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20.—Inequalities for the Capacity of an Electrified Conducting Annular Disc

Published online by Cambridge University Press:  14 February 2012

E. R. Love  and
W. D. Collins
E. R. Love
Affiliation:
Department of Mathematics, University of Melbourne, Australia
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Synopsis

The simultaneous integral equations of Noble and Cooke, for inter alia the Dirichlet problem for an annular disc, are transformed into equations closely analogous to those recently given by Clements and Love for the corresponding Neumann problem. The transformed equations are relatively simple, and do not involve any artificial preliminary dissection of the known functions. They are also uncoupled, and admit iterative solution for virtually all radius ratios.

In the case of the conducting annular disc, iteration of these equations isused to obtain two series for the capacity, an interim one with all terms positive, and a final one with all terms after the first negative. The final series is more rapidly convergent as well as neater. It is used to provide a family of inequalities which enclose the capacity and determine it to high accuracy with very few terms, failing only when the radius ratio is close to 1.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 74 , 1976 , pp. 257 - 270
Copyright
Copyright © Royal Society of Edinburgh 1976

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References

1Copson, E. T.. On the problem of the electrified disc. Proc. Edinburgh Math. Soc. 8 (1947), 14–19.CrossRefGoogle Scholar
2Smythe, W. R.. The capacitance of a circular annulus. J. Appl. Phys 22 (1951), 1499–1501.CrossRefGoogle Scholar
3Noble, B.. Certain dual integral equations. J. Math. Phys. 37 (1958), 128–136.CrossRefGoogle Scholar
4Gubenko, V. S. and Mossakovskii, V. I.. Pressure of an axially symmetric circular die on an elastic half space. Prikl. Mat. Meh. 24 (1960), 334–340. (Transl. J. Appl. Math. Mech. 24 (1960), 477–486).Google Scholar
5Tranter, C. J.. Some triple integral equations. Proc. Glasgow Math. Assoc. 4 (1960), 200–203.CrossRefGoogle Scholar
6Collins, W. D.. On some triple series equations and their applications. Arch. Rational Mech Anal. 11 (1962), 122–137.CrossRefGoogle Scholar
7Noble, B.. The solution of Bessel function dual integral equations by a multiplying factor method. Proc. Cambridge Philos. Soc. 59 (1963), 351–362.CrossRefGoogle Scholar
8Collins, W. D.. On the solution of some axisymmetric boundary value problems by means of integral equations: VIII. Potential problems for a circular annulus. Proc. Edinburgh Math. Soc. 13 (1963), 235–246.CrossRefGoogle Scholar
9Cooke, J. C.. Triple integral equations. Quart. J. Mech. Appl. Math. 16 (1963), 193–203.CrossRefGoogle Scholar
10Williams, W. E.. Note on the electrostatic problem for a circular annulus. Quart. J. Mech Appl. Math. 16 (1963), 205–207.CrossRefGoogle Scholar
11Williams, W. E.. Integral equation formulation of some three part boundary value problems. Proc. Edinburgh Math. Soc. 13 (1963), 317–323.CrossRefGoogle Scholar
12Cooke, J. C.. Some further triple integral equation solutions. Proc. Edinburgh Math. Soc 13 (1963), 303–316.CrossRefGoogle Scholar
13Spence, D. A.. A Wiener-Hopf solution to the triple integral equations for the electrified disc in a coplanar gap. Proc. Cambridge Philos. Soc 68 (1970), 529–545.CrossRefGoogle Scholar
14Leppington, F. G. and Levine, H.. Some axially symmetric potential problems. Proc. Edinburgh Math. Soc 18 (1972), 55–76.CrossRefGoogle Scholar
15Clements, D. L. and Love, E. R.. Potential problems involving an annulus. Proc. Cambridge Philos. Soc 76 (1974), 313–325.CrossRefGoogle Scholar