The problem of disc electrodes is discussed. Riemann's solution and the solution by the method of images are shown to be unacceptable, and a method based on an integral equation is devised to acquire a more satisfactory solution. In the second half of this paper, the problem of two tangent spheres is studied, using tangent-sphere coordinates. Potential distirbutions due to charged tangent spheres and also due to tangent spheres in an external electric field are analytically obtained

KIYONO, Takeshi

TSUDA, Takao

English

Kyoto University Research Information Repository

RIGHT:URL:CITATION:AUTHOR(S):ISSUE DATE:TITLE:Some Contributions to the Theoryof Electrostatic FieldKIYONO, Takeshi; TSUDA, TakaoKIYONO, Takeshi ...[et al]. Some Contributions to the Theory of Electrostatic Field.Memoirs of the Faculty of Engineering, Kyoto University 1968, 30(3): 315-3261968-09-25http://hdl.handle.net/2433/280737Some Contributions to the Theory of Electrostatic Field By Takeshi K1v0No* and Takao TsuoA** (Received March 4, 1968) The problem of disc electrodes is discussed. Riemann's solution and the solution by the method of images are shown to be unacceptable, and a method based on an integral equation is devised to acquire a more satisfactory solution. In the second half of this paper, the problem of two tangent spheres is studied, using tangent-sphere coordinates. Potential distirbutions due to charged tangent spheres and also due to tangent spheres in an external electric field are analytically obtained. Introduction 315 Although any problems of electrostatic fields can be reduced to the Laplace-type equations, rigorous solutions to these problems are in most cases extremely difficult. The motivation behind the present paper is to give some examples, if not typical, that show how one can get around such difficulties. The first problem to be treated is a problem of disc electrodes. Riemann's solution quoted in the well-known book of Gray and Mathews [1] is shown to be incorrect, since it leads to a physical contradiction. Even the method of images is found misleading for this problem. After indicating these pitfalls, we then proceed to obtain a more satisfactory solution. The other problem discussed in this paper is a problem of tangent spheres that are electrically conducting. Particular coordinates suited to this problem are in-troduced, and solutions are presented for the case of two tangent charged spheres and also for the case of tangent spheres immersed in an external electric field. The result obtained may find some application in the theory of electrostatic separation of ores. Numerical results are scheduled for a companion paper. 2. Disc Electrodes Problem 2.1 Statement of the Problem. As shown in Fig. 1, we consider a plate of thickness 2h and of finite electrical • Department of Electrical Engineering II. •• Department of Electronics. 316 Takeshi KIYONO and Takao TsuoA z I I Fig. I. Setup of the problem of two disc electrodes. Two disc electrodes are mounted coaxially on a conducting plate (shaded) of infinite extent. conductivity a. This plate is bounded by vacuum at z= ±h. We use the cylindri-cal coordinates of Fig. 1. On this plate of infi~ite extent, we coaxially arrange two identical perfectly-conducting discs of radius a as electrodes. Through these electrodes we force a constant electric current I to flow, and we attempt to find the distribution of electric potential Vin the conducting plate. The boundary conditions to be invoked at z= ±h are: av= 0 (r>a); az ' V = ± V0 , (r<a) . ( I ) ( 2) Here V0, a constant, is the potential of the electrode at z=h, and -V0 is that of the other electrode at z= -h. Condition (1) follows from the fact that at the plate surface away from the disc there is no electric current flowing out into vacuum. Condition (2) is simply the requirement that the perfectly conducting electrode be an equipotential body. 2.2 Riemann's Solution and its Fallacy Gray and Mathews [l] quoted in their book on