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Logo of nihpaAbout Author manuscriptsSubmit a manuscriptHHS Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
 
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.2009.06.007
PMCID: PMC2764543
NIHMSID: NIHMS132122

Electrostatic potential of point charges inside dielectric oblate spheroids

Abstract

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.

Keywords: Electrostatic potential, Oblate spheroid, Poisson equation, Linearized Poisson-Boltzmann equation, Spheroidal wave function

1. Introduction

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 Ψin(r) inside the cavity is given by the solution of the Poisson equation

·(εiΨin(r))=4πqδ(rrs),

where εi is the electric permittivity of the dielectric cavity, rs 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 Ψout(r) is then given by the solution of the linearized Poisson-Boltzmann equation (LPBE)

2Ψout(r)λ2Ψout(r)=0,
(1)

where λ is the inverse Debye screening length determined by the ionic concentration and the dielectric constant εo 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

ΨoutΓ=ΨinΓandεoΨoutnΓ=εiΨinnΓ,
(2)

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

2. An oblate spheroid immersed in the pure water solvent

Let us first consider a dielectric oblate spheroid of permittivity εi centered at the origin and embedded in a homogeneous dielectric medium of permittivity εo, 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

Fig. 1
A point charge and a dielectric oblate spheroid.
x2+y2a12+z2a22=1,a1>a2.

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

x=a2(1+ξ2)(1η2)cosφ,y=a2(1+ξ2)(1η2)sinφ,z=a2ξη,

where a=2a12a22 is the interfocal distance of the oblate spheroid, ξ [set membership] [0, ∞) is the radial variable, η [set membership] [−1, 1] is the angular variable, and [var phi] [set membership] [0, 2π] 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.

{whenz0:ξ=(4r2a2)+(4r2a2)2+16a2z22a,η=22z(4r2a2)+(4r2a2)2+16a2z2,whenz=0:ξ=4r2a21,η=0,ifra2,η=±14r2a2,ξ=0,ifra2,wherer2=x2+y2+z2,andinallcasesφ=arctanyx.

Without loss of generality, let a point charge q be located at the point rs = (ξ0, η0, [var phi]0=0) in the plane y = 0 inside the dielectric spheroid defined by the equation ξ = ξ1 (so ξ1 ξ0 ≥ 0), as shown in Fig. 1. Outside the spheroid (ξ1ξ), the total electrostatic potential takes the form

Ψout(r)=qεian=0m=0n(Amncosmφ+Bmnsinmφ)Pnm(η)Qnm(iξ),
(3)

and inside the spheroid (0 ≤ ξξ1), it is written as

Ψin(r)=qεirrs+qεian=0m=0n(Cmncosmφ+Dmnsinmφ)Pnm(η)Pnm(iξ),
(4)

where i=1,Pnm(x) and Qnm(x) are the associated Legendre functions of the first kind and the second kind, respectively, and Amn, Bmn, Cmn and Dmn are the unknown expansion coefficients.

Note that the general expansion of 1/|rrs| in the oblate spheroidal coordinates is given by [35]

1rrs=1an=0m=0nHmncosmφPnm(η0)Pnm(η){Pnm(iξ0)Qnm(iξ),ξ>ξ0,Pnm(iξ)Qnm(iξ0),ξ<ξ0,

where

Hmn=2i(2n+1)(2δm0)(1)m[(nm)!(n+m)!]2.

