Home | About | Journals | Submit | Contact Us | Français |

**|**HHS Author Manuscripts**|**PMC2764543

Formats

Article sections

- Abstract
- 1. Introduction
- 2. An oblate spheroid immersed in the pure water solvent
- 3. An oblate spheroid immersed in an ionic solvent
- 4. Numerical examples
- 5. Conclusions
- References

Authors

Related links

J Electrostat. Author manuscript; available in PMC 2010 September 1.

Published in final edited form as:

J Electrostat. 2009 September; 67(5): 807–814.

doi: 10.1016/j.elstat.2009.06.007PMCID: PMC2764543

NIHMSID: NIHMS132122

Shaozhong Deng^{*}

Shaozhong Deng, Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC 28223-0001, USA;

See other articles in PMC that cite the published article.

As a sequel to a previous paper on electrostatic potential of point charges inside dielectric prolate spheroids [J. Electrostatics 66 (2008) 549-560], this note further presents the exact solution to the electrostatic problem of finding the electric potential of point charges inside a dielectric oblate spheroid that is embedded in a dissimilar dielectric medium. Numerical experiments have demonstrated the convergence of the proposed series solutions.

As in [1], this note concerns the exact solution to the electrostatic problem of finding the electric potential of point charges inside a dielectric cavity that is embedded in a dissimilar dielectric medium. Such a problem may find its application in hybrid explicit/implicit solvent biomolecular simulations, in which biomolecules and a part of solvent molecules within a dielectric cavity are explicitly modeled while a surrounding dielectric continuum is used to model bulk effects of the solvent beyond the cavity [2]. In [1], the exact solution of the problem when the dielectric cavity is either a sphere or a prolate spheroid has been obtained by using the classical electrostatic theory. Following the same procedure as used in [1], this note will present the exact solution of the problem when the dielectric cavity is an oblate spheroid.

By the principle of linear superposition, the electrostatic problem with a single point source charge *q* inside a dielectric cavity only needs to be considered. It is then well-known that the total electric potential Ψ_{in}(**r**) inside the cavity is given by the solution of the Poisson equation

$$\nabla \xb7({\epsilon}_{\text{i}}\nabla {\mathrm{\Psi}}_{\text{in}}(\mathbf{r}))=-4\pi q\delta (\mid \mathbf{r}-{\mathbf{r}}_{\text{s}}\mid ),$$

where *ε*_{i} is the electric permittivity of the dielectric cavity, **r**_{s} represents the location of the point charge, and *δ*(*r*) denotes the Dirac delta function. Outside the cavity, on the other hand, by assuming that the mobile ion concentration follows the Debye-Hückel theory, the electric field Ψ_{out}(**r**) is then given by the solution of the linearized Poisson-Boltzmann equation (LPBE)

$${\nabla}^{2}{\mathrm{\Psi}}_{\text{out}}(\mathbf{r})-{\lambda}^{2}{\mathrm{\Psi}}_{\text{out}}(\mathbf{r})=0,$$

(1)

where *λ* is the inverse Debye screening length determined by the ionic concentration and the dielectric constant *ε*_{o} of the solvent medium exterior to the cavity (*λ* = 0 for the pure water solvent). On the interface Γ of the dielectric oblate spheroid and its surrounding dielectric medium, the following two boundary conditions are to be satisfied for the continuities of the potential and the fluxes along the normal direction

$${\mathrm{\Psi}}_{\text{out}}{\mid}_{\mathrm{\Gamma}}={\mathrm{\Psi}}_{\text{in}}{\mid}_{\mathrm{\Gamma}}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\text{and}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}{{\epsilon}_{\text{o}}\frac{\partial {\mathrm{\Psi}}_{\text{out}}}{\partial \mathbf{n}}\mid}_{\mathrm{\Gamma}}={{\epsilon}_{\text{i}}\frac{\partial {\mathrm{\Psi}}_{\text{in}}}{\partial \mathbf{n}}\mid}_{\mathrm{\Gamma}},$$

(2)

where **n** is the unit outward vector normal to the surface of the cavity.

Let us first consider a dielectric oblate spheroid of permittivity *ε*_{i} centered at the origin and embedded in a homogeneous dielectric medium of permittivity *ε*_{o}, as shown in Fig. 1. Let further the *z*-axis be the non-focal axis of symmetry, and the equation for the oblate spheroid in Cartesian coordinates be

$$\frac{{x}^{2}+{y}^{2}}{{a}_{1}^{2}}+\frac{{z}^{2}}{{a}_{2}^{2}}=1,\phantom{\rule{0.38889em}{0ex}}{a}_{1}>{a}_{2}.$$

There are several definitions of oblate spheroidal coordinates, and in this note, the oblate spheroidal coordinates (*ξ, η, *) are defined through

$$x=\frac{a}{2}\sqrt{(1+{\xi}^{2})(1-{\eta}^{2})}cos\phi ,\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}y=\frac{a}{2}\sqrt{(1+{\xi}^{2})(1-{\eta}^{2})}sin\phi ,\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}z=\frac{a}{2}\xi \eta ,$$

where
$a=2\sqrt{{a}_{1}^{2}-{a}_{2}^{2}}$ is the interfocal distance of the oblate spheroid, *ξ* [0, ∞) is the radial variable, *η* [−1*,* 1] is the angular variable, and [0*,* 2*π*] is the azimuthal variable, respectively. Under this definition, the surface of constant *ξ* is an oblate spheroid with interfocal distance *a*. In particular, when *ξ* = 0, the spheroid is degenerate and flattens to the circular disk in the plane *z* = 0 with radius *a*/2. The inverse transformation from Cartesian coordinates to the oblate spheroidal coordinates is also reported here for convenience.

$$\{\begin{array}{l}\text{when}\phantom{\rule{0.16667em}{0ex}}z\ne 0:\hfill \\ \xi ={\scriptstyle \frac{\sqrt{(4{r}^{2}-{a}^{2})+\sqrt{{(4{r}^{2}-{a}^{2})}^{2}+16{a}^{2}{z}^{2}}}}{\sqrt{2}\phantom{\rule{0.16667em}{0ex}}a}},\hfill \\ \eta ={\scriptstyle \frac{2\sqrt{2}\phantom{\rule{0.16667em}{0ex}}z}{\sqrt{(4{r}^{2}-{a}^{2})+\sqrt{{(4{r}^{2}-{a}^{2})}^{2}+16{a}^{2}{z}^{2}}}}},\hfill \\ \text{when}\phantom{\rule{0.16667em}{0ex}}z=0:\hfill \\ \xi =\sqrt{{\scriptstyle \frac{4{r}^{2}}{{a}^{2}}}-1},\phantom{\rule{0.16667em}{0ex}}\eta =0,\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\text{if}\phantom{\rule{0.16667em}{0ex}}r\ge {\scriptstyle \frac{a}{2}},\hfill \\ \eta =\pm \sqrt{1-{\scriptstyle \frac{4{r}^{2}}{{a}^{2}}}},\phantom{\rule{0.16667em}{0ex}}\xi =0,\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\text{if}\phantom{\rule{0.16667em}{0ex}}r\le {\scriptstyle \frac{a}{2}},\hfill \\ \text{where}\phantom{\rule{0.16667em}{0ex}}{r}^{2}={x}^{2}+{y}^{2}+{z}^{2},\phantom{\rule{0.16667em}{0ex}}\text{and}\phantom{\rule{0.16667em}{0ex}}\text{in}\phantom{\rule{0.16667em}{0ex}}\text{all}\phantom{\rule{0.16667em}{0ex}}\text{cases}\hfill \\ \phi =arctan\frac{y}{x}.\hfill \end{array}$$

