# Dielectric polarizability of circular cylinder

## Dielectric polarizability of circular cylinder

ARTICLE IN PRESS Journal of Electrostatics 63 (2005) 101–117 www.elsevier.com/locate/elstat Dielectric polarizability of circular cylinder Jukka Ven...

ARTICLE IN PRESS

Journal of Electrostatics 63 (2005) 101–117 www.elsevier.com/locate/elstat

Dielectric polarizability of circular cylinder Jukka Venermo, Ari Sihvola Electromagnetics Laboratory, Helsinki University of Technology, P.O. Box 3000, FIN–02015 HUT, Finland Received 26 March 2004; received in revised form 18 June 2004; accepted 9 September 2004 Available online 12 October 2004

Abstract This paper presents results from a numerical study of the polarizability characteristics of dielectric circular cylinders. Because no closed-form solution exists for the polarizability of cylinder, this quantity has to be calculated numerically. In the present article, this is done by solving the appropriate integral equation for the scalar potential when the cylinder is exposed to a static, uniform electric ﬁeld. In the evaluation, Method of Moments solution is used along with third-order basis functions. The polarizability is calculated as function of the permittivity  and the length-to-diameter ratio l=d of the cylinder, and it has two components, axial and transversal polarizabilities. The polarizability of the cylinder is close to that of a spheroid (ellipsoid of revolution) with the same  and l=d: Since the polarizability of the spheroid is known and easily calculable in closed form, a very effective way of presenting the numerical results for the cylinder is to express the polarizability difference between these two objects as a function of the two parameters. The accuracy of the resulting estimates is better than 1 per cent. r 2004 Elsevier B.V. All rights reserved. Keywords: Cylinder; Ellipsoid; Polarizability; Dipole moment; Depolarization

1. Introduction This article focuses on the electrostatic response of dielectric cylinders with circular cross-section when it is exposed to a uniform ﬁeld. There are few geometrical Corresponding author.

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shapes which allow a closed-form solution for the Laplace equation. Spheres and ellipsoids are such forms. But circular cylinders are not, and still they are basic shapes which are encountered in many canonical problems. The aim of the present paper is to provide accurate estimates for the polarizability components of such cylinders. It is of interest to note that also Platonic polyhedra (cube, tetrahedron, etc.) lack a closed-form solution as far as their dielectric polarizability is concerned. Numerical estimates, some of them quite accurate, have been presented in the recent literature for the polarizabilities of regular polyhedra [1–3]. However, only scattered results for special cases of cylinders have appeared in the literature [4,5]. Here we wish to give a uniﬁed and accurate presentation for the polarizability characteristics of circular cylinders. We solve numerically the electrostatic problem where a homogeneous dielectric cylinder is in a uniform ﬁeld. When a dielectric inclusion is put into a homogeneous electric ﬁeld, it causes a perturbation to the total electric ﬁeld distribution. The perturbation is concentrated in the neighborhood of the inclusion. In connection with this problem in electrostatics, the concept of polarizability of such an inclusion is important. The main component of the ‘‘scattered ﬁeld,’’ in other words the difference between the total ﬁeld when the scatterer is present and the uniform incident ﬁeld, is a dipolar ﬁeld. This electrostatic dipolar ﬁeld (which decays according to the inverse cube of the distance from the scatterer) can be identiﬁed as arising from a point dipole. Subsequently, the polarizability aabs is the ratio between the dipole moment p and the incident ﬁeld Ee : p ¼ aabs Ee :

(1)

A simple example of an isotropic scatterer is a homogeneous sphere. If the sphere has permittivity i ; and it is embedded in an environment with permittivity e ; the polarizability is well known  aabs ¼ 3e V

i  e ; i þ 2e

(2)

where V is the volume of the sphere. In the following, let us use normalized quantities: the permittivity of the inclusion is measured relative to the background (t ¼ i =e ), and the polarizability is normalized with respect to e and the volume of the inclusion a¼

aabs e V

(3)

which gives the (normalized) polarizability for the dielectric sphere a¼3

t1 : tþ2

(4)

