An application of toroidal functions in electrostatics

Abstract

Quantitative analysis of the damping of magnet oscillations by eddy currents in aluminum foil Am. J. Phys. 80, 804 (2012) Rolling magnets down a conductive hill: Revisiting a classic demonstration of the effects of eddy currents Am. J. Phys. 80, 800 (2012) A semiquantitative treatment of surface charges in DC circuits Am. J. Phys. 80, 782 (2012) Relation between Poisson and Schrödinger equations Am. J. Phys. 80, 715 (2012) Magnetic dipole moment of a moving electric dipole Am.A method employing the use of toroidal functions is introduced for calculating the scalar potential and the electric field from a charged conducting ring. This method is an alternative to the well-known elliptic integral formulation and is usually easier to formulate than the elliptic integral solution.

Peculiarities in the gravitational field of a filamentary ring

“…In connection to our interests here, the sketch of the electric field in the [xz] plane implicitly showed that the field, and thus the force on a point charge is not central. Selvaggi et al employed 22 a different approach and demonstrated -without detailed introduction of the toroidal functions-that the electric potential can be expressed in a compact form using the Fourier cosine expansion of the r − ξ −1 , in which expansion the toroidal functions appear as coefficients of the cos(mϕ) functions. This approach has some numerically attractive features: any non-uniform charge distribution can be systematically analyzed after calculating the Fourier expansion of the charge distribution on the ring, and for each term in this expansion there is a single corresponding toroidal function.…”

“…However, the choice of toroidal functions of first or second kind may require further thoughts. In Selvaggi et al 22 only the second kind is used, while in the mathematical literature it is used predominantly for the interior problem, r ⊥ < R.…”

Legendre functions of fractional degree: transformations and evaluations

“…It should be noted that restricted to (−1, 1), the functions P µ ν , 2Q µ ν are Hilbert transforms of each other (via 'Neumann's integral') when µ = 0, and are related by more complicated integral transforms when µ = 0 [35]. This relationship, which is suggested by (9), is why Q µ ν , Q µ ν and Q µ ν should really be defined to be twice as large. But to maintain compatibility with the past, the factor 1 2 implicit in their definitions will be kept.…”

“…By (1b), it can be expressed in terms of the Legendre functions P ±m −s , and thus in terms of complete elliptic integrals. Legendre (rather than Ferrers) functions with ν a half-odd-integer and µ an integer are commonly called toroidal or 'anchor ring' functions, since harmonics including factors of the form P m n−1/2 (cosh ξ), Q m n−1/2 (cosh ξ) appear when solving boundary value problems in toroidal domains, upon separating variables in toroidal coordinates [9,10]. The efficient calculation of values of toroidal functions, employing recurrences or other numerical schemes, is well understood [11,12].…”

A Toroidal Harmonic Representation of the Yukawa-potential Kernel for a Circular Cylindrical Source

“…The right hand side of Eq. 8is the zeroth order toroidal harmonic expansion for the inverse distance function in circular cylindrical coordinates [21][22][23][24][25][26][27] and can be derived directly from the free-space Green's function in circular cylindrical coordinates [28][29][30]. Figures 2 and 3 are plots which illustrate a comparison between Eqs.…”