Without loss of generality, let a point charge *q* be located at the point **r**_{s} = (*ξ*_{0}*, η*_{0}*, *_{0}=0) in the plane *y* = 0 inside the dielectric spheroid defined by the equation *ξ* = *ξ*_{1} (so *ξ*_{1} *ξ*_{0} ≥ 0), as shown in Fig. 1. Outside the spheroid (*ξ*_{1} ≤ *ξ*), the total electrostatic potential takes the form

$${\mathrm{\Psi}}_{\text{out}}(\mathbf{r})=\frac{q}{{\epsilon}_{\text{i}}a}\sum _{n=0}^{\infty}\sum _{m=0}^{n}({A}_{mn}cosm\phi +{B}_{mn}sinm\phi ){P}_{n}^{m}(\eta ){Q}_{n}^{m}(i\xi ),$$

(3)

and inside the spheroid (0 ≤ *ξ* ≤ *ξ*_{1}), it is written as

$${\mathrm{\Psi}}_{\text{in}}(\mathbf{r})=\frac{q}{{\epsilon}_{\text{i}}\mid \mathbf{r}-{\mathbf{r}}_{\text{s}}\mid}+\frac{q}{{\epsilon}_{\text{i}}a}\sum _{n=0}^{\infty}\sum _{m=0}^{n}({C}_{mn}cosm\phi +{D}_{mn}sinm\phi ){P}_{n}^{m}(\eta ){P}_{n}^{m}(i\xi ),$$

(4)

where
$i=\sqrt{-1},{P}_{n}^{m}(x)$ and
${Q}_{n}^{m}(x)$ are the associated Legendre functions of the first kind and the second kind, respectively, and *A _{mn}*,

Note that the general expansion of 1/|**r** − **r**_{s}| in the oblate spheroidal coordinates is given by [3–5]

$$\frac{1}{\mid \mathbf{r}-{\mathbf{r}}_{\text{s}}\mid}=\frac{1}{a}\sum _{n=0}^{\infty}\sum _{m=0}^{n}{H}_{mn}cosm\phi {P}_{n}^{m}({\eta}_{0}){P}_{n}^{m}(\eta )\{\begin{array}{l}{P}_{n}^{m}(i{\xi}_{0}){Q}_{n}^{m}(i\xi ),\phantom{\rule{0.38889em}{0ex}}\xi >{\xi}_{0},\hfill \\ {P}_{n}^{m}(i\xi ){Q}_{n}^{m}(i{\xi}_{0}),\phantom{\rule{0.38889em}{0ex}}\xi <{\xi}_{0},\hfill \end{array}$$

where

$${H}_{mn}=2i(2n+1)(2-{\delta}_{m0}){(-1)}^{m}{\left[\frac{(n-m)!}{(n+m)!}\right]}^{2}.$$

Here *δ _{m}*

$$\begin{array}{l}{\mathrm{\Psi}}_{\text{in}}(\mathbf{r})=\frac{q}{{\epsilon}_{\text{i}}a}\sum _{i=0}^{\infty}\sum _{m=0}^{n}({C}_{mn}cosm\phi +{D}_{mn}sinm\phi ){P}_{n}^{m}(\eta ){P}_{n}^{m}(i\xi )\\ +\frac{q}{{\epsilon}_{\text{i}}a}\sum _{n=0}^{\infty}\sum _{m=0}^{n}{H}_{mn}cosm\phi {P}_{n}^{m}({\eta}_{0}){P}_{n}^{m}(\eta )\{\begin{array}{l}{P}_{n}^{m}(i{\xi}_{0}){Q}_{n}^{m}(i\xi ),\phantom{\rule{0.38889em}{0ex}}\xi >{\xi}_{0},\hfill \\ {P}_{n}^{m}(i\xi ){Q}_{n}^{m}(i{\xi}_{0}),\phantom{\rule{0.38889em}{0ex}}\xi <{\xi}_{0}.\hfill \end{array}\end{array}$$

(5)

The expansion coefficients *A _{mn}, B_{mn}, C_{mn}* and

$${\mathrm{\Psi}}_{\text{out}}{\mid}_{\xi ={\xi}_{1}}={\mathrm{\Psi}}_{\text{in}}{\mid}_{\xi ={\xi}_{1}}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\text{and}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}{{\epsilon}_{\text{o}}\frac{\partial {\mathrm{\Psi}}_{\text{out}}}{\partial \xi}\mid}_{\xi ={\xi}_{1}}={{\epsilon}_{\text{i}}\frac{\partial {\mathrm{\Psi}}_{\text{in}}}{\partial \xi}\mid}_{\xi ={\xi}_{1}}.$$

(6)

Then, by using the orthogonality of cos *m* and sin *m* as well as that of
${P}_{n}^{m}(x)$, one can obtain *B _{mn}* =

$${Q}_{n}^{m}(i{\xi}_{1}){A}_{mn}-{P}_{n}^{m}(i{\xi}_{1}){C}_{mn}={H}_{mn}{P}_{n}^{m}({\eta}_{0}){P}_{n}^{m}(i{\xi}_{0}){Q}_{n}^{m}(i{\xi}_{1}),$$

(7)

$${\epsilon}_{\text{o}}{\widehat{Q}}_{n}^{m}(i{\xi}_{1}){A}_{mn}-{\epsilon}_{\text{i}}{\widehat{P}}_{n}^{m}(i{\xi}_{1}){C}_{mn}={\epsilon}_{\text{i}}{H}_{mn}{P}_{n}^{m}({\eta}_{0}){P}_{n}^{m}(i{\xi}_{0}){\widehat{Q}}_{n}^{m}(i{\xi}_{1}),$$

