# Total Ponderomotive Force on an  Extended Test Body

Authors: D. Langemann
Journal: International Journal of Mathematics and Mathematical Sciences
 (2009)

---

## Abstract

Droplets on insulating material suffer a nonvanishing total ponderomotive force because of the inhomogeneity of the surrounding electric field. A series expansion of this total force is proven in a two-dimensional setting by determining the line charge density at the boundary of the test body via a Fredholm integral equation, which is solved by Fourier techniques. The influence of electric charges in the neighborhood of the test body can be estimated as well as the convergence speed of the series expansion. In all realistic applications the series converges very fast. The numerical effort in the simulation of the motion of rainwater droplets on outdoor insulators reduces considerably.


---

## Body

## 1. Introduction

The total forceF acting on an uncharged test body in an inhomogeneous electric field is called ponderomotive force [1]. It is the reason why an uncharged droplet moves within an electric field in experiments [2], and it is an important influence factor for the motion of realistic rainwater droplets on outdoor high-voltage insulating equipment [3, 4].The simulation of moving rainwater droplets requires the determination of the ponderomotive force, and the concentration on the total forceF is a reasonable simplification with respect to other influences like the weather. The computation of the electric fieldaround the droplet is a computationally expensive task, which is dealt with finite integration techniques in [5, 6] or by finite elements on an adaptive triangular grid in [4]. Similarly, a related problem is solved in [7] inthe investigation of ferromagnetic fluids.In plasma physics, the ponderomotive force plays an important role [8], and therefore the ponderomotive force sometimes seems to be strictly related to oscillating electric fields. However, the same inhomogeneity of the electric field causes a nonvanishing divergence of the Maxwell stress tensor and thus a force in classical electrodynamics as well as in plasma physics.In a previous work, the droplet was idealized to a round conductive and undeformable test body, and an explicit expression of the total forceF could be derived [9–11]. This explicit expression is a fast-converging series expansion in inhomogeneity indicators Ik(Ψ), which are computed in terms of the undisturbed potential Ψ in the absence of the test body.Until now, the series expansion was proven by introducing an auxiliary domainΘ containing the domain Ω of the round test body. The potential Ψ at the boundary ∂Θ of the auxiliary domain was used as Dirichlet boundary condition for the potential Φ disturbed by the presence of the test body. After having derived the series expansion of F in a fixed auxiliary domain Θ, an argumentation for extending the domain Θ was used.This procedure is insatisfactory, because boundary conditions ofΨ and additional charges in the vicinity of the test body are not regarded, and they would let fail the argument of an extending Θ.Here, we give a new proof of the series expansion in inhomogeneity indicators. This new proof concentrates on the line chargeσΓ at the boundary Γ=∂Ω of the test body, and it does not need any auxiliary domain. Basing on the proof, a new estimation of the damped influence of neighbored charges on the test body is given. By similar investigations, the diminishing behavior of the terms in the series for F is estimated. It can be shown that the series converges at least as fast as a geometric series.The present paper is organized as follows. It starts with preliminaries and notations where the undisturbed electric potentialΨ and the potential Φ disturbed by the round, conductive, and charge-free test body are presented. Here, the generated line charge density σΓ at the boundary Γ of the test body is introduced. Section 3 solves a Fredholm integral equation of first order by Fourier techniques and shows some facts which are needed to give the series expansion of F in inhomogeneity indicators. Then, Section 4 deals with the convergence behavior and the convergence speed of the series. Furthermore, it discusses the diminishing speed of the summands in the series expansion and the decreasing influence of more remote electric charges.The paper ends with a short conclusion resuming the results and giving an outlook to further investigations like, for example, the analogous investigation in higher dimensions or the extension of the results to more general shapes of the test body.


