Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-27T11:27:47.355Z Has data issue: false hasContentIssue false
  • English
  • Français

E. R. Love's integral equation for the circular plate condenser

Published online by Cambridge University Press:  17 February 2009

Edgar Reich
Edgar Reich
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455, USA; e-mail: reich@math.umn.edu.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In a classical paper, E. R. Love considered a certain function defined by a singular integral which is harmonic outside a circular disk. Love's objective was to derive a simple integral equation whose solution leads to a useful formula for the capacitance of the condenser consisting of two parallel circular plates. We close a gap in Love's derivation by finding a new nonsingular representation of Love's singular integral which permits one to draw the required conclusions about its boundary values and thereby establishes the correctness of Love's expression for the capacitance.

Type
Research Article
Information
The ANZIAM Journal , Volume 46 , Issue 1 , July 2004 , pp. 85 - 93
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Adamchik, V., Personal communication via V. H. Moll, March 2002.Google Scholar
[2]Hutson, V., “The circular plate condenser at small separation”, Proc. Cambridge Phil. Soc. 59 (1963) 211–225.CrossRefGoogle Scholar
[3]Jäger, G., “Über das elektrische Feld eines ellipsoidischen Leiters”, S. B. Kaiserl. Akad. Wiss. Math. Naturw. Classe Abth. 2a, Wien 110 (1901) 449–453.Google Scholar
[4]Jeans, J. H., The mathematical theory of electricity and magnetism, 5th ed. (Cambridge Univ. Press, New York, 1925).Google Scholar
[5]Kühnau, R., “Randeffekte beim elektrostatischen Kondensator”, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 254 (1998) 132–144.Google Scholar
[6]Kühnau, R., “Die Kapazität dünner Kondensatoren”, Math. Nachr. 203 (1999) 125–130.CrossRefGoogle Scholar
[7]Love, E. R., “The electrostatic field of two equal circular co-axial conducting disks”, Quart. J. Mech. Appl. Math. 2 (1949) 428–451.CrossRefGoogle Scholar
[8]Nicholson, J. W., “The electrification of two parallel circular disks”, Roy. Soc. Phil. Trans. A (2) 224 (1924) 303–369.Google Scholar
[9]Reich, E., “A random walk related to the capacitance of the circular plate condenser”, Quart. Appl. Math. 11 (1953) 342–345.CrossRefGoogle Scholar
[10]Sneddon, I. N., Mixed boundary value problems in potential theory (Wiley, New York, 1966).Google Scholar
[11]Soibelman, Ya. S., “Asymptotic behavior of the capacitance of a condenser with plates of arbitrary form”, Sibirsk. Mat. Zh. 25 (1984) 167–181.Google Scholar
[12]Soibelman, Ya. S., “Condenser capacity and invariants of Riemannian submanifolds”, Selecta Math. 2 (1996) 653–667.CrossRefGoogle Scholar
[13]Stefan, J., “Über das Gleichgewicht der Electricität auf einer Scheibe und einem Ellipsoid”, S. B. Math. Naturw. Classe Kaiserl. Akad. Wiss., Wien 101 (1892) 1583–1588.Google Scholar