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Edge corrections for parallel-plate capacitors

Part of: Harmonic analysis in several variables Two-dimensional theory Partial differential equations Equations of mathematical physics and other areas of application Approximations and expansions

Published online by Cambridge University Press:  30 April 2020

EHUD YARIV Open the ORCID record for EHUD YARIV [Opens in a new window]
EHUD YARIV*
Affiliation:
Department of Mathematics, Technion—Israel Institute of Technology, Haifa32000, Israel, email: [email protected]
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Abstract

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As a result of field fringing, the capacitance of a parallel-plate capacitor differs from that predicted by the textbook formula. Using singular perturbations and conformal mapping techniques, we calculate the leading-order correction to the capacitance in the limit of large aspect ratio. We additionally obtain a comparable approximation for the electrostatic attraction between the plates.

Keywords

Singular perturbationsconformal mappings

MSC classification

Primary: 35B25: Singular perturbations 35Q60: PDEs in connection with optics and electromagnetic theory 41A60: Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 31A99: None of the above, but in this section
Secondary: 42B37: Harmonic analysis and PDE
Type
Papers
Information
European Journal of Applied Mathematics , Volume 32 , Issue 2 , April 2021 , pp. 226 - 241
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Footnotes

†

This work was supported by the Israel Science Foundation (Grant no. 1081/16).

References

Atkinson, C. & Leppington, F. G. (1983) The asymptotic solution of some integral equations. IMA J. Appl. Math. 31, 169–182.CrossRefGoogle Scholar
Atkinson, C. & Sherwood, J. D. (2017) Added mass of a pair of disks at small separation. Eur. J. Appl. Math. 28, 687–706.CrossRefGoogle Scholar
Binns, K. J. & Lawrenson, P. J. (1973) The Analytical and Numerical Solution of Electric and Magnetic Field Problems, Pergamon Press, New York.Google Scholar
Bromwich, T. J. I’A. Note on condenser problems. Mess. Math. 31, 184 (1902).Google Scholar
Brown, J. W. & Churchill, R. V. (2003) Complex Variables and Applications, McGraw-Hill, New York.Google Scholar
Carlson, G. T. & Illman, B. L. (1994) The circular disk parallel plate capacitor. Am. J. Phys. 62, 1099–1105.CrossRefGoogle Scholar
Chew, W. C. & Kong, J. A. (1981) Asymptotic formula for the capacitance of two oppositely charged discs. Math. Proc. Cambridge Philos. Soc. 89, 373–384.Google Scholar
Cooke, J. C. (1958) The coaxial circular disc problem. ZAMM 38, 349–356.CrossRefGoogle Scholar
Feynman, R. P., Leighton, R. B. & Sands, M. (1979) The Feynman Lectures on Physics, Vol. 2: Mainly Electromagnetism and Matter, Addison–Wesley, Massachusetts.Google Scholar
Grove, T. T., Masters, M. F. & Miers, R. E. (2005) Determining dielectric constants using a parallel plate capacitor. Am. J. Phys. 73, 52–56.CrossRefGoogle Scholar
Harwood, R. J. & Kinkaid, D. E. (1975) Use of a Mettler balance and a parallel plate capacitor to measure the permittivity of free space. Am. J. Phys. 43, 924–925.CrossRefGoogle Scholar
Hinch, E. J. (1991) Perturbation Methods, Cambridge University Press, Cambridge.CrossRefGoogle Scholar
Hutson, V. (1963) The circular plate condenser at small separations. Math. Proc. Cambridge Philos. Soc. 59, 211–224.Google Scholar
Ignatowsky, W. (1932) Kreisscheibenkondensator. Akad. Sci. U.R.S.S. Trav. Inst. Stekloff. 3, 1–104.Google Scholar
Jackson, J. D. (1999) A curious and useful theorem in two-dimensional electrostatics. Am. J. Phys. 67, 107–115.CrossRefGoogle Scholar
Jackson, J. D. (2002) Charge density on a thin straight wire: the first visit. Am. J. Phys. 70, 409–410.CrossRefGoogle Scholar
Jeffrey, D. J. (1978) The temperature field or electric potential around two almost touching spheres. J. Inst. Math. Appl. 22, 337–351.CrossRefGoogle Scholar
Ref, In. [17], the latter is denoted “inner”.Google Scholar
Kirchhoff, G. R. (1877) Zur theorie des condensators. Monatsb. Deutsch. Acad. Wiss. Berlin, 144–172.Google Scholar
Kromhout, R. A. & Moulton, W. G. (1956) Effect of fringing field on capacitance and a measurement of ε 0. Am. J. Phys. 24, 631–632.CrossRefGoogle Scholar
Landau, L. D. & Lifshitz, E. M. (1960) Course of Theoretical Physics: Electrodynamics of Continuous Media, Permagon Press.Google Scholar
Leppington, F. & Levine, H. (1970) On the capacity of the circular disc condenser at small separation. Math. Proc. Cambridge Philos. Soc. 68, 235–254.Google Scholar
Love, A. E. H. (1924) Some electrostatic distributions in two dimensions. P. Lond. Math. Soc. 2, 337–369.CrossRefGoogle Scholar
Naini, A. & Green, M. (1977) Fringing fields in a parallel-plate capacitor. Am. J. Phys. 45, 877–879.CrossRefGoogle Scholar
Nishiyama, H. & Nakamura, M. (1990) Capacitance of a strip capacitor. IEEE Trans. Compon. Hybrids Manuf. Technol. 13, 417–423.CrossRefGoogle Scholar
Parker, G. W. (2002) Electric field outside a parallel plate capacitor. Am. J. Phys. 70, 502–507.CrossRefGoogle Scholar
Purcell, E. M. (1985) Electricity and Magnetism, Berkeley Physics Course, Vol. 2, McGraw-Hill.Google Scholar
Shaw, S. J. N. (1970) Circular-disk viscometer and related electrostatic problems. Phys. Fluids 13, 1935–14.CrossRefGoogle Scholar
Sneddon, I. N. (1966) Mixed Boundary Value Problems in Potential Theory, Wiley, New York).Google Scholar
Thomson, J. J. (1893) Recent Researches in Electricity and Magnetism, Clarendon, Oxford, UK, pp. 208–250.Google Scholar
Wintle, H. J. & Kurylowicz, S. (1985) Edge corrections for strip and disc capacitors. IEEE Trans. Instrum. Meas. 34, 41–47.CrossRefGoogle Scholar
Yan, F. N. & Wong, H. K. (1993) Force between the plates of a parallel-plate capacitor. Am. J. Phys. 61, 1153–1153.CrossRefGoogle Scholar