Thin-wire approximation

When the radius of a perfecdy conducting wire is much smaller than the wavelength of interest, the current I and charge q could be assumed to be confined to the wire axis, as shown in Figure 5.2. This assumption is called thin-wire approximation. With this assumption, the electric-field integral equation. Equation 5.47 or Equation 5.49, for a perfectly conducting thin wire in air, is simplified to ... 

Using this formulation, we calculate the radiation from the bent fiber in Section 23-4 by treating it as if it were a wire of negligible thickness located along the fiber axis and carrying the current of Eq. (23-2). This calculation isolates the dominant contribution to radiation loss. The accuracy of the thin-wire approximation is established in Section 23-7, where the effect of the finite cross-section of the fiber is included. 

The thin-wire approximation to radiation from the fundamental mode on a bent step-profile fiber is given in terms of the power attenuation coefficient of Eq. (23-12) by setting/ = 0 over 0 < R < 1 and/= 1 elsewhere. The / = 0 integral and eigenvalue equation in Table 14-6, page 319, together with the integral of Eq. (37-91), lead to... 

The thin-wire approximation of Section 23-2 superposes the equivalent core currents of the bent fiber onto its axis. Thus the results of Section 23-4 ignore ects due to thefinite core cross-section. Here we derive the power attenuation coefficient of the fundamental mode allowing for the volume distribution of currents, and thus determine corrections to the above results [1-3]. However, we show below that the additional effect due to the finite cross-section can usually be ignored since the attenuation coefficient is exponentially small. 

The Zero Creep Technique. The zero creep technique was developed by Udin, Shaler, and Wulff(8) to measure the surface energy of Cu wires. The technique was later extended for use with thin foils by Hondros(16). Very thin foils, approximating a surface, are readily available. When shaped into a cylinder, the sample will tolerate large loads without necking. Since necking does not occur, the stress can be considered constant throughout the experiment. Figure 1 shows a schematic of a foil and the associated stress under an applied load... 

In practice, 20 pm wires are rather easy to handle, but 10 pm wires become very fragile and an accidental high voltage breakdown unavoidedly breaks these thin wires. So, practically, one is limited to minimum wire distances of approximately 1 mm. 

Field ionization occurs when gas-phase sample molecules are inteijected in a strong electrical field that is on the order of 10 Vcm The field distorts the electron cloud around the sample molecule and lowers the barrier for the removal of an electron. The quantum mechanical tunneling of this electron from the molecule to the conduction bands of the emitter produces M+ ions [10]. The heart of the FI ion source is an emitter electrode made fi om a sharp metal object such as a razor blade or thin wire. The emitter electrode is placed approximately 1 mm away from the cathode. The field is produced by applying a high potential (10 to 20 kV) to the tip of the emitter electrode. FI is a very soft ionization technique that produces primarily a molecular ion signal. It is applicable to volatile samples only. 

For modeling of thin-wire antennas, that is, wires with radii a O.OIA., the reduced kernel approximation of Eq. (13.31) is often used. This approximation is... 

It should be noted that Eq. (13.33) is exact for a vanishingly thin wire (a = 0) and becomes approximate as soon as the wire radius begins to increase (a > 0). On the other hand, for moderately thick wires with radii in the range O.OIA. a O.IA, the complete kernel expression ofEq. (13.31) must be considered in order to achieve an accurate solution. Several techniques for evaluating Eq. (13.31) have been discussed in Werner (1995). These techniques include numerical integration schemes, as well as analytical methods. Among the more useful techniques is a recently found exact representation of Eq. (13.31) given by (Werner, 1993, 1995)... 

The MoM formulations used in these codes are essentially valid for the analysis of electrically thin wires since they are based on the reduced kernel approximation of Eq. (13.33). A thin-wire antenna analysis capability is sufficient for many practical applications. However, there are some situations that may require the analysis of moderately thick-wire antennas, especially at higher frequencies (100 MHz < / < 1 GHz). For these cases, a MoM code has been developed at the Pennsylvania State University, AppHed Research Laboratory, which is capable of accurately analyzing moderately thick- as weU as thin-wire antennas. This code uses a robust treatment of the cylindrical wire kernel Eq. (13.31) in its analysis of thicker wires. Figure 13.14 graphically illustrates the improvements in input impedance predictions for moderately thick wires when using a robust treatment of the kernel as compared to a strictly reduced kernel thin-wire treatment. The code predictions as well as measured values of input resistance and reactance for various moderately thick monopoles are plotted in Fig. 13.14(a) and Fig.l3.14(b), respectively. 

Terms such as A, Q, Cp, and w can be found experimentally (i.e.. Fig. 11-12) Cp and h values are derived from the literature. The h values can be taken from the work of h eller [51], who studied heat transfer to long thin wires in parallel airstreams. Ti can be closely approximated in the first z segment since the temperature profile is flat at z = 0. This, however, still leaves T and Ts to be determined. If Eq. (56) can be averaged over the cross section to give an expression for T,... 

The data for liquid xenon were obtained in a coaxial test cell consisting of a thin wire as anode (2.0 to 5.0 pm diameter) and an outer cylinder (8 mm diameter) as cathode. For the production of the initial electrons, a radioactive ° Hg source was used. It emits quanta of 85 and 279 keV energy. When a 279-keV quant interacts with a xenon atom it produces an energetic photoelectron which dissipates its energy in approximately 12,500 ionization events. These ionization events are distributed around the point of interaction in a radius of 240 pm. It can be assumed that each photoelectron produces a sphere filled with electrons (and positive ions) at some distance r from the wire. Neglecting for the moment recombination and attachment, these electrons drift toward the anode wire and at some distance their mean energy is sufficient for collisional ionization. At that point they drift with the saturation velocity of 3 x lO cm/s. The number of electrons formed up to a distance r is obtained from integration of Equation 1 as... 

A thin wire of isotropic material is of length 2L and area of cross-section a. It is stretched between two rigid supports so that it is horizontal and has a tension Fq. When a mass M is suspended from the midpoint of the wire the equilibrium position of that point is a distance y below its original position. If the effects of the mass of the wire can be neglected, show that, approximately ... 

In the first of these techniques an approximation to uniform rate of shear throughout the sample is achieved by shearing a thin film of the liquid between concentric cylinders. The outer cylinder can be rotated (or oscillated) at a constant rate and the shear stress measured in terms of the deflection of the inner cylinder, which is suspended by a torsion wire (Figure 9.2) or the inner cylinder can be rotated (or oscillated) with the outer cylinder stationary and the resistance offered to the motor measured. 

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