Theory of Perturbation of Electrostatic Field by an Anisotropic Dielectric Sphere Open Access

Summary

The boundary-value problem for the perturbation of an electric potential by a homogeneous anisotropic dielectric sphere in vacuum was formulated. The total potential in the exterior region was expanded in series of radial polynomials and tesseral harmonics, as is standard for the Laplace equation. A bijective transformation of space was carried out to formulate a series representation of the potential in the interior region. Boundary conditions on the spherical surface were enforced to derive a transition matrix that relates the expansion coefficients of the perturbation potential in the exterior region to those of the source potential. Far from the sphere, the perturbation potential decays as the inverse of the distance squared from the center of the sphere, as confirmed numerically when the source potential is due to either a point charge or a point dipole.

1. Introduction

When an object made of a certain linear homogeneous dielectric material is exposed to a time-invariant electric field, the atoms constituting the object interact with that electric field until an electrostatic steady state is reached a short time later (1, 2). This interaction is described in terms of the polarization $\mathbf{P}$ which is the volumetric density of electric dipoles induced inside the object. The polarization $\mathbf{P}$ is linearly proportional to the electric field $\mathbf{E}$⁠, the proportionality constant being the dielectric susceptibility of the material multiplied by the free-space permittivity. Coulomb’s law then leads to the definition of the electric displacement field $\mathbf{D}$ which is linearly related to $\mathbf{E}$ by the permittivity of the material. The permittivity is scalar for isotropic materials but dyadic for anisotropic materials (3, 4).

Perturbation of an electrostatic field by a linear homogeneous dielectric object in free space has been studied for a long time (5–10), presently with biological (11), biomedical (12), electrochemical (13) and manufacturing (14) applications. Let us refer to the electric field present in the absence of the object as the source field, the difference between the electric field present at any location outside the object and the source electric field as the perturbation field, and the electric field present at any location inside the object as the internal field. The boundary-value problem for the electrostatic steady state is solved analytically by (i) expanding the source, perturbation and internal fields in terms of suitable basis functions and (ii) imposing appropriate boundary conditions at the surface of the perturbing object. The basis functions suitable for representing the source and the perturbation fields emerge from the eigenfunctions of the Laplace equation (5, 15, 16). When the object is made of an isotropic material, the basis functions suitable for representing the internal field also emerge from the eigenfunctions of the Laplace equation.

The positive definiteness (21) of $\underset{―}{\underset{―}{A}}$ allows for an affine transformation of space in which the governing equation from which the eigenfunctions are obtained is transformed into the Laplace equation. After solving the Laplace equation and obtaining the eigenfunctions in the transformed space, an inverse transformation of space is effected to obtain eigenfunctions in the original space for the material described by (1). We apply the procedure here to analytically investigate the perturbation of an electrostatic field by the anisotropic dielectric sphere.

The plan of this article is as follows. Section 2 contains the formulation of the boundary-value problem in terms of the electric potential. The expansion of the potential in the exterior region, which is vacuous, is presented in section 2.1; two illustrative examples of the source potential are provided in section 2.2; the expansion of the potential inside the anisotropic dielectric sphere is derived in section 2.3; boundary conditions are enforced in sections 2.4–2.6 to derive a transition matrix that relates the expansion coefficients of the perturbation potential to those of the source potential; the symmetries of the transition matrix are presented in section 2.7; and an asymptotic expression for the perturbation potential is derived in section 2.8. Section 3 presents illustrative numerical results. Conclusions are summarized in section 4.

2. Boundary-value problem

2.1 Potential in the region $r>a$

2.2 Source potential

We proceed with the assumption that the coefficients ${\mathcal{A}}_{\mathrm{s}\mathrm{m}\mathrm{n}}$ are known but the coefficients ${\mathcal{B}}_{\mathrm{s}\mathrm{m}\mathrm{n}}$ are not. Furthermore, (8) is required to hold in some sufficiently large open region that contains the spherical region $r\le a$ but not the region containing the source of ${\mathrm{\Phi }}_{\mathrm{s}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{c}\mathrm{e}}(\mathbf{r})$⁠.

2.3 Potential in the region $r<a$

The bijective transformation (16) maps a point $\mathbf{r}$ on a spherical surface to a point ${\mathbf{r}}_q$ on an ellipsoidal surface, since ${\alpha }_{\mathrm{x}}>0$ and ${\alpha }_{\mathrm{y}}>0$⁠, with ${\theta }_q$ lying in the same quadrant as $\theta$ and ${\varphi }_q$ in the same quadrant as $\varphi$⁠; see Fig. 1.

2.4 Boundary conditions

2.5 Transition matrix

The transition matrix is a diagonal matrix when the sphere is made of an isotropic dielectric material (that is, $\underset{―}{\underset{―}{A}}=\underset{―}{\underset{―}{I}}$⁠, where $\underset{―}{\underset{―}{I}}=\hat{\mathbf{x}}\hat{\mathbf{x}}+\hat{\mathbf{y}}\hat{\mathbf{y}}+\hat{\mathbf{z}}\hat{\mathbf{z}}$ is the identity dyadic) because $\left[\mathcal{N}\right]{\left[\mathcal{K}\right]}^{-1}$ is then a diagonal matrix. In general, $\left[\mathcal{N}\right]{\left[\mathcal{K}\right]}^{-1}$ is not a diagonal matrix when the sphere is made of an anisotropic dielectric material, so that $\left[\mathcal{T}\right]$ is not a diagonal matrix either.

2.6 Reduction of computational effort

2.7 Symmetries of the transition matrix

By virtue of its definition through (30), the transition matrix $\left[\mathcal{T}\right]$ does not depend on the source potential. This matrix depends only on the radius $a$ and the constitutive parameters ${\alpha }_{\mathrm{x}}$⁠, ${\alpha }_{\mathrm{y}}$ and ${\epsilon }_{\mathrm{r}}$ of the perturbing sphere.

Let us denote each element of the transition matrix defined in (31) by ${\mathcal{T}}_{\mathrm{s}{\mathrm{s}}^{\prime }\mathrm{m}{\mathrm{m}}^{\prime }\mathrm{n}{\mathrm{n}}^{\prime }}$⁠. It was verified numerically that ${\mathcal{T}}_{\mathrm{s}{\mathrm{s}}^{\prime }\mathrm{m}{\mathrm{m}}^{\prime }\mathrm{n}{\mathrm{n}}^{\prime }}\ne 0$ if the following three conditions are satisfied:

• (i)

  $s=s^{\prime }$⁠,

• (ii)

  $m$ and $m^{\prime }$ have the same parity (that is, even or odd), and

• (iii)

  $n$ and $n^{\prime }$ have the same parity.

Finally, let the transition matrix elements be denoted as ${\mathcal{T}}_{\mathrm{s}{\mathrm{s}}^{\prime }\mathrm{m}{\mathrm{m}}^{\prime }\mathrm{n}{\mathrm{n}}^{\prime }}^{(a)}$ for a specific choice $\{{\alpha }_{\mathrm{x}},{\alpha }_{\mathrm{y}}\}$ of the anisotropy parameters, but as ${\mathcal{T}}_{\mathrm{s}{\mathrm{s}}^{\prime }\mathrm{m}{\mathrm{m}}^{\prime }\mathrm{n}{\mathrm{n}}^{\prime }}^{(b)}$ after ${\alpha }_{\mathrm{x}}$ and ${\alpha }_{\mathrm{y}}$ have been interchanged without changing ${\epsilon }_{\mathrm{r}}$⁠. In other words, ${\mathcal{T}}_{\mathrm{s}{\mathrm{s}}^{\prime }\mathrm{m}{\mathrm{m}}^{\prime }\mathrm{n}{\mathrm{n}}^{\prime }}^{(a)}$ changes to ${\mathcal{T}}_{\mathrm{s}{\mathrm{s}}^{\prime }\mathrm{m}{\mathrm{m}}^{\prime }\mathrm{n}{\mathrm{n}}^{\prime }}^{(b)}$ when the sphere is rotated about the $z$ axis by $\pi /2$⁠. Then, the following relationships exist between the pre- and post-rotation transition matrixes:

• ${\mathcal{T}}_{\mathrm{s}\mathrm{s}\mathrm{m}\mathrm{m}\mathrm{n}{\mathrm{n}}^{\prime }}^{(a)}={\mathcal{T}}_{\mathrm{s}\mathrm{s}\mathrm{m}\mathrm{m}\mathrm{n}{\mathrm{n}}^{\prime }}^{(b)}$ when $m$ is even,

• ${\mathcal{T}}_{\mathrm{e}\mathrm{e}\mathrm{m}\mathrm{m}\mathrm{n}{\mathrm{n}}^{\prime }}^{(a)}={\mathcal{T}}_{\mathrm{o}\mathrm{o}\mathrm{m}\mathrm{m}\mathrm{n}{\mathrm{n}}^{\prime }}^{(b)}$ and ${\mathcal{T}}_{\mathrm{o}\mathrm{o}\mathrm{m}\mathrm{m}\mathrm{n}{\mathrm{n}}^{\prime }}^{(a)}={\mathcal{T}}_{\mathrm{e}\mathrm{e}\mathrm{m}\mathrm{m}\mathrm{n}{\mathrm{n}}^{\prime }}^{(b)}$ when $m$ is odd,

• ${\mathcal{T}}_{\mathrm{s}\mathrm{s}\mathrm{m}{\mathrm{m}}^{\prime }\mathrm{n}{\mathrm{n}}^{\prime }}^{(a)}=-{\mathcal{T}}_{\mathrm{s}\mathrm{s}\mathrm{m}{\mathrm{m}}^{\prime }\mathrm{n}{\mathrm{n}}^{\prime }}^{(b)}$ when $m\ne m^{\prime }$ and both are even, and

• ${\mathcal{T}}_{\mathrm{e}\mathrm{e}\mathrm{m}{\mathrm{m}}^{\prime }\mathrm{n}{\mathrm{n}}^{\prime }}^{(a)}=-{\mathcal{T}}_{\mathrm{o}\mathrm{o}\mathrm{m}{\mathrm{m}}^{\prime }\mathrm{n}{\mathrm{n}}^{\prime }}^{(b)}$ and ${\mathcal{T}}_{\mathrm{o}\mathrm{o}\mathrm{m}{\mathrm{m}}^{\prime }\mathrm{n}{\mathrm{n}}^{\prime }}^{(a)}=-{\mathcal{T}}_{\mathrm{e}\mathrm{e}\mathrm{m}{\mathrm{m}}^{\prime }\mathrm{n}{\mathrm{n}}^{\prime }}^{(b)}$ when $m\ne m^{\prime }$ and both are odd.

2.8 Asymptotic expression for perturbation potential

Accordingly, the first term on the right side of (37) does not exist in the equatorial plane (i.e., $\theta =\pi /2$⁠) whereas the second term is absent on the $z$ axis (that is $\theta \in \{0,\pi \}$⁠).

Now, the perturbation potential ${\mathrm{\Phi }}_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{t}}(r,\theta ,\varphi )$ must depend on the source as well as on the radius $a$ of the sphere. For the two sources chosen for illustrative results, $f_{\text{pert}}$ must depend linearly on both the sign and the magnitude of $Q$ or $p$ (as appropriate). The location of either of the two sources enters the potential expressions only by means of the source-potential coefficients ${\mathcal{A}}_{\mathrm{s}\mathrm{m}\mathrm{n}}$⁠, which are proportional to $r_{\mathrm{o}}^{-(n+1)}$ for the point charge and to $r_{\mathrm{o}}^{-(n+2)}$ for the point dipole, according to (12b) and (14b). Since the transition matrix $\left[\mathcal{T}\right]$ does not depend on the source, the perturbation-potential coefficients ${\mathcal{B}}_{\mathrm{e}01}$⁠, ${\mathcal{B}}_{\mathrm{e}11}$ and ${\mathcal{B}}_{\mathrm{o}11}$ increase/decrease as $r_{\mathrm{o}}$ decreases/increases. Accordingly, the magnitude of $f_{\text{pert}}$ increases/decreases as $r_{\mathrm{o}}$ decreases/increases.

