A problem of disc electrodes is discussed on the basis of dual integral equations both for an electrostatic problem and for a steady current field problem in the unified manner. It is shown that two well-known problems, 1) the plate condenser problem and 2) the disc electrode problem for steady current may be treated as special cases

KIYONO, Takeshi

SHIMASAKI, Masaaki

English

Kyoto University Research Information Repository

RIGHT:URL:CITATION:AUTHOR(S):ISSUE DATE:TITLE:On a Generalized Problem of DiscElectrodes IKIYONO, Takeshi; SHIMASAKI, MasaakiKIYONO, Takeshi ...[et al]. On a Generalized Problem of Disc Electrodes I. Memoirs of theFaculty of Engineering, Kyoto University 1969, 31(2): 231-2361969-06-30http://hdl.handle.net/2433/280774On a Generalized Problem of Disc Electrodes. I. By Takeshi K1v0No* and Masaaki SHIMASAKI* (Received December 20, 1968) A problem of disc electrodes is discussed on the basis of dual integral equations both for an electrostatic problem and for a steady current field problem in the unified manner. It is shown that two well-known problems, 1) the plate condenser problem and 2) the disc electrode problem for steady current may be treated as special cases. 1. Introduction 231 Although the method of series expansion of potential m eigenfunctions is powerful for the Dirichlet and Neumann problem of Laplace's equation, it is usually difficult to determine coefficients of series expansion for mixed boundary value problems in which two or more boundary conditions are involved. Some important classes of mixed boundary value problems may be reduced to dual integral equations. The general treatment of mixed boundary value problems by dual integral equations is given by Sneddon1>. At first we describe an electrostatic problem of disc electrodes as a general case. The solution may be easily interpreted as that of an appropriate steady current field problem. After the general treatment, two important problems are discussed as special cases. 2. Statement of the Problem As shown in Fig. 1, we consider an infinite plate of thickness 2t and of dielectric constant c1 in uniform space of dielectric constant c2' We use the cylindrical coordinates of Fig. 1. Two identical conducting discs of radius 1 are mounted coaxially as electrodes on this plate. The electrodes are maintained at prescribed potential V0 and -V0 respectively. We attempt to find the distribution of potential m the whole space and compute the capacity between two electrodes. If we denote the potential for z~t, t~z~ -t and -t~z by v+, Vand v-* DeJ'artment of Electrical Engineering II. 232 T AKESHI KIYONO and MASAAKI SHIMASAKI z ; X Fig. 1. Setup of two disc electrodes for an electro-static problem. Two disc electrodes of radius I are mounted coaxially on an infinite plate. For a steady current prob-lem, c; should be exchanged into a;. respectively, the boundary conditions to be satisfied are: V± =V, z = ±t V± = ±V0 , z = ±t, O::;;p<I av± av c--=c-2az 1 az' z = ±t, p>l. 3. Integral Equations ( 1 ) ( 2) ( 3 ) In view of symmetric setup of the problem, the potential may be described in the following integral forms: v+ = Vo [ u~ 1 vz-t)U -e~(Z+f)U} L(u) lo (pu) du' V = Vo r u~ 1 {eCZ-t)U _e~(Z+f)U} L(u) lo (Pu) du' v- = V0 J~ u~1 {ecz-tiu_eCz+fJ"}L(u)lo(pu) du, These functions have the property on the plane z=t and z=-t ( 4) On a Generalized Problem of Disc Electrodes. I. 233 Therefore condition (1) is satisfied for any function L(u). The function L(u) should be determined so that conditions (2) and (3) may be satisfied. Substitution of Eqs. (4) into Eq. (2) gives [ u-1 (1-e-2'") L(u) J 0 (pu) du = I , Since - = -V0 (l-e- 2'")L(u)J0(pu)du av±I 1= 8z z=±t o av 11 = Vo r= (1 +e- 2'") L(u) lo(Pu) du, az z=±t Jo condition (3) is expressed as where Ifwe write and e1 -e2 IC=--e1 +ez p>l, ( 5 ) ( 6) ( 7 ) ( 8) ( 9) we have {l-exp(-2tu)}L(u)={l+k(u)}M(u) and Eqs. (5) and (6) lead to dual integral equations, [ u- 1{1 +k(u)} M(u) J 0(pu) du = I , [ M(u)J 