For an ellipsoid, polarizability is dependent on the ﬁeld direction. The polarizability component in the direction of the ith axis of the ellipsoid with three

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t1 ; 1 þ N i ðt  1Þ

(5)

where the important parameter in the geometry of an ellipsoid is its depolarization factor N i : If the semi-axes of an ellipsoid in the three orthogonal directions are ax ; ay ; and az ; the depolarization factor N x (the factor in the ax -direction) is [7,8] Z ax ay az 1 ds qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : Nx ¼ (6) 2 0 ðs þ a2x Þ ðs þ a2x Þðs þ a2y Þðs þ a2z Þ For the other depolarization factor N y ðN z Þ; interchange ay and ax (az and ax ) in the above integral. The depolarization factor is a measure how much the internal ﬁeld within the ellipsoid is weakened by the polarization. If it is zero, the internal ﬁeld is the same as the external ﬁeld. The three depolarization factors for any ellipsoid satisfy N x þ N y þ N z ¼ 1:

(7)

A sphere has three equal depolarization factors of 13 because of rotational symmetry. The other two special cases are a disk (depolarization factors 1; 0; 0) and a needle (0; 12; 12). For ellipsoids of revolution, prolate and oblate ellipsoids, various closed-form expressions for integral (6) can be found in [7,8]. Prolate spheroids ðax 4ay ¼ az Þ have   1  e2 1þe  2e (8) Nx ¼ ln 1e 2e3 and N y ¼ N z ¼ 12ð1  N x Þ;

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ where the eccentricity of ellipsoid is e ¼ 1  a2y =a2x : For oblate spheroids ðax ¼ ay 4az Þ; Nz ¼

1 þ e2 ðe  tan1 eÞ; e3

(9)

(10)

N x ¼ N y ¼ 12ð1  N z Þ; (11) pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ where e ¼ a2x =a2z  1: The focus of the present study, circular cylinders, are also objects with uniaxially anisotropic polarizability. The polarizability is a diagonal matrix which can also be written as a dyadic, with axial component and transversal component a ¼ az uz uz þ at I t ;

(12)

where z-axis is taken to be along the cylinder axis. In the following, we present the numerical results for the polarizability components as functions of the permittivity t and the shape of the cylinder (the length-to-diameter ratio).

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2. Numerical evaluation of the polarizability The determination of the polarizability of an inclusion requires solving the electrostatic problem where the inclusion is located in a uniform external ﬁeld Ee : The principle and the numerical method used to calculate the polarizability for arbitrary scatterers is described in . An integral equation for the unknown potential function on the surface of the inclusion f in this electrostatic problem reads as follows :   Z tþ1 t1 1 0 @ fe ðrÞ ¼ fðrÞ þ fðr Þ 0 (13) dS 0 ; r on S: 2 4p S @n jr  r0 j In this equation, S is the surface of the inclusion, fe ¼ E e z is the potential of the incident ﬁeld, and f is the total potential on the surface. Once the potential is known on the surface, the dipole moment p can be calculated from the polarization density P of the inclusion Z Z Z P dV ¼ ði  e Þ Ei dV ¼ ði  e Þ rfðrÞ dV (14) p¼ V

V

V

which can be transformed from the volume integral into a surface integral using Gauss’s identity Z (15) p ¼ ðt  1Þe fðrÞn dS: S

In the above expressions, V is volume of the inclusion, Ei is electric ﬁeld within the inclusion and n is the outward unit normal vector to S. Finally, when the value for the dipole moment is known from (13) and (14), the polarizability a can be calculated from the connection given in Eqs. (1) and (3). In the evaluation of the potential, third-order basis functions are applied. The mesh density is around 1800 elements with reﬁnement of the meshing near corners as illustrated in Fig. 1. The results were calculated using a PC machine equipped with AMD XP 1800þ processor and 512 MB of RAM. Calculation of a single polarizability value took around 10–15 min with mesh consisting of about 1800 elements and 8000 unknowns with third-order basis functions. Generating the mesh took about 60 s.