Here δm0 is the Kronecker delta. Therefore, the total potential inside the oblate spheroid can also be written as

Ψin(r)=qεiai=0m=0n(Cmncosmφ+Dmnsinmφ)Pnm(η)Pnm(iξ)+qεian=0m=0nHmncosmφPnm(η0)Pnm(η){Pnm(iξ0)Qnm(iξ),ξ>ξ0,Pnm(iξ)Qnm(iξ0),ξ<ξ0.
(5)

The expansion coefficients Amn, Bmn, Cmn and Dmn in (3) and (4) are to be determined by the boundary conditions of (2) which, under the oblate spheroidal coordinates, become

Ψoutξ=ξ1=Ψinξ=ξ1andεoΨoutξξ=ξ1=εiΨinξξ=ξ1.
(6)

Then, by using the orthogonality of cos m[var phi] and sin m[var phi] as well as that of Pnm(x), one can obtain Bmn = Dmn = 0, and

Qnm(iξ1)AmnPnm(iξ1)Cmn=HmnPnm(η0)Pnm(iξ0)Qnm(iξ1),
(7)

εoQ^nm(iξ1)AmnεiP^nm(iξ1)Cmn=εiHmnPnm(η0)Pnm(iξ0)Q^nm(iξ1),
(8)

where

P^nm(iξ1)=(nm+1)Pn+1m(iξ1)(n+1)iξ1Pnm(iξ1),Q^nm(iξ1)=(nm+1)Qn+1m(iξ1)(n+1)iξ1Qnm(iξ1).

Further solving (7) and (8) leads to

Amn=εiHmnPnm(η0)Pnm(iξ0)Kmn1(Pnm(iξ1)Q^nm(iξ1)Qnm(iξ1)P^nm(iξ1)),

and

Cmn=(εiεo)HmnPnm(η0)Pnm(iξ0)Kmn1Qnm(iξ1)Q^nm(iξ1),

where

Kmn=εoPnm(iξ1)Q^nm(iξ1)εiQnm(iξ1)P^nm(iξ1).

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

Ψin(r)=qεirrs+qεian=0m=0n(εiεo)HmnPnm(η0)Pnm(iξ0)Kmn1×Qnm(iξ1)Q^nm(iξ1)cosmφPnm(η)Pnm(iξ),
(9)

while the potential outside the oblate spheroid is

Ψout(r)=qεian=0m=0nεiHmnPnm(η0)Pnm(iξ0)Kmn1×(Pnm(iξ1)Q^nm(iξ1)Qnm(iξ1)P^nm(iξ1))cosmφPnm(η)Qnm(iξ).
(10)

Similarly, when the point charge q is located at the point rs = (ξ0, η0, [var phi]0=0) outside the dielectric spheroid defined by the equation ξ = ξ1 (so ξ0 > ξ1 > 0), the electrostatic potential outside the spheroid takes the form

Ψout(r)=qεorrs+qεoan=0m=0nAmncosmφPnm(η)Qnm(iξ),
(11)

or

Ψout(r)=qεoan=0m=0nAmncosmφPnm(η)Qnm(iξ)+qεoan=0m=0nHmncosmφPnm(η0)Pnm(η){Pnm(iξ0)Qnm(iξ),ξ>ξ0,Pnm(iξ)Qnm(iξ0),ξ<ξ0.
(12)

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

Ψin(r)=qεoan=0m=0nCmncosmφPnm(η)Pnm(iξ).
(13)

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

Qnm(iξ1)AmnPnm(iξ1)Cmn=HmnPnm(η0)Pnm(iξ1)Qnm(iξ0),
(14)
εoQ^nm(iξ1)AmnεiP^nm(iξ1)Cmn=εoHmnPnm(η0)P^nm(iξ1)Qnm(iξ0).
(15)

Solving (14) and (15) leads to

Amn=(εiεo)HmnPnm(η0)Qnm(iξ0)Kmn1Pnm(iξ1)P^nm(iξ1),

and

Cmn=εoHmnPnm(η0)Qnm(iξ0)Kmn1(Pnm(iξ1)Q^nm(iξ1)Qnm(iξ1)P^nm(iξ1)).