(8)

where

$$\begin{array}{l}{\widehat{P}}_{n}^{m}(i{\xi}_{1})=(n-m+1){P}_{n+1}^{m}(i{\xi}_{1})-(n+1)i{\xi}_{1}{P}_{n}^{m}(i{\xi}_{1}),\\ {\widehat{Q}}_{n}^{m}(i{\xi}_{1})=(n-m+1){Q}_{n+1}^{m}(i{\xi}_{1})-(n+1)i{\xi}_{1}{Q}_{n}^{m}(i{\xi}_{1}).\end{array}$$

Further solving (7) and (8) leads to

$${A}_{mn}={\epsilon}_{\text{i}}{H}_{mn}{P}_{n}^{m}({\eta}_{0}){P}_{n}^{m}(i{\xi}_{0}){K}_{mn}^{-1}\left({P}_{n}^{m}(i{\xi}_{1}){\widehat{Q}}_{n}^{m}(i{\xi}_{1})-{Q}_{n}^{m}(i{\xi}_{1}){\widehat{P}}_{n}^{m}(i{\xi}_{1})\right),$$

and

$${C}_{mn}=({\epsilon}_{\text{i}}-{\epsilon}_{\text{o}}){H}_{mn}{P}_{n}^{m}({\eta}_{0}){P}_{n}^{m}(i{\xi}_{0}){K}_{mn}^{-1}{Q}_{n}^{m}(i{\xi}_{1}){\widehat{Q}}_{n}^{m}(i{\xi}_{1}),$$

where

$${K}_{mn}={\epsilon}_{\text{o}}{P}_{n}^{m}(i{\xi}_{1}){\widehat{Q}}_{n}^{m}(i{\xi}_{1})-{\epsilon}_{\text{i}}{Q}_{n}^{m}(i{\xi}_{1}){\widehat{P}}_{n}^{m}(i{\xi}_{1}).$$

In summary, the total potential inside the oblate spheroid is given by

$$\begin{array}{c}{\mathrm{\Psi}}_{\text{in}}(\mathbf{r})=\frac{q}{{\epsilon}_{\text{i}}\mid \mathbf{r}-{\mathbf{r}}_{\text{s}}\mid}+\frac{q}{{\epsilon}_{\text{i}}a}\sum _{n=0}^{\infty}\sum _{m=0}^{n}({\epsilon}_{\text{i}}-{\epsilon}_{\text{o}}){H}_{mn}{P}_{n}^{m}({\eta}_{0}){P}_{n}^{m}(i{\xi}_{0}){K}_{mn}^{-1}\\ \times {Q}_{n}^{m}(i{\xi}_{1}){\widehat{Q}}_{n}^{m}(i{\xi}_{1})cosm\phi {P}_{n}^{m}(\eta ){P}_{n}^{m}(i\xi ),\end{array}$$

(9)

while the potential outside the oblate spheroid is

$$\begin{array}{l}{\mathrm{\Psi}}_{\text{out}}(\mathbf{r})=\frac{q}{{\epsilon}_{\text{i}}a}\sum _{n=0}^{\infty}\sum _{m=0}^{n}{\epsilon}_{\text{i}}{H}_{mn}{P}_{n}^{m}({\eta}_{0}){P}_{n}^{m}(i{\xi}_{0}){K}_{mn}^{-1}\\ \times \left({P}_{n}^{m}(i{\xi}_{1}){\widehat{Q}}_{n}^{m}(i{\xi}_{1})-{Q}_{n}^{m}(i{\xi}_{1}){\widehat{P}}_{n}^{m}(i{\xi}_{1})\right)cosm\phi {P}_{n}^{m}(\eta ){Q}_{n}^{m}(i\xi ).\end{array}$$

(10)

Similarly, when the point charge *q* is located at the point **r**_{s} = (*ξ*_{0}*, η*_{0}*, *_{0}=0) outside the dielectric spheroid defined by the equation *ξ* = *ξ*_{1} (so *ξ*_{0} > *ξ*_{1} > 0), the electrostatic potential outside the spheroid takes the form

$${\mathrm{\Psi}}_{\text{out}}(\mathbf{r})=\frac{q}{{\epsilon}_{\text{o}}\mid \mathbf{r}-{\mathbf{r}}_{\text{s}}\mid}+\frac{q}{{\epsilon}_{\text{o}}a}\sum _{n=0}^{\infty}\sum _{m=0}^{n}{A}_{mn}cosm\phi {P}_{n}^{m}(\eta ){Q}_{n}^{m}(i\xi ),$$

(11)

or

$$\begin{array}{l}{\mathrm{\Psi}}_{\text{out}}(\mathbf{r})=\frac{q}{{\epsilon}_{o}a}\sum _{n=0}^{\infty}\sum _{m=0}^{n}{A}_{mn}cosm\phi {P}_{n}^{m}(\eta ){Q}_{n}^{m}(i\xi )\\ +\frac{q}{{\epsilon}_{\text{o}}a}\sum _{n=0}^{\infty}\sum _{m=0}^{n}{H}_{mn}cosm\phi {P}_{n}^{m}({\eta}_{0}){P}_{n}^{m}(\eta )\{\begin{array}{l}{P}_{n}^{m}(i{\xi}_{0}){Q}_{n}^{m}(i\xi ),\phantom{\rule{0.38889em}{0ex}}\xi >{\xi}_{0},\hfill \\ {P}_{n}^{m}(i\xi ){Q}_{n}^{m}(i{\xi}_{0}),\phantom{\rule{0.38889em}{0ex}}\xi <{\xi}_{0}.\hfill \end{array}\end{array}$$

(12)

And on the other hand, the potential inside the spheroid can be written as

$${\mathrm{\Psi}}_{\text{in}}(\mathbf{r})=\frac{q}{{\epsilon}_{\text{o}}a}\sum _{n=0}^{\infty}\sum _{m=0}^{n}{C}_{mn}cosm\phi {P}_{n}^{m}(\eta ){P}_{n}^{m}(i\xi ).$$

(13)

Then using the boundary conditions of (6), one can obtain

$${Q}_{n}^{m}(i{\xi}_{1}){A}_{mn}-{P}_{n}^{m}(i{\xi}_{1}){C}_{mn}=-{H}_{mn}{P}_{n}^{m}({\eta}_{0}){P}_{n}^{m}(i{\xi}_{1}){Q}_{n}^{m}(i{\xi}_{0}),$$

(14)

$${\epsilon}_{\text{o}}{\widehat{Q}}_{n}^{m}(i{\xi}_{1}){A}_{mn}-{\epsilon}_{\text{i}}{\widehat{P}}_{n}^{m}(i{\xi}_{1}){C}_{mn}=-{\epsilon}_{\text{o}}{H}_{mn}{P}_{n}^{m}({\eta}_{0}){\widehat{P}}_{n}^{m}(i{\xi}_{1}){Q}_{n}^{m}(i{\xi}_{0}).$$