3. Numerical results and discussion

3.1 Preliminaries

• (i)

  is independent of $Q$ or $p$ (as appropriate),

• (ii)

  increases/decreases as $r_{\mathrm{o}}$ decreases/increases, and

• (iii)

  is directly proportional to $a$⁠.

The theory described in section 2 was validated by comparing its results for the perturbation of the source potential by an isotropic dielectric sphere with the corresponding exact solutions available in the literature. First, the source was taken to be a point charge located on the $+z$ axis (i.e., ${\theta }_{\mathrm{o}}=0$⁠) at $r_{\mathrm{o}}=10a$⁠; note that ${\varphi }_{\mathrm{o}}$ is irrelevant when $\sin {\theta }_{\mathrm{o}}=0$⁠. Excellent agreement was obtained with respect to the exact solution (26) for all examined values of $a$ and ${\epsilon }_{\mathrm{r}}$⁠. Next, the point charge was replaced by a point dipole. Again, excellent agreement was found with respect to the corresponding exact solutions (25, 27).

3.2 Normalized asymptotic perturbation $F_{\text{pert}}(\theta ,\varphi )$

Having clarified in section 3.1 the effects of the parameters $Q$⁠, $p$⁠, $r_{\mathrm{o}}$⁠, and $a$ on the normalized asymptotic perturbation $F_{\text{pert}}(\theta ,\varphi )$⁠, we present numerical results for the variations of $F_{\text{pert}}(\theta ,\varphi )$ as a function of the sphere’s anisotropy parameters ${\alpha }_{\mathrm{x}}$ and ${\alpha }_{\mathrm{y}}$ for the following three cases:

• Case 1: ${\alpha }_{\mathrm{x}}\ne {\alpha }_{\mathrm{y}}=1$⁠,

• Case 2: ${\alpha }_{\mathrm{y}}\ne {\alpha }_{\mathrm{x}}=1$⁠, and

• Case 3: ${\alpha }_{\mathrm{x}}={\alpha }_{\mathrm{y}}\ne 1$⁠.

Plots of the perturbation-potential coefficients ${\mathcal{B}}_{\mathrm{e}01}$⁠, ${\mathcal{B}}_{\mathrm{e}11}$ and ${\mathcal{B}}_{\mathrm{o}11}$ as functions of ${\alpha }_{\mathrm{x}}$ and ${\alpha }_{\mathrm{y}}$ are examined in conjunction with the corresponding plots of $F_{\text{pert}}(\theta ,\varphi )$ versus $\theta$ and $\varphi$ for $a=5$ cm, $r_{\mathrm{o}}=2a$⁠, ${\theta }_{\mathrm{o}}=\pi /4$ and ${\varphi }_{\mathrm{o}}=\pi /3$⁠. All calculations were made for either a point charge of magnitude $Q={10}^{-10}$ C or a point dipole of moment $p={10}^{-10}$ C m.

3.2.1 Case 1 (⁠${\alpha }_{\mathrm{x}}=\overline{\alpha }$⁠, ${\alpha }_{\mathrm{y}}=1$⁠)

We varied ${\alpha }_{\mathrm{x}}=\overline{\alpha }\in [0.5,1.5]$ but kept ${\alpha }_{\mathrm{y}}=1$ fixed. Figure 2 shows plots of ${\mathcal{B}}_{\mathrm{e}01}$⁠, ${\mathcal{B}}_{\mathrm{e}11}$ and ${\mathcal{B}}_{\mathrm{o}11}$ versus $\overline{\alpha }$ for a point-charge source as well as for a point-dipole source with $\hat{\mathbf{p}}\in \{\hat{\mathbf{x}},\hat{\mathbf{y}},\hat{\mathbf{z}}\}$⁠. Angular profiles of $F_{\text{pert}}(\theta ,\varphi )$ for the same sources are depicted in Fig. 3 for $\overline{\alpha }=0.5$⁠, and in Fig. 4 for $\overline{\alpha }=1.5$⁠.

