1 Introduction

Levitation of samples in electrostatic field is one of possible method for arising objects in gravitational field by means of electrostatic forces (Table 1). Nowadays, technology is often focused on electrostatic separations in medicament treatments, levitations of liquid drops and particle, alloys and some laboratory investigations [3, 4, 6, 7, 14]. There are also technologies that apply static levitations arising without capacitances [8]. The paper considers electrostatic levitation driven only by material force acting on dielectric samples. Nevertheless, the electrostatic levitation force may act together with another electromagnetic forces [10, 11].

Table 1 Forces densities in electromagnetic field
Full size table

The material forces of electric nature act at the boundary of spheroid (Table 1). While material forces are acting on dielectric spheroid, they could raise it in gravitational field [1, 5, 9]. The spheroid levitation is feasible if electrostatic force is greater than gravitation force mg.

In this paper are carried out the analyses of electrostatic levitation forces for oblate and prolate dielectric spheroids, that extends the analysis for the ball [13]. The analytical solutions are derived by means of variable separation method for electric potential. The distributions of electric field are given by Legendre functions of first and second kinds. The Legendre functions are given by means of hypergeometric series [3].

The analytical solutions are clearly revealing the influence of spheroids parameters (the most interesting is anisotropy) on the field distributions and electrostatic forces. The anisotropy can appear as a result of crystalline feature, rapid solidification, layering, ball crushing or 3D printing of produced spheroid. The permittivity may be adjusted due to saturation or nonlinearity for particular work point, too. The analytical solution presented could be regarded as a default solution. Moreover, the analytical solution is a benchmark tests for numerical algorithms. It may constitute start points for multistep design and can be also a part of hybrid, i.e. analytical–numerical algorithms [2, 5].

The approaches present analytical solutions of electrostatic levitations for diagonally anisotropic dielectric spheroids. The approaches cover analysis of electric field distribution inside and outside the spheroids and multiple levitation force evaluations. The levitations forces are evaluated by the Maxwell stress tensor generalized method, material force, coenergy and equivalent dipole formula (valid only for isotopic spheroids).

The novelty of the approaches lies in the fact that are analytically described:

  • The electric field distributions inside and outside of both oblate and prolate anisotropic dielectric spheroids, and

  • The force arising at the surface of anisotropic dielectric both oblate and prolate spheroids by four methods.

The three semi-axes of spheroid are denoted as follows for either oblate (R, R, h < R) or prolate (R, R, h > R) spheroids, respectively. Both spheroids are geometrically different thus appropriate coordinate systems used are different, too. Field distributions for oblate and prolate spheroids are given by different series and will be separately analysed.

2 Electrostatic levitation of oblate spheroid

Let imposed static electric field be along to z-axis oriented upwards—Fig. 1. In the field is set non-charged, nonconductive, homogeneous and diagonally anisotropic dielectric oblate spheroid.

Fig. 1
figure 1

Anisotropic dielectric and nonconductive oblate spheroid in imposed electrostatic field

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The imposed electric field strength (far from the spheroid) parallel component to axis z is as given below

$$ E_{z} = E_{0} + \sum\limits_{k = 1}^{N - 1} {F_{k} z^{k} } , $$
(1)

where E0 and Fk are given constants (z = 0 for the centre of spheroid). For N > 2 the imposed field gradient is spatially variable. If N = 2, thus the field gradient is constant.

In case E0 ≠ 0 and Fk = 0 (k = 1, 2, 3, …) the field is symmetrical above and below the spheroid, thus force does not appear. In such a case, the presented below solutions constitute solutions for the academic problem of spheroid in a uniform electrostatic field.

The most appropriate orthogonal coordinate system for oblate spheroid electric field analysis is oblate spheroidal coordinates (Fig. 2).

Fig. 2
figure 2

Oblate spheroidal coordinates and semi-axes assignment

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For ce > 0 the oblate spheroidal coordinates η, θ are as follows

$$ \eta = a\cosh \left( {\frac{{d_{2} + d_{1} }}{{2c_{{\text{e}}} }}} \right), $$
(2)
$$ \theta = a\sin \left( {\frac{{d_{2} - d_{1} }}{{2c_{{\text{e}}} }}} \right), $$
(3)

where \(d_{1} = \sqrt {(\rho - c_{{\text{e}}} )^{2} + z^{2} }\), \(d_{2} = \sqrt {(\rho + c_{{\text{e}}} )^{2} + z^{2} }\) are distances of the point P(ρ, z) from the two focuses (+ ce, 0) and (– ce, 0), respectively. The coordinate ηmax = acosh(R/ce) describes the ellipse (d1 + d2 = 2R) at which is located a certain point P(ηmax,θ), and θ is the angle between hyperbole asymptote and axis z (Fig. 2). These coordinates must be always supplemented by the linear eccentricity ce.