## 2. Preliminaries and Notations

We denote the points of the two-dimensional Euclidean spaceℝ2 by x=(x1,x2)T. The polar co-ordinates are named x=x(r,φ). The two-dimensional Laplacian operator -∆:H1(ℝ2)→H-1(ℝ2) maps H-1-functions in a weak or distributional sense.The undisturbed electric potentialΨ∈H1(ℝ2) is generated by a charge density ρ∈H-1(ℝ2). The dielectricity constant is denoted by ε0. We normalize any possible relative dielectricity, and the potential equation is(2.1)-ε0∆Ψ(x)=ρ(x)forx∈ℝ2,lim∥x∥→∞Ψ(x)=0,
where the boundary condition at infinity assures uniqueness of the solution [12]. It means that the potential tends to zero for any unbounded sequence of points in ℝ2. In two dimensions, that is, in ℝ2, the boundary condition at infinity contains the realistic condition that the sum of all charges in a bounded domain tends to zeros if the domain is increased onto the whole space. This is an artefact of the two-dimensional setting. In the natural case of three dimensions, it is not necessary to require a vanishing sum of the electric charge [13].The formulation (2.1) includes possible boundary conditions at bounded domains because they are effected by suitable charge densities too. The fundamental solution g of the two-dimensional Laplacian -∆ is(2.2)g(z)=-12πln∥z∥,
and Green's formula for  (2.1) reads(2.3)Ψ(x)=1ε0∫ℝ2ρ(y)g(x-y)dy,
where dy denotes an area element here.The test body occupies the round domainΩ={x:∥x∥<rΓ} with radius rΓ, where ∥·∥ is the Euclidean norm. The boundary of Ω is named Γ=∂Ω={x:∥x∥=rΓ}. The outer normal is denoted by n=x/rΓ for x∈Γ. We denote the projection of x∈ℝ2∖{0} onto the boundary Γ as xΓ=rΓx/∥x∥.The test body influences the undisturbed potential, and the resulting potential is called the potentialΦ∈H1(ℝ2) disturbed by the presence of the test body. Since the test body is conductive, the potential Φ is constant in Ω. We denote Φ(x)=c for x∈Ω̅, where the constant c is unknown until now. Furthermore outside Ω, it is generated by the charge density ρ, too. The conductivity condition of the test body requires(2.4)Ω̅∩suppρ=∅.
The potential Φ bends on Γ, and thus it generates an additional charge density σ∈H-1(ℝ2) concentrated at Γ due to the separation of charges there, which obeys σ(x)=-ε0∆Φ(x) for x∈Γ in a weak sense and is identical to zero elsewhere. Since σ is concentrated on Γ, it can be expressed as the product(2.5)σ(x)=σΓ(xΓ)δ(∥x∥-rΓ)
of the line charge density σΓ∈C(Γ) and the one-dimensional Dirac distribution δ. We show a lemma about the line density.Lemma 2.1.
The generated line charge density isσΓ(xΓ)=-ε0∇Φ(xΓ)n with the unilateral outer gradient ∇Φ(xΓ) at xΓ∈Γ.Proof.
We denoteσ(x)=σ(r,φ) in polar coordinates, Φ(x)=Φ(r,φ) and σΓ(xΓ)=σΓ(φ), respectively. With α>0,  (2.5) gives
(2.6)σΓ(xΓ)=∫rΓ-αrΓ+ασ(r,φ)dr=limα→0+∫rΓ-αrΓ+ασ(r,φ)dr.
The Laplacian of the disturbed potential Φ is
(2.7)ΔΦ(x)=[∂2∂r2+1r∂∂r+1r2∂2∂φ2]Φ(r,φ),
in polar coordinates. With the Heaviside function H, the Laplacian for the potential Φ bending at Γ reads(2.8)ΔΦ(r,φ)=δ(r-rΓ)[∂∂r+-∂∂r-]Φ(r,φ)+H(r-rΓ)[∂2∂r+2+1r∂∂r+]Φ(r,φ)+1r2∂2∂φ2Φ(r,φ),
where the unilateral derivatives are marked by the indices + and -, respectively. Since Φ is constant on Γ and thus independent of the angle φ,  (2.6) together with the constance of Φ inside Ω gives
(2.9)σΓ(xΓ)=-ε0∂∂r+Φ(rΓ,φ)=-ε0∂Φ(xΓ)∂n
because Φ is constant in the orthogonal, angular direction.Since the test body is free of charge, the integral overσ and hence over σΓ vanishes, and the potential equation for the disturbed potential Φ reads(2.10)-ε0ΔΦ(x)=ρ(x)forx∈ℝ2∖Ω̅,Φ(x)=cforx∈Ω̅,∫Γ∇Φ(x)ndx=0,lim∥x∥→∞Φ(x)=0,
where the first equation is Poisson's equation with the charge density ρ outside the test body. The second equation encodes the conductivity of the test body and, therefore, the constance of the potential Φ in the test body and particularly at its boundary Γ. The third relation in  (2.10) contains the fact that the test body is free of charge, (cf. Lemma 2.1). Again, the boundary condition of Φ at infinity assures uniqueness. The line element on Γ is denoted by dx too. The constant c is determined by the charge-free condition in  (2.10) [9, 10].Finally, the total ponderomotive force is given by(2.11)F=ε02∫Γ|∇Φ(x)|2ndx=12∫ΓσΓ(xΓ)|∇Φ(xΓ)|ndxΓ=12ε0∫ΓσΓ(xΓ)2ndxΓ,
where the second equivalence follows from Lemma 2.1. The existence of the integral in the defining  (2.11) is not immediately obvious because the trace theorem [13] assures only ∇Φ∣Γ∈H-1/2(Γ) in general. However, due to the smooth boundary Γ, it holds true that ∇Φ∣Γ∈L2(Γ) [14], and the integral in  (2.11) is meaningful.Now, the disturbed potentialΦ is determined by the charge density ρ and by the generated charge density σ. Thus, Green's formula reads(2.12)Φ(x)=1ε0∫ℝ2[ρ(y)+σ(y)]g(x-y)dy=Ψ(x)+1ε0∫ΓσΓ(y)g(x-y)dy,
where dy in the first term denotes an area element, and in the second term, it denotes the line element on Γ, respectively, to the integration domain.By the way,  (2.12) includes the known fact that the influence of the test body onto the neighborhood diminishes at least with the decreasing behavior of the fundamental solution. Since the test body is charge-free and the sum of the line charge σΓ is vanishing, the influence actually diminishes like the reciprocal of the distance in the two-dimensional setting.


