

tinuously differentiable throughout A∪∂A and ad-

heres to the boundary conditions outlined in equation

(15)

F(r) = 1 r∈∂A1, F (r) = 0 r∈∂A2,

(15)

In such a scenario, the inequality relation expressed

in equation (16) holds true

C≤CU=

∇F·ε· ∇FdA. (16)

This theorem establishes a connection between the

continuously differentiable function Fand the capac-

itance C, affirming the validity of the inequality rela-

tion in terms of the gradient of Fand the permittivity

tensor ε.

Proof. The foundation for the inequality relation can

be traced back to the Cauchy-Schwarz inequality, en-

capsulated in equation (17)

(

∇F·ε· ∇udA)2≤



∇F·ε· ∇FdA

∇u·ε· ∇udA. (17)

A straightforward computation yields



∇F·ε· ∇udA=

∂A

Fn·ε· ∇uds−



F∇ · (ε· ∇u)dA=

∂A1

n·ε· ∇uds. (18)

The combination of inequality (17) with equation

(18), along with the utilization of the formula (14),

results in the establishment of the upper bound ex-

pressed in (16). A concise examination reveals that

equality in (16) is only valid if F(r) = u(r). This

crucial insight enhances the significance of the de-

rived inequality relation.

4 Lower%ound for C

Theorem 2. Consider a vector field q=q(r)defined

in the hollow two-dimensional domain A=A∪∂A.

Suppose qsatisfies the equation

∇ · (ε·q)r∈A. (19)

In this scenario, the following lower-bound formula

holds

C≥CL=

(

∂A1

n·ε·qds)2



q·ε·qdA,q2dV= 0.

(20)

Equality in the lower bound formula (20) is achieved

only if

q=λ∇u, (21)

where λis an arbitrary constant that is different from

zero. This result establishes a connection between

the vector field qand the lower bound capacitance

CL, providing insight into the conditions under which

equality is attained.

Proof. The proof of Theorem 2 is based on the fol-

lowing Cauchy-Schwarz inequality relation

(

p·ε·qdA)2≤

p·ε·pdA

q·ε·qdA. (22)

Let

p=∇u(23)

be in (22). A simple calculation yields the result



∇u·ε·qdA=

∂A

un·ε·qds−



u∇ · (ε·q)dA=

∂A1

n·ε·qds. (24)

Substitution equation (24) into inequality (22)

gives

(

∂A1

n·ε·qds)2≤C

q·ε·qdA, (25)

which shows the validity of Theorem 2.

Theorem 3. Consider a non-identically constant func-

tion f=f(r)within the hollow two-dimensional

domain A=A∪∂A. Assume that fsatisfies the

Laplace equation

∇ · ∇f=△f= 0 r∈A∪∂A. (26)

In this context, the following lower bound formula

holds for the capacitance C

C≥CL=

(

∂A1

∂f

∂n ds)2



∇f·ε−1· ∇fdA.(27)

This lower bound formula establishes a connection

between the non-identically constant function fand

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the capacitance C, shedding light on the lower limits

of capacitance under the given conditions.

Proof. The verification of the lower bound (27) can

be derived from Theorem 2 by considering the vector

q=q(r)defined as

q(r) = ε−1· ∇f. (28)

In equations (27) and (28), ε−1represents the inverse

of ε, given by

ε−1=1

εx0

εy.(29)

This choice of qsatisfies the conditions of Theorem

2, and by applying it to the theorem’s lower bound

formula, we establish the validity of the equation (27).

5 Geometric&apacitance

The geometric capacitance C0, characterizing the

two-dimensional hollow domain depicted in Figure 1,

is defined as follows

C0=

∥∇u0∥2dA, (30)

where u0satisfies the Laplace equation

△u0= 0 r∈A, (31)

with boundary conditions

u0(r) = 1 r∈∂A1(32)

u0(r) = 0 r∈∂A2.(33)

It is worth noting the validity of the following for-

mula, expressing C0in terms of the boundary

C0=

∂A1

n· ∇u0ds=

∂A1

∂u0

∂n ds. (34)

By invoking Theorem 1 and Theorem 3, along with

equations (30) and (31), it can be inferred that



∇u0·ε· ∇u0dA

≥C

≥C0



∇u0·ε−1· ∇u0dA.

(35)

These inequalities establish a relationship between

the capacitance Cand the geometric capacitance C0,

offering insights into the interplay between the elec-

trical properties of the domain and its geometric char-

acteristics.

6 Example. The capacitance of

HollowCircular Domain

The hollow circular domain, depicted in Figure 2, fea-

tures a cross-section bounded by two concentric cir-

cles with inner and outer radii denoted as R1and R2,

respectively. The equations of the boundary circles

are given by

x2+y2−R2

1= 0 x2+y2−R2

2= 0.(36)

P(r, ϕ)

r=OP

∂A2

∂A1

Fig.2: Two-dimensional hollow circular domain.