3. Results 3.1. Cylinder vs spheroid The parameters that we use to characterize the cylinder are the ratio of the axial length to the diameter l=d (corresponding to the axis ratio of an ellipsoid), and the permittivity contrast between the inclusion and the external material t ¼ i =e : When it comes to polarizability, the behavior of a cylinder, as a scatterer, is quite close to an ellipsoid. A long cylinder (l=d ! 1) is like a long ellipsoid (i.e., a needle),

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Fig. 1. Example of the mesh of 2100 triangles covering the circular cylinder.

and thus saturates to the same polarizability value, at least according to our intuition, which is supported also by the numerical results. Also with the complementary case, a plate, the polarizabilities of a cylinder and an ellipsoid seem to saturate to the same value. Between the extremes the polarizabilities differ, but not by very much. Fig. 2 compares the polarizabilities of a cylinder and an ellipsoid having permittivities 5 and 15; as a function of the shape. They obviously are similar to each other which motivates us to present the numerical results for the cylinder by relating them to the corresponding result for the ellipsoid which is known analytically. Instead of trying to ﬁnd some general formula for polarizability of a cylinder (a ¼ aðt; l=dÞ), we estimate the difference (D ¼ Dðl=dÞ) of polarizability between cylinder and ellipsoid with certain t values as a function of l=d: The obvious reason is that the difference D is a much softer function of t: The polarizability can be now calculated using the formula acylinder ¼ aellipsoid þ D;

(16)

where aellipsoid is the polarizability (either az or at ) of an ellipsoid with axis ratio corresponding to the l=d of the cylinder and the same permittivity contrast t: 3.2. Accuracy of the numerical results The calculation time and accuracy in the computational evaluation of the polarizability depend on many factors, such as the number of elements, the order of basis functions, numerical integration, reﬁnement of the mesh, etc. The critical restriction was the number of unknowns due to the ﬁnite amount of computer memory. Especially, when the l=d ratio was high or low, a concentration of elements around the corners was used to give more accurate results. The accuracy of the numerical solver was tested by calculating the polarizability of an ellipsoid having a similar meshing to that of the cylinder. Since we know the

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4 3.5

α

t

cylinder ellipsoid

α / Vεe

3 2.5 2 1.5 1 0.5 −3 10

αz −2

−1

10

10

0

10 l/d

1

10

2

3

10

10

−0.5

α

−1

t

α / Vεe

−1.5

cylinder ellipsoid

−2

−2.5 −3

αz

−3.5 −4 −3 10

−2

10

−1

10

0

10

1

10

2

3

10

10

l/d Fig. 2. Polarizabilities of cylinder and ellipsoid with t values 5 (upper ﬁgure) and

1 5

(lower ﬁgure).

polarizability of an ellipsoid, the error is easy to calculate. Errors in a few cases are presented in Tables 1 and 2, where e ¼ anumerical  aanalytical

(17)

    e : e% ¼  aanalytical 

(18)

and

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Table 1 The error of numerical az value of an ellipsoid Axis ratio 1 50

