Multiple Image Method for the Two Conductor Spheres in a Uniform Electrostatic Field

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1 Commun. Theor. Phys. 57 (2012) Vol. 57, No. 6, June 15, 2012 Multiple Image Method for the Two Conductor Spheres in a Uniform Electrostatic Field GAO Xin (Ô ), 1,2 HU Lin ( ), 1, and SUN Gang (ê ) 2, 1 Department of Physics, Guizhou University, Guiyang , China 2 Beijing National Laboratory for Condensed Matter Physics and Key Laboratory of Soft Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing , China (Received April 16, 2012) Abstract A method using multiple mirror images of point charges is put forward to analyze the polarization of two identical conductor spheres in a uniform electrostatic field. By use of the method, the electric field distribution and the interaction force between two spheres can be calculated accurately even for very small gap between two spheres. Our results show that the magnitude of the product of the gap between two spheres and the local electric field in the center of the gap is approximately in the same order and the interaction between two spheres increases very fast as the two spheres are close to each other. We also show that the interaction force between two conductor spheres is almost same with that between two dielectric spheres with high permittivity. PACS numbers: Cv, b, De Key words: electric field, polariztion, mirror image method 1 Introduction Interaction between spheres under electrostatic fields is an attractive basic problem, which concerns broad fields, such as electrorheological fluids, [1 4] electrophoresis, [5 6] and other fields. [7 8] The electric field induced interaction force between two dielectric spheres is studied both experimentally [9] and theoretically. [10 14] The experimental work shows that the measured force is not only much higher than that predicted by the dipole approximation [10] when the two spheres are closed up, but also prominently higher than that calculated by the improved dipole model (E corrections dipole model) [11] and the finite-element approach (FEA). [12] To explain the abnormal strong interaction force when the spheres are closed up, Cox et al. have developed a calculation method by including higher order of the polarization moment [13] and shown that the results obtained by their method agree well with that of the experiment. These works imply that it is absolutely needed to include the higher order of polarization in the calculation. Thereafter, Jiao et al. have developed a socalled multiple scattering method, [14] which also include higher order of polarization by a numerical algorithm and can be widely used in almost all kinds of system composed of spheres. In that work, the convergence of the calculated results with the order of the polarization moment is also discussed in detail, and it is found that the gap larger than sphere radius is a limitation for the calculation by use of the spherical harmonics up to the order of 30. For a sphere of ideal conductor, the polarization and hence the electric field distribution can be calculated exactly by mirror image method. [15 17] By use of this method, the polarization of an ideal conductor sphere induced by a single charge is described exactly by an effective image charge (in some circumstances a compensation charge is also needed). The description of the effective image charge can automatically include the multipole polarization of the sphere exactly, but avoid using the general scheme of the multipole moment, which is very difficult to treat for higher order. For this reason, the calculation of the mirror image method is much faster than multiple scattering method or re-expansion method and can be carried out almost without the limitation of the iterative number for a current computer. The mirror image method is applicable for the system composed of the ideal conductor spheres in principle. However, we shall show later that the force between two dielectric spheres with high permittivity is almost same with that between two conductor spheres under uniform electrostatic field. Therefore, this method is also useful in study of the system composed of the dielectric spheres, such as, the electrorheological fluid system. In the next section, the theory of the multiple image method is introduced in detail, which includes the methods to calculate the electric field distribution and the electric induced force. In Sec. 3, the calculated results are shown, and they are compared to the previous results for Supported by Guizhou Provincial Science and Technology Foundation (Z103167), Youth Foundation of Guizhou University (X092012), the National Basic Research Program of China under Grant No. 2009CB930800, and National Natural Science Foundation of China under Grant Nos and hulin53@sina.com gsun@aphy.iphy.ac.cn c 2011 Chinese Physical Society and IOP Publishing Ltd

