Hostname: page-component-848d4c4894-nmvwc Total loading time: 0 Render date: 2024-06-28T23:30:30.092Z Has data issue: false hasContentIssue false
  • English
  • Français

End effects in a two dimensional potential problem for closely spaced rectangular plates

Published online by Cambridge University Press:  17 February 2009

N. G. Barton
N. G. Barton
Affiliation:
C.S.I.R.O. Division of Mathematics and Statistics, P. O. Box 218, Lindfield, N.S.W., 2070.
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.

This paper presents a conformal mapping solution of Laplace's equation in the two dimensional region exterior to two rectangular plates or electrodes at different potentials. Plates with finite and semi-infinite lengths are considered separately and particular emphasis is placed upon the case when the separation between the plates is small. The key results of the paper are expressions for the integral of the square of the normal field along the mid-line between the plates. This integral is of importance in certain gaseous conductor experiments that are sufficiently accurate for a consideration of end effects to be necessary. For small gaps, the dominant end correction to the integral is linear with the gap width. It is also shown that, for small gaps, the simplified (semi-infinite) geometry gives essentially the same value for the integral as the full (finite plate length) geometry.

Type
Research Article
Information
The ANZIAM Journal , Volume 26 , Issue 4 , April 1985 , pp. 404 - 414
Copyright
Copyright © Australian Mathematical Society 1985

References

[1]Abramowitz, M. and Stegun, I. A. (eds.), Handbook of mathematical functions (Dover, New York, 1965).Google Scholar
[2]Carrier, G. F., Krook, M. and Pearson, C. E., Functions of a complex variable (McGraw-Hill, New York, 1966).Google Scholar
[3]Cobine, J. D., Gaseous conductors (Dover, New York, 1941).Google Scholar
[4]Davy, N., “The field between equal semi-infinite rectangular electrodes or magnetic pole pieces”, Philos. Mag. (7) 35 (1944), 819–340.CrossRefGoogle Scholar
[5]Fenton, J. D. and Gardiner-Garden, R. S., “Rapidly convergent method for evaluating elliptic integrals and theta and elliptic functions”, J. Austral. Math. Soc. Ser. B 24 (1982), 47–58.CrossRefGoogle Scholar
[6]Gradshteyn, I. S. and Ryzhik, I. M., Table of integrals, series and products (Academic, New York, 1965).Google Scholar
[7]Kober, H., Dictionary of conformal representations (Dover, New York, 1957).Google Scholar
[8]Rayleigh, Lord, “On the theory of resonance”, Philos. Trans. Roy. Soc. London 161 (1871), 77–118.Google Scholar
[9]Riemann, G. F. B. and Weber, H., Differentialgleichungen der Physik (Brunswick, 1927).Google Scholar