An Electrostatics Problem on the Sphere Arising from a Nearby Point Charge

For a positively charged insulated d-dimensional sphere we investigate how the distribution of this charge is affected by proximity to a nearby positive or negative point charge when the system is governed by a Riesz s-potential 1/r^s, s>0, where r denotes Euclidean distance between point charges...

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Bibliographic Details
Published in:arXiv.org 2014-02
Main Authors: Brauchart, Johann S, Dragnev, Peter D, Saff, Edward B
Format: Article
Language:English
Subjects:
Charge distribution
Complex variables
Electrostatics
Euclidean geometry
Point charge
Polynomials
Spherical caps
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Summary:For a positively charged insulated d-dimensional sphere we investigate how the distribution of this charge is affected by proximity to a nearby positive or negative point charge when the system is governed by a Riesz s-potential 1/r^s, s>0, where r denotes Euclidean distance between point charges. Of particular interest are those distances from the point charge to the sphere for which the equilibrium charge distribution is no longer supported on the whole of the sphere (i.e. spherical caps of negative charge appear). Arising from this problem attributed to A. A. Gonchar are sequences of polynomials of a complex variable that have some fascinating properties regarding their zeros.
ISSN:2331-8422