0 (pu)du = 0, p>l. (10) ( 11) According to Sneddon2>, the dual integral equations may be reduced to a more convenient form. If we write u f' M(u) = V 11: Jo v cos uvh,(v) dv, ( 12) the dual integral equations ( 10) and ( 11) are reduced to a Fredholm integral equation of the second kind (13) 234 T AKEsm K1YONO and MAsAAKI SHIMASAKI where (14) and Kc(/;) denotes the Fourier cosine transform of k(u): ( 15) Ifwe definef(x) by ( 16) we have from Eq. (13) ( 17) Substitution of Eq. ( 16) into Eq. (12) gives 2 ~1 M(u) = - u f(v) cos uvdv. 7r: 0 (18) Taking both sides of an electrode into consideration, the charge density a(p) on the upper electrode is expressed by a(p) = -€2- +c1-av+l avl 8z z=t oz z=t = (c1+c2)V0 [ M(u)j0 (pu)du If we denote the total electric charge on the upper electrode by Q, we have Q= ~>1r:pa(p)dp = 4(c1+c2)V0 ~>(v) dv, where use is made of formulas such that ~:J1 (u) cos uvdu = 1, v<l . ( 19) (20) On a Generalized Problem ef Disc Electrodes. I. 235 The capacity C between the two electrodes is given by (21) The integral equation for M(u)/u may be derived from the integral equation (17). Multiplying both sides ofEq. (17) by (2/n) cos ux and integrating from O to 1, we have So far we have considered an electrostatic problem but we can treat an appropriate steady current field problem in the same way by exchanging c; into a;. In this case we can compute the inverse of the resistance between the two electrodes by Eq. (21). 4. Special Cases We now show that two well-known disc electrode problems may be treated as special cases. 1) K=O. We consider the case in which the whole space is filled with uniform medium of dielectric constant c (=c1 =c2). This problem has been discussed as that of a plate condenser by many authors. From Eqs. (22) and (9) with K=O, we have where y(u) =M(u)/u. This is identical with the result by Nicholson3 >. Substitu-tion of Eq. (9) with K=O into Eq. (15) gives K (,) - - . I 1__2t_ (24) C - 'V 7I: e+4t20 From Eqs. (24) and (17), we have 1 ~I 2t f(x) -- ---f(u) du = I . 11: -1 (x-u)2+4t2 (25) This is identical with the result by Love4 >. 2) K= 1. We now consider the steady current problem. We assume that the plate of finite conductivity a, is bounded by vacuum at z= ±t. Thus we have 0"2=0. 236 'r AKESHI K1v0No and MAsAAKI SmMAsAKl Kiyono and Tsuda5> pointed out that the well-known solution of Riemann to this problem was incorrect because it led to a physical contradiction. Substitution of JC= 1 into Eq. (9) gives Ifwe write C-tU k(u) =- --. cosh tu B(u) = _l _. M(u) , cosh tu u we have from Eqs. ( 4) and (8) V = V0 [ B(u) sinh zuj0(pu) du, From Eqs. (22) and (27), we have (26) (27) izl ~t. (28) B(u) = ~ sin u +_!_ f= ~ {sin (u+J) +sin (u-J)} B(J) dJ. (29) TC u cosh tu TC Jo cosh tu u+i u-J This is identical with the result established by Kiyono and Tsuda. Eq. (29) may be solved numerically with the aid of the Laguerre-Gauss quadrature formula. The numerical computation of the resistance between the two electrodes and the difference between Riemann's solution are discussed in 6). I) 2) References I.N. Sneddon: "Mixed Boundary Value Problems Puhl. Co. (1966). Ibid. pp. 106-113. in Potential Theory" North-Holland 3) J.W. Nicholson: The Electrification of Two Parallel Circular Discs, Phil. Trans. Roy. Soc. London A, 224, pp. 303-369 (1924). 4) E.R. Love: The Electrostatic Field of Two Equal Circular Co-axial Conducting Disks, Quart. J. Mech. and Appl. Math., 2, pp. 428-451 (1949). 5) T. Kiyono and T. Tsuda: Some Contributions to the Theory of Electrostatic Field, THIS MEMOIRS, Vol. XXX, Part 3 (Sept. 1968) pp. 315-326. 6) T. Kiyono and M. Shimasaki: On the Resistance between the Two Disc Electrodes Applied to an Infinite Plate Conductor, THIS MEMOIRS, Vol. XXX, Part 4 (Oct. 1968) pp. 463-479. 

On a Generalized Problem of Disc Electrodes I

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On a Generalized Problem of Disc Electrodes I

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