Bessel functions Riemann's solution to the present disc-electrode porblem. The argument is as follows. In place of condition (2), the condition av 1 8z - 211:oa .../ a2-r2 (z=±h, r<a) ( 3) Some Contributions to the Theory of Electrostatic Field 317 is considered. Condition ( 1) must of course be retained. The above condition (3) results from the assumption that the normal component of the electric field at the electrode surface is identical with that of the single-electrode case where a single disc electrode, providing a current source of 21, is immersed in an unlimited medium of conductivity a. In view of the symmetrical setup of the problem, one can assume V = [ L(J..) sinh J..zj0 (J..r)dJ.., ( 4) where L(J..) is to be determined by the boundary conditions. By means of the formula one finds that L(J..) assumes the form / 1 sinJ..a L(J..) - 2n-aa cosh J..h J.. Thus, the solution is: (r>a) (r<a) V - I (00 sinh AZ sin J..a lo(J..r)dJ.. • - 2n-aa Jo cosh J..h J.. ( 5) ( 6) ( 7) This solution certainly fulfils conditions ( 1) and (3), whereas it does not meet the requirement (2), i.e., V=constant at the electrode surface. Therefore, Riemann's solution (7) does not describe the actual physical situation of two disc electroces facing each other. The fallacy of Riemann's solution arises from the inappropriate condition (3), and one should start with conditions ( 1) a~d· (2). Note that in the book by Gray and Mathews [1] Riemann's solution is quoted as ifit were an exact solution to the present problem of two disc -electrodes. '~, 2.3 Method of Images Suppose that an infinite number of .disc electrodes, identical in shape, are arranged in an infinite space of conductivity a. The arrangement is shown in Fig. 2. We assign current intensities ±2/ to each of these electrodes, so that the poten-tial distribution in -h<z<h (shaded in Fig. 2) can be regarded as the same as that in the conducting region of Fig. 1. The potential due to one of the electrodes e,. (n=O, I, 2, •··) is given by ( 1 )"2/ !00 dJ.. v = - e-c2n+i)>-h+>-z 1 (J..r) sin J..a - • " 4n-aa o Jo J.. ( 8) 318 Takeshi KIYoNo and Takao TsuDA z 82 +21 -----------------------z=51 e, -21 ----------------------z=3h e', ---- -------+-+2-1 ------- ---z=--3h Fig. 2. Method of images for the problem of two disc electrodes. Potential distribution in the shaded region is identical with that in the conducting region of Fig. I. Similarly, the potential due to one of the electrodes e,.' (n=0, I, 2, ···) is given by ( 9) Therefore, the potential Vin the region -h<z<h is: = V = ~ (v,.+v,.') n=o xj0(,l.r) sin ).a~). _ I ~= sinh ).z 1 (, ) • , d). - -- ----JO Ar Sln AQ - . 2iraa o cosh ).h ). (10) Thus, here again, the simple application of the method yeilds an incorrect result, since in Eq. (10) we have recovered Eq. (7). The fallacy in the method of images, as well as in Riemann's solution previously discussed, lies in the neglect of mutual interactions between the electrodes. Solutions (7) and (10) indicate that the superposition principle breaks down in the Some Contributions to the Theory of Electrostatic Field 319 present problem. Each of the electrodes does not contribute to the resultant potential as if it were in isolation, but rather, these electrodes mutually rearrange the potential distribution in between the electrodes in order that condition (2) be not violated. 2,4 Proposed Exact Solution by an Integral Equation To determine the function L(J.) ofEq. (4), we start with the exact boundary conditions at z= ±h. for for r<a, r>a. Also we perform a scaling of variables as follows: ). = u/a, z = a( , r = ap , h = at, } VfV0 = U, L(J.)/(aV0 ) = B(u) Eq. (4) then takes the form U = r B(u) sinh (uj0 (pu)du, while the boundary conditions (11) and (12) are expressed as r B(u) sinh tu lo(Pu)du = 1 for O<p< 1, r B(u)u cosh tu lo(pu)du = 0 for 1 <P . Conditions (15) and (16) may further be simplified to r F(u)j0(pu)du = g(P) for O<p< 1 , r F(u)uj0(pu)du = 0 for 1 <P, where F(u) = B(u) cosh tu. (11) (12) (13) (14) (15) (16) (17) (18) (19) Eqs. (17) and (18) may be called dual integral equations, the solution of which is given by Bus bridge [2] as follows: 320 Takeshi KIYoNo and Takao TsuoA F(u) = ~ cos u f1 yg(y) dy 7r Jo Vl-y2 2 11 d 11 +- j ry g(xy)xu sin (xu)dx . 