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

Ψout(r)=qεorrs+qεoan=0m=0n(εiεo)HmnPnm(η0)Qnm(iξ0)Kmn1×Pnm(iξ1)P^nm(iξ1)cosmφPnm(η)Qnm(iξ),
(16)

while the potential inside the oblate spheroid is

Ψin(r)=qεoan=0m=0nεoHmnPnm(η0)Qnm(iξ0)Kmn1×(Pnm(iξ1)Q^nm(iξ1)Qnm(iξ1)P^nm(iξ1))cosmφPnm(η)Pnm(iξ).
(17)

3. An oblate spheroid immersed in an ionic solvent

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

ξ[(ξ2+1)Ψξ]+η[(1η2)Ψη]+ξ2+η2(ξ2+1)(1η2)2Ψφ2c2(ξ2+η2)Ψ=0,

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

Ψ(ξ,η,φ)=Smn(c,η)Rmn(c,iξ)(cosmφsinmφ),

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

ddη[(1η2)dSmn(c,η)dη]+[λmn(c)c2η2m21η2]Smn(c,η)=0,
(18)

ddξ[(ξ2+1)dRmn(c,iξ)dξ][λmn(c)+c2ξ2m2ξ2+1]Rmn(c,iξ)=0,
(19)

where those values λmn(c) for which (18) admits solutions that are finite at η = ±1 are eigenvalues of the differential equation (18), and Smn(c, η) and Rmn(c, iξ) are the eigenfunctions associated with the eigenvalue λmn(c).

3.1 The angular function Smn(c, η)

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 Smn(1)(c,η) 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

Smn(1)(c,η)=r=0,1drmn(c)Pm+rm(η).
(20)

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

The eigenvalues λmn(c) and the angular expansion coefficients drmn(c) in (20) satisfy the following recursion formula for r ≥ 0 [6]

αrm(c)dr+2mn(c)+(βrm(c)λmn(c))drmn(c)+γrm(c)dr2mn(c)=0,
(21)

in which

αrm(c)=(2m+r+2)(2m+r+1)c2(2m+2r+3)(2m+2r+5),
(22)
βrm(c)=(m+r)(m+r+1)+2(m+r)(m+r+1)2m21(2m+2r1)(2m+2r+3)c2,
(23)
γrm(c)=r(r1)c2(2m+2r3)(2m+2r1).
(24)

The eigenvalues are determined by the condition that drmn(c)0 as r → ∞, and various methods have been proposed to calculate the eigenvalues and the expansion coefficients; see [614] and references therein.

3.2 The radial function Rmn(c, iξ)

In the standard theory, the prolate spheroidal radial wave function Rmn(c, ξ) for the Helmholtz equation satisfies the differential equation

ddξ[(ξ21)dRmn(c,ξ)dξ][λmn(c)c2ξ2+m2ξ21]Rmn(c,ξ)=0.
(25)

Note that Eq. (19) can be obtained from Eq. (25) by replacing ξ with . The standard prolate radial functions are usually expanded in a basis of spherical Bessel functions. Accordingly, the radial solution Rmn(c, iξ) to (19) can be expanded in the modified spherical Bessel functions [15]

Rmn(1)(c,iξ)=1ρmn(c)(ξ2+1ξ2)m2r=0,1drmn(c)(2m+r)!r!im+r(cξ),

and

Rmn(3)(c,iξ)=1ρmn(c)(ξ2+1ξ2)m2r=0,1drmn(c)(2m+r)!r!km+r(cξ),
(26)

where in(r) and kn(r) are the modified spherical Bessel functions of the first kind and the third kind, respectively, defined as

in(r)=π2rIn+1/2(r),kn(r)=π2rKn+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 ρmn(c) is given by

ρmn(c)=r=0,1drmn(c)(2m+r)!r!.

When expressing the electric potential outside the oblate spheroid where ξξ1 > 0, only Rmn(3)(c,iξ) needs to be used since it is finite outside the oblate spheroid and vanishes at ξ = ∞. For convenience, drmn(c) is simply denoted by drmn,Smn(1)(c,η) by Smn(η), and Rmn(3)(c,iξ) by Rmn(ξ), respectively, for the rest of this note. In particular, i is dropped since Rmn(3)(c,iξ) is actually real-valued.