Inversely, cylindrical coordinates are as follows

$$ \rho (\eta ,\theta ) = c_{{\text{e}}} {\text{ch}}(\eta )\sin (\theta ), $$
(4)

and

$$ z(\eta ,\theta ) = c_{{\text{e}}} {\text{sh}}(\eta )\cos (\theta ). $$
(5)

The assumption ce > 0 implicates that analytical solutions for neither oblate nor prolate spheroids can be applied for ball (ce ≡ 0). However, the solutions for both spheroids, in limit ce → 0, converge to the solution of the ball [13].

Axially symmetrical electrostatic field problem describes electric scalar potential that is independent from longitudinal angle φ

$$ V(\eta ,\theta ,\varphi ) = V(\eta ,\theta ). $$
(6)

Hence, according to relation

$$ \vec{E} = - \vec{i}_{\eta } \frac{\partial V}{{L_{\eta } \partial \eta }} - \vec{i}_{\theta } \frac{\partial V}{{L_{\theta } \partial \theta }} - \vec{i}_{\varphi } \frac{\partial V}{{L_{\varphi } \partial \varphi }}, $$
(7)

all electric field components are determined, e.g.

$$ E_{\varphi } = 0. $$
(8)

For diagonal (normal, principal axes) anisotropy of medium, the electric displacement field in oblate spheroidal coordinates is as follows

$$ \left[ {\begin{array}{*{20}c} {D_{\eta } } \\ {D_{\theta } } \\ {D_{\varphi } } \\ \end{array} } \right] = \hat{\varepsilon } \left[ {\begin{array}{*{20}c} {E_{\eta } } \\ {E_{\theta } } \\ {E_{\varphi } } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\varepsilon_{\eta \eta } } & 0 & 0 \\ 0 & {\varepsilon_{\theta \theta } } & 0 \\ 0 & 0 & {\varepsilon_{\varphi \varphi } } \\ \end{array} } \right] \left[ {\begin{array}{*{20}c} {E_{\eta } } \\ {E_{\theta } } \\ {E_{\varphi } } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\varepsilon_{\eta \eta } E_{\eta } } \\ {\varepsilon_{\theta \theta } E_{\theta } } \\ {\varepsilon_{\varphi \varphi } E_{\varphi } } \\ \end{array} } \right]. $$
(9)

Gauss’s law for divergence-free electric field is as follows

$$ {\text{div}}\vec{D} = {\text{div}}\left( {\varepsilon_{\eta \eta } D_{\eta } \vec{i}_{\eta } + \varepsilon_{\theta \theta } D_{\theta } \vec{i}_{\theta } } \right) = 0. $$
(10)

From (10) results partial differential equation as follows

$$ \frac{1}{{L_{\eta } L_{\theta } L_{\varphi } }}\left( {\varepsilon_{\eta \eta } \frac{{\partial \left( {L_{\varphi } \frac{\partial V}{{\partial \eta }}} \right)}}{\partial \eta } + \varepsilon_{\theta \theta } \frac{{\partial \left( {L_{\varphi } \frac{\partial V}{{\partial \theta }}} \right)}}{\partial \theta }} \right) = 0. $$
(11)

For variable separation in the form of

$$ V(\eta ,\theta ) = H(\eta )\Theta (\theta ), $$
(12)

is satisfied

$$ \frac{{\varepsilon_{\eta \eta } }}{{\varepsilon_{\theta \theta } {\text{ch}}(\eta )}}\frac{{\partial \left( {{\text{ch}}(\eta )\frac{\partial H}{{\partial \eta }}} \right) }}{H\partial \eta } = \frac{ - 1}{{ \sin (\theta )}}\frac{{\partial \left( {\sin (\theta )\frac{\partial \Theta }{{\partial \theta }}} \right)}}{{\Theta \partial \theta }}. $$
(13)

The Legendre polynomials Pn(cos(θ)), for integer n > 0, are solutions Θ(θ) of the Lagrange equation in the form of

$$ n(n + 1)\Theta (\theta ) + \frac{1}{\sin \theta }\frac{{\text{d}}}{{{\text{d}}\theta }}\left( {\sin \theta \frac{{{\text{d}}(\Theta (\theta ))}}{{{\text{d}}\theta }}} \right) = 0. $$
(14)

The second linearly independent solution constitutes Legendre functions denoted by Qn(cos(θ)) ([3], pp. 958–960). Subsequently, it can be written equation

$$ \frac{1}{{{\text{ch}}(\eta )}}\frac{{{\text{d}}\left( {{\text{ch}}(\eta )\frac{\partial H}{{\partial \eta }}} \right) }}{{H{\text{d}}\eta }} = - \frac{{\varepsilon_{\theta \theta } }}{{\varepsilon_{\eta \eta } }}n(n + 1) = - \nu (\nu + 1), $$
(15)

where

$$ \nu = \nu (n) = - 1/2 \pm \sqrt {1/4 + (\varepsilon_{\theta \theta } /\varepsilon_{\eta \eta } )n(n + 1)} . $$
(16)

While the sign minus in the front of square root in (16) is being rejected, then ν is always positive for natural n.