2 No. 6 Communications in Theoretical Physics 1067 the system composed of two dielectric spheres. The last section is dedicated to the conclusion and discussions. 2 Calculation Method When an idea conductor sphere with radius of R is positioned in an applied uniform electrostatic field E 0, a dipole moment, p 0 = 4πǫR 3 E0, (1) will be induced, [15] where ǫ is the permittivity of the medium around. For convenience of the calculation below, the dipole moment is approximately described by a pair of point charges with Q 0 and Q 0 separated by a small distance of l 0 in the direction of p 0, which satisfies, p 0 = Q 0 l0. (2) We shall show later that the calculation results will be almost unchanged if a small enough distance l 0 is used. Based on the theory of image method, [15] the polarization of an idea conductor sphere by a point charge q positioned at z can be described by an effective imaging charge, at q = Rq z, (3) r = z, (4) and a compensation charges q c in the center of the sphere. The value of compensation charge is related to the connection of the sphere to other device, such as, isolated, earthed, or connected to certain fixed voltage. For an isolated sphere, the compensation charges are determined by keeping the net charge on the sphere in zero. Fig. 1 Illustration of the image charges induced by a uniform electrostatic field for a couple of conductor spheres. Now we consider two idea conductor spheres with same radius R align along the direction of uniform electrostatic field Ê0 as shown in Fig. 1. The centers of two spheres are separated by d = 2R + δ, where δ is the gap between two spheres. The left and right spheres are marked as A and B, respectively. According to the previous statement of the polarization of a sphere by the uniform field, the polarization of each sphere induced by the uniform field can be described by two effective charges q o X,1 and qe X,1 at r o X,1 and r e X,1, respectively, where X = A or B denotes variables for sphere A or B. According to Eqs. (1) and (2), q e A,1 = qo A,1 = 4πǫR3 E 0 l 0, (5) q e B,1 = qo B,1 = 4πǫR3 E 0 l 0, (6) r o X,1 = l 0 2, (7) r e X,1 = l 0 2. (8) For two spheres system, the polarization of each sphere is not only induced by the uniform field but also induced by the field resulting from the polarization of other sphere. The field results from other sphere can be considered effectively by the fields of several point charges, and the response for these fields can be calculated by the image method. For example, the point charge induced by the uniform field, i.e., Eqs. (5), (6), (7), and (8), will further cause polarization in sphere A, which can be describe by effective charges, at qa,2 λ = Rqλ B,1 d rb,1 λ, (9) ra,2 λ = d rb,1 λ, (10) where the superscript λ = o or e denotes the variables for the odd or even series of image charges. At this stage,

3 1068 Communications in Theoretical Physics Vol. 57 qa,2 o + qe A,2 0, so it is needed to add a compensation charge, qa,2 c = qo A,2 qe A,2, (11) at the center of the sphere A to keep the net charge on the sphere being zero. For sphere B, similar polarization can be calculated. For our system, we can easily find that the variable has the symmetries, qb,i λ = qλ A,i and rb,i λ = rλ A,i. By use of these symmetries, Eqs. (9) and (10) can be rewritten as, qx,i+1 λ = Rqλ X,i λ, (12) rx,i+1 λ = λ. (13) Equations (12) and (13) can be easily extended to higher order calculations. For the compensation charge, the situation is somewhat different with the odd or even image charge sequences. The compensation charge occurs firstly at the second order calculation, and an image charge concerned with it occurs at the third order calculation, and so on, which generate another image charge sequence qx,i c at rc X,i. However, for this image charge sequence, the first element qx,1, c which is at the center of the sphere, need to be updated after every order of calculation by the formula, i i qx,1 c = qx,j c (qx,j o + qe X,j ). (14) j=2 j=1 Because of this, the whole image charge qx,i c in the sequence will be also updated according to qx,i+1 c = Rqc X,i c, (15) rx,i+1 c = c. (16) Fortunately, the position of the image charge is not related to the amount of charge, so the number of the image charges only increase by one for one order higher calculation. This iterative calculation can be easily carried out numerically