## 3. Line Charge Density and a Fredholm Integral Equation

We use the preliminaries for deriving a Fredholm integral equation for the line charge density. Further, we will solve it in dependence of the undisturbed potentialΨ. This solution will enable us to express the total ponderomotive force F in terms of Ψ.Theorem 3.1.
LetrΓ∈(0,1). If the line charge density σΓ fulfills
(3.1)∫ΓσΓ(y)g(x-y)dy=ε02πrΓ∫ΓΨ(y)dy-ε0Ψ(x)
for all x∈Γ, then σΓ generates Φ with ∫Γ∇Φ(x)ndx=0 and with Φ(x)=c for all x∈Ω̅.Proof.
Starting with a vanishing difference and using condition (3.1), it holds true that
(3.2)0=∫Γ(ε02πrΓ∫ΓΨ(y)dy-ε0Ψ(x))dx=∫Γ∫ΓσΓ(y)g(x-y)dydx.
Hence, one finds after changing the integration order
(3.3)0=∫ΓσΓ(y)∫Γg(x-y)dxdy=C∫ΓσΓ(y)dy
with C=∫Γg(x-y)dx, which is independent of y because of the rotational symmetry of Γ. With ∥x(rΓ,φ)-y(rΓ,0)∥=2rΓsin(φ/2), the constant C is calculated by
(3.4)C=-12π∫02πln(2rΓsinφ2)rΓdφ=-rΓlnrΓ.
Hence, rΓ∈(0,1) implies C≠0, and  (3.3) together with Lemma 2.1 is the first proposition. Again the requirement rΓ<1 is an artefact of the two-dimensional setting, which does not occur in three dimensions because the fundamental solution does not have any zeros then.
Next,  (2.12) with the condition (3.1) gives
(3.5)Φ(x)=12πrΓ∫ΓΨ(y)dy=cforallx∈Γ,
whichis independent of x. Since Φ is constant on Γ and since there is no electric charge in Ω (cf.  (2.4)) it is true that Φ is constant in Ω.Let us remark that the restrictionrΓ<1 does not occur in higher dimension because the fundamental solutions do not change sign then. However, even rΓ<1 can be overcome by the transformation of coordinates.Corollary 3.2.
It holds true thatc=Ψ(0) for every undisturbed potential Ψ.Proof.
The undisturbed potentialΨ is a potential function in Ω̅ because of  (2.4), and  (3.5) yields the proposition.Using the proof of Corollary3.2, condition (3.1) reads(3.6)∫ΓσΓ(y)g(x-y)dy=ε0[Ψ(0)-Ψ(x)]
for all x∈Γ. This is a Fredholm integral equation of first order [15] for the determination of σΓ. In the following, the integral equation (3.6) or (3.1), respectively, is solved by Fourier techniques with the aim to determine σΓ and the total ponderomotive force F in  (2.11).Since the domainΩ̅ is free of charge (cf.  (2.4)) Ψ is a potential function, and it can be given as(3.7)Ψ(x)=Ψ(r,φ)=∑k=-∞∞ãkr|k|eikφ
for all x∈Ω̅, that is, for r∈[0,rΓ]. The notation ak=ãkrΓ|k|, k∈ℤ gives(3.8)Ψ(rΓ,φ)=∑k=-∞∞akeikφ.
In  (3.8), we find the Fourier coefficients ak=ck(Ψ(rΓ,·)) which are defined by(3.9)ck(f)=12π∫02πf(φ)e-ikφdφ
for a 2π-periodic function f. We identify the point xΓ=xΓ(rΓ,φ) with the angle φ of the polar coordinates, and  (3.6) reads(3.10)∫02πσΓ(ψ)g(|φ-ψ|)rΓdψ=ε0[Ψ(0)-Ψ(rΓ,φ)]=-ε0∑k=-∞,k≠0∞akeikφ