The first solutions of (15) take the form of

$$ H(\eta ) = P_{\nu } (j \cdot {\text{sh}}(\eta )), $$
(17)

and are built of associated Legendre functions Pνµ(z) ([3] Eq. 8.820.1). Pνµ(z) are convergent for |1 – z|< 2. While µ = 0—electrostatic problem described by (13), and it is denoting for simplicity Pν0(z) = Pν(z).

The second linearly independent solutions are the functions Qν(j·sh(η)) ([3] Eq. 8.820.2) where Qν(z) = Qν0(z). Functions Qνµ(z) are convergent for |z|> 1.

Electric scalar potential outside the oblate spheroid (isotropic region, ν = n) takes the form of

$$ V_{{{\text{out}}}} (\eta , \theta ) = \sum\limits_{n = 1}^{N} {\left( {a_{n} P_{n} (j \cdot {\text{sh}}(\eta )) + b_{n} Q_{n} (j \cdot {\text{sh}}(\eta ))} \right) P_{n} (\cos (\theta ))} $$
(18)

and inside oblate spheroid is as follows

$$ V(\eta ,\theta ) = \sum\limits_{n = 1}^{N} {\left( {c_{n} P_{\nu } (j \cdot {\text{sh}}(\eta )) + d_{n} Q_{\nu } (j \cdot {\text{sh}}(\eta ))} \right) P_{n} (\cos (\theta ))} , $$
(19)

where dn = 0 due to singularities of Qν(z) and its derivatives ([3], p. 975).

The real parts of complex solutions are approved as physical results and further are applied for force calculations.

Basing on electric scalar potential the electric field strength (7) components can be easily derived. Exemplary, inside the spheroid

$$ E_{\eta } (\eta , \theta ) = - jL_{\eta }^{ - 1} {\text{ch}}(\eta )\sum\limits_{n = 1}^{N} {c_{n} P^\prime_{\nu } (j \cdot {\text{sh}}(\eta )) P_{n} (\cos (\theta ))} , $$
(20)

and

$$ E_{\theta } (\eta , \theta ) = L_{\theta }^{ - 1} \sin (\theta )\sum\limits_{n = 1}^{N} {c_{n} P_{\nu } (j \cdot {\text{sh}}(\eta )) P^\prime_{n} (\cos (\theta ))} . $$
(21)

The electric field strength distribution must be described by (1) far from the spheroid. Thus, the constants an result directly from the imposed field (1) and depend on polynomial coefficients as follows

$$ a_{1} = jc_{{\text{e}}} E_{0} /(1 - j\pi S_{1} /2), $$
(22)

and for n = 2, …, N

$$ a_{n} = - ( - j)^{n} c_{{\text{e}}}^{n} k_{n} F_{n - 1} /(1 - j\pi S_{n} /2), $$
(23)

where constants kn = 1/3, 2/15, 2/35, …, gmax = j·sh(ηmax), and

$$ S_{n} = \frac{{\frac{{\varepsilon_{\eta \eta } }}{{\varepsilon_{{{\text{out}}}} }}P_{\nu }^{^{\prime}} (g_{\max } )P_{n} (g_{\max } ) - P_{\nu } (g_{\max } )P_{n}^{^{\prime}} (g_{\max } )}}{{\frac{{\varepsilon_{\eta \eta } }}{{\varepsilon_{{{\text{out}}}} }}P_{\nu }^{^{\prime}} (g_{\max } )Q_{n} (g_{\max } ) - P_{\nu } (g_{\max } )Q_{n}^{^{\prime}} (g_{\max } )}}. $$
(24)

The electric field must obey two boundary conditions, i.e. continuities of both tangential electric strength Eθ and normal electric displacement Dη at the spheroid surface η = ηmax (outside the spheroid εθθ = εηη = εout). The continuity conditions constitute two equations for unknown constants bn, cn. Hence, the constants bn, cn (n = 1, 2, …, N) are as follows

$$ b_{n} = - S_{n} a_{n} , $$
(25)
$$ c_{n} = a_{n} \frac{{P_{n}^{^{\prime}} (g_{\max } )Q_{n} (g_{\max } ) - P_{n} (g_{\max } )Q_{n}^{^{\prime}} (g_{\max } )}}{{\frac{{\varepsilon_{\eta \eta } }}{{\varepsilon_{{{\text{out}}}} }}P_{\nu }^{^{\prime}} (g_{\max } )Q_{n} (g_{\max } ) - P_{\nu } (g_{\max } )Q_{n}^{^{\prime}} (g_{\max } )}}, $$
(26)

