**5. Channel length and scaling laws for Larmor radius**

The ratio of electric and magnetic field is such that the ion's gyro radius is much larger than electron's gyro radius, so that ions are not magnetized inside the channel. The channel length is calculated by determining the Larmor radius of both the plasma species [10]. The motion of the moving charged particle under the electromagnetic fields defined as follows

$$\frac{\left(m\nu\_{\perp}\right)^{2}}{r} = q(\nu\_{\perp} \times B) \tag{7}$$

This gives the radius of gyration

$$r = \frac{mv\_\perp}{qB} = \frac{v\_\perp}{\Omega\_c} \tag{8}$$

Here, <sup>Ω</sup>*<sup>c</sup>* <sup>¼</sup> *qB <sup>m</sup>* , is called the cyclotron frequency. If ions are accelerated through electrical potential *V*<sup>⊥</sup> (perpendicular to the magnetic field), then energy balance leads to *<sup>m</sup>υ*⊥<sup>2</sup> <sup>2</sup> ¼ *qV*⊥.

This implies to

$$
v\_{\perp} = \sqrt{\frac{2qV\_{\perp}}{m}}\tag{9}$$

These above equations gives the larmor radius in terms of the applied potential

$$r = \frac{1}{B} \sqrt{\frac{2mV\_\perp}{q}}\tag{10}$$

The magnitudes of the fields are such that *ri* > >*L*> > *re*, where *L* (�6 cm) is the length of the acceleration channel. For a typical Hall thruster, Larmor radius for electrons (�0.13 cm) and ions (�180 cm) corresponds to the radial magnetic field strength of order 150 G and energy 300 eV.

Despite many successful applications of Hall thrusters, some aspects of their operation are still poorly understood. One notable problem is the anomalous electron mobility [16, 17] and plasma sheath, which is far above the classical collisional values. It has been established that the inhomogeneous plasma under the electromagnetic fields is not in the thermodynamically equilibrium state [17–20]. The equilibrium E � B electron drift, ion flow rate, plasma and magnetic field gradients are all sources of plasma instabilities in Hall plasmas. Hall plasmas devices with ExB electron drift demonstrate wide range of turbulent fluctuations. These fluctuations are probably the reason of the observed anomaly in the electron transport across the magnetic field [21–23] and other nonlinear phenomena (coherent rotating spoke) [24–26]. Understanding of the mechanisms of the coherent structures and anomalous transport requires the detailed study of linear instabilities in Hall plasma devices. These instabilities are also considered to be a principal source of anomalous transport in toroidal magnetic confinement devices [2].

## **6. Oscillations and instabilities in Hall thrusters**

If free energy is available in the system and even though system is in equilibrium in the sense that all the forces are in balance, then these oscillations can grow at the cost of free energy and hence instabilities can take place. There are different types of instabilities that depend on different conditions. For proper description of a particular instability, one should be able to define the mode of the growing wave, the nature of the growth and the source of the free energy. Instabilities are mainly classified into four groups, namely streaming instabilities, Rayleigh–Taylor instabilities, universal instