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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

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