Magnetic Energy of Surface Currents on a Torus

Abstract

Abstract-The magnetic energy and inductance of current distributions on the surface of a torus are considered. Specifically, we investigate the influence of the aspect ratio of the torus, and of the pitch angle for helical current densities, on the energy. We show that, for a fixed surface area of the torus, the energy experiences a minimum for a certain pitch angle. New analytical relationships are presented as well as a review of results scattered in the literature. Results for the ideally conducting torus, as well as for thin rings are given.

Calculation of the inductance of conductive nonmagnetic conductors by means of finite element method simulations

“…There is no exact closed-form solution for the inductance of a round conductor forming a circular loop, which takes into account the effect of the eddy currents induced. However, an approximate formula for the total inductance which assumes an azimuthal current in a ring of major radius R with circular cross section of radius a , as shown in Figure 7, is given by 34,35 where y = 0 for uniform current distribution, i.e. low-frequency operation, whereas y = −1 corresponds to the natural current distribution.…”

Algebraic motion of vertically displacing plasmas

“…The expression and interpretation of self-inductance via Neumann formula is somewhat subtle because it formally diverges [32]. The normalised self-inductance of a thin circular coil of perimeter l = 2πR is given for a uniform current density by [33]…”

Low‐frequency electromagnetic scattering by a metal torus in a lossless medium with magnetic dipolar illumination

“…Taking into account the harmonic functions (2), we initially solve independently the boundary value problems (23) and (26), to obtain H s 0 and H s 3 , respectively and then proceed to the more complicated problems (24) and (25), to calculate H s 2 (thus E s 1 / and E s 3 . The appropriate non-penetrable boundary conditions for the total electromagnetic fields on the surface D s of the metallic object given by (28) and followed by the radiation conditions (30) fit the aforementioned boundary value problems. The scattering toroidal domain of propagation Á r 2 V R 3 fr 0 g is depicted by (8), in which the low-frequency magnetic and the electric fields must be built at each n D 0, 1, 2, 3, recalling that based on (34), there are no electromagnetic fields inside the torus.…”

“…In summary, we are ready to apply the toroidal geometry that fits our case, where the boundary value problems to be solved for the electromagnetic fields at low frequencies are provided by relationships (23)-(26) with (22), accompanied by the appropriate boundary (28) and radiation (30) conditions.…”

“…Nevertheless, seeking for more complicated body‐structures, we arrived at the conclusion that physical applications such as developments in determining the gravitational potential using toroidal functions are involved with the toroidal geometry and appear frequently in the literature, in acoustics , electromagnetics , and elasticity . Although research on quasi‐static fields around tori or torus‐like structures is vast, there are still elementary properties to be discovered, see, for example, ; hence, the demand for such investigations remains continuous nowadays. Consequently, in the present work, the aforementioned analysis for two spheres is extended to a perfectly reflective ring torus, embedded in a lossless ambient, where the appropriate and best fitting system for the particular modeling implementation purposes is the toroidal coordinate system .…”