Let the function F(x, y)be defined as

F(x, y) = ln x2+y2

ln R2

(x, y)∈A∪∂A (37)

Applying the bounding formula (16) for F=F(x, y)

yields

C≤CU=εx+εy

2π

ln R2

.(38)

To establish a lower bound for the capacitance using

inequality relation (27), the harmonic function (see

Figure 2)

f(r) = ln r2=ln(x2+y2)(39)

will be employed. It can be verified that the following

equations hold:

∂f

∂x =2x

x2+y2,∂f

∂y =2y

x2+y2,(40)





∂A1

∂f

∂n ds



= 16π2,(41)

∇f·ε−1· ∇f=1

εx∂f

∂x 2

εy∂f

∂y 2

r2cos2φ

εx

+sin2φ

εy.(42)

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Substituting equations (41) and (42) into the lower

bound formula (27) leads to the results

C > CL=2π

21

εx+1

εyln R2

.(43)

These findings establish a relationship between the

capacitance Cand the geometric properties of the cir-

cular domain, offering valuable insights into its elec-

trical behavior.

Let

εx= 8 ×10−12 F

m, εy= 9 ×10−12 F

m,(44)

R1= 0.015 m, R2= 0.02 m (45)

be in the upper and lower bound formulae (38) and

(43). In this case, we have

1.850 037 965 ×10−10 F

m≤C

C≤1.907 395 540 ×10−10 F

m.(46)

We introduce parameter λaccording to the follow-

ing definition

λ=R2

>1.(47)

The expressions of CUand CLas function of λcan

be represented as

CU(λ) = λ2+ 1

λ2−1(εx+εy)π, (48)

CL(λ) = 4πεxεy

(εx+εy)ln λ.(49)

The plots of CU(λ)and CL(λ)as a function of λ

are shown in Figure 3 for 1.1< λ ≤6.

1e-10

2e-10

3e-10

4e-10

5e-10

6e-10

1 2 3 4 5 6

λ[−]

CUCL[F

Fig.3: The plots of CU(λ)and CL(λ)as a function

of λfor 1.1≤λ≤6.

We note that, from the upper bound formula (16)

by the application of following function

F(x, y) =

ln r

R22

ln R1

R22=ln x2+y2

ln R1

R22(50)

we obtain

C≤CU=2π

ln R2

εx+εy

2.(51)

In the present numerical example the use of formula

(51) gives

C≤1.856 461 708 ×10−10 F

m.(52)

In the case of isotropic dielectric material, the

proven upper and lower bounds give the same result.

7 Conclusions

This study introduces upper and lower bounds for the

capacitance of a two-dimensional capacitor, showcas-

ing a rigorous analysis within the context of homoge-

neous Cartesian orthotropic dielectric materials. The

formulation of these bounds relies on the application

of the Cauchy-Schwarz inequality relation, offering a

robust foundation for proving the bounding formulae.

By presenting these bounding formulae, the re-

search not only contributes to the theoretical under-

standing of capacitance in orthotropic dielectric mate-

rials but also establishes practical tools for validation.

An illustrative example is provided to demonstrate the

practical application of the derived upper and lower

bounds, offering insights into the real-world implica-

tions of the formulated formulas.

These proven bounding formulae hold signifi-

cance beyond theoretical considerations, providing

a valuable means for validating numerical computa-

tions. Specifically, these bounds can be utilized to

assess and verify results obtained through numerical

methods such as the finite element method and other

computational solutions. This not only enhances the

reliability of numerical simulations but also fosters

a deeper understanding of the intricate interplay be-

tween material properties and the electrical character-

istics of two-dimensional capacitors.

In essence, this research bridges theoretical in-

sights with practical applications, offering a com-

prehensive exploration of capacitance bounds in the

realm of homogeneous Cartesian orthotropic dielec-

tric materials. The developed formulae not only ex-

pand our theoretical understanding but also serve as

valuable tools for quality assurance in numerical sim-

ulations, reinforcing the robustness of computational

results in the study of two-dimensional capacitors.

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Contribution of Individual Authors to the

Creation of a Scientific Article

(Ghostwriting Policy)

István Ecsedi and Attila Baksa carried out the inves-

tigation and the formal analysis. István Ecsedi has

implemented the algorithm for the example. Attila

Baksa was responsible for the validation and for the

visualization of the results. All authors have been

writing the paper with original draft, review, and edit-

ing.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

The author(s) received no financial support for the re-

search, authorship, and/or publication of this article.

Conflicts of Interest

Furthermore, on behalf of all authors, the correspond-

ing author states that there is no conflict of interest.

Creative Commons Attribution License 4.0

(Attribution 4.0 International , CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en_US

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