Ellipsoidal harmonics and their application in electrostatics

Johan Sten (Corresponding Author)

Research output: Contribution to journalArticleScientificpeer-review

23 Citations (Scopus)

Abstract

Ellipsoidal functions and ellipsoidal co-ordinates, naturally adapted to the treatment of potential problems involving ellipsoidal boundaries, are due to a relatively complicated mathematical appearance avoided in the majority of textbooks on electromagnetism, leading to an unjustified ignorance of the whole system. The paper at hand is devoted to showing that the system is worthwhile using and investigating. By first recollecting the key results of the co-ordinate system and the related potential functions, including the indispensable orthogonality result and the reciprocal distance formula, basic potential problems of electrostatics are reviewed. Closed-form expressions are thereupon derived for the induced charge and its centroid in the case of a point charge influencing a grounded ellipsoid. An image source interpretation, generalising the image solution for a sphere, follows easily from these expressions, as well as an approximate expression for the mutual capacitance between two ellipsoids or a parallel disk capacitor. An expression is also given for the total polarisation induced in a dielectric ellipsoid by the influence of a point dipole.
Original languageEnglish
Pages (from-to)647-654
Number of pages8
JournalJournal of Electrostatics
Volume64
Issue number10
DOIs
Publication statusPublished - 2006
MoE publication typeA1 Journal article-refereed

Fingerprint

Textbooks
Static Electricity
Electrostatics
Hand
ellipsoids
electrostatics
Electromagnetism
harmonics
Capacitors
Capacitance
Polarization
textbooks
orthogonality
centroids
capacitors
capacitance
electromagnetism
dipoles
polarization

Keywords

  • ellipsoidal particles
  • ellipsoidal harmonics
  • ellipsoidal systems
  • capacitance
  • polarizability
  • electromagnetic theory
  • electrostatics

Cite this

Sten, Johan. / Ellipsoidal harmonics and their application in electrostatics. In: Journal of Electrostatics. 2006 ; Vol. 64, No. 10. pp. 647-654.
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Ellipsoidal harmonics and their application in electrostatics. / Sten, Johan (Corresponding Author).

In: Journal of Electrostatics, Vol. 64, No. 10, 2006, p. 647-654.

Research output: Contribution to journalArticleScientificpeer-review

TY - JOUR

T1 - Ellipsoidal harmonics and their application in electrostatics

AU - Sten, Johan

PY - 2006

Y1 - 2006

N2 - Ellipsoidal functions and ellipsoidal co-ordinates, naturally adapted to the treatment of potential problems involving ellipsoidal boundaries, are due to a relatively complicated mathematical appearance avoided in the majority of textbooks on electromagnetism, leading to an unjustified ignorance of the whole system. The paper at hand is devoted to showing that the system is worthwhile using and investigating. By first recollecting the key results of the co-ordinate system and the related potential functions, including the indispensable orthogonality result and the reciprocal distance formula, basic potential problems of electrostatics are reviewed. Closed-form expressions are thereupon derived for the induced charge and its centroid in the case of a point charge influencing a grounded ellipsoid. An image source interpretation, generalising the image solution for a sphere, follows easily from these expressions, as well as an approximate expression for the mutual capacitance between two ellipsoids or a parallel disk capacitor. An expression is also given for the total polarisation induced in a dielectric ellipsoid by the influence of a point dipole.

AB - Ellipsoidal functions and ellipsoidal co-ordinates, naturally adapted to the treatment of potential problems involving ellipsoidal boundaries, are due to a relatively complicated mathematical appearance avoided in the majority of textbooks on electromagnetism, leading to an unjustified ignorance of the whole system. The paper at hand is devoted to showing that the system is worthwhile using and investigating. By first recollecting the key results of the co-ordinate system and the related potential functions, including the indispensable orthogonality result and the reciprocal distance formula, basic potential problems of electrostatics are reviewed. Closed-form expressions are thereupon derived for the induced charge and its centroid in the case of a point charge influencing a grounded ellipsoid. An image source interpretation, generalising the image solution for a sphere, follows easily from these expressions, as well as an approximate expression for the mutual capacitance between two ellipsoids or a parallel disk capacitor. An expression is also given for the total polarisation induced in a dielectric ellipsoid by the influence of a point dipole.

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KW - ellipsoidal harmonics

KW - ellipsoidal systems

KW - capacitance

KW - polarizability

KW - electromagnetic theory

KW - electrostatics

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