1

2

10

50*

t¼2 e e%

0.00022 0.0003

0.0002 0.0002

0.00004 0.00004

0.00002 0.00001

0.00004 0.00007

t ¼ 10 e e%

0.00018 0.0008

0.0037 0.001

0.0030 0.0004

0.0007 0.00008

0.00015 0.0002

t ¼ 50 e e%

0.0035 0.0012

0.0083 0.0016

0.03344 0.0014

0.012 0.0003

0.00019 0.0002

1 t ¼ 10 e e%

0.00053 0.0004

0.00025 0.0002

0.00002 0.00002

0.00001 0.00001

0.0037 0.0005

Table 2 The error of numerical at value of an ellipsoid Axis ratio 1

1 2

1 10

1 50*

t¼2 e e%

0.00008 0.0001

0.0001 0.0001

0.00005 0.00006

0.00002 0.00002

t ¼ 10 e e%

0.00064 0.0003

0.0011 0.0004

0.0011 0.0002

0.0028 0.0004

0.000007 0.000004

t ¼ 50 e e%

0.00094 0.0003

0.0019 0.0005

0.0035 0.0003

0.0414 0.0015

0.00002 0.00001

1 t ¼ 10 e e%

0.00034 0.0003

0.00029 0.0002

0.00011 0.0001

0.00007 0.00008

50

0.000003 0.000005

0.00008 0.00005

Note that the error in cases marked with asterisk (*) differs in its behavior from other cases as can be noted in the sign which is opposite to other cases. It seems that there are two kind of errors that may partially cancel each other. When the axis ratio 1 grows very high (over 100) or very small (below 100 ), the numerical result becomes

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inaccurate due to the low quality and the insufﬁcient number of elements in the mesh as the triangles become very elongated. Nevertheless, the tables seem to indicate that the error in the numerically calculated polarizability of the ellipsoid is less than 0.001, and often very much smaller. We therefore believe that the relative error of the numerical values for the polarizability of a cylinder can be estimated to be approximately one part per thousand. For practical use of the numerical results, we ﬁtted an approximation curve  3  2  A ln l=d þ B ln l=d þ C ln l=d þ D D¼ (19) 4  3  2  ln l=d þ E ln l=d þ F ln l=d þ G ln l=d þ H to the computed D points to give useful interpolation formulas. The accuracy of 1 and 50. The estimated relative these formulas is optimum when l=d is between 50 error within this range is around one percent. In the following subsection, we present the parameters of these formulas for various t values and illustrate the result with ﬁgures. In each ﬁgure, we have drawn the actual computed values (marked with asterisks) and the approximation curve for both Dz and Dt : In addition to the polarizability components az ; at we have plotted the trace of the dyadic polarizability trD ¼ Dz þ 2Dt : The trace of the polarizability is a very essential parameter for random mixtures of anisotropic particles, because each of the polarizability components contribute equally to the macroscopic dielectric response. 3.3. Polarizability curves and interpolation formulas In this section, easy-to-use approximation formulas for the cylinder polarizabil1 ities are given for the cases t ¼ 2; 5; 10; 20; 50; 12; 15; and 10 : These formulas are all of form (19), and the coefﬁcients A; B; H for all the cases are collected in Table 3. Graphically these differences Dz and Dt are shown in Figs. 3–10. 3.3.1. Unit cylinder As a special case, the ‘‘unit cylinder’’ is given more attention. This means a cylinder where the length equals the diameter: l=d ¼ 1: For this case, some numerical results have been given earlier in literature [4,5]. We have calculated polarizability of a unit cylinder as a function of t: The polarizability values for az are given by az ðtÞ ¼ a1 ðt  1Þ

t4

t3  0:0519t2 þ 0:9427t  a0 ; þ 3:2226t3  0:0021t2 þ 5:3391t þ a1

(20)

where a1 ¼ 3:8662 and a0 ¼ 1:5840; and for at at ðtÞ ¼ a1 ðt  1Þ

t4

t3 þ 2:0283t2 þ 1:9821t  a0 ; þ 4:4453t3 þ 6:2134t2 þ 6:0500t þ a1

(21)

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Table 3 The coefﬁcients to use the approximation formula (19) for the different cases of the permittivity ratio t A

