A random walk related to the capacitance of the circular plate condenser
Author:
E. Reich
Journal:
Quart. Appl. Math. 11 (1953), 341-345
MSC:
Primary 65.0X
DOI:
https://doi.org/10.1090/qam/57625
MathSciNet review:
57625
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Abstract: It is shown that the solution of Love’s equation for the capacitance of the circular plate condenser can be expressed in terms of the mean duration of a certain one-dimensional random walk with absorbing barriers. The interpretation as a random walk makes it possible to confirm the fact that the actual capacitance of the condenser is always larger than the value given by the standard approximation for small separations, and yields an upper bound as well. In addition to its theoretical interest, the random walk appears to provide a practical means for the calculation of the capacitance by a Monte Carlo technique.
J. H. Jeans, The mathematical theory of electricity and magnetism, 5th edition, Cambridge Univ. Press, London, 1925, pg. 249.
- Mark Kac and Harry Pollard, The distribution of the maximum of partial sums of independent random variables, Canad. J. Math. 2 (1950), 375–384. MR 36465, DOI https://doi.org/10.4153/cjm-1950-034-9
G. Kirchhoff, Gesammelte Abhandlungen, Barth, Leipzig, 1882, 101–117.
- E. R. Love, The electrostatic field of two equal circular co-axial conducting disks, Quart. J. Mech. Appl. Math. 2 (1949), 428–451. MR 34700, DOI https://doi.org/10.1093/qjmam/2.4.428
J. W. Nicholson, The electrification of two parallel circular disks, Roy. Soc. Phil. Trans. (2) 224, 303-369 (1924).
- Wolfgang Wasow, On the mean duration of random walks, J. Research Nat. Bur. Standards 46 (1951), 462–471. MR 0047962
J. H. Jeans, The mathematical theory of electricity and magnetism, 5th edition, Cambridge Univ. Press, London, 1925, pg. 249.
M. Kac and H. Pollard, Partial sums of independent random variables, Can. J. Math. 2, 375-384 (1950).
G. Kirchhoff, Gesammelte Abhandlungen, Barth, Leipzig, 1882, 101–117.
E. R. Love, The electrostatic field of two equal circular conducting disks, Q. J. Mech. Appl. Math. 2, 428-451 (1949).
J. W. Nicholson, The electrification of two parallel circular disks, Roy. Soc. Phil. Trans. (2) 224, 303-369 (1924).
W. Wasow, On the mean duration of random walks, J. Res. Nat. Bur. Standards 46, 462-471 (1951).
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Article copyright:
© Copyright 1953
American Mathematical Society