(15)

Solving (14) and (15) leads to

$${A}_{mn}=({\epsilon}_{\text{i}}-{\epsilon}_{\text{o}}){H}_{mn}{P}_{n}^{m}({\eta}_{0}){Q}_{n}^{m}(i{\xi}_{0}){K}_{mn}^{-1}{P}_{n}^{m}(i{\xi}_{1}){\widehat{P}}_{n}^{m}(i{\xi}_{1}),$$

and

$${C}_{mn}={\epsilon}_{\text{o}}{H}_{mn}{P}_{n}^{m}({\eta}_{0}){Q}_{n}^{m}(i{\xi}_{0}){K}_{mn}^{-1}\left({P}_{n}^{m}(i{\xi}_{1}){\widehat{Q}}_{n}^{m}(i{\xi}_{1})-{Q}_{n}^{m}(i{\xi}_{1}){\widehat{P}}_{n}^{m}(i{\xi}_{1})\right).$$

In summary, the total potential outside the oblate spheroid is given by

$$\begin{array}{l}{\mathrm{\Psi}}_{\text{out}}(\mathbf{r})=\frac{q}{{\epsilon}_{\text{o}}\mid \mathbf{r}-{\mathbf{r}}_{\text{s}}\mid}+\frac{q}{{\epsilon}_{\text{o}}a}\sum _{n=0}^{\infty}\sum _{m=0}^{n}({\epsilon}_{\text{i}}-{\epsilon}_{\text{o}}){H}_{mn}{P}_{n}^{m}({\eta}_{0}){Q}_{n}^{m}(i{\xi}_{0}){K}_{mn}^{-1}\\ \times {P}_{n}^{m}(i{\xi}_{1}){\widehat{P}}_{n}^{m}(i{\xi}_{1})cosm\phi {P}_{n}^{m}(\eta ){Q}_{n}^{m}(i\xi ),\end{array}$$

(16)

while the potential inside the oblate spheroid is

$$\begin{array}{l}{\mathrm{\Psi}}_{\text{in}}(\mathbf{r})=\frac{q}{{\epsilon}_{\text{o}}a}\sum _{n=0}^{\infty}\sum _{m=0}^{n}{\epsilon}_{\text{o}}{H}_{mn}{P}_{n}^{m}({\eta}_{0}){Q}_{n}^{m}(i{\xi}_{0}){K}_{mn}^{-1}\\ \times \left({P}_{n}^{m}(i{\xi}_{1}){\widehat{Q}}_{n}^{m}(i{\xi}_{1})-{Q}_{n}^{m}(i{\xi}_{1}){\widehat{P}}_{n}^{m}(i{\xi}_{1})\right)cosm\phi {P}_{n}^{m}(\eta ){P}_{n}^{m}(i\xi ).\end{array}$$

(17)

Using the oblate spheroidal coordinates, the LPBE of (1) can be written as

$$\frac{\partial}{\partial \xi}\left[({\xi}^{2}+1)\frac{\partial \mathrm{\Psi}}{\partial \xi}\right]+\frac{\partial}{\partial \eta}\left[(1-{\eta}^{2})\frac{\partial \mathrm{\Psi}}{\partial \eta}\right]+\frac{{\xi}^{2}+{\eta}^{2}}{({\xi}^{2}+1)(1-{\eta}^{2})}\frac{{\partial}^{2}\mathrm{\Psi}}{\partial {\phi}^{2}}-{c}^{2}\left({\xi}^{2}+{\eta}^{2}\right)\mathrm{\Psi}=0,$$

where *c* = (*a*/2) *λ* is the spheroidal parameter. By means of the separation of variables with

$$\mathrm{\Psi}(\xi ,\eta ,\phi )={S}_{mn}(c,\eta ){R}_{mn}(c,i\xi )\left(\begin{array}{c}cosm\phi \\ sinm\phi \end{array}\right),$$

where *m* and *n* are zero or positive integers with *n* ≥ *m*, one can obtain the ordinary differential equations

$$\frac{\text{d}}{\text{d}\eta}\left[(1-{\eta}^{2})\frac{\text{d}{S}_{mn}(c,\eta )}{\text{d}\eta}\right]+\left[{\lambda}_{mn}(c)-{c}^{2}{\eta}^{2}-\frac{{m}^{2}}{1-{\eta}^{2}}\right]\phantom{\rule{0.16667em}{0ex}}{S}_{mn}(c,\eta )=0,$$

(18)

$$\frac{\text{d}}{\text{d}\xi}\left[({\xi}^{2}+1)\frac{\text{d}{R}_{mn}(c,i\xi )}{\text{d}\xi}\right]-\left[{\lambda}_{mn}(c)+{c}^{2}{\xi}^{2}-\frac{{m}^{2}}{{\xi}^{2}+1}\right]\phantom{\rule{0.16667em}{0ex}}{R}_{mn}(c,i\xi )=0,$$

(19)

where those values *λ _{mn}*(

Note that Eq. (18) is exactly the same as that solved by prolate spheroidal angular wave functions for the Helmholtz equation in the standard theory, whose solutions are usually called the prolate angular functions of the first kind and the second kind. However, only the angular functions of the first kind
${S}_{mn}^{(1)}(c,\eta )$ are used in this paper since they are regular at *η* = *±*1. Moreover, we use the following expansion in terms of the associated Legendre functions of the first kind

$${S}_{mn}^{(1)}(c,\eta )=\sum _{r=0,1}^{\infty}\prime {d}_{r}^{mn}(c){P}_{m+r}^{m}(\eta ).$$

(20)

Here and in the sequel, the prime over the summation sign indicates that the summation is over only even values of *r* when *n* − *m* is even, and over only odd values of *r* when *n* − *m* is odd.

The eigenvalues *λ _{mn}*(

$${\alpha}_{r}^{m}(c){d}_{r+2}^{mn}(c)+({\beta}_{r}^{m}(c)-{\lambda}_{mn}(c)){d}_{r}^{mn}(c)+{\gamma}_{r}^{m}(c){d}_{r-2}^{mn}(c)=0,$$

(21)

in which

$${\alpha}_{r}^{m}(c)=\frac{(2m+r+2)(2m+r+1){c}^{2}}{(2m+2r+3)(2m+2r+5)},$$

(22)

$${\beta}_{r}^{m}(c)=(m+r)(m+r+1)+\frac{2(m+r)(m+r+1)-2{m}^{2}-1}{(2m+2r-1)(2m+2r+3)}{c}^{2},$$

(23)

$${\gamma}_{r}^{m}(c)=\frac{r(r-1){c}^{2}}{(2m+2r-3)(2m+2r-1)}.$$

(24)

The eigenvalues are determined by the condition that
${d}_{r}^{mn}(c)\to 0$ as *r* → ∞, and various methods have been proposed to calculate the eigenvalues and the expansion coefficients; see [6–14] and references therein.