t

B

C

D

E

F

G

H

2

Dz Dt

0.0067 0.0024

0.0026 0.0414

0.8005 0.5028

0.7013 0.0760

0.2298 2.8392

6.0730 7.9423

7.5300 4.8578

30.4746 37.7240

5

Dz Dt

0.0701 0.0644

0.0996 0.4469

5.5778 5.0532

7.2680 1.3535

3.4771 5.5788

13.3447 20.1085

22.9063 32.8823

39.1290 70.0492

10

Dz Dt

0.1288 0.2079

0.5391 1.9117

10.8352 14.6474

17.6914 6.9272

5.5852 7.3051

19.5114 31.4648

35.9069 68.5857

45.4889 118.208

20

Dz Dt

0.4597 0.4891

2.1890 3.7350

22.1582 29.4807

43.4874 17.4925

8.1471 10.1336

33.6054 49.2250

69.7067 121.966

75.8259 178.452

50

Dz Dt

0.1361 0.9596

6.4710 13.7299

30.7899 68.2364

86.3886 46.7680

9.9039 12.6335

46.1000 74.2474

1 2

Dz Dt

0.0047 0.0024

0.0648 0.0147

0.7881 0.2776

0.0768 0.2043

4.0415 0.8951

11.2520 5.3434

11.3483 3.9461

42.9262 29.6973

1 5

Dz Dt

0.0815 0.0228

1.3418 0.0277

7.9548 1.3255

2.4299 1.2602

6.2414 0.3785

29.1386 8.9586

68.6888 5.0747

138.607 39.5211

1 10

Dz Dt

0.1104 0.0286

1.8058 0.0308

12.5252 1.7652

6.3383 1.8307

9.5249 0.1611

43.3180 8.3199

102.346 5.3355

153.062 36.2308

107.949 226.337

117.425 353.086

Note that for each case two sets of coefﬁcients are needed: one for the axial ﬁeld and the other for the transversal exciting ﬁeld.

where a1 ¼ 3:1691 and a0 ¼ 1:5799: The relative inaccuracy of these formulas is below 1 part per thousand in the worst case, but it is generally better. Fig. 11 illustrates the polarizabilities of the unit cylinder relative to sphere. A closer look at the cylinder polarizabilities is provided in Fig. 12. It is worth noting that the polarizability normalized to sphere can be smaller than 1 (i.e., the induced dipole moment is smaller). But this is for one of the polarizability components. In such a case the other component is larger, and the balance is such that the trace of the polarizability dyadic is larger than that of the sphere, as can be seen in Fig. 12. This is in agreement with well-known observation that sphere is a shape with variational minimum when it comes to polarizability.

3.3.2. The polarizability as a function of l=d and t Finally, we illustrate in Figs. 13–15 the behavior of the difference of polarizability of cylinder as a function of both l=d and t: Again the polarizability can be obtained by using Eq. (16).

ARTICLE IN PRESS J. Venermo, A. Sihvola / Journal of Electrostatics 63 (2005) 101–117

110 0.04

0.03

∆ / Vεe

0.02

0.01

∆z

0

−0.01

∆t

∆ + 2∆ z

t

−0.02

−0.03 −3 10

−2

10

−1

10

0

10

1

10

2

10

3

10

l/d

Fig. 3. The polarizability differences Dz ; Dt and dyadic trace trðDÞ ¼ Dz þ 2Dt between cylinder and ellipsoid, with permittivity contrast t ¼ 2: The asterisks are the actual numerically calculated values and the curves are the interpolated functions from (19). 0.3

0.2

∆ / V εe

0.1

∆z

0

∆t

−0.1

−0.2

−0.3

−0.4 −3 10

∆z + 2∆t −2

10

−1

10

0

10

1

10

2

10

3

10

l/d

Fig. 4. The polarizability differences Dz ; Dt and dyadic trace trðDÞ ¼ Dz þ 2Dt between cylinder and ellipsoid, with permittivity contrast t ¼ 5:

4. Conclusions and analysis This paper has presented results of a thorough numerical study of the polarizability characteristics of dielectric circular cylinders. The polarizability dyadic, in other words the axial and transversal components of the dipole moment

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111

0.6 0.4 0.2

∆z

∆ / Vεe

0 −0.2 −0.4

∆t

−0.6 −0.8 −1 −1.2 −3 10

∆z + 2∆t −2

10

−1

10

0

10

1

10

2

10

3

10

l/d

Fig. 5. The polarizability differences Dz ; Dt and dyadic trace trðDÞ ¼ Dz þ 2Dt between cylinder and ellipsoid, with permittivity contrast t ¼ 10:

1 0.5

∆z

0

∆ / Vεe

−0.5 −1

∆t

−1.5 −2 −2.5 −3 −3.5 −3 10

∆z + 2∆t −2

10

−1

10

0

10

1

10

2

10

3

10

l/d

Fig. 6. The polarizability differences Dz ; Dt and dyadic trace trðDÞ ¼ Dz þ 2Dt between cylinder and ellipsoid, with permittivity contrast t ¼ 20:

induced in the cylinder in an electric ﬁeld, are given as functions of the permittivity and the shape of the cylinder. The closest shape for which a closed-form solution for the polarizability is found is a spheroid (an ellipsoid of revolution), and therefore it is natural that the calculated results of the present study are related to the polarizability components of the ellipsoid with the same axis ratio.

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112 2

∆z 0

∆ / Vεe

−2

∆t

−4

−6

−8

−10 −3 10

∆z + 2∆t −2

−1

10

10

0

10

1

10

2

10

3

10

l/d

Fig. 7. The polarizability differences Dz ; Dt and dyadic trace trðDÞ ¼ Dz þ 2Dt between cylinder and ellipsoid, with permittivity contrast t ¼ 50:

0.04

∆z

0.03

∆ / Vε

e

0.02

∆ + 2∆

0.01

z

t

0

t

−0.01

−0.02 −3 10

−2

−1

10

10

0

10

1

10

2

10

3

10

l/d

Fig. 8. The polarizability differences Dz ; Dt and dyadic trace trðDÞ ¼ Dz þ 2Dt between cylinder and ellipsoid, with permittivity contrast t ¼ 12 :

The limiting cases when the length-to-diameter ratio l=d goes to inﬁnity (a needle) or zero (a disk) are given by az ¼ t  1;

at ¼ 2

t1 ; tþ1

needle

(22)

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113

0.3

∆z

0.25

0.2

∆z + 2∆t

∆ / Vεe

0.15

0.1

0.05

0

t

−0.05 −0.1 −3 10

−2

10

−1

10

0

10

1

10

2

10

3

10

l/d

Fig. 9. The polarizability differences Dz ; Dt and dyadic trace trðDÞ ¼ Dz þ 2Dt between cylinder and ellipsoid, with permittivity contrast t ¼ 15 :

0.9

z

0.8 0.7 0.6

∆ / Vε

e

0.5 0.4 0.3 0.2

∆z + 2∆t

0.1 0

∆t

−0.1 −3

10

−2

10

−1

10

0

10

1

10

2

10

3

10

l/d

Fig. 10. The polarizability differences Dz ; Dt and dyadic trace trðDÞ ¼ Dz þ 2Dt between cylinder and 1 ellipsoid, with permittivity contrast t ¼ 10 :

and az ¼

t1 ; t

at ¼ t  1;

disk

(23)

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114 1.3

αz 1.25

α

1.15

ave

1.1

α

cylinder

sphere

1.2

1.05

αt

1 0.95 −3 10

−2

10

−1

10

0

10

τ

1

2

10

3

10

10

Fig. 11. Normalized polarizabilities of a unit cylinder (relative to sphere) as function of the relative permittivity. aave is one-third of the trace of the polarizability dyadic aave ¼ 13 ðaz þ 2at Þ:

1.07

αz

1.06

αcylinder /αsphere

1.05 1.04

α

ave

1.03 1.02

α

t

1.01 1 0.99 0.1

1

τ

10

Fig. 12. A closer look of normalized polarizabilities of the unit cylinder. aave is one-third of the trace of the polarizability dyadic aave ¼ 13 ðaz þ 2at Þ:

as functions of the relative permittivity t: These values apply to the limiting cases of both spheroids and cylinders. Therefore, the difference D ¼ acylinder  aellipsoid vanishes at both ends of the l=d axis. The behavior of the difference D has been studied in Section 3.3 which show the numerical results. It is of interest to note that the axial and transversal polarizabilities vary differently as the cylinder is compared to the spheroid. When the axial polarizability is greater for the cylinder ðDz 40), the transversal is smaller, and vice versa. It also seems that Dz o0 when the cylinder is elongated, i.e., a

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Fig. 13. Polarizability difference Dz between cylinder and ellipsoid.

