Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-26T14:24:54.810Z Has data issue: false hasContentIssue false
  • English
  • Français

The circular plate condenser at small separations

Published online by Cambridge University Press:  24 October 2008

V. Hutson
V. Hutson
Affiliation:
Department of Applied Mathematics, Sheffield University
Get access Rights & Permissions [Opens in a new window]

Abstract

The problem considered is the oppositely charged, circular disk condenser when the disks are very close together. An integral equation due to Love(5) is used as the governing equation of the problem. This equation is solved asymptotically for small separations by splitting the field into regions, one being an annulus containing the edges, the other being the rest of the domain, and combining these solutions. Bounds for the error in the solution of the integral equation are obtained rigorously; the error is shown to approach zero as the separation approaches zero. The capacity of the system is deduced from this solution. The problem of finding the capacity has previously been attempted by various authors, whose results have differed. The present treatment establishes which of these results is correct.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 59 , Issue 1 , January 1963 , pp. 211 - 224
Copyright
Copyright © Cambridge Philosophical Society 1963

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

REFERENCES

(1)Courant, R. and Hilbert, D.Methods of mathematical physics, vol. 1 (Interscience; New York, 1953).Google Scholar
(2)Ignatowsky, W.Kreisscheibenkondensator. Acad. Sci. U.R.S.S. Trav. Inst. Stekloff. (2), 3 (1932), 1–104.Google Scholar
(3)Kac, M. and Pollard, H.The distribution of the maximum of partial sums of independent random variables. Canadian J. Math. 2 (1956), 375–384.Google Scholar
(4)Kirchhoff, G.Zur Theorie des Condensators. Monatsb. Acad. Wiss. Berlin (1877), 144–162.Google Scholar
(5)Love, E. R.The electrostatic field of two circular coaxial conducting disks. Quart. J. Mech. Appl. Math. 2 (1949), 428–451.Google Scholar
(6)Morse, P. M. and Feshbach, H.Methods of theoretical physics, vol. 2 (McGraw-Hill; New York, 1953).Google Scholar
(7)Noble, B.The Wiener—Hopf technique (Pergamon Press; London, 1958).Google Scholar
(8)Pólya, G. and Szegö, G.Isoperimetric inequalities in mathematical physics (Princeton, 1951).Google Scholar
(9)Reich, E.A Random walk related to the capacitance of the circular plate condenser. Quart. Appl. Math. 11 (1953), 341–345.Google Scholar
(10)Sparenberg, J. A.Application of the theory of sectionally holomorfic functions to Wiener—Hopf type integral equations. Indag. Math. 18 (1957), 29–34.Google Scholar
(11)Tricomi, F. G.Integral equations (Interscience; New York, 1957).Google Scholar