1r o l-y2 o (20) Considering ( Jl ) Jl ydy sin u cos u+ xu sin (xu)dy V = --o o l-y2 u (21) and using g1(P), Eq. (20) is written as F( ) _ 2 sin u + 2 J1 yg1(u) d u - - -- - cos u ~~- ry 1C U 1C O Vl-:I 2 JI d Jl + - j ry g1 (xy)xu sin (xu)dx . 1r o 1-:/ o (22) After some algebra and with the use of formulas such as f 1 , (z ) ydy = sin z JoJo ry Vl-y2 Z ' f · . d 1 [sin (u-u)x sin (u+u)x] j smxusmxu x = 2 u-u u+u , we finally arrive at the following Fredholm integral equation of the second kind: B(u) = f(u) + [ B(J.)K(J., u)dJ. , where f u = ~ sin u () 1rucoshtu' K(J., u) =_!_~[sin (J.+u)+sin (J.-u)] 1r cosh tu J.+u J.-u . (23) (24) (25) B(u) given as the solution of the single intergral equation (23) is the solution of the dual integral equations (15) and (16). Once B(u), hence L(J.), is known, the potential is given by Eq. (4), and the electric field component normal to the electrode surface is calculated from E,. = -(Ez)z=h = [ L(J.) cosh J.hJ0 (J.r)J.d). . (26) Obviously, I = a J: E,.21rrdr = 21raaV0 [ B(u) cosh tuj1(u)du. (27) Some Contributions to the Theory of Electrostatic Field 321 The resistance, R, between the two electrodes is known from R=2V0//. One feasible way of solving Eq. (23) is the successive approximation. Namely, since B(u) is expected to differ slightly fromf(u) we may takef(u) for the zeroth order solution. Expanding B(u) as (28) we have (n=l, 2, •··), (29) where B0 (u)=f(u). Riemann's solution previously discussed merely provides the zeroth order approximation to the present problem. Unless we calculate the residual sum B1+B2+•·•+Bn+··· of (28), we cannot obtain a physically con-sistent solution. Numerical solutions along this line will be presented in a separate paper in the near future. 3. Problem of Tangent Conducting Spheres 3.1 Tangent Sphere Coordinates and Laplace's Equation For , boundary-value problems of two similar spheres mutually m contact, tangent-sphere coordinates [3] are the most appropriate. Orthogonal coordinate system is defined, as shown in Fig. 3, by a set of three surfaces, i.e., a toroidal surface w::const>O w=const <O z Fig. 3. Tangent-sphere coordinates. u=constant (with no peephole at the center), two tangent spheres (w=constant>0 for the upper sphere, and w=constant<0 for the lower sphere), the centers of which lie on the z-axis, and a plane (g,=constant), comprising the z-axis, to determine 322 Takeshi K1v0No and Takao TsunA the azimuthal angle <p. The relation between the Cartesian coordinates (x, y, z) and the tangent-sphere coordinates (u, w, <p) is: x _ au cos <p - u2+w2 ' au sin <p y = u2+w2 ' aw z---~ - u2+w2 . (30) Here a is a constant that represents the radius of a reference spherical surface. Laplace's equation in terms of (u, w, <p) is I a ( u a V) + a ( I a V) + I a2 V - 0 ( 31) u au u2+w2 au aw u2+w2 aw u2(u2+ w2) acp2 which is known to be R-separable [4]. Namely, decomposing the unknown func-tion V into products such as V(u, w, <p) = yu2+w2U(u)W(w)<D(<p), (32) we have three ordinary differential equations one of which is Bessel's equation. Thus, the general solution V that takes finite values along the z-axis is given by = V(u, w, <p) = y u2+w2 ~ (am cos m<p+bm sin m<p) m=O X [ (L cosh J.w+M sinh J.w)Jm(J.u)dJ. , (33) where am and bm are arbitrary constants, and L and M are arbitrary functions of ). or arbitrary constants. In case of axial symmetry, we have, for (33), a much simpler expression as V(u, w) = yu2+w2 [ (L cosh J.w+M sinh J.w)J0 (J.u)dJ.. (34) 3.2 Two Conducting Charged Spheres Here we consider the case of two identical tangent spheres, as shown in Fig. 4, placed in vacuum. Each of these conducting spheres is given charge Q, ( = Q/2). Using Eq. (34) and considering the potential Vis an even function of w, we have V = yu2+w2 [ L(J.) cosh J.w J 0 (J.u)dJ.. (35) To determine L(J.) we assume the potential Vat the spherical surfaces (w=±w0 ; Some Contributions to the Theory of Electrostatic Field z X Fig. 4. Two identical conducting spheres mutually in contact. The radius of the sphere is a. Each of the spheres has electrostatic charge Q 1 (=Q/2). 