Finally, as pointed out in [16], kn() approaches infinity and the expansion (26) diverges and thus becomes inadequate when the argument is fairly small. This could be the case when the inverse Debye screening length λ, which is proportional to the square root of the ionic concentration of the solvent medium, is small, or when the aspect ratio of the oblate spheroid a2/a1 is high (so the interfocal distance a is small). This may also happen even when the aspect ratio of the oblate spheroid is low because ξ1, which defines the oblate spheroidal, may be small in this case. Therefore, it is desirable to deduce appropriate expressions for Rmn(3)(c,iξ) which will be more conveniently used when the argument is small [1518]. We use an expansion modified from that given in [16], which, as far as we are aware, seems to be new. The constant multipliers are so chosen that the modified expansion is consistent with (26) when they both converge.

Rm(m+2r)(3)(c,iξ)=2m+12(m)!d0m(m+2r)cm·(ξ2+1)m2·q=0ω2qkm+q(c2(ξ2+1+ξ))im+q(c2(ξ2+1ξ)),
(27)

Rm(m+2r+1)(3)(c,iξ)=2m+3(m)!d1m(m+2r+1)cm+1·ξ(ξ2+1)m2·q=0ω2q+1km+q+1(c2(ξ2+1+ξ))im+q+1(c2(ξ2+1ξ)),
(28)

where the coefficients ω2q’s and ω2q+1’s are

ω2q=(2m+2q+1)s=0qt=sq(1)t+q2m+2s+1·(2m+q+t)!(2m+2s)!(2t)!(t+m+s)!t!(m+t)!(qt)!(2s)!(ts)!(2t+2m+2s+1)!d2sm(m+2r),ω2q+1=(2m+2q+3)s=0qt=sq(1)t+q2m+2s+2·(2m+q+t+2)!(2m+2s+1)!(2t+1)!(t+m+s+1)!t!(m+t+1)!(qt)!(2s+1)!(ts)!(2t+2m+2s+3)!d2s+1m(m+2r+1).

3.3 Series solution of the electric potential

Let us consider again a point charge q located at the point rs = (ξ0, η0, [var phi]0 = 0) in the plane y = 0 inside the dielectric oblate spheroid defined by the equation ξ = ξ1 (so ξ1 > ξ0 ≥ 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 (ξ1ξ) must be finite and vanishes at ξ = ∞, it should take on the form

Ψout(r)=qεian=0m=0n(Amncosmφ+Bmnsinmφ)Smn(η)Rmn(ξ),
(29)

where Amn and Bmn are the unknown expansion coefficients.

Once again, by using the orthogonality of cos m[var phi] and sin m[var phi] together with the boundary conditions of (6), one can obtain Bmn = Dmn = 0, and

n=m[Smn(η)Rmn(ξ1)AmnPnm(η)Pnm(iξ1)Cmn]=n=mHmnPnm(η)Pnm(η0)Pnm(iξ0)Qnm(iξ1),
(30)
n=m[εoSmn(η)Rmn(ξ1)AmniεiPnm(η)Pnm(iξ1)Cmn]=in=mεiHmnPnm(η)Pnm(η0)Pnm(iξ0)Qnm(iξ1).
(31)

Using (20) to rewrite Smn(η) in series of the associated Legendre polynomials, and then applying the orthogonality of the associated Legendre polynomials, one can arrive at the following equations for the expansion coefficients

r=0,1dnmm(m+r)Rm(m+r)(ξ1)Am(m+r)Pnm(iξ1)Cmn=HmnPnm(η0)Pnm(iξ0)Qnm(iξ1),
(32)
εor=0,1dnmm(m+r)Rm(m+r)(ξ1)Am(m+r)iεiPnm(iξ1)Cmn=iεiHmnPnm(η0)Pnm(iξ0)Qnm(iξ1).
(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 Amm, Am(m+2), ···, Cmm, Cm(m+2), ···, and the other for Am(m+1), Am(m+3), ···, Cm(m+1), Cm(m+3), ···.