in a computer. As long as large enough number of image charge is obtained, the electric field out of the two conductor spheres can be calculated by summing up all fields produced by these image charges and the uniform applied field E 0. As a demonstration, we calculate the field strength in the middle point of the connecting line between the centers of two conducting spheres. The electric field at this point can be expressed as E c = E n 4πǫ i=0 λ=o,e,c [ qa,i λ (d/2 ra,i λ Ê0 + )2 = E πǫ n i=0 λ=o,e,c q λ B,i (d/2 rb,i λ ( Ê0) )2 q λ A,i (d/2 r λ A,i )2 Ê0, (17) ] where n is the iterative number in calculation, i.e., the largest order of image operation. In general, when the electric field is known, the electric force acting on a body can be obtained by the following integral, [18] F = ǫ( E n) E 1 2 ǫe2 nda, (18) Σ where n is the unit normal vector of surface, da is area element, and Σ is the enveloping surface surrounding the body. When the body is made by metal, Eq. (18) is reduced to, 1 F = 2 ǫe2 nda, (19) Σ because the direction of electric field near metal surface is just along n. Other than Eq. (19), we can calculate electric force more convenient and accurately by employing the image charges. The electric force exerts on sphere B can be considered as the summation of all forces of image charges in sphere B caused by the field of all image charges in other sphere A, F BA = 1 n [ n qb,i λ qa,j τ ]Ê0 4πǫ d rb,i λ. (20) rτ A,i i=1 λ=o,e,c j=1 τ=o,e,c The calculation of Eq. (20) is apparently more easier than Eq. (19), because it avoids the very time consuming integral calculation. Equation (20) also shows that the force is proportional to E 2 0, because all image charge is proportional to applied electric field E 0. 3 Results To comparing with experiment and other theoretical work, two conductor spheres with identical radius R = 3.15 mm are chosen in this paper. The magnitude of applied field is V/m, and value of l 0 is chosen as m tentatively. Figure 2 plots the product of the gap δ between two spheres and the local electric field E c in the center of the gap as a function of the iterative number. The figure shows that the convergence of E c is seriously related to the gap size. The smaller the gap size is, the larger the iterative number is needed. For the smallest gap δ = 0.1 nm we calculated, it needs about iterative to obtain a convergent results. However, when the gap increases to δ = 1.0 nm, or further to δ = 10.0 nm, iterative or further 5000 iterative is enough to obtain convergent results. The multiple scatter method can ensure convergence only for δ/r > 0.005, [14] while the multiple image method can be applied to much smaller gap. Figure 2 shows that the multiple image method can obtain convergent result at about δ/r = 10 7, which should be enough for all realistic calculation. For the gap is larger than 10 µm, 1000 iterative is enough to obtain convergent results. We shall use this iterative number in the following calculation. From Fig. 2, we can also find that the magnitude of the product E c δ is approximately in the same order, though the gap size δ is changed seriously.

4 No. 6 Communications in Theoretical Physics 1069 This means that the local electric field strength will be divergent when the gap size tends to zero with 1/δ. Fig. 2 The product of the gap between spheres and the local electric field in the middle point between two spheres as a function of iterative number for several fixed gap sizes. As mention previously, we have describe the polarization of the sphere induced by the uniform electric field by a pair of point charges with Q 0 and Q 0 separated by a small distance of l 0, which is originally a pure dipole moment. To show how the parameter l 0 affects the final results, the calculated electric field strength in the middle of connecting line between two spheres centers are plotted in Fig. 3 for a wide range of l 0. Figure 3 shows that the value is almost unchanged with l 0 for all the gap size, which means if a small enough l 0 is used, the final calculated results are reliable. spheres is proportional to E0 2, so it is generally to use the normalized force (F/E0 2 ) as an independent quantity. The normalized forces for different gap size are calculated by both Eqs. (19) and (20), and the results are shown in Fig. 4. From Fig. 4, we can find that the integral method [Eq. (19)] and the effective point charge method [Eq. (20)] are almost equivalent. Figure 4 also shows that the field induced force increases very faster as the two spheres are close to each other, and it seems to be divergent when the gap size tends to