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A Wiener-Hopf solution to the triple integral equations for the electrified disc in a coplanar gap

Published online by Cambridge University Press:  24 October 2008

D. A. Spence
D. A. Spence
Affiliation:
Department of Engineering Science, Parks Road, Oxford
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Abstract

The axisymmetric potential problem for a plane circular electrode of radius a in a concentric hole of radius b in a coplanar earthed sheet is formulated in terms of triple integral equations for the Hankel transform of the potential, and reduced to a single Fredholm equation by use of the Erdélyi-Kober fractional operators.

In the limit of small gap width (b − a)/b, the equation takes the form

which is solved by applying the Wiener-Hopf technique to the Mellin transform of f(x). This leads to the asymptotic expression

for the capacity of the disc; for the opposite limit the expression

is derived. Numerical integration of the governing Fredholm equation has been carried out for a range of intermediate values of b/a.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 68 , Issue 2 , September 1970 , pp. 529 - 545
Copyright
Copyright © Cambridge Philosophical Society 1970

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References

REFERENCES

(1)Collins, W. D.Arch. Rational Mech. Anal. 11 (1962), 122.CrossRefGoogle Scholar
(2)Cooke, J. C.Quart. J. Mech. Appl. Math. 16 (1963), 193.CrossRefGoogle Scholar
(3)Cooke, J. C.Quart. J. Mech. Appl. Math. 18 (1965), 57.CrossRefGoogle Scholar
(4)Erdélyi, A. (ed.) Higher transcendental functions, vol. I (New York: McGraw Hill, 1953).Google Scholar
(5)Erdélyi, A. and Sneddon, I. N.Canad. J. Math. 14 (1962), 685.CrossRefGoogle Scholar
(6)Grinberg, G. A. and Kuritsyn, V. N.Soviet Physics Tech. Phys. 6 (9) (1962), 743.Google Scholar
(7)Smythe, W. R.J. Appl. Phys. 22 (1951), 1499.CrossRefGoogle Scholar
(8)Sneddon, I. N.Mixed boundary value problems in potential theory (Amsterdam: North Holland, 1966).Google Scholar
(9)Tassicker, O. J.Electronics letters 5 (13) (1969), 285.CrossRefGoogle Scholar
(10)Weber, H.J. Math. 75 (1873), 75.Google Scholar
(11)Whittaker, E. T. and Watson, G. N.A course of modern analysis (4th edn.) (Cambridge University Press, 1946).Google Scholar
(12)Williams, W. E.Quart. J. Mech. Appl. Math. 16 (1963), 204.CrossRefGoogle Scholar