In the standard theory, the prolate spheroidal radial wave function *R _{mn}*(

$$\frac{\text{d}}{\text{d}\xi}\left[({\xi}^{2}-1)\frac{\text{d}{R}_{mn}(c,\xi )}{\text{d}\xi}\right]-\left[{\lambda}_{mn}(c)-{c}^{2}{\xi}^{2}+\frac{{m}^{2}}{{\xi}^{2}-1}\right]\phantom{\rule{0.16667em}{0ex}}{R}_{mn}(c,\xi )=0.$$

(25)

Note that Eq. (19) can be obtained from Eq. (25) by replacing *ξ* with *iξ*. The standard prolate radial functions are usually expanded in a basis of spherical Bessel functions. Accordingly, the radial solution *R _{mn}*(

$${R}_{mn}^{(1)}(c,i\xi )=\frac{1}{{\rho}_{mn}(c)}{\left(\frac{{\xi}^{2}+1}{{\xi}^{2}}\right)}^{{\scriptstyle \frac{m}{2}}}\sum _{r=0,1}^{\infty}\prime {d}_{r}^{mn}(c)\frac{(2m+r)!}{r!}{i}_{m+r}(c\xi ),$$

and

$${R}_{mn}^{(3)}(c,i\xi )=\frac{1}{{\rho}_{mn}(c)}{\left(\frac{{\xi}^{2}+1}{{\xi}^{2}}\right)}^{{\scriptstyle \frac{m}{2}}}\sum _{r=0,1}^{\infty}\prime {d}_{r}^{mn}(c)\frac{(2m+r)!}{r!}{k}_{m+r}(c\xi ),$$

(26)

where *i _{n}*(

$${i}_{n}(r)=\sqrt{\frac{\pi}{2r}}{I}_{n+1/2}(r),\phantom{\rule{0.38889em}{0ex}}{k}_{n}(r)=\sqrt{\frac{\pi}{2r}}{K}_{n+1/2}(r).$$

Here, *I*(*r*) and *K*(*r*) are the modified Bessel functions of the first kind and the third kind, respectively. The normalization factor *ρ _{mn}*(

$${\rho}_{mn}(c)=\sum _{r=0,1}^{\infty}\prime {d}_{r}^{mn}(c)\frac{(2m+r)!}{r!}.$$

When expressing the electric potential outside the oblate spheroid where *ξ* ≥ *ξ*_{1} > 0, only
${R}_{mn}^{(3)}(c,i\xi )$ needs to be used since it is finite outside the oblate spheroid and vanishes at *ξ* = ∞. For convenience,
${d}_{r}^{mn}(c)$ is simply denoted by
${d}_{r}^{mn},{S}_{mn}^{(1)}(c,\eta )$ by *S _{mn}*(

Finally, as pointed out in [16], *k _{n}*(

$$\begin{array}{l}{R}_{m(m+2r)}^{(3)}(c,i\xi )=\frac{2m+1}{2(m)!{d}_{0}^{m(m+2r)}{c}^{m}}\xb7{({\xi}^{2}+1)}^{{\scriptstyle \frac{m}{2}}}\\ \xb7\sum _{q=0}^{\infty}{\omega}_{2q}{k}_{m+q}\left(\frac{c}{2}(\sqrt{{\xi}^{2}+1}+\xi )\right){i}_{m+q}\left(\frac{c}{2}(\sqrt{{\xi}^{2}+1}-\xi )\right),\end{array}$$

(27)

$$\begin{array}{l}{R}_{m(m+2r+1)}^{(3)}(c,i\xi )=\frac{2m+3}{(m)!{d}_{1}^{m(m+2r+1)}{c}^{m+1}}\xb7\xi {({\xi}^{2}+1)}^{{\scriptstyle \frac{m}{2}}}\\ \xb7\sum _{q=0}^{\infty}{\omega}_{2q+1}{k}_{m+q+1}\left(\frac{c}{2}(\sqrt{{\xi}^{2}+1}+\xi )\right){i}_{m+q+1}\left(\frac{c}{2}(\sqrt{{\xi}^{2}+1}-\xi )\right),\end{array}$$

(28)

where the coefficients *ω*_{2}* _{q}*’s and

$$\begin{array}{l}{\omega}_{2q}=(2m+2q+1)\sum _{s=0}^{q}\sum _{t=s}^{q}{(-1)}^{t+q}{2}^{m+2s+1}\\ \xb7\frac{(2m+q+t)!(2m+2s)!(2t)!(t+m+s)!}{t!(m+t)!(q-t)!(2s)!(t-s)!(2t+2m+2s+1)!}{d}_{2s}^{m(m+2r)},\\ {\omega}_{2q+1}=(2m+2q+3)\sum _{s=0}^{q}\sum _{t=s}^{q}{(-1)}^{t+q}{2}^{m+2s+2}\\ \xb7\frac{(2m+q+t+2)!(2m+2s+1)!(2t+1)!(t+m+s+1)!}{t!(m+t+1)!(q-t)!(2s+1)!(t-s)!(2t+2m+2s+3)!}{d}_{2s+1}^{m(m+2r+1)}.\end{array}$$

Let us consider again a point charge *q* located at the point **r**_{s} = (*ξ*_{0}*, η*_{0}*, *_{0} = 0) in the plane *y* = 0 inside the dielectric oblate spheroid defined by the equation *ξ* = *ξ*_{1} (so *ξ*_{1} > *ξ*_{0} ≥ 0); see Fig. 1. Inside the spheroid, the total potential still takes the forms (4) and (5). On the other hand, since the potential outside the oblate spheroid (*ξ*_{1} ≤ *ξ*) must be finite and vanishes at *ξ* = ∞, it should take on the form

$${\mathrm{\Psi}}_{\text{out}}(\mathbf{r})=\frac{q}{{\epsilon}_{\text{i}}a}\sum _{n=0}^{\infty}\sum _{m=0}^{n}({A}_{mn}cosm\phi +{B}_{mn}sinm\phi ){S}_{mn}(\eta ){R}_{mn}(\xi ),$$

(29)

where *A _{mn}* and

Once again, by using the orthogonality of cos *m* and sin *m* together with the boundary conditions of (6), one can obtain *B _{mn}* =

$$\sum _{n=m}^{\infty}[{S}_{mn}(\eta ){R}_{mn}({\xi}_{1}){A}_{mn}-{P}_{n}^{m}(\eta ){P}_{n}^{m}(i{\xi}_{1}){C}_{mn}]=\sum _{n=m}^{\infty}{H}_{mn}{P}_{n}^{m}(\eta ){P}_{n}^{m}({\eta}_{0}){P}_{n}^{m}(i{\xi}_{0}){Q}_{n}^{m}(i{\xi}_{1}),$$

(30)

$$\sum _{n=m}^{\infty}[{\epsilon}_{\text{o}}{S}_{mn}(\eta ){R}_{mn}^{\prime}({\xi}_{1}){A}_{mn}-i{\epsilon}_{\text{i}}{P}_{n}^{m}(\eta ){{P}_{n}^{m}}^{\prime}(i{\xi}_{1}){C}_{mn}]=i\sum _{n=m}^{\infty}{\epsilon}_{\text{i}}{H}_{mn}{P}_{n}^{m}(\eta ){P}_{n}^{m}({\eta}_{0}){P}_{n}^{m}(i{\xi}_{0}){{Q}_{n}^{m}}^{\prime}(i{\xi}_{1}).$$

(31)

Using (20) to rewrite *S _{mn}*(

$$\sum _{r=0,1}^{\infty}\prime {d}_{n-m}^{m(m+r)}{R}_{m(m+r)}({\xi}_{1}){A}_{m(m+r)}-{P}_{n}^{m}(i{\xi}_{1}){C}_{mn}={H}_{mn}{P}_{n}^{m}({\eta}_{0}){P}_{n}^{m}(i{\xi}_{0}){Q}_{n}^{m}(i{\xi}_{1}),$$

(32)

$${\epsilon}_{\text{o}}\sum _{r=0,1}^{\infty}\prime {d}_{n-m}^{m(m+r)}{R}_{m(m+r)}^{\prime}({\xi}_{1}){A}_{m(m+r)}-i{\epsilon}_{\text{i}}{{P}_{n}^{m}}^{\prime}(i{\xi}_{1}){C}_{mn}=i{\epsilon}_{\text{i}}{H}_{mn}{P}_{n}^{m}({\eta}_{0}){P}_{n}^{m}(i{\xi}_{0}){{Q}_{n}^{m}}^{\prime}(i{\xi}_{1}).$$

(33)

For a fixed value of *m*, (32) and (33) for all *n* = *m, m* + 1, ···, constitute a system of an infinite set of equations, but the system can be completely decoupled into two independent systems, one for *A _{mm}*,

In practice, the series expansion of the angular function *S _{mn}*(

$${\mathrm{\Psi}}_{\text{out}}(\mathbf{r})\approx \frac{q}{{\epsilon}_{\text{i}}a}\sum _{n=0}^{N}\sum _{m=0}^{n}{A}_{mn}cosm\phi {S}_{mn}(\eta ){R}_{mn}(\xi ),$$

(34)

and

$${\mathrm{\Psi}}_{\text{in}}(\mathbf{r})\approx \frac{q}{{\epsilon}_{\text{i}}\mid \mathbf{r}-{\mathbf{r}}_{\text{s}}\mid}+\frac{q}{{\epsilon}_{\text{i}}a}\sum _{n=0}^{N}\sum _{m=0}^{n}{C}_{mn}cosm\phi {P}_{n}^{m}(\eta ){P}_{n}^{m}(i\xi ),$$

(35)

where *N*, the truncation number of the summation over *n*, must be large enough to make the truncation error negligible for a certain physical application. Accordingly, for each fixed *m* (0 ≤ *m* ≤ *N*), the regular oblate spheroidal wave functions *S _{mn}*(

$${S}_{mn}(\eta )\approx \sum _{r=0,1}^{{r}_{mn}}\prime {d}_{r}^{mn}{P}_{m+r}^{m}(\eta ),$$

(36)

where *r _{mn}* = (

Then for each fixed *m*, the expansion coefficients *A _{m,m}*,

$${\stackrel{\u2323}{\mathcal{K}}}_{\sigma}{\stackrel{\u2323}{\mathbf{A}}}_{\sigma}={\stackrel{\u2323}{\mathbf{F}}}_{\sigma},\phantom{\rule{0.38889em}{0ex}}\sigma =0,1,$$

(37)

where

$${\stackrel{\u2323}{\mathcal{K}}}_{\sigma}=\left[\begin{array}{cccc}{k}_{\sigma ,\sigma}& {k}_{\sigma ,\sigma +2}& \cdots & {k}_{\sigma ,\sigma +2{N}_{m}^{(\sigma )}}\\ {k}_{\sigma ,2,\sigma}& {k}_{\sigma +2,\sigma +2}& \cdots & {k}_{\sigma +2,\sigma +2{N}_{m}^{(\sigma )}}\\ \vdots & \vdots & \ddots & \vdots \\ {k}_{\sigma +2{N}_{m}^{(\sigma )},\sigma}& {k}_{\sigma +2{N}_{m}^{(\sigma )},\sigma +2}& \cdots & {k}_{\sigma +2{N}_{m}^{(\sigma )},\sigma +2{N}_{m}^{(\sigma )}}\end{array}\right],$$

and

$${\stackrel{\u2323}{\mathbf{A}}}_{\sigma}=\left(\begin{array}{c}{A}_{m,m+\sigma}\\ {A}_{m,m+\sigma +2}\\ \vdots \\ {A}_{m,m+\sigma +2{N}_{m}^{(\sigma )}}\end{array}\right),\phantom{\rule{0.38889em}{0ex}}{\stackrel{\u2323}{\mathbf{F}}}_{\sigma}=\left(\begin{array}{c}{f}_{\sigma}\\ {f}_{\sigma +2}\\ \vdots \\ {f}_{\sigma +2{N}_{m}^{(\sigma )}}\end{array}\right).$$

Here

$$\begin{array}{l}{k}_{st}={\epsilon}_{\text{i}}{a}_{st}{\widehat{b}}_{s}-{\epsilon}_{\text{o}}{\widehat{a}}_{st}{b}_{s},\\ {f}_{s}={\epsilon}_{\text{i}}({g}_{s}{\widehat{b}}_{s}-{\widehat{g}}_{s}{b}_{s}),\\ {a}_{sr}={d}_{s}^{m(m+r)}{R}_{m(m+r)}({\xi}_{1}),\\ {\widehat{a}}_{sr}={d}_{s}^{m(m+r)}{R}_{m(m+r)}^{\prime}({\xi}_{1}),\\ {b}_{s}={P}_{m+s}^{m}(i{\xi}_{1}),\\ {\widehat{b}}_{s}=i{P}_{m+s}^{{m}^{\prime}}(i{\xi}_{1}),\\ {g}_{s}={H}_{m(m+s)}{P}_{m+s}^{m}({\eta}_{0}){P}_{m+s}^{m}(i{\xi}_{0}){Q}_{m+s}^{m}(i{\xi}_{1}),\\ {\widehat{g}}_{s}=i{H}_{m(m+s)}{P}_{m+s}^{m}({\eta}_{0}){P}_{m+s}^{m}(i{\xi}_{0}){Q}_{m+s}^{{m}^{\prime}}(i{\xi}_{1}).\end{array}$$

And ${N}_{m}^{(\sigma )}$ is defined as

$${N}_{m}^{(0)}={N}_{m}^{(1)}=\frac{N-m-1}{2}$$

if *N* − *m* is odd, and as

$${N}_{m}^{(0)}=\frac{N-m}{2},\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}{N}_{m}^{(1)}=\frac{N-m-2}{2}$$

if *N* − *m* is even.