zero. In Fig. 4, we also plot the experimental and the theoretical (calculated by multiple scattering method) results of the normalized forced between two same size dielectric spheres with permittivity ǫ s = 294. [9,14] Figure 4 shows that the normalized force between two dielectric spheres with high permittivity is almost same with that between two metal spheres under uniform electrostatic field. This suggests that the multiple image method introduced in this paper can also be used for calculating the interactions between two dielectric spheres with high permittivity under electrostatic field, and by this method we can successfully treat almost all realized system (we have shown that the reliable results can be obtained for a gap size with 1 Å), while by the multiple scattering method, which is considered to be the best method for the system composed of dielectric spheres, we are limited to treat the system with δ > 0.005R. Fig. 3 The local electric field in the middle point between two spheres as a function of an approximate size of the dipole moment in Eq. (2). Now, let us consider the field induced force between two spheres. All theories, including Eqs. (19) and (20), have predicted that field induced force between two Fig. 4 The normalized force (F/E 2 0) as a function of the gap between spheres. The line is for the results obtained by the effective point charge method [Eq. (20)] and the closed circles are for that by the integral method [Eq. (19)] for the conductor spheres. The open squares and the diamonds are theoretical (by the multiple scattering method) and experimental results for the dielectric spheres with permittivity ǫ s = 294, respectively. 4 Conclusion A method of multiple images has been advanced to analyze induction of two identical conductor spheres in a uniform electrostatic field. This method can converge for almost all realized small gap size. By the multiple image

5 1070 Communications in Theoretical Physics Vol. 57 method, we can not only obtain the electric field distribution, but can also calculate field induced interaction force. In the calculation of the field induced force, we have developed an effective point charge method, which employ the picture of the image charges. We have shown that the effective point charge method is equivalent to the integral method, but it avoids the very time consuming integral calculation. The calculation results of the multiple image method have shown that the magnitude of the product of the gap between two spheres and the local electric field in the center of the gap is approximately in the same order. This means that the local electric field strength in the gap can be very high if the gap size is very small. The normalized field induce force between two metal spheres has also been calculated for different gap size, the results show that the field induced force increases very fast as the two spheres are close to each other, and it seems to be divergent when the gap size tends to zero. We have shown that the normalized force between two dielectric spheres with high permittivity is almost same with that between two metal spheres under uniform electrostatic field. Therefore, this method is also useful in the study of the system composed of the dielectric spheres. References [1] W.M. Winslow, J. Appl. Phys. 20 (1949) [2] H. Ma, W. Wen, W.Y. Tam, and P. Sheng, Adv. Phys. 52 (2003) 343. [3] W.J. Wen, X.X. Huang, S.H. Yang, K.Q. Lu, and P. Sheng, Nature Materials 2 (2003) 727. [4] K.Q. Lu, R. Shen, X.Z. Wang, G. Sun, W.J. Wen, and J.X. Liu, Chin. Phys. 15 (2006) [5] Z. Qiu, J. Appl. Phys. 92 (2002) [6] H.J. Keh and S.B. Chen, J. Colloid Interface Sci. 130 (1989) 542. [7] W. Chen, S. Tan, T.K. Ng, W.T. Ford, and P. Tong, Phys. Rev. Lett. 95 (2005) [8] J.H. Cloete and J. van der Merwe, IEEE Transactions on Education 41 (1998) 141. [9] Z. Wang, Z. Peng, K. Lu, and W. Wen, Appl. Phys. Lett. 82 (2003) [10] C.A. Coulson, Electricity, Wiley Interscience, New York (1961). [11] L.C. Davis, J. Appl. Phys. 72 (1992) [12] R. Tao, Q. Jiang, and H.K. Sim, in Proceedings of the Fifth International Conference on ER Fluids, ed. R. Tao, World Scientific, Singapore (1996) p. 47. [13] B.J. Cox, N. Thamwattana, and J.M. Hill, Appl. Phys. Lett. 88 (2006) [14] M.C. Jiao, G. Sun, Q. Wang, and K.Q. Lu, Mod. Phys. Lett. 26 (2012) [15] T.B. Jones, Electromechanics of Particles, Cambridge University Press, New York (1995). [16] J.C.E. Sten and K.I. Nikoskinen, J. Electrostat. 35 (1995) 267. [17] B. Techaumnata and T. Takuma, J. Electrostat. 64 (2006) 165. [18] J.A. Stratton, Electromagnetic Theory, McGraw-Hill, New York and London (1941).