Vol. 92

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2021-03-19

Capacitance Matrix Revisited

By Ivica Smolić and Bruno Klajn
Progress In Electromagnetics Research B, Vol. 92, 1-18, 2021
doi:10.2528/PIERB21011501

Abstract

The capacitance matrix relates potentials and charges on a system of conductors. We review and rigorously generalize its properties, block-diagonal structure and inequalities, deduced from the geometry of system of conductors and analytic properties of the permittivity tensor. Furthermore, we discuss alternative choices of regularization of the capacitance matrix, which allow us to find the charge exchanged between the conductors having been brought to an equal potential. Finally, we discuss the tacit approximations used in standard treatments of the electric circuits, demonstrating how the formulae for the capacitance of capacitors connected in parallel and series may be recovered from the capacitance matrix.

Citation


Ivica Smolić and Bruno Klajn, "Capacitance Matrix Revisited," Progress In Electromagnetics Research B, Vol. 92, 1-18, 2021.
doi:10.2528/PIERB21011501
http://jpier.org/PIERB/pier.php?paper=21011501

References


    1. Maxwell, J. C., A Treatise on Electricity and Magnetism, 1873, Dover Publications, New York, 2007.

    2. Smythe, W., Static and Dynamic Electricity, Hemisphere Pub. Corp, New York, 1989.

    3. Landau, L. D. and E. M. Lifshitz, Electrodynamics of Continuous Media, Pergamon, Oxford Oxfordshire New York, 1984.

    4. Jackson, J., Classical Electrodynamics, Wiley, New York, 1999.

    5. Durand, E., Électrostatique et magnétostatique, Masson et Cie, 1953.

    6. Chirgwin, B., C. Plumpton, and C. W. Kilmister, Elementary Electromagnetic Theory. Volume 1: Steady Electric Fields and Currents, Pergamon Press, Oxford, New York, 1971.

    7. Schwartz, M., Principles of Electrodynamics, Dover Publications, New York, 1987.

    8. Wangsness, R., Electromagnetic Fields, Wiley, New York, 1986.

    9. Nayfeh, M. H. and M. Brussel, Electricity and Magnetism, Dover Publications, Inc., Mineola, New York, 2015.

    10. Ohanian, H., Classical Electrodynamics, Infinity Science Press, Hingham, Mass, 2007.

    11. Greiner, W., Classical Electrodynamics, Springer, New York, 1998.
    doi:10.1007/978-1-4612-0587-6

    12. Popović, Z. and B. D. Popović, Introductory Electromagnetics, Prentice Hall, Upper Saddle River, NJ, 2000.

    13. Müller-Kirsten, H. J. W., Electrodynamics: An Introduction Including Quantum Effects, World Scientific, Hackensack, NJ Singapore, 2004.
    doi:10.1142/5510

    14. Vanderlinde, J., Classical Electromagnetic Theory, Kluwer Academic Publishers, Dordrecht London, 2004.

    15. Zangwill, A., Modern Electrodynamics, Cambridge University Press, Cambridge, 2013.

    16. Garg, A., Classical Electromagnetism in a Nutshell, Princeton University Press, Princeton N.J., 2012.

    17. Toptygin, I. N., Electromagnetic Phenomena in Matter: Statistical and Quantum Approaches, Wiley-VCH, Weinheim Germany, 2015.

    18. Schwinger, J., L. L. DeRaad, K. A. Milton, W. Tsai, and J. Norton, Classical Electrodynamics, Perseus Books, Reading, Mass, 1998.

    19. Herrera, W. J. and R. A. Diaz, "The geometrical nature and some properties of the capacitance coefficients based on Laplace's equation," Am. J. Phys., Vol. 76, 55-59, 2008.
    doi:10.1119/1.2800355

    20. Diaz, R. A. and W. J. Herrera, "The positivity and other properties of the matrix of capacitance: Physical and mathematical implications," J. Electrostat., Vol. 69, 587-595, 2011.
    doi:10.1016/j.elstat.2011.08.001

    21. Lee, J. M., Introduction to Smooth Manifolds, Springer, New York, 2003.
    doi:10.1007/978-0-387-21752-9

    22. Federer, H., Geometric Measure Theory, Springer, Berlin New York, 1996.
    doi:10.1007/978-3-642-62010-2

    23. Morgan, F., Geometric Measure Theory: A Beginner's Guide, Elsevier Ltd., Amsterdam, 2016.

    24. Guillemin, V. and V. Pollack, Differential Topology, Prentice-Hall, Englewood Cliffs, N.J., 1974.

    25. Lima, E. L., "The Jordan-Brouwer separation theorem for smooth hypersurfaces," Amer. Math. Monthly, Vol. 95, 39-42, 1988.
    doi:10.1080/00029890.1988.11971963

    26. McGrath, P., "On the smooth jordan brouwer separation theorem," Amer. Math. Monthly, Vol. 123, 292-295, 2016.
    doi:10.4169/amer.math.monthly.123.3.292

    27. Perles, M. A., H. Martini, and Y. S. Kupitz, "A Jordan-Brouwer separation theorem for polyhedral pseudomanifolds," Disrete Comput. Geom., Vol. 42, 277-304, 2009.
    doi:10.1007/s00454-009-9192-0

    28. Gilbarg, D. and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore Tokyo, 2001.
    doi:10.1007/978-3-642-61798-0