In practice, the series expansion of the angular function Smn(η), Eq. (20), is presumed to be a convergent representation. Hence, one can perform the computation based on a finite set of equations. More specifically, let us assume that the series expansions of the potentials are truncated as

Ψout(r)qεian=0Nm=0nAmncosmφSmn(η)Rmn(ξ),
(34)

and

Ψin(r)qεirrs+qεian=0Nm=0nCmncosmφPnm(η)Pnm(iξ),
(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 ≤ mN), the regular oblate spheroidal wave functions Smn(η), n = m, m + 1, ···, N, are truncated as

Smn(η)r=0,1rmndrmnPm+rm(η),
(36)

where rmn = (Nm) − σ. Here σ = 0 if Nn is even, and σ = 1 if Nn is odd.

Then for each fixed m, the expansion coefficients Am,m, Am,m+1, ···, Am,N are obtained by solving the following two independent systems

KσAσ=Fσ,σ=0,1,
(37)

where

Kσ=[kσ,σkσ,σ+2kσ,σ+2Nm(σ)kσ,2,σkσ+2,σ+2kσ+2,σ+2Nm(σ)kσ+2Nm(σ),σkσ+2Nm(σ),σ+2kσ+2Nm(σ),σ+2Nm(σ)],

and

Aσ=(Am,m+σAm,m+σ+2Am,m+σ+2Nm(σ)),Fσ=(fσfσ+2fσ+2Nm(σ)).

Here

kst=εiastb^sεoa^stbs,fs=εi(gsb^sg^sbs),asr=dsm(m+r)Rm(m+r)(ξ1),a^sr=dsm(m+r)Rm(m+r)(ξ1),bs=Pm+sm(iξ1),b^s=iPm+sm(iξ1),gs=Hm(m+s)Pm+sm(η0)Pm+sm(iξ0)Qm+sm(iξ1),g^s=iHm(m+s)Pm+sm(η0)Pm+sm(iξ0)Qm+sm(iξ1).

And Nm(σ) is defined as

Nm(0)=Nm(1)=Nm12

if Nm is odd, and as

Nm(0)=Nm2,Nm(1)=Nm22

if Nm is even.

The expansion coefficients Cm,m, Cm,m+1, ···, Cm,N can then be calculated using

BσCσ=NσAσGσ,σ=0,1,

where

Bσ=diag(bσ,bσ+2,,bσ+2Nm(σ)),Nσ=[aσ,σaσ,σ+2aσ,σ+2Nm(σ)aσ+2,σaσ+2,σ+2aσ+2,σ+2Nm(σ)aσ+2Nm(σ),σaσ+2Nm(σ),σ+2aσ+2Nm(σ),σ+2Nm(σ)],

and

Cσ=(Cm,m+σCm,m+σ+2Cm,m+σ+2Mm(σ)),Gσ=(gσgσ+2gσ+2Nm(σ)).

4. Numerical examples

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 εi = 2 be embedded in a dielectric medium of dielectric constant εo = 8. The oblate spheroid is defined by (x2 +y2)/4+z2 = 1, which leads to ξ1 = 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 x0 = (1, 0, 0) (ξ0 = 0) or x0 = (1, 0, 0.5) (ξ0 = 0.344532), respectively. As been exhibited, the electric potential is continuous but its normal derivative is discontinuous across the spheroidal surface.