The electric scalar potential leads to electric field strength by means of gradient formula (7) and electric displacement field by means of (9), subsequently. The electric field strength and electric displacement determine the force acting on the spheroid, i.e. electrostatic levitation force. Levitation force physically acts at the boundary of the spheroid, because only there appears permittivity step change in electrostatic field.

For oblate spheroidal orthogonal coordinates electric force density components are as follows

$$ f_{\eta } = - L_{\eta }^{{ - 1}} {\text{div}}\left( {L_{\eta } \vec{\sigma }_{\eta } } \right) - \varDelta_{\eta } , $$
(27)

and

$$ f_{\theta } = - L_{\theta }^{ - 1} {\text{div}}\left( {L_{\theta } \vec{\sigma }_{\theta } } \right) - \varDelta_{\theta } , $$
(28)

where (“Appendix I”)

$$ \varDelta_{\eta } = e_{\eta } \frac{{\partial \ln (L_{\eta }^{2} /L)}}{{L_{\eta } \partial \eta }} + e_{\theta } \frac{{\partial \ln (L_{\theta }^{2} /L)}}{{L_{\eta } \partial \eta }}, $$
(29)
$$ \varDelta_{\theta } = e_{\eta } \frac{{\partial \ln (L_{\eta }^{2} /L)}}{{L_{\theta } \partial \theta }} + e_{\theta } \frac{{\partial \ln (L_{\theta }^{2} /L)}}{{L_{\theta } \partial \theta }}. $$
(30)

It should be underlined that the above equations are derived from the Maxwell electromagnetic field equations and Lorentz force formula ([10, 12], Table 1).

Force density along z-axis is given as below

$$ f_{z} = f_{\eta } \cos (\alpha_{n} ) - f_{\theta } \sin (\alpha_{n} ) $$
(31)

where αn is angle between z-axis and normal to the oblate spheroid surface. It should be underline that the angle αn is not equal to the coordinate θ (Fig. 2). The angle αn is described by the trigonometric functions as follows

$$ \cos (\alpha_{n} ) = \frac{{{\text{ch}}(\eta )\cos (\theta )}}{{\sqrt {{\text{ch}}^{2} (\eta ) - \sin^{2} (\theta )} }},\quad \sin (\alpha_{n} ) = \frac{{{\text{sh}}(\eta )\sin (\theta )}}{{\sqrt {{\text{ch}}^{2} (\eta ) - \sin^{2} (\theta )} }}. $$
(32)

Equivalently, the force density formula can be derived by means of coordinate transformation as it is given below

$$ f_{z} = \frac{{f_{\eta } \partial z}}{{L_{\eta } \partial \eta }} + \frac{{f_{\theta } \partial z}}{{L_{\theta } \partial \theta }} = \frac{{f_{\eta } {\text{ch}}(\eta )\cos (\theta )}}{{\sqrt {{\text{ch}}^{2} (\eta ) - \sin^{2} (\theta )} }} - \frac{{f_{\theta } {\text{sh}}(\eta )\sin (\theta )}}{{\sqrt {{\text{ch}}^{2} (\eta ) - \sin^{2} (\theta )} }}. $$
(33)

Hence, the force density along z axis (31) describes the following relation

$$ \begin{aligned} f_{z} & = - {\text{div}}(\vec{\sigma }_{\eta } \cos (\alpha_{n} ) - \vec{\sigma }_{\theta } \sin (\alpha_{n} )) \\ & \quad - \frac{{(\sigma_{\eta \theta } - \sigma_{\theta \eta } )}}{{L_{\eta } }}\frac{{{\text{sh}}(\eta ){\text{ch}}(\eta )\sin (\alpha_{n} ) - \sin (\theta )\cos (\theta )\cos (\alpha_{n} )}}{{{\text{ch}}(\eta )^{2} - \sin (\theta )^{2} }}, \\ \end{aligned} $$
(34)

where vectors are built of the Maxwell stress tensor components [1, 5, 11, 12] as follows

$$ \vec{\sigma }_{u} = - E_{u} \vec{D} + \vec{i}_{u} (\vec{E}\vec{D})/2. $$
(35)

