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Mathematical Physics By Pk Chattopadhyay Pdf Free Download
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Mathematical Physics By Pk Chattopadhyay Pdf Free Download
In plasmas, the motion of the charged particles can sometimes generate the internal current and the resultant magnetic field even in current-free plasmas. Measurement of the internal-current-induced magnetic field can give important information to discuss electromagnetic plasma acceleration processes induced by a Lorentz force and plasma-induced modification of the magnetic field structure, e.g., as performed in Tobari et al. (2007) and Roberson et al. (2011). Hence the measurement of the plasma-induced magnetic field is also an important technique to discuss the thrust-generation physics and the plasma dynamics in the magnetic nozzle. The magnetic field induced by the plasma current can be measured by a B-dot probe and a Hall element probe in pulsed and steady-state plasmas, respectively (Stenzel and Urrutia 2000; Corr and Boswell 2007). The internal plasma-induced current density \(\mathbfJ\) can be obtained by taking a rotation of the magnetic field, i.e., \(\nabla \times \mathbfB = \mu \mathbfJ\).
A helicon source attached to a diffusion chamber and not immersed in vacuum as shown in Fig. 7a is easier to be operated on than that immersed in vacuum, since the antenna immersed in vacuum frequently induces anomalous and parasitic discharges due to the capacitive coupling and the high-voltage breakdown at the antenna. This type of experiments has clarified and discovered many aspects of physics in the magnetic nozzle. Measurements of the plasma potential (Fig. 7a) and the IEDF downstream of the source using the RFEA (Fig. 7b) and LIF techniques (Fig. 7c) have shown the presence of the two components of the ions consisting of the supersonic beam and the thermal ones in low-pressure operations (Cohen et al. 2003, 2006; Charles and Boswell 2003; Sutherland et al. 2005; Sun et al. 2005; Takahashi and Fujiwara 2011; Wiebold et al. 2011). Charles and Boswell identified that the energy of the supersonic ion beam corresponds to the rapid potential drop with a thickness of about a few tens-hundreds of Debye length (Charles and Boswell 2004). This structure is observed to be sustained in steady state and called the current-free double layer (CFDL). The similar acceleration by ambipolar electric fields having a gradual potential decrease has also been observed in experiments (Charles et al. 1991; Takahashi et al. 2009; Volynets et al. 2006; Corr et al. 2008; Longmier et al. 2011). The accelerated ion flow has also been observed downstream of a high-power helicon source (Prager et al. 2008). Two-dimensional nature of the ion dynamics relating to such electrostatic ion acceleration in the magnetically and/or geometrically expanding plasmas have been investigated in a number of laboratory experiments. The radial measurement of the ion beam has revealed the generation of the collimated supersonic ion beam accelerated by the CFDL (Charles 2005; Cox et al. 2008; Takahashi et al. 2011), where the result in Takahashi et al. (2011) has shown the slight expansion of the ion beam radius along the magnetic field lines near the thruster exit and the deviation downstream of the magnetic nozzle. When changing the gas pressure or gas species, the two-dimensional structure of the potential drop changes from plane to hemispherical structures; then the divergence of the ion beam is simultaneously changed (Takahashi and Fujiwara 2009; Takahashi et al. 2010). Charles et al. have observed a U-shape CFDL having a equipotential surface oblique to the magnetic field lines (Charles et al. 2009). Parametric studies in the laboratories have revealed some features of the CFDLs in the magnetically expanding plasmas. Lieberman and Charles have identified the pressure ranges of the CFDL appearance (Lieberman and Charles 2006). This type of the structure is also observed for various propellant gases, e.g., Ar, Xe, \(\hbox N_2\), \(\hbox N_2\hbox O\), \(\hbox CH_4\), \(\hbox CO_2\),and so on (Charles et al. 2008). After the magnetic field range for the CFDL formation was experimentally observed (Charles and Boswell 2007), Takahashi et al. have performed the ion-beam measurement when changing the magnetic field and the source diameter. Contour color plots in Fig. 8 show the IEDFs normalized by the maximum value as functions of the magnetic field strength for three different diameter source tubes. The measurements were performed downstream of the source tube; the local plasma potentials downstream of the source are plotted by open circles. The IEDF contains the single peak around the local plasma potential, corresponding to the thermal ions, for the weak magnetic field strengths. When increasing the magnetic field strength, the additional peak at higher potential side appears for all the three cases and implying the formation of the CFDL, where the discriminator voltage giving the second peak is defined as beam potential and plotted by open squares in Fig. 8. It was clearly observed that the threshold of the magnetic field providing the CFDL formation and the ion-beam generation is changed by the source tube diameter, showing that the CFDL ion acceleration is triggered when the ion Larmor radius calculated with the ion thermal velocity becomes smaller than the source tube radius (Takahashi et al. 2010).