The expansion coefficients *C _{m,m}*,

$${\stackrel{\u2323}{\mathcal{B}}}_{\sigma}{\stackrel{\u2323}{\mathbf{C}}}_{\sigma}={\stackrel{\u2323}{\mathcal{N}}}_{\sigma}{\stackrel{\u2323}{\mathbf{A}}}_{\sigma}-{\stackrel{\u2323}{\mathbf{G}}}_{\sigma},\phantom{\rule{0.38889em}{0ex}}\sigma =0,1,$$

where

$$\begin{array}{l}{\stackrel{\u2323}{\mathcal{B}}}_{\sigma}=\text{diag}({b}_{\sigma},{b}_{\sigma +2},\cdots ,{b}_{\sigma +2{N}_{m}^{(\sigma )}}),\\ {\stackrel{\u2323}{\mathcal{N}}}_{\sigma}=\left[\begin{array}{cccc}{a}_{\sigma ,\sigma}& {a}_{\sigma ,\sigma +2}& \cdots & {a}_{\sigma ,\sigma +2{N}_{m}^{(\sigma )}}\\ {a}_{\sigma +2,\sigma}& {a}_{\sigma +2,\sigma +2}& \cdots & {a}_{\sigma +2,\sigma +2{N}_{m}^{(\sigma )}}\\ \vdots & \vdots & \ddots & \vdots \\ {a}_{\sigma +2{N}_{m}^{(\sigma )},\sigma}& {a}_{\sigma +2{N}_{m}^{(\sigma )},\sigma +2}& \cdots & {a}_{\sigma +2{N}_{m}^{(\sigma )},\sigma +2{N}_{m}^{(\sigma )}}\end{array}\right],\end{array}$$

and

$${\stackrel{\u2323}{\mathbf{C}}}_{\sigma}=\left(\begin{array}{c}{C}_{m,m+\sigma}\\ {C}_{m,m+\sigma +2}\\ \vdots \\ {C}_{m,m+\sigma +2{M}_{m}^{(\sigma )}}\end{array}\right),\phantom{\rule{0.38889em}{0ex}}{\stackrel{\u2323}{\mathbf{G}}}_{\sigma}=\left(\begin{array}{c}{g}_{\sigma}\\ {g}_{\sigma +2}\\ \vdots \\ {g}_{\sigma +2{N}_{m}^{(\sigma )}}\end{array}\right).$$

To illustrate the simulated electric potential of point charges inside oblate spheroids using the direct series expansions (9) and (10), we let an oblate spheroid of dielectric constant *ε*_{i} = 2 be embedded in a dielectric medium of dielectric constant *ε*_{o} = 8. The oblate spheroid is defined by (*x*^{2} +*y*^{2})/4+*z*^{2} = 1, which leads to *ξ*_{1} = 0.57735. Figs. 2(a)–(b) show the equipotential contours in the plane *y* = 0 for the computed total potential with using *N* = 20 due to a unit point charge inside this oblate spheroid located at the point **x**_{0} = (1*,* 0*,* 0) (*ξ*_{0} = 0) or **x**_{0} = (1*,* 0*,* 0.5) (*ξ*_{0} = 0.344532), respectively. As been exhibited, the electric potential is continuous but its normal derivative is discontinuous across the spheroidal surface.

Equipotential contours in the plane *y* = 0 for the computed total potential due to a unit point charge located at the point **x**_{0} = (1*,* 0*,* 0) or **x**_{0} = (1*,* 0*,* 0.5) inside the oblate spheroid (*x*^{2} + *y*^{2})/4 + *z*^{2} = 1, respectively.

To investigate the convergence of the series expansions (9) and (10), the electric potentials due to point charges at four typical locations computed with using different *N* values are compared to those computed with using *N* = 20. Fig. 3 plots the relative error *E* of the computed electric potential (**x*** _{i}*) at 101

$$E={\left(\frac{{\sum}_{i=1}^{{101}^{3}}\mid \mathrm{\Psi}({\mathbf{x}}_{i})-\overline{\mathrm{\Psi}}({\mathbf{x}}_{i}){\mid}^{2}}{{\sum}_{i=1}^{{101}^{3}}\mid \mathrm{\Psi}({\mathbf{x}}_{i}){\mid}^{2}}\right)}^{1/2}.$$

Here Ψ (**x*** _{i}*) represents the potential at

Similarly, to illustrate the simulated electric potential due to point charges outside oblate spheroids using the direct series expansions (16) and (17), Figs. 4(a)-(b) plot the equipotential contours in the plane *y* = 0 for the computed total potential with using *N* = 20 due to a unit point charge located at the point **x**_{0} = (3*,* 0*,* 0) (*ξ*_{0} = 0) or **x**_{0} = (3*,* 0*,* 1.5) (*ξ*_{0} = 1.732051) outside the same oblate spheroid, respectively.

Equipotential contours in the plane *y* = 0 for the computed total potential due to a unit point charge located at the point **x**_{0} = (3*,* 0*,* 0) or **x**_{0} = (3*,* 0*,* 1.5) outside the oblate spheroid (*x*^{2} + *y*^{2})/4 + *z*^{2} = 1, respectively.

We now consider the situation that the dissimilar dielectric medium outside a dielectric oblate spheroid is an ionic solvent, in which the electrostatic potential due to a point charge inside the oblate spheroid is approximated using the direct expansions (34) and (35). The inverse Debye screening length in the LPBE of (1) is assumed to be *λ* = 4/*a* so the spheroidal parameter is *c* = 2. Again we let an oblate spheroid of dielectric constant *ε*_{i} = 2 be embedded in a dielectric medium of dielectric constant *ε*_{o} = 8, but we first consider an oblate spheroid with relatively low ellipticity of 1/4 defined by (*x*^{2}+*y*^{2})/4+4*z*^{2}/9 = 1, which leads to *ξ*_{1} = 1.133893. Therefore, the expansion (26) for
${R}_{mn}^{(3)}(c,i\xi )$ can be and is actually used in programming since in this case *cξ* > 2. Figs. 5(a)–(b) again show the equipotential contours in the plane *y* = 0 for the computed total potential with using *N* = 20 due to a unit point charge inside this oblate spheroid located at the point **x**_{0} = (1*,* 0*,* 0) (*ξ*_{0} = 0) or **x**_{0} = (1*,* 0*,* 0.75) (*ξ*_{0} = 0.718262), respectively.

