This paper presents a closed form solution for the electrical potential perturbation of a perfectly insulating flat circular disc embedded in a homogeneous half-space in a uniform primary electric field. It is an adaptation of Weber's method for the potential around a charged conducting disk. It yields closed form analytic solutions for the electric and magnetic fields and, by straightforward numerical integration, an easily evaluated numerical solution for the electric potential, and an explicit solution for the electrical resistivity of a composite material consisting of a dilute concentration of such embedded disks in an otherwise uniform conductor.
2. Anderson, W. L., "Numerical integration of related hankel transforms of orders 0 and 1 by adaptive digital filtering," Geophysics, Vol. 44, 1287-1305, 1979.
3. Anderson, W. L., "Fast hankel transforms using related and lagged convolutions," ACM Trans. on Math. Software, Vol. 8, 344-368, 1982.
4. Atkinson, W. J., J. H. Young, and I. A. Brezovich, "An analytic solution for the potential due to a circular parallel plate capacitor," J. Phys. A: Math. Gen., Vol. 16, 2837-2841, 1983.
5. Chave, A. D., "Numerical integration of related hankel transforms by quadrature and continued fraction expansion," Geophysics, Vol. 48, 1671-1686, 1983.
6. Chew, W. C., "A current source emitting in the presence of an insulating and a conducting disk over stratified media," Journal of Electrostatics, Vol. 18, 273-287, 1986.
7. Clyne, T. W., Comprehensive Composite Material, Vol. 3, No. 16, 447, Elsevier, 2000.
8. Cornille, P., "Computation of hankel transforms," SIAM Review, Vol. 14, No. 2, 278-285, 1972.
9. Desclos, L. and P. Pouliguen, "Physical optics applied to discs and square plates in near-field leads to a simple formula," Microwave and Optical Technology Letters, Vol. 9, 278-283, 1995.
10. Guizar-Sicairos, M. and J. C. Gutierrez-Vega, "Computation of quasi-discrete hankel transforms of integer order for propagating optical wave fields," J. Opt. Soc. Am. A, Vol. 21, No. 1, 53-58, 2004.
11. Guptasarma, D. and B. Singh, "New digital linear filters for hankel j0 and j1 transforms," Geophysical Prospecting, Vol. 45, 745-762, 1997.
12. Higgins, W. E. and D. C. Munson, "An algorithm for computing general integer-order hankel transforms," IEE Transactions on Acoustics, Speech, and Signal Processing, Vol. 35, No. 1, 86-97, 1987.
13. Hongo, K. and Q. A. Naqvi, "Diffraction of electromagnetic wave by disk and circular hole in a perfectly conducting plane," Progress In Electromagnetics Research, Vol. 68, 113-150, 2007.
14. Jackson, J. D., Classical Electrodynamics, John Wiley and Sons, 1962.
15. Johansen, H. K. and K. Sorensen, "Fast hankel transforms," Geophysical Prospecting, Vol. 27, 876-901, 1979.
16. Mahmoud, S. F., "Full wave analysis of electromagnetic wave scattering from an imperfectly conducting circular disc," Journal of Electromagnetic Waves and Applications, Vol. 15, 917-932, 2001.
17. Prudnikov, A. P., Y. A. Brychkov, and O. I. Marichev, Integrals and Series, Special Functions, Vol. 2, Gordon and Breach Science, 1986.
18. Prudnikov, A. P., Y. A. Brychkov, and O. I. Marichev, Integrals and Series, Direct Laplace Transforms, Vol. 4, Gordon and Breach Science, 1986.
19. Sneddon, I. N., Mixed Boundary Value Problems in Potential Theory, North-Holland, Amsterdam, 1966.
20. Weber, H. S., "Ueber die besselschen functionen und ihre anwendung auf die theorie der elektrischen strme," J. fr. Math., Vol. 75, 75-105, 1873.
21. Withers, P. J., Encyclopedia of Materials Science and Technology, Elsevier, 2001.