For a point-charge source, ${\mathcal{B}}_{\mathrm{e}01}$ decreases with $\overline{\alpha }\in [0.5,1.5]$ in Fig. 2(a). Thus, $F_{\text{pert}}(0,\varphi )$ decreases but $F_{\text{pert}}(\pi ,\varphi )$ increases as ${\alpha }_{\mathrm{x}}$ changes from $0.5$ to $1.5$⁠, as can be gathered from Figs. 3(a) and 4(a). Also, ${\mathcal{B}}_{\mathrm{e}11}$ increases and ${\mathcal{B}}_{\mathrm{o}11}$ decreases with increasing ${\alpha }_{\mathrm{x}}$⁠. Therefore, $F_{\text{pert}}(\pi /2,0)$ increases but $F_{\text{pert}}(\pi /2,\pi /2)$ decreases as ${\alpha }_{\mathrm{x}}$ changes from $0.5$ to $1.5$⁠.

Next, for the point-dipole sources, ${\mathcal{B}}_{\mathrm{e}01}$ increases with increasing $\overline{\alpha }\in [0.5,1.5]$ for all three dipole orientations, as is clear from Figs. 2(b)–(d); the largest increase is observed for $\hat{\mathbf{p}}=\hat{\mathbf{y}}$⁠. Hence, a comparison of Figs. 3(b)–(d) and 4(b)–(d) reveals that $F_{\text{pert}}(0,\varphi )$ increases but $F_{\text{pert}}(\pi ,\varphi )$ decreases as ${\alpha }_{\mathrm{x}}$ changes from $0.5$ to $1.5$⁠. The rate of these increases or decreases is highest for $\hat{\mathbf{p}}=\hat{\mathbf{y}}$⁠, moderate for $\hat{\mathbf{p}}=\hat{\mathbf{x}}$⁠, and lowest for $\hat{\mathbf{p}}=\hat{\mathbf{z}}$⁠.

Besides, for $\hat{\mathbf{p}}=\hat{\mathbf{x}}$⁠, both ${\mathcal{B}}_{\mathrm{e}11}$ and ${\mathcal{B}}_{\mathrm{o}11}$ increase with ${\alpha }_{\mathrm{x}}$ in Fig. 2(b) and, thus, $F_{\text{pert}}(\pi /2,0)$ and $F_{\text{pert}}(\pi /2,\pi /2)$ also increase with ${\alpha }_{\mathrm{x}}$ in Figs. 3(b) and 4(b). On the other hand, for $\hat{\mathbf{p}}=\hat{\mathbf{y}}$ and $\hat{\mathbf{p}}=\hat{\mathbf{z}}$⁠, ${\mathcal{B}}_{\mathrm{e}11}$ decreases in Fig. 2(c) but ${\mathcal{B}}_{\mathrm{o}11}$ increases in Fig. 2(d) as ${\alpha }_{\mathrm{x}}$ increases. Therefore, $F_{\text{pert}}(\pi /2,0)$ decreases and $F_{\text{pert}}(\pi /2,\pi /2)$ increases with ${\alpha }_{\mathrm{x}}$⁠, as can be gathered from comparing Figs. 3(c) and (d) with Figs. 4(c) and (d), respectively.

3.2.2 Case 2 (⁠${\alpha }_{\mathrm{x}}=1$⁠, ${\alpha }_{\mathrm{y}}=\overline{\alpha }$⁠)

Next, we fixed ${\alpha }_{\mathrm{x}}=1$ but varied ${\alpha }_{\mathrm{y}}\in [0.5,1.5]$⁠. The dependencies of the coefficients ${\mathcal{B}}_{\mathrm{e}01}$⁠, ${\mathcal{B}}_{\mathrm{e}11}$⁠, and ${\mathcal{B}}_{\mathrm{o}11}$ on ${\alpha }_{\mathrm{y}}$ are depicted in Fig. 5, whereas the angular profiles of $F_{\text{pert}}(\theta ,\varphi )$ are depicted in Fig. 6 for ${\alpha }_{\mathrm{y}}=0.5$ and Fig. 7 for ${\alpha }_{\mathrm{y}}=1.5$⁠.