Fig. 2
Equipotential contours in the plane y = 0 for the computed total potential due to a unit point charge located at the point x0 = (1, 0, 0) or x0 = (1, 0, 0.5) inside the oblate spheroid (x2 + y2)/4 + z2 = 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 [Psi] (xi) at 1013 observation points xi uniformly located inside the rectangular box [−2.5, 2.5]2 × [−1.5, 1.5] for various N values, where

Fig. 3
Illustration of the convergence rate of the series solutions (9) and (10).
E=(i=11013Ψ(xi)Ψ¯(xi)2i=11013Ψ(xi)2)1/2.

Here Ψ (xi) represents the potential at xi obtained by using the series expansions with N = 20. First of all, for all four cases the series solutions (9) and (10) have been shown numerically convergent, and the error decreases monotonically as N increases. Second, the convergence rate clearly depends on where the point charge is located at. In particular, when the point charge is close to the spheroidal boundary, the convergence rate shall be slow, requiring a large number of terms to achieve high accuracy in the potential field. For instance, the series solutions converge much faster for the x0 = (0, 0, 0) (ξ0 = 0) and x0 = (1, 0, 0) (ξ0 = 0) cases than for the x0 = (0, 0, 0.5) (ξ0 = 0.288675) and x0 = (1, 0, 0.5) (ξ0 = 0.344532) cases.

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 x0 = (3, 0, 0) (ξ0 = 0) or x0 = (3, 0, 1.5) (ξ0 = 1.732051) outside the same oblate spheroid, respectively.

Fig. 4
Equipotential contours in the plane y = 0 for the computed total potential due to a unit point charge located at the point x0 = (3, 0, 0) or x0 = (3, 0, 1.5) outside the oblate spheroid (x2 + y2)/4 + z2 = 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 εi = 2 be embedded in a dielectric medium of dielectric constant εo = 8, but we first consider an oblate spheroid with relatively low ellipticity of 1/4 defined by (x2+y2)/4+4z2/9 = 1, which leads to ξ1 = 1.133893. Therefore, the expansion (26) for Rmn(3)(c,iξ) can be and is actually used in programming since in this case > 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 x0 = (1, 0, 0) (ξ0 = 0) or x0 = (1, 0, 0.75) (ξ0 = 0.718262), respectively.

Fig. 5
Equipotential contours in the plane y = 0 for the computed total potential due to a unit point charge located at the point x0 = (1, 0, 0) or x0 = (1, 0, 0.75) inside the oblate spheroid (x2 + y2)/4 + 4z2/9 = 1, respectively.

Next we consider the oblate spheroid with relatively high ellipticity of 1/2 defined by (x2 + y2)/4 + z2 = 1, which leads to ξ1 = 0.57735. For this case, the expansion (26) for Rmn(3)(c,iξ) 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 x0 = (1, 0, 0) (ξ0 = 0) or x0 = (1, 0, 0.5) (ξ0 = 0.344532), respectively.

Fig. 6
Equipotential contours in the plane y = 0 for the computed total potential due to a unit point charge located at the point x0 = (1, 0, 0) or x0 = (1, 0, 0.5) inside the oblate spheroid (x2 + y2)/4 + z2 = 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 (x2 + y2)/4 + 4z2/9 = 1 and (x2 + y2)/4 + z2 = 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 1013 observation points uniformly located inside the rectangular box [−1.25a1, 1.25a1]2 × [−1.5a2, 1.5a2] 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 (x2 + y2)/4 + 4z2/9 = 1, the series solutions converge much faster for the x0 = (0, 0, 0) (ξ0 = 0) and x0 = (1, 0, 0) (ξ0 = 0) cases than for the x0 = (0, 0, 0.75) (ξ0 = 0.566947) and x0 = (1, 0, 0.75) (ξ0 = 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.

Fig. 7
Illustration of the convergence rate of the series solutions (34) and (35) for oblate spheroids of different ellipticity. (a) The oblate spheroid is (x2 + y2)/4 + 4z2/9 = 1 and the expansion (26) is used for Rmn(3)(c,iξ); (b) The oblate spheroid ...

5. Conclusions

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.

Acknowledgments

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

Footnotes

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