Hostname: page-component-7479d7b7d-jwnkl Total loading time: 0 Render date: 2024-07-12T02:56:49.689Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic formula for the capacitance of two oppositely charged discs

Published online by Cambridge University Press:  24 October 2008

W. C Chew  and
J. A Kong
W. C Chew
Affiliation:
Massachusetts Institute of Technology
J. A Kong
Affiliation:
Massachusetts Institute of Technology
Get access Rights & Permissions [Opens in a new window]

Abstract

Asymptotic formulae for the capacitances of two oppositely charged, identical, circular, coaxial discs and two identical, infinite parallel strips separated by dielectric slabs are derived from the dual integral equations approach. The formulation in terms of dual integral equations using transforms gives rise to relatively simple Green's functions in the transformed space and renders the derivation of the asymptotic formulae relatively easy. The solution near the edge of the plates, a two-dimensional problem previously thought not solvable by the Wiener-Hopf technique, is solved indirectly using the method. Solution of the Helmholtz wave equation is first sought, and the solution to Laplace's equation is obtained by letting the wavenumber go to zero. The solution away from the edge is obtained by solving the dual integral equations approximately. The total charge on the plate is obtained by matching the solution near the edge and away from the edge giving the capacitance.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 89 , Issue 2 , March 1981 , pp. 373 - 384
Copyright
Copyright © Cambridge Philosophical Society 1981

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

(1)Borkar, S. R. and Yang, R. F. M.Capacitance of a circular disk for applications in microwave integrated circuits. IEEE Trans. Micro. Theory Tech., MTT-23, 583 (1975).Google Scholar
(2)Chew, W. C. and Kong, J. A.Effects of fringing fields on the capacitance of circular microstrip disk. IEEE Trans. Micro. Theory Tech., MTT-28, 98 (1980).Google Scholar
(3)Cooke, J. C.A solution of Tranter's dual integral equations problem. Quart. J. Mech. Appl. Math. 9 (1956), 103.CrossRefGoogle Scholar
(4)Cooke, J. C.The coaxial circular disc problem. Z. angew. Math. Mech. 38 (1958), 349.CrossRefGoogle Scholar
(5)Gradshteyn, I. S. and Rhyshik, I. W.Tables of integrals, series and products (Academic Press, New York, 1965).Google Scholar
(6)Harrington, R. F.Field computation by moment methods (Macmillan, New York, 1968).Google Scholar
(7)Hutson, V.The circular plate condenser at small separations. Proc. Cambridge Philos. Soc. 59 (1963), 211.CrossRefGoogle Scholar
(8)Ignatowsky, W.Akad. Nctuk SSSR, Mat. Inst. Trudy (2), 3 (1932), p. 1.Google Scholar
(9)Itoh, T. and Mittra, R.A new method for calculating the capacitance of microwave integrated circuits. IEEE Trans. Micro. Theory Tech., MTT-21, 431 (1973).Google Scholar
(10)Kirchhoff, G.Monatsb. Deutsch Akad. Wiss., Berlin (1877), p. 144.Google Scholar
(11)Leppington, F. and Levine, H.On the capacity of the circular disk condenser at small separation. Proc. Cambridge Philos. Soc. 68 (1970), 235.CrossRefGoogle Scholar
(12)Love, A. E. H.Some electrostatic distributions in two dimensions. Proc. London Math. Soc. 22 (1924), 337.CrossRefGoogle Scholar
(13)Nomura, Y.The electrostatic problems of two equal parallel circular plates. Proc. Phys. Math. Soc. Japan 3, no. 23 (1941), 168.Google Scholar
(14)Noble, B.The Wiener–Hopf Technique (Pergamon Press, London, 1958).Google Scholar
(15)Pólya, G. and Szegö, G.Isoperimetric inequalities in mathematical physics (Princeton University Press, 1951).Google Scholar
(16)Serini, R.R. Accad. Naz. Lin. Rend. 29, no. 34 (1920), p. 257.Google Scholar
(17)Shaw, S.Circular-disk viscometer and related electrostatic problems. Phys. Fluids 13, no. 8 (1970), 1935.CrossRefGoogle Scholar
(18)Sparenberg, J. A.Application of the theory of sectionally holomorphic functions to Wiener–Hopf type integral equation. Koninkl. Ned. Akad. Wetenschap AA (59), (1956) 29–34.CrossRefGoogle Scholar
(19)Tranter, C. J.On some dual integral equations occurring in potential problems with axial symmetry. Quart. J. Mech. Applied Math. 3 (1950), 411.CrossRefGoogle Scholar
(20)Watson, G. N.A Treatise on the theory of Bessel functions (Cambridge University Press, 1944).Google Scholar
(21)Wigglesworth, L. A.Comments on ‘circular disk viscometer and related electrostatic problems’. Phys. Fluids 15, no. 4 (1972), 718.CrossRefGoogle Scholar