On the generalized potential problem for a surface of revolution

Authors:
V. I. Fabrikant, T. S. Sankar and M. N. S. Swamy

Journal:
Proc. Amer. Math. Soc. **90** (1984), 47-56

MSC:
Primary 31B20; Secondary 45E10

DOI:
https://doi.org/10.1090/S0002-9939-1984-0722414-3

MathSciNet review:
722414

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Abstract: The following generalized potential problem is considered: given an arbitrary potential distribution on a surface of revolution, the charge density is to be determined. The problem is called generalized due to the assumption that the point charge potential diminishes with the distance as , . The particular case of corresponds to the electrostatic potential problem. A closed form solution to the problem is obtained for a certain class of surfaces of revolution due to a special integral representation of the kernel of the governing integral equation. Three examples are considered: the uniform potential distribution over a spherical cap, the case of an earthed conducting spherical cap in a uniform external field acting in two different directions, namely along the axis and the axis, respectively. The expressions for the charge density distributions are determined for each of the examples. The general results presented in this paper may also be applied to the solution of the mathematically identitical problems in hydrodynamics, thermal conductivity, etc.

**[1]**W. D. Collins,*On the solution of some axisymmetric boundary value problems by means of integral equations. I. Some electrostatic and hydrodynamic problems for a spherical cap*, Quart. J. Mech. Appl. Math.**12**(1959), 232–241. MR**0106009**, https://doi.org/10.1093/qjmam/12.2.232**[2]**D. Homentcovschi,*On the electrostatic potential problem for a spherical cap*, Z. Angew. Math. Mech.**60**(1980), 636-637.**[3]**T. S. Sankar and V. Fabrikant,*Asymmetric contact problem including wear for nonhomogeneous half space*, Trans. ASME Ser. E J. Appl. Mech.**49**(1982), no. 1, 43–46. MR**653974****[4]**Granino A. Korn and Theresa M. Korn,*Mathematical handbook for scientists and engineers*, Second, enlarged and revised edition, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1968. MR**0220560****[5]**I. S. Gradshteyn and I. M. Ryzhik,*Table of integrals, series, and products*, 6th ed., Academic Press, Inc., San Diego, CA, 2000. Translated from the Russian; Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger. MR**1773820**

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DOI:
https://doi.org/10.1090/S0002-9939-1984-0722414-3

Article copyright:
© Copyright 1984
American Mathematical Society