Fig. 14. Transversal polarizability difference Dt between cylinder and ellipsoid.

‘‘prolate-type’’ cylinder. In other words the axial polarizability of such a shape is smaller than that of the spheroid having the same volume and axis ratio. Likewise, for ‘‘oblate-type’’, ﬂattened cylinders the transversal polarizability is smaller than

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1

∆ / Vε

e

0 −1 −2 −1

10

0

10

−1

10

τ

0

10

1

10

1

10

l/d

Fig. 15. Trace of the polarizability difference dyadic trðDÞ ¼ Dz þ 2Dt :

that of the corresponding spheroid ðDt o0Þ: Especially, interesting is the fact that this behavior is observed independently whether t41 or to1: In other words, it does not matter if the inclusion is dielectrically ‘‘heavier’’ than the background or ‘‘lighter’’. Of course the maximum of the peaks in the Dz and Dt curves appears at different places as t varies. In materials science and modelling of random media the polarizabilities are essential parameters with which one can make a model to estimate effective, macroscopic permittivity of the matter. If the inclusions are randomly oriented and distributed, the important parameter affecting the macroscopic response is the trace (the trace of a matrix or a dyadic is the sum of its diagonal components) of the polarizability dyadic, in other words the sum az þ 2at : In previous ﬁgures, the trace has also been calculated (relative to the trace of the spheroid). From the trace ﬁgures one can make an interesting observation: The trace of the cylinder is not always greater than the trace of the spheroid. It is known that a sphere is a variational minimum in terms of its polarizability properties, meaning that any deviation of the spherical shape will increase in average its polarizability [10,11]. Evidently, our results seem to conﬁrm this, as for all cases shown in Section 3.3 the quantity Dz þ 2Dt is positive for the axis ratio l=d ¼ 1 (this refers to a comparison of a circular cylinder of unit ratio of diameter to length with a sphere). But the results show that such a ‘‘minimum-character’’ is not present for spheroids. Deviation from this behavior is especially visible for l=d values around 101 102 :

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Acknowledgements The authors would like to express their thanks to Dr. P. Yla¨-Oijala for his valuable help with the numerical software. References  A. Sihvola, P. Yla¨-Oijala, S. Ja¨rvenpa¨a¨, J. Avelin, Polarizabilities of platonic solids, IEEE Trans. Antenn. Propag. 52 (9) (2004) 2226–2233.  M.L. Mansﬁeld, J.F. Douglas, E.J. Garboczi, Intrinsic viscosity and the electrical polarizability of arbitrarily shaped objects, Phys. Rev. E 64 (6) (2001) 61401–61416.  J. Avelin, A. Sihvola, Polarizability of polyhedral dielectric scatterers, Microw. Opt. Technol. Lett. 32 (1) (2002) 60–64.  A. Sihvola, J. Venermo, P. Yla¨-Oijala, Dielectric response of matter with cubic, circular, and spherical microstructure, Microw. Opt. Technol. Lett. 41 (4) (2004) 245–248.  M. Fixman, Variational method for classical polarizabilities, J. Chem. Phys. 5 (8) (1981) 4040–4047.  J.D. Jackson, Classical Electrodynamics, third ed., Wiley, New York, 1999.  L.D. Landau, E.M. Lifshitz, Electrodynamics of Continuous Media, second ed., Pergamon Press, Oxford, 1984, Section 4.  O.D. Kellogg, Foundations of Potential Theory, Dover Publications, New York, 1953, Chapter VII.  J. Van Bladel, Electromagnetic Fields, revised ed., Hemisphere Publishing Corporation, New York, 1985.  D.S. Jones, Low-frequency electromagnetic radiation, J. Inst. Math. Appl. 23 (1979) 421–447.  M. Schiffer, G. Szego¨, Virtual mass and polarization, Trans. Am. Math. Soc. 67 (1949) 130–205.