On the electrostatic potential for the two-hyperboloid and double-cone of a single sheet with elliptic cross-section. Issue 1 (26th February 2021)

• Record Type: Journal Article
• Authors: Vafeas, Panayiotis; Sten, Johan C -E; Chatjigeorgiou, Ioannis K
• Is Part Of: Quarterly journal of mechanics and applied mathematics. Volume 74:Issue 1(2021)
• Page Start: 117
• Page End: 135
• Publication Date: 2021-02-26
• Subjects: Mechanics -- Mathematics -- Periodicals; Applied mathematics -- Periodicals; 530.1
• DOI: 10.1093/qjmam/hbaa021
• Languages: English
• ISSNs: 0033-5614

Abstract

Summary: The study of the response of divergence-free electric fields near corners and edges, resembling singularities that accumulate charges, is significant in modern engineering technology. A sharp point can mathematically be modelled with respect to the tip of the one sheet of a double cone. Here, we investigate the behaviour of the generated harmonic potential function close to the apex of a single-sheeted two-hyperboloid with elliptic cross-section, whose asymptote is the corresponding elliptic double cone with one sheet present. Hence, the electrostatic potential problem, involving a single sheet of a two-hyperboloid, is developed using the theory of ellipsoidal-hyperboloidal harmonics, wherein the particular consideration enforces as solution in terms of generalised Lamé functions of non-integer order. A numerical method to determine these functions is outlined and tested. We demonstrate our technique to the solution of a classical boundary value problem in electrostatics, referring to a metallic and charged single-sheeted elliptic two-hyperboloid and its double-cone limit. Semi-analytical expressions for the related fields are derived, all cases being accompanied by the necessary numerical implementation.