323 W0 is taken to be½ hereafter) to be V0 • Thus, at the spherical surfaces, we must have It follows from this that Therefore, the required solution is from which the electric field can be calculated by E = u u2 +w2 av -a-au' (36) (37) (38) (39) To find the relationship between the given charge, Qi, and the potential of the con-ducting spheres, V0 , we proceed to evaluate the electric field component, En, directed normal to, and outward from, the spherical surface. Concerning this 324 Takeshi K1v0No and Takao TsuoA field component, we have The total electric charge of the upper sphere is given by Q1 = e f E,.dS, where We finally arrive at the simple formula where use is made of formulas such as Therefore, [ e-P>.. Jo(J.q)d). = (p2)q2)1/z> [ e-P>.. Jo(J.q)J.d). = (p2:q2)3/2, = I I:! (-1)"-1 = log.2. •=O n+ V. - Q . 0 - 81rea log., 2 ' and the electrostatic capacitance C of the two tangent spheres is given by C = i = 41rea(2 log., 2) . 0 2,3 Two Tangent Spheres in an External Electric Field (40) (41) (42) (43) (44) (45) (46) (47) (48) Consider two identical conducting spheres of radius a mutually in contact and Some Contributions to the Theory of Electrostatic Field z X Fig. 5. Two tangent spheres immersed in a uniform electric field E0• The direction of E0 is along the z-axis. immersed in a uniform electric field E0 as shown in Fig. 5. The potential distribution V of our interest can be assumed where Ve is the potential in the absence of the spheres, i.e., 325 (49) (50) In view of the symmetry with respect to the xy-plane, the induced potential V* is given by V* = \f u2 +w2 [ M(,l) sinh ,lwj0 (,lu)d,l. (51) The function M(,l) is to be determined by the boundary condition on the spherical surfaces, i.e., (52) By means of formula (45), Ve ofEq. (50) is written as (53) so that (54) 326 Takeshi K1v0No and Takao TsuoA Thus the solution we need is found as follows: (w>O) , (55) from which the electric field components Eu and Ew can be calculated using Eqs. (38) and (39). Normal component of the electric field on the sphere (w=w0=½) is given by En = -(Ew)w=wo = E0 (u2 +w/)312 [ e->-wo(l +coth J.w0 ) J0 (J.u)J. 2d). _ 2E ( 2+ 2)s;2 [ 3 ~ (2n+ l )2w/ - 0 u Wo ~ (u2 + (2n+ l )2w/)s;2 ~ 1 ] - ~ (u2+(2n+l)2 w/) 312 • (56) Since the present tangent spheres are not initially given any electric charge, the total induced charge on the two spheres vanishes. However, it will be of some interest to know the amount of the induced charge on either of these two contacting spheres. This may be of some help when one estimates the efficiency of the method of electro-static separation of ores, namely, by what electric field conducting particles can most effectively be separated from non-conducting particles. When the induced electric charge on the upper sphere is Q;, then the induced charge on the lower sphere is -Q;, For this Q;, we have (57) Using Eq. (56) for the above equation (57), where w0 =½, we have the final result that 2 ~ l Q; = 4n ca E0 ~ 2 •=1 n (58) References 1) A. Gray and G.B. Mathews, A Treatise on Bessel Functions and their Applications to Physics (MacMillan and Co., London. 1895), pp. 128-131. 2) 1.W. Busbridge, Proc. London Math. Soc. 44 (1937), 115-129. 3) P. Moon and D.E. Spener, Field Theory Handbook (Springer, 1961), pp. 104--106. 4) For discussions on R-separability, see P. Moon and D.E. Spencer, Field Theory for Engineers (Van Nostrand Co., 1961). 

Some Contributions to the Theory of Electrostatic Field

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Some Contributions to the Theory of Electrostatic Field

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