The main Eq. (34) leads to the Maxwell stress tensor generalized method

$$ F_{z} = F_{Mz} + \varDelta F_{z} , $$
(36)

where only the Maxwell stress tensor is integrated over the spheroid surface at η = ηmax+ as follows

$$ \begin{aligned} F_{Mz} & = - 2\pi Rh\int\limits_{0}^{\pi } {{\text{Re}} \{ E_{\theta } D_{\eta } \} \sin^{2} \theta {\text{d}}\theta } \\ & \quad - \pi R^{2} \int\limits_{0}^{\pi } {{\text{Re}} \{ - E_{\eta } D_{\eta } + e_{\varepsilon } \} \sin (2\theta ){\text{d}}\theta } . \\ \end{aligned} $$
(37)

The additional term of the Maxwell stress tensor generalized method ΔFz is a volume integral. For oblate spheroidal coordinates value of ΔFz results directly from (34) and takes the following extended forms given by (38) or (39)

$$ \begin{aligned} \varDelta F_{z} & = - 2\pi \int\limits_{0}^{\pi } {\int\limits_{0}^{{\eta_{\max } }} {{\text{Re}} \{ \sigma_{\eta \theta } - \sigma_{\theta \eta } \} ({\text{sh}}(\eta ){\text{ch}}(\eta )\sin (\alpha_{n} )} } \\ & \quad - \sin (\theta )\cos (\theta )\cos (\alpha_{n} ))\frac{{c_{{\text{e}}}^{2} {\text{ch}}(\eta )\sin (\theta )}}{{\sqrt {{\text{ch}}^{2} (\eta ) - \sin^{2} (\theta )} }}\sin (2\theta ){\text{d}}\eta {\text{d}}\theta , \\ \end{aligned} $$
(38)

or equivalently

$$\begin{aligned} \varDelta F_{z} &= - \frac{\pi }{2}c_{{\text{e}}}^{2} \int\limits_{0}^{\pi } \int\limits_{0}^{{\eta_{\max } }} {\text{Re}} \{ \sigma_{\eta \theta } - \sigma_{\theta \eta } \} \nonumber \\ &\quad \times \frac{{{\text{sh}}^{2} (2\eta )\sin^{2} (\theta ) - \sin^{2} (2\theta ){\text{ch}}^{2} (\eta )}}{{{\text{ch}}^{2} (\eta ) - \sin^{2} (\theta )}}{\text{d}}\eta {\text{d}}\theta.\end{aligned} $$
(39)

Physically, the electrostatic levitation force, i.e. material force FNz is acting only at surface of the spheroid where permittivity changes in step way. The material force density is equal to zero inside the homogeneous spheroid. Mathematically, the material force describes inhomogeneity component [5, 10] as follows

$$ \vec{N} = - \tfrac{1}{2}E_{u} E_{w} {\text{grad}}(\varepsilon_{uw} ), $$
(40)

where repeated indices are implicitly summed over. For diagonally (normally) anisotropy of spheroid permittivity (9) force can be presented as follows

$$ F_{Nz} = \pi \int\limits_{0}^{\pi } {\left[ {\left( {\frac{1}{{\varepsilon_{0} }} - \frac{1}{{\varepsilon_{\eta \eta } }}} \right)\left| {D_{\eta } } \right|^{2} + (\varepsilon_{\theta \theta } - \varepsilon_{0} )\left| {E_{\theta } } \right|^{2} } \right]\cos (\alpha_{n} )L_{\varphi } L_{\theta } {\text{d}}\theta } . $$
(41)

where appear only continuous field components Dη and Eθ at the boundary η = ηmax.

In order to confirm the value of the electrostatic force coenergy method [1, 5] is applied as follows

$$ F_{Cz} = \frac{{\partial W_{C} }}{\partial z} = - \int\limits_{V} {\vec{D}\frac{{\partial \vec{E}}}{\partial z}{\text{d}}V} , $$
(42)

hence

$$ F_{Cz} = \pi R\int\limits_{0}^{\pi } {\left\{ {\left( {\frac{1}{{\varepsilon_{0} }} - \frac{1}{{\varepsilon_{\eta \eta } }}} \right)\left| {D_{\eta } } \right|^{2} + (\varepsilon_{\theta \theta } - \varepsilon_{0} )\left| {E_{\theta } } \right|^{2} } \right\} L_{\varphi } \cos \theta {\text{d}}\theta } . $$
(43)

Formulas (41) and (42) are mathematically equivalent.

For isotropic spheroid, the force can be also calculated by means of equivalent electric dipole model [1, 12, 13] in constant gradient field (1), i.e. for N = 2. Namely, for a effective dipole moment as follows

$$ p_{{{\text{eff}}}} = - \frac{{4\pi c_{{\text{e}}}^{3} }}{3} \varepsilon_{0} b_{1} , $$
(44)

the force acting on the dipole given by the formula

$$ F_{Dz} = p_{{{\text{eff}}}} F_{1} , $$
(45)

is equal to electrostatic force.