for all φ∈[0,2π) and with g=g(η)=ln(2rΓsin(η/2)) for η∈[0,2π] like in  (3.4).Since  (3.10) is a convolution of the 2π-periodic functions σΓ=σΓ(ψ) and g=g(η), it holds true [16] that(3.11)2πck(σΓ)ck(g)=ck(σΓ∘g)=-ε0akrΓ
for k≠0 and c0(σΓ)c0(g)=0. We compute ck(g)=1/(4π|k|) for k≠0, and we get(3.12)ck(σΓ)=-2ε0|k|rΓak
for k≠0. The term c0(σΓ) is not determined by the integral equations (3.6) or (3.10), respectively (cf. Fredholm's alternative [15]). However,  (2.10) yields c0(σΓ)=0 in the charge-free condition. Therefore, we find(3.13)σΓ(φ)=-2ε0rΓ∑k=-∞∞|k|akeikφ.
Finally, the total ponderomotive force in  (2.11) is(3.14)F=2ε0rΓ2∫02π(∑k=-∞∞|k|akeikφ)2nrΓdφwithn=(cosφsinφ).
In fact (3.14) is already an expression for F in terms of the undisturbed potential Ψ, in particular, in the Fourier coefficents of its restriction to the boundary of the round test body. In the following, we develop a more convenient expression in terms of Ψ at the origin 0.We consider the components of the forceF=(F1,F2), and we have(3.15)F1=2ε0rΓ2∑k=-∞∞∑ℓ=-∞∞|k||ℓ|aka-ℓ∫02πeiφ+e-iφ2ei(k-ℓ)φrΓdφ.
The evaluation of the integral gives(3.16)F1=2ε0πrΓ∑k=-∞∞∑ℓ=-∞∞|k||ℓ|aka-ℓ(δk,ℓ+1+δk,ℓ-1)
with the Kronecker symbol δkℓ, which is δkℓ=1 if and only if k=ℓ and vanishing else. Since Ψ(rΓ,·) is real-valued, it holds true a̅-k=ak, and the double sum reduces to(3.17)F1=4πε0rΓ∑k=1∞k(k+1)(a̅kak+1+aka̅k+1).
Analogously, we find(3.18)F2=4πiε0rΓ∑k=1∞k(k+1)(a̅kak+1-aka̅k+1).
On the other hand, the formula of Moivre gives  (3.7) in Cartesian co-ordinates as(3.19)Ψ(x)=Ψ(x1,x2)=ã0+∑k=1∞ãk(x1+ix2)k+∑k=1∞ã-k(x1-ix2)k.
As defined in [9, 10] the inhomogeneity indicators are(3.20)Ik(Ψ)(x)=∇[∇kΨ(x):∇kΨ(x)],
where “:” denotes the full tensor contraction of the kth derivatives ∇kΨ. The inhomogeneity indicators encode the deviation of the undisturbed potential Ψ from the potential of a homogeneous electric field, which has vanishing inhomogeneity indicators. In [9], it is shown that ak=ãkrΓ|k| in  (3.19) implies(3.21)Ik(Ψ)(0)=2k+1k!(k+1)!rΓ2k+1(a̅kak+1+aka̅k+1ia̅kak+1-iaka̅k+1).
The comparison of this result with (3.17) and (3.18) yields the series(3.22)F=2πε0∑k=1∞kk!2rΓ2k2kIk(Ψ)(0),
which is the proposed relation between the derivatives of the undisturbed potential Ψ at the middle of the test body, which was set to 0 without loss of generality here.  Equation (3.22) allows us to separate the computation of the electric potential Ψ and the determination of the total ponderomotive force F on an uncharged conductive body. So, the motion of the body inside the electric field can be determined with a single computation of the undisturbed electric potential. The following section will answer the question of the convergence speed of the series (3.22).


