# Fields and charges near the apex of a hyperbolic cone

Johan Stén, Päivi Koivisto

Research output: Contribution to journalArticleScientificpeer-review

1 Citation (Scopus)

### Abstract

The sharp point of a cone is known to accumulate charges, thus representing a singularity for the electric field. However, a real pointed object has a finite radius of curvature at the apex. Thus, we investigate in this paper how the 'roundness' affects the behaviour of fields and charges at the tip of a cone. Two cases are considered: a double cone being the asymptote of a two-sided hyperboloid of two sheets and a one-sided cone being the asymptote of a similar one-sided hyperboloid. Our analysis employs the prolate spheroidal system of coordinates and uses Legendre functions of the first kind of non-integer degree. The behaviour of the surface charge density is illustrated graphically in terms of the distance to the apex. Simple approximate formulas for the charge density valid near the apex are given
Original language English 015019 European Journal of Physics 35 1 https://doi.org/10.1088/0143-0807/35/1/015019 Published - 2014 A1 Journal article-refereed

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cones
apexes
asymptotes
Legendre functions
curvature
electric fields

### Cite this

Stén, Johan ; Koivisto, Päivi. / Fields and charges near the apex of a hyperbolic cone. In: European Journal of Physics. 2014 ; Vol. 35, No. 1.
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Fields and charges near the apex of a hyperbolic cone. / Stén, Johan; Koivisto, Päivi.

In: European Journal of Physics, Vol. 35, No. 1, 015019, 2014.

Research output: Contribution to journalArticleScientificpeer-review

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AU - Stén, Johan

AU - Koivisto, Päivi

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AB - The sharp point of a cone is known to accumulate charges, thus representing a singularity for the electric field. However, a real pointed object has a finite radius of curvature at the apex. Thus, we investigate in this paper how the 'roundness' affects the behaviour of fields and charges at the tip of a cone. Two cases are considered: a double cone being the asymptote of a two-sided hyperboloid of two sheets and a one-sided cone being the asymptote of a similar one-sided hyperboloid. Our analysis employs the prolate spheroidal system of coordinates and uses Legendre functions of the first kind of non-integer degree. The behaviour of the surface charge density is illustrated graphically in terms of the distance to the apex. Simple approximate formulas for the charge density valid near the apex are given

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