The oblate spheroid shape is similar to a ball for h → R (ce → 0, ηmax → ∞). For h → 0 (ce → R, ηmax → 0) spheroid is similar to a thin disc (is coin-like). In this second case the force (41) decreases to zero almost linearly for h → 0, and tangent line has got a slope of

$$ \mathop {\lim }\limits_{h \to 0} \frac{{F_{Nz} (h)}}{h} = \pi R^{2} \left( {1 - \frac{{\varepsilon_{0} }}{{\varepsilon_{\eta \eta } }}} \right)\varepsilon_{0} E_{0} F_{1} . $$
(46)

The tangent for thin spheroid is drawn in Fig. 5 (and Fig. 11).

The forces evaluated for electrostatic field by means of the Maxwell stress tensor generalized method Fz + ΔFz, material force FNz (inhomogeneity component), coenergy FCz and equivalent dipole FDz (only for isotropic dielectric spheroid) are equal one another

$$ F_{z} = F_{Nz} = F_{Cz} \mathop = \limits^{{{\text{isotr}}}} F_{Dz} . $$
(47)

In Figs. 3 and 4, the electrostatic levitation forces are presented vs. relative axis permittivity change + /–20% of permittivity for the isotropic case. The greater axis permittivity the greater force. The increase is greater for latitudinal permittivity εθθ rises than for hyperbolic permittivity εηη rises. It should be also pointed out that the residual component ΔFz (the Maxwell stress tensor generalized method volume integral [12, 13]) vanishes for isotropic dielectric spheroid (central abscissa point of each curve).

Fig. 3
figure 3

Electrostatic levitation force for anisotropic dielectric oblate spheroid evaluated by Maxwell stress tensor generalized method (points), material force (dash-dot line) and residual component ΔFz (line near abscissa) versus relative hyperbolic permittivity εηηrel (R = 2 mm, h = R/2, εηηrrel = (0.8 ÷ 1.2) εθθrel, εθθrel = 5, E0 = 5 MV m–1, F1 = 1.5 GV m–2, F2 = 0, F3 = 40 GV m–4, N = 4)

Full size image
Fig. 4
figure 4

Electrostatic levitation force for anisotropic dielectric oblate spheroid evaluated by Maxwell stress tensor generalized method (points), material force (dash-dot line) and residual component ΔFz (line near abscissa) versus relative latitudinal permittivity εθθrel (R = 2 mm, h = R/2, εηηrel = 5, εθθrel = (0.8 ÷ 1.2)εηηrel, E0 = 5 MV m–1, F1 = 1.5 MV m–2, F2 = 0, F3 = 40 GV m–4, N = 4)

Full size image

Figure 5 presents the electrostatic force vs. oblate spheroid height h = (0, R]. The tangent for thin disc h → 0 (coin-like) is drawn due to (46). The star * in the right-upper corner of Fig. 5 is incorporated basing on analytical solution obtained independently in [13] for the ball, i.e. h ≡ R (analogously, the star * is set in Fig. 9). It can be seen that the solution for spheroid at limit h → R is convergent to that for the ball h ≡ R.

Fig. 5
figure 5

Electrostatic levitation force for anisotropic dielectric oblate spheroid evaluated by Maxwell stress tensor generalized method (points), material force (dash-dot line) and residual component ΔFz (line near abscissa) versus height h = (0, R] (R = 2 mm, εηηrel = 100, εθθrel = 90, E0 = 5 MV m–1, F1 = 1.5 MV m–2, F2 = 0, F3 = 40 GV m–4, N = 4)

Full size image

The force evaluated by both the Maxwell stress tensor generalized and material force methods always leads to equal results

$$ F_{Mz} + \varDelta F_{z} = \varDelta F_{Nz} . $$
(48)

For checking, the global average relative error is defined as follows

$$ {\text{err}} = \frac{1}{n + 1}\sum\limits_{i = 0}^{n} {\left| {\frac{{(F_{Mzi} + \varDelta F_{zi} ) - (F_{Nzi} )}}{{(F_{Mzi} + \varDelta F_{zi} )}}} \right|} . $$
(49)

Exemplary, for forces in Fig. 5 (n = 20 intervals per curve) the average relative error is equal to 4.414E−003.

If ΔFzi were neglected for anisotropic region, thus maximal error would be equal to 5.1E−002 (Fig. 5).

3 Electrostatic levitation of prolate spheroid

Let us consider consecutively prolate spheroid at the same conditions of electric field excitation (Fig. 6).

