Toroidal approximation

FIGURE 14.1 Toroidal approximation of a liquid bridge formed between two spherical particles. Adapted from Ref. [12] with permission. Copyright 2013, Elsevier. 

In order to facilitate the calculation of capillary forces, several approximations on the meniscus shape have been proposed. They are mainly applied for experimental conditions where the radius of curvature of the meniscus interface is much smaller than the radius of curvature of the solid surfaces. This is relevant for the surface force apparatus where the surface has centimetric radius, while the meniscus is typically tens of hundreds of nanometers. The most used approximation is the toroidal approximation assuming the liquid interface has a circular profile. Obviously, such a meniscus does not exhibit a constant curvature. Nevertheless, this approximation gave good results, in particular for small contact angles, and is therefore widespread (see Ref. 15 for its application in various geometries and section 9.3.1.1 for an example of its application in atomic force microscopy [AFM]). In the case of capillary condensation between a plane and a sphere with a large radius of curvature R, in contact, the tension term of the capillary force is negligible and the Laplace term leads to the simple formula F = AnRy cos 9 A parabolic profile is also sometimes used to eliminate some numerical difficulties inherent in circle approximation. 

An interpretation of the origin of these behaviors can be obtained by a geometrical approach. If one considers the so-called toroidal approximation for which the meniscus profile is assumed circular and if one neglects the tension force, the capillary force can simply be written as a Laplace term ... 

A ferrite toroid or E core can be used for a drive transformer. No gap is needed since the input coupling capacitor guarantees that the core will operate in a bipolar fashion. A high permeability core is also suitable for this purpose. The wire that is going to be used will be in the range of 32 to 36 AWG. The core size will be approximately 0.4 to 0.6 inches (10 to 15mm). 

For production considerations, the typical mounting approach is to place the finished toroid on a terminal board and pot the toroid assembly. This makes the assembly impervious to handling damage and allows the assembly to be easily placed into a PCB. Estimated parts cost approximately 2.50 U.S. 

Fig. 2. Model of a toroid-like complex between plasmid DNA and dendrimers. Dendrimer molecules (spheres) are located both inside and outside the coiled DNA molecules, and more than one plasmid DNA molecule can be present in one toroid-like complex. The left part of the graphic shows the most probable situation the dendrimers on the outside of the complex are not shown in the remainder of the graphic in order to give a better view of the inner part of the complex. The diameter of such a complex is approximately 50 to 100 nm...

We expect an efficient a — Q-dynamo to be at work in the merger remnant. The differential rotation will wind up initial poloidal into a strong toroidal field ( Q-effect ), the fluid instabilities/convection will transform toroidal fields into poloidal ones and vice versa ( a—effect ). Usually, the Rossby number, Ro = is adopted as a measure of the efficiency of dynamo action in a star. In the central object we find Rossby numbers well below unity, 0.4, and therefore expect an efficient amplification of initial seed magnetic fields. A convective dynamo amplifies initial fields exponentially with an e-folding time given approximately by the convective overturn time, rc ss 3 ms the saturation field strength is thereby independent of the initial seed field (Nordlund et al. 1992). 

The above figures are often approximated by the simpler cylindrical or toroidal mirrors, but these are much less efficient and give significantly increased aberrations... 

The vacancy diffusion field around the toroidal loop will be quite complex, but at distances from it greater than about 2RL, it will appear approximately as shown in Fig. 11.12a. A reasonably accurate solution to this complex diffusion problem may be obtained by noting that the total flux to the two flat surfaces in Fig. 11.12a will not differ greatly from the total flux that would diffuse to a spherical surface of radius d centered on the loop as illustrated in Fig. 11.126. Furthermore, when d > Rl, the diffusion field around such a source will quickly reach a quasi-steady state [20, 26], and therefore... 

Measurements of permeability and associated magnetic properties are usually made on toroids of uniform section when, to a close approximation, the flux density B is uniform throughout the material and lies entirely within it. In most practical applications the magnetic circuit is more complex, and variations in component section and permeability give rise to variations in flux density. Important effects arise from air gaps, which may be intentionally introduced or may be cracks or porosity. 

This coil is, in principle, a ring-shaped core with toroidal winding. It is made of an alloy of nickel and iron (in approximate proportions of 1 1) which is characterized by a nearly rectangular hysteresis loop. The alloy is cold rolled... 

Measurements of the poloidal distribution of the ELM energy deposition at the divertor target, such as those in Fig. 3.18, are usually taken at only at one toroidal location. Few experiments have been carried out with measurements of the divertor ELM energy and particle fluxes (with infra-red cameras and Langmuir probes) at various toroidal positions. These experiments show that the divertor ELM energy flux is, approximately, toroidally... 

Since the toroidal globule can only be formed as a result of a strong attraction of the parts of the macromolecule (in any case, it must be (e/T) > (e/T)3), the volume of the torus is approximately equal to the selfvolume of the macromolecule, i.e. 

The reversed field pinch (RFP) is a closely related configuration, since the plasma formation and ohmic heating are essentially identical to the tokamak. However, in the tokamak the average poloidal field is limited by stability requirements to approximately an order of magnitude smaller than the toroidal field, whereas the two are of similar magnitude in the RFP, both being typically less than 1 Tesla. 

A properly folded chain molecule can form a cavity with a shape globally complementing the shape of an enclosed, quasi-spherical molecule. If, however, the enclosed molecule has a toroidal or more complicated topology, where the hole of the torus is small so the surrounding chain molecule cannot enter this hole, then an approximate global complementarity by a chain molecule is unlikely to reflect the correct topology of the enclosed molecule. 

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