    29. Kittel, C., Elementary Statistical Physics, Dover Publications, Mineola, N.Y., 2004.

    30. Batygin, V. and I. N. Toptygin, Problems in Electrodynamics, Academic Press, London New York, 1978.

    31. Love, R. R., "The electrostatic field of two equal circular co-axial conducting disks," Q. J. Mech. Appl. Math., Vol. 2, No. 4, 428-451, 1949.
    doi:10.1093/qjmam/2.4.428

    32. Hutson, V., "The circular plate condenser at small separations," Math. Proc. Camb. Philos. Soc., Vol. 59, 211-224, 1963.
    doi:10.1017/S0305004100002152

    33. Rao, T. V., "Capacity of the circular plate condenser: Analytical solutions for large gaps between the plates," J. Phys. A, Vol. 38, No. 46, 10037-10056, 2005.
    doi:10.1088/0305-4470/38/46/010

    34. Paffuti, G., E. Cataldo, A. Di Lieto, and F. Maccarrone, "Circular plate capacitor with different discs," Proc. R. Soc. A, Vol. 472, No. 2194, 20160574, 2016.
    doi:10.1098/rspa.2016.0574

    35. Paffuti, G., "Numerical and analytical results for the two discs capacitor problem," Proc. R. Soc. A, Vol. 73, No. 2197, 20160792, 2017.
    doi:10.1098/rspa.2016.0792

    36. Erma, V. A., "Perturbation approach to the electrostatic problem for irregularly shaped conductors," J. Math. Phys., Vol. 4, 1517-1526, 1963.
    doi:10.1063/1.1703933

    37. Pólya, G. and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Princeton University Press, Princeton, 1951.
    doi:10.1515/9781400882663

    38. Sloggett, G. J., N. G. Barton, and S. J. Spencer, "Fringing fields in disc capacitors," J. Phys. A, Vol. 19, No. 14, 2725-2736, 1986.
    doi:10.1088/0305-4470/19/14/012

    39. James, M. C. and J. R. Solheim, "The effect of trapped charge on series capacitors," Am. J. Phys., Vol. 83, No. 7, 621-627, 2015.
    doi:10.1119/1.4916888

    40. Olyslager, F., Electromagnetic Waveguides and Transmission Lines, Oxford University Press, Oxford New York, 1999.

    41. Bhunia, S., S. Mukhopadhyay, and ed., Low-power Variation-tolerant Design in Nanometer Silicon, Springer, New York, 2011.
    doi:10.1007/978-1-4419-7418-1

    42. Cardoso, D. B., E. T. de Andrade, R. A. A. Calderón, M. H. S. Rabelo, C. de A. Dias, and I. Á. Lemos, "Determination of thermal properties of coffee beans at different degrees of roasting," Coffee Science, Vol. 13, No. 4, 498-509, 2018.
    doi:10.25186/cs.v13i4.1491

    43. Zaremba, S., "Sur le principe de dirichlet," Acta Math., Vol. 34, 293-316, 1911.
    doi:10.1007/BF02393130

    44. Lebesgue, H., "Sur des cas d'impossibilité du problème de Dirichlet ordinaire," C.R. Séances Soc. Math. France, 17, 1913.

    45. Armitage, D. H. and S. J. Gardiner, Classical Potential Theory, Springer, London, 2001.
    doi:10.1007/978-1-4471-0233-5

    46. Van Bladel, J. G., Electromagnetic Fields, Wiley-Interscience John Wiley, Distributor, Hoboken, N.J. Chichester, 2007.
    doi:10.1002/047012458X

    47. Salsa, S., Partial Differential Equations in Action: From Modelling to Theory, Springer, Cham, 2015.
    doi:10.1007/978-3-319-15093-2_2

    48. Evans, L., Partial Differential Equations, American Mathematical Society, Providence, R.I, 2010.

    49. Grisvard, P., Elliptic Problems in Nonsmooth Domains, Society for Industrial and Applied Mathematics, Philadelphia, Pa, 2011.
    doi:10.1137/1.9781611972030

    50. Serrin, J. and H. F. Weinberger, "Isolated singularities of solutions of linear elliptic equations," Am. J. Math., Vol. 88, 258-272, 1966.
    doi:10.2307/2373060

    51. Mitrea, D. and I. Mitrea, "On the Besov regularity of conformal maps and layer potentials on nonsmooth domains," J. Funct. Anal., Vol. 201, No. 2, 380-429, 2003.
    doi:10.1016/S0022-1236(03)00086-7

    52. Meyers, N. and J. Serrin, "The exterior dirichlet problem for second order elliptic partial differential equations," J. Math. Mech., Vol. 9, 513-538, 1960.

    53. Moser, J., "On Harnack's theorem for elliptic differential equations," Commun. Pure Appl. Math., Vol. 14, 577-591, 1961.
    doi:10.1002/cpa.3160140329

    54. Simon, B., Harmonic Analysis. A Comprehensive Course in Analysis, Part 3, American Mathematical Society, Providence, Rhode Island, 2015.

    55. Han, Q. and F. Lin, Elliptic Partial Differential Equations, American Mathematical Society, New York, N.Y. Providence, R.I, 2011.

    56. Morrey, Jr., C. B. and L. Nirenberg, "On the analyticity of the solutions of linear elliptic systems of partial differential equations," Commun. Pure Appl. Math., Vol. 10, 271-290, 1957.
    doi:10.1002/cpa.3160100204

    57. Morrey, Jr., C. B., "On the analyticity of the solutions of analytic non-linear elliptic systems of partial differential equations: Part I. Analyticity in the interior," Am. J. Math., Vol. 10, 198-218, 1958.
    doi:10.2307/2372830

    58. Morrey, C., Multiple Integrals in the Calculus of Variations, Springer, Berlin, 2008.