## 4. Convergence Speed of the Series

The domain of the test body itself is free of charge. The charge densityρ has a support which is strictly remote of the test body (cf.  (2.4)).We investigate how the terms of the series (3.22) or rather (3.17) and (3.18), respectively, depend on the charge density ρ. We remark that the undisturbed potential Ψ depends linearly on ρ in  (2.3), and hence the coefficients ak in  (3.8) do so. Finally, the line charge σΓ depends linearly on ρ via  (3.13).From a physical viewpoint, it is obvious – and we see it in the formulas too – that the influence of the charges on the total force diminishes with the distance of the charge from the test body. So, we will start with the investigation of a single point charge in the distancet>rΓ from the origin 0. Since this setting is rotationally symmetric, this means, for example,  ρ(r,φ)=δ(t-r)δ(φ).The undisturbed electric potential generated by this single point charge is(4.1)Ψ(r,φ)=-12πε0lnt2+r2-2rtcosφ
with the Fourier coefficients(4.2)ak=ck(Ψ(rΓ,·))=-1(2π)2ε0∫02πe-ikφlnt2+rΓ2-2rΓtcosφdφ
as in  (3.8) at the boundary Γ of the test body. With this abbreviation τ=trΓ/(t2+rΓ2)<1/2 holds true, that is(4.3)lnt2+rΓ2-2rΓtcosφ=12ln(t2+rΓ2)+12ln(1-2τcosφ).
Thus in  (4.2), the Fourier coefficients with k>0 are(4.4)ak=-18π2ε0∫02πe-ikφln(1-2τcosφ)dφ=i4kπ2ε0∫02πe-ikφτsinφ1-2τcosφdφ.
Since |2τcosφ|<1, the term (1-2τcosφ)-1 can be written as geometric series, and we find the relation(4.5)ak=τ8kπ2ε0∫02π(e-i(k-1)φ-e-i(k+1)φ)∑j=0∞[τ(eiφ+e-iφ)]jdφ.
The binomials in the sum do not vanish only in the cases that the exponents coincide with the exponents i(k-1)φ and -i(k+1)φ in the sinus-term. Hence, we get(4.6)ak=τk+14kπε0[∑n=0∞(2n-1+kn)τ2n-1-∑n=0∞(2n+1+kn)τ2n+1].
After separation of the first summand for n=0 and an index shift in the first sum, we find(4.7)ak=14πε0[τkk+τk+2A(k,τ)]withA(k,τ)=∑n=0∞τ2nn+1(2n+1+kn).
The hypergeometric expression A(k,τ) is monotonously increasing in τ for every k>0, and it holds true that(4.8)A(k,0)=1,A(k,12)=4(2k-1)k.
However, (4.7) shows that the coefficients ak are positive for k>0, and  (4.8) yields the estimation(4.9)ak≤14kπε0[τk+4(2k-1)τk+2]<14kπε0[τk+(2τ)k+2].
Finally, we see that the modulus of the series in  (3.17) and (3.18)fulfils(4.10)|Fj|≤8πε0rΓ∑k=1∞k(k+1)akak+1<12πε0rΓ∑k=1∞[τk+(2τ)k+2][τk+1+(2τ)k+3]
for j∈{1,2},whichleads to convergent geometric series because of 2τ<1. After the evaluation of the geometric series in the right-hand expression in  (4.10), we get the relation(4.11)|Fj|<12πε0rΓ[τ31-τ2+24τ51-2τ2+32τ71-4τ2].
In realistic applications, electric charges are remote from the test body compared to the size of the test body, for example, droplets on insulating material, and thus often we have τ≪1/2. Then, the series (3.17) and (3.18) and hence the series in  (3.22) converges fast.At the same time,  (4.10) estimates the influence of remote charges to its neighborhood. For falling τ, that is, for an increasing distance t of the charge to the test body, it holds true that(4.12)|Fj|~𝒪(τ3)forτ→0withτ<rΓt.
The discussion of this section is accomplished by the apprehensible fact that the influence of a charge distribution can be estimated by a concentrated absolute charge distribution at the nearest point of suppρ to the test body. By  (4.7) we know that |ak|≤q(t) with a positive and monotonously decreasing function q in the case of a concentrated normed charge at distance t. Consequently, a distributed charge density gives(4.13)|ak|≤|∫suppρρ(x)q(∥x∥)dx|≤∫suppρ|ρ(x)|q(∥x∥)dx≤minx∈suppρq(∥x∥)∫suppρ|ρ(x)|dx.
In fact (4.13) shows that the above investigation about the decreasing behavior of the summands in the series expansion in  (3.22) are valid for a distributed charge density too. Furthermore, such a distributed charge density implies an even faster convergence, in particular, in the realistic case of a vanishing total charge.


## 5. Conclusion

We have developed a series expansion for the total ponderomotive force acting on a round, conductive, and charge-free test body in an homogeneous media. This is a reasonable approximation for rainwater droplets on insulating material in outdoor high-voltage equipment, because the total ponderomotive force only gives a tendency of their motion, which is additionally influenced by the weather, further external causes, and by the surface properties of the insulating material.The motion of a rainwater droplet on insulating material [3–6] can be simulated by the determination of the time-dependent position of the test body, which moves under the influence of the ponderomotive force. Now, the series expansion in inhomogeneity indicators considerately reduces the numerical effort in this simulation. It requires only one solution of the field equation, and the derivatives needed in the determination of the inhomogeneity indicators can bereadout for every position. Compared to the determination of the disturbed electric field around the test body, for example, the rainwater droplet, in its present position, which changes in every time instant and time step, this single solution of the partial differential equation is of a great advantage.In the present paper, a new proof for the series expansion is given which argues with the line density at the boundary of the test body in two dimensions. So, it does not need any additional, non-physical domain in the neighborhood of the test body. The application of this idea for higher dimensional settings, in particular, for three dimensions, is straightforward if spherical harmonics [17] are used in the Fourier approach.A mathematical much more challenging topic is the generalization of the series expansion in inhomogeneity indicators for more generally shaped test bodies. For small deviations from the circular form, the ideas in [18] about partial differential equations with perturbated boundaries are a starting point.



---

*Source: 232768-2009-09-22.xml*