For all four types of source considered, ${\mathcal{B}}_{\mathrm{e}01}$ varies with $\overline{\alpha }$ in Case 1 in the same way as it varies with $\overline{\alpha }$ in Case 2. Hence, the characteristics of $F_{\text{pert}}(0,\varphi )$ and $F_{\text{pert}}(\pi ,\varphi )$ in Case 2 replicate those in Case 1. Also, if ${\mathcal{B}}_{\mathrm{e}11}$ or ${\mathcal{B}}_{\mathrm{o}11}$ is an increasing (decreasing) function of $\overline{\alpha }$ in Case 1, then it is a decreasing (increasing) function of $\overline{\alpha }$ in Case 2. Therefore, the characteristics of $F_{\text{pert}}(\pi /2,0)$ and $F_{\text{pert}}(\pi /2,\pi /2)$ in Case 2 are opposed to those in Case 1.

3.2.3 Case 3 (⁠${\alpha }_{\mathrm{x}}={\alpha }_{\mathrm{y}}=\overline{\alpha }$⁠)

Finally, we set ${\alpha }_{\mathrm{x}}={\alpha }_{\mathrm{y}}=\overline{\alpha }$⁠. The corresponding plots for ${\mathcal{B}}_{\mathrm{s}\mathrm{m}\mathrm{n}}$ vs $\overline{\alpha }\in [0.5,1.5]$ are depicted in Fig. 8, and for $F_{\text{pert}}(\theta ,\varphi )$ versus $\theta$ and $\varphi$ are depicted in Fig. 9 for $\overline{\alpha }=0.5$⁠, and in Fig. 10 for $\overline{\alpha }=1.5$⁠.

The curves of ${\mathcal{B}}_{\mathrm{e}01}$ versus $\overline{\alpha }\in [0.5,1.5]$ have the same increasing/decreasing tendencies with the respective ones in Cases 1 and 2; however, the values of ${\mathcal{B}}_{\mathrm{e}01}$ and the rate of increase/decrease w.r.t. $\overline{\alpha }$ are definitely different. Furthermore, in Case 3, the increasing/decreasing tendencies of ${\mathcal{B}}_{\mathrm{e}11}$ and ${\mathcal{B}}_{\mathrm{o}11}$ with $\overline{\alpha }$ are as those in Cases 1 and 2, respectively.

3.3 Normalized perturbation potential ${\tilde{\mathrm{\Phi }}}_{\text{pert}}(r,\theta ,\varphi )$

Finally, on comparing Figs. 11–13, we observe that the portions of the $\theta$-$\varphi$ plane corresponding to the maximum/minimum increase in $|{\tilde{\mathrm{\Phi }}}_{\text{pert}}(r,\theta ,\varphi )|$ remain almost the same as $\tilde{r}$ increases, for each of the four source potentials considered.

4. Concluding remarks

We formulated and solved the boundary-value problem for the perturbation of an electric potential by a homogeneous anisotropic dielectric sphere in vacuum. As is commonplace for the exterior region, the source potential and the perturbation potential were represented in terms of the standard solutions of the Laplace equation in the spherical coordinate system. A bijective spatial transformation was implemented for the interior region in order to formulate the series representation of the internal potential. Boundary conditions on the spherical surface were enforced and then the orthogonality of tesseral harmonics was employed to derive a transition matrix that relates the expansion coefficients of the perturbation potential in the exterior region to those of the source potential. The angular profile of the perturbation profile changes with distance from the center of the sphere, but eventually it settles down with the perturbation potential decaying as the inverse of the distance squared from the center of the sphere.

Acknowledgement

A. Lakhtakia is grateful to the Charles Godfrey Binder Endowment at Penn State for ongoing support of his research activities.

References

Appendix

Equation (18) follows from (A.4) and (A.5).