Equipotential contours in the plane *y* = 0 for the computed total potential due to a unit point charge located at the point **x**_{0} = (1*,* 0*,* 0) or **x**_{0} = (1*,* 0*,* 0.75) inside the oblate spheroid (*x*^{2} + *y*^{2})/4 + 4*z*^{2}/9 = 1, respectively.

Next we consider the oblate spheroid with relatively high ellipticity of 1/2 defined by (*x*^{2} + *y*^{2})/4 + *z*^{2} = 1, which leads to *ξ*_{1} = 0.57735. For this case, the expansion (26) for
${R}_{mn}^{(3)}(c,i\xi )$ has been found inadequate so the expansion described by (27)–(28) is instead used. Figs. 6(a)–(b) plot the equipotential contours in the plane *y* = 0 for the computed total potential with using *N* = 20 due to a unit point charge inside this oblate spheroid located at the point **x**_{0} = (1*,* 0*,* 0) (*ξ*_{0} = 0) or **x**_{0} = (1*,* 0*,* 0.5) (*ξ*_{0} = 0.344532), respectively.

Equipotential contours in the plane *y* = 0 for the computed total potential due to a unit point charge located at the point **x**_{0} = (1*,* 0*,* 0) or **x**_{0} = (1*,* 0*,* 0.5) inside the oblate spheroid (*x*^{2} + *y*^{2})/4 + *z*^{2} = 1, respectively.

Finally, the series expansions (34) and (35) can be demonstrated to be numerically convergent with the convergence rate depending on the location of the source charge. To this end, for the foregoing oblate spheroids defined by (*x*^{2} + *y*^{2})/4 + 4*z*^{2}/9 = 1 and (*x*^{2} + *y*^{2})/4 + *z*^{2} = 1 respectively, the electric potentials due to point charges at four typical locations computed with using different *N* values are again compared to those computed with using *N* = 20. Figs. 7(a)-(b) plot the relative error *E* of the computed electric potential at 101^{3} observation points uniformly located inside the rectangular box [−1.25*a*_{1}*,* 1.25*a*_{1}]^{2} *×* [−1.5*a*_{2}*,* 1.5*a*_{2}] for various *N* values. As shown in Fig. 7, for both oblate spheroids, the series solutions converge for all four charge locations, but they converge much slower when the source charge is closer to the spheroidal boundary. For example, in the case of (*x*^{2} + *y*^{2})/4 + 4*z*^{2}/9 = 1, the series solutions converge much faster for the **x**_{0} = (0*,* 0*,* 0) (*ξ*_{0} = 0) and **x**_{0} = (1*,* 0*,* 0) (*ξ*_{0} = 0) cases than for the **x**_{0} = (0*,* 0*,* 0.75) (*ξ*_{0} = 0.566947) and **x**_{0} = (1*,* 0*,* 0.75) (*ξ*_{0} = 0.718262) cases. In addition, the results shown in Fig. 7 appear to have demonstrated that the convergence rate is slower for oblate spheroids of higher ellipticity, agreeing with the common understanding that oblate spheroids of higher ellipticity or oblateness are more difficult to handle.

In this paper, the series expansions of the electric potential of point charges inside a dielectric oblate spheroid are presented in terms of the associated Legendre functions or the spheroidal wave functions, depending on whether the surrounding dissimilar dielectric medium of the spheroid is ionic or non-ionic. Numerical experiments have demonstrated the convergence of the series solutions for both cases.

The author thanks the support of the National Institutes of Health (grant number: 1R01GM083600-01) for the work reported in this note.

**Publisher's Disclaimer: **This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

1. Deng S. Electrostatic potential of point charges inside dielectric prolate spheroids. Journal of Electrostatics. 2008;66:549–560. [PMC free article] [PubMed]

2. Okur A, Simmerling C. Hybrid explicit/implicit solvation methods. In: Spellmeyer D, editor. Annu Rep Comput Chem. 2 chap 6. 2006. pp. 97–109.

3. Morse PM, Feshbach H. Methods of Theoretical Physcs. McGraw-Hill; New York: 1953.

4. Hobson EW. The Theory of Spherical And Ellipsoidal Harmonics. Cambridge University Press; 1931.

5. Smythe WR. Static and Dynamic Electricity. Hemisphere; New York: 1989.

6. Flammer C. Spheroidal Wave Functions. Stanford University Press; Stanford, CA: 1957.

7. Stratton J, Morse PM, Chu LJ, et al. Spheroidal Wave Functions. Wiley; New York: 1956.

8. Kirby P. Calculation of spheroidal wave functions. Comput Phys Commun. 2006;175:465–472.

9. Falloon PE, Abbott PC, Wang JB. Theory and computation of spheroidal wavefunctions. J Phys A: Math Gen. 2003;36:5477–5495.

10. Li L-W, Kang X-K, Leong M-S. Spheroidal Wave Functions in Electromagnetic Theory. Wiley; New York: 2002.

11. Li LW, Leong MS, Yeo TS, et al. Computations of spheroidal harmonics with complex arguments: A review with an algorithm. Phys Rev E. 1998;58:6792–6806.

12. Thompson WJ. Spheroidal wave functions. Comput Sci Eng. 1999;1:84–87.

13. Beu TA, Cmpeanu RI. Prolate radial spheroidal wave functions. Comput Phys Commun. 1983;30:177–185.

14. Hodge DB. Eigenvalues and eigenfunctions of the spheroidal wave equation. J Math Phys. 1970;11:2308–2312.

15. Yoon BJ, Kim S. Electrophoresis of spheroidal particles. J Colloid Interface Sci. 1989;128:275–288.

16. Hsu JP, Liu BT. Solution to the linearized Poisson-Boltzmann equation for a spheroidal surface under a general surface condition. J Colloid Interface Sci. 1996;183:214–222.

17. Hsu JP, Liu BT. Exact solution to the linearized Poisson-Boltzmann equation for spheroidal surfaces. J Colloid Interface Sci. 1996;178:785–788.

18. Aoi T. On spheroidal functions. J Phys Soc Japan. 1955;10:130–141.

PubMed Central Canada is a service of the Canadian Institutes of Health Research (CIHR) working in partnership with the National Research Council's national science library in cooperation with the National Center for Biotechnology Information at the U.S. National Library of Medicine(NCBI/NLM). It includes content provided to the PubMed Central International archive by participating publishers. |