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Capacitance Matrix Revisited

By Ivica Smolić and Bruno Klajn
Progress In Electromagnetics Research B, Vol. 92, 1-18, 2021


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.


Ivica Smolić and Bruno Klajn, "Capacitance Matrix Revisited," Progress In Electromagnetics Research B, Vol. 92, 1-18, 2021.


    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.

    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.

    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.

    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.

    21. Lee, J. M., Introduction to Smooth Manifolds, Springer, New York, 2003.

    22. Federer, H., Geometric Measure Theory, Springer, Berlin New York, 1996.

    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.

    26. McGrath, P., "On the smooth jordan brouwer separation theorem," Amer. Math. Monthly, Vol. 123, 292-295, 2016.

    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.

    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.

    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.

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

    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.

    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.

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

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

    37. Pólya, G. and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Princeton University Press, Princeton, 1951.

    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.

    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.

    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.

    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.

    43. Zaremba, S., "Sur le principe de dirichlet," Acta Math., Vol. 34, 293-316, 1911.

    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.

    46. Van Bladel, J. G., Electromagnetic Fields, Wiley-Interscience John Wiley, Distributor, Hoboken, N.J. Chichester, 2007.

    47. Salsa, S., Partial Differential Equations in Action: From Modelling to Theory, Springer, Cham, 2015.

    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.

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

    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.

    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.

    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.

    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.

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