Fig. 6
figure 6

Anisotropic dielectric and nonconductive prolate spheroid in imposed electrostatic field

Full size image

The partial differential Eq. (11) for prolate spheroid takes slightly different form, due to different Lame coefficients, than for oblate spheroid problem. Subsequently, the separation Eq. (13) leads to the following relations

$$ \frac{{\varepsilon_{\eta \eta } }}{{\varepsilon_{\theta \theta } {\text{sh}}(\eta )}}\frac{{\partial \left( {{\text{sh}}(\eta )\frac{\partial H}{{\partial \eta }}} \right) }}{H\partial \eta } = \frac{ - 1}{{\sin (\theta )}}\frac{{\partial \left( {\sin (\theta )\frac{\partial \Theta }{{\partial \theta }}} \right)}}{\Theta \partial \theta } = n(n + 1). $$
(50)

The first solution for H() function in prolate spheroidal coordinates is as follows

$$ H(\eta ) = P_{\nu } ({\text{ch}}(\eta )). $$
(51)

The second linearly independent solutions constitute Legendre functions of second kind Qνµ(ch(η)) (for µ = 0 are shortly denoted as Qν(ch(η)) [3]). The solutions for angular function Θ(θ) are Legendre polynomials Pn(cos(θ)).

Electric scalar potential outside the prolate spheroid (isotropic region) takes the form of

$$ V_{{{\text{out}}}} (\eta , \theta ) = \sum\limits_{n = 1}^{N} {\left( {a_{n} P_{n} ({\text{ch}}(\eta )) + b_{n} Q_{n} ({\text{ch}}(\eta ))} \right) P_{n} (\cos (\theta ))} . $$
(52)

and inside prolate spheroid is as follows below

$$ V(\eta ,\theta ) = \sum\limits_{n = 1}^{N} {\left( {c_{n} P_{\nu } ({\text{ch}}(\eta )) + d_{n} Q_{\nu } ({\text{ch}}(\eta ))} \right) P_{n} (\cos (\theta ))} . $$
(53)

where dn = 0 due to irregularity of function Qν() = Qν0().

Basing on solutions of electric scalar potential the electric field strength is derived by gradient operator for prolate spheroidal coordinates. Subsequently, the electric field force can be evaluated by analogous formulas as those for the oblate spheroid. The mathematical differences in oblate and prolate formulas lay in different relations for Lame coefficients and normal angle βn (Fig. 7). Exemplary, for prolate spheroid the additional term for the Maxwell stress tensor generalized method takes the form as follows

$$ \varDelta F_{z} = - \frac{\pi }{2}c_{{\text{e}}}^{2} \int\limits_{0}^{\pi } {\int\limits_{0}^{{\eta_{\max } }} {{\text{Re}} \{ \sigma_{\eta \theta } - \sigma_{\theta \eta } \} {\text{sh}}(2\eta )\sin (2\theta ) {\text{d}}\eta } \;{\text{d}}\theta } . $$
(54)
Fig. 7
figure 7

Prolate spheroidal coordinates and semi-axes assignment

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The relations for material force, stress tensor surface integral, coenergy method formula and equivalent dipole moment are analogous.

In Figs. 8 and 9, electrostatic forces for prolate spheroid vs. relative axis permittivity change ± 20% of permittivity for the isotropic case. Force increase is greater for hyperbolic permittivity εηη changes than for latitudinal permittivity εθθ changes. In Fig. 10 the force vs. prolate spheroid height is shown.

Fig. 8
figure 8

Electrostatic levitation force for anisotropic dielectric prolate spheroid evaluated by Maxwell stress tensor generalized method (points), material force (dash-dot line) and residual component ΔFz (line near abscissa) vs. relative hyperbolic permittivity εηηrel (R = 2 mm, h = 2R, εηηrel = 5, εθθrel = (0.8 ÷ 1.2) εηηrel, E0 = 5 MV m–1, F1 = 0.3 MV m–2, F2 = 0, F3 = 40 GV m–4, N = 4)

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Fig. 9
figure 9

Electrostatic levitation force for anisotropic dielectric prolate spheroid evaluated by Maxwell stress tensor generalized method (points), material force (dash-dot line) and residual component ΔFz (line near abscissa) vs. relative latitudinal permittivity εθθrel (R = 2 mm, h = 2R, εηηrel = 5, εθθrel = (0.8 ÷ 1.2) εηηrel, E0 = 5 MV m–1, F1 = 0.3 MV m–2, F2 = 0, F3 = 40 GV m–4, N = 4)

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Fig. 10
figure 10

Electrostatic levitation force for anisotropic dielectric prolate spheroid evaluated by Maxwell stress tensor generalized method (points), material force (dash-dot line) and residual component ΔFz (line near abscissa) vs. height h = (R, 2R] (R = 2 mm, εηηrel = 100, εθθrel = 90, E0 = 5 MV m–1, F1 = 0.3 MV m–2, F2 = 0, F3 = 40 GV m–4, N = 4)

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4 Oblate and prolate spheroids

Electrostatic forces, that may cause electrostatic levitation in gravitation field, appears at the boundary of spheroids. The forces are of material nature and are appearing when the inside permittivity is different from the outside permittivity.

Among others, the forces depend on spheroid axes length R and h. The relative differences between these length enable to present the geometry influence on forces for each spheroid height h. Relative axes difference (a—the greater semi-axis = R, b—the smaller semi-axis = h) for the oblate spheroid is defined as given below

$$ \frac{b - a}{a} = \frac{h - R}{R} \in ( - 1,\,0), $$
(55)

and converges to minus unity for infinitely thin disc (coin-like).

For prolate spheroid relative axes difference (a—the greater semi-axis = h, b—the smaller semi-axis = R) is defined as follows

$$ - \frac{b - a}{a} = \frac{h - R}{h} \in (0,\,1), $$
(56)

and converges to plus unity for the infinitely high prolate spheroid (stick-like). Relative axes difference is equal to zero for the ball. While relative axes difference is converging to zero (h → R), the shape of each spheroid is converging to the shape of ball (h ≡ R).

The definitions (55) and (56) enable to present forces for any spheroid h = 0 ÷ ∞ at one common curve shown in Fig. 11.

Fig. 11
figure 11

Electrostatic levitation force for anisotropic dielectric spheroids both oblate and prolate (h = 0 ÷ ∞, R = const) evaluated by Maxwell stress tensor generalized method (points), material force (dash-dot line) and residual component ΔFz (line near abscissa) vs. relative axes difference ± (b − a)/a (R = 2 mm, εηηrel = 100, εθθrel = 90, E0 = 5 MV m–1, F1 = 1.0 MV m–2, F2 = 0, F3 = 40 GV m–4, N = 4)

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Obviously, the curve presented in Fig. 11 does not include points h = 0 and h = ∞.

For h → 0 the slope of force curve is equal to tangent given by the limit relation (46).

Figure 11 proves that the forces for both oblate (h → R–) and prolate (h → R +) spheroids are converging to the force of the ball (h ≡ R). The electrostatic force for the ball is the greatest one (for given imposed electrostatic field).

It should be pointed out that the analytical solutions given by series (especially by functions Qνµ() [3], pp. 958–960) are weakly converging for prolate spheroid at limits both h → R + and h → ∞.

5 Conclusions

The analytical solution of oblate and prolate spheroids in electrostatic field has been presented. The presence of electrostatic field may result in levitation of dielectric spheroids either oblate or prolate. The considered spheroids are nonconductive and diagonal (normal) anisotropic dielectric in oblate and prolate spheroidal coordinates

$$ \hat{\varepsilon } = \left[ {\begin{array}{*{20}c} {\varepsilon_{\eta \eta } } & 0 & 0 \\ 0 & {\varepsilon_{\theta \theta } } & 0 \\ 0 & 0 & {\varepsilon_{\varphi \varphi } } \\ \end{array} } \right] , $$

or equivalently

$$ \hat{\varepsilon } = {\text{diag}}[\varepsilon_{\eta \eta } ,\varepsilon_{qq} ,\varepsilon_{\varphi \varphi } ]. $$

The electric fields distributions for oblate and prolate spheroids are obtained by the variable separation method in adequate co-ordinate system.

It should be pointed out that the differential equations for oblate and prolate spheroids are different due to different formulas for Lame coefficients. Hence, mathematical formulas for field components and forces are slightly different. As a consequence, the forces for oblate and prolate spheroids at the same ratio of axes length are different (Fig. 11).

The correctness of the methodology and calculations are confirmed among others by multiple force calculations for both spheroids.

Particularly, four methods for each spheroid are applied. There are the Maxwell stress tensor generalized, material force, coenergy and equivalent dipole methods. The first three methods lead always to the same values of forces. The fourth method, i.e. equivalent dipole method, is valid only for isotropic dielectric spheroid.

The analytical solutions obtained for dielectric spheroids are valid over a large range of parameters, e.g. permittivities, height, radius and imposed field parameters that is not so easy to obtain by numerical analysis.

The presented solutions for oblate and prolate dielectric spheroids can be applied for rapid design, testing numerical algorithms and prototyping.

The presented solution for electrostatic problem can be applied also for no-current magnetostatic problem. The introducing relations (6) and (7) can be applied directly for magnetic scalar potential, too. However, such a solution for scalar magnetic potential cannot be applied when current appears [13].