On the problem of the electrified disc

E. T. Copson

doi:10.1017/S0013091500027644

On the problem of the electrified disc

Published online by Cambridge University Press: 20 January 2009

E. T. Copson

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E. T. Copson: Affiliation:
University College, University of St Andrews, Dundee.

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When a perfectly conducting uniform thin circular disc is kept at a potential V0 in an external electrostatic field of potential Φ, electric charge is induced on the surface of the disc; the problem is to find the surface-density σ of this induced charge and its potential V so that the total potential V + Φ has the constant value V0 on the surface of the disc. This problem was first discussed by Green in 1832, and the solution in the case when there is no external field was deduced by Lord Kelvin from the known formula for the gravitational potential of an elliptic homoeoid. The problem is still of interest since similar ideas occur in the theory of diffraction by a circular disc and in the theory of the generation of sound waves by a vibrating disc when the wave-length is large compared with the radius of the disc.

Type: Research Article
Information: Proceedings of the Edinburgh Mathematical Society , Volume 8 , Issue 1 , October 1947 , pp. 14 - 19

DOI: https://doi.org/10.1017/S0013091500027644 [Opens in a new window]
Copyright: Copyright © Edinburgh Mathematical Society 1947

References

^{page 14 note 1} By σ we mean the total surface density on the two faces of the disc.

^{page 14 note 2} Trails. Gamb. Phil. Soc., (11. 12, 1832), reprinted in Green's Mathematical Papers (London, 1871), 172–183.Google Scholar

^{page 14 note 3} Papers on Electricity and Magnetism, (Cambridge, 1884), 175–191.Google Scholar

^{page 14 note 4} Journal f. Math., 75 (1873), 75–105.Google Scholar

^{page 15 note 1} Weber, H., loc. cit.Google Scholar

^{page 15 note 2} Quarterly Journal of Mathematics, 21 (1886), 229–256.Google Scholar

^{page 15 note 3} Proc. Oamb. Phil. Soc., 5 (1886), 425–443.Google Scholar

^{page 15 note 4} Proc. London Math. Soc. (1), 26 (1895), 257–260.Google Scholar

^{page 15 note 5} Bologna Memork (4), 2 (1880), 461–505.Google Scholar

^{page 15 note 6} Proc. Boy. Soc. (A), 153 (1936), 1–16.Google Scholar

^{page 15 note 7} Cf.Watson, , Theory of Bessel Functions (Cambridge, 1944), 398–406.Google Scholar

^{page 15 note 8} The Theory of Fourier Integrals (Oxford, 1937), 334–339.Google Scholar

^{page 17 note 1} CfWhittaker, and Watson, , Modern Analysis (Cambridge, 1920), p. 229.Google Scholar

^{page 19 note 1} CfWatson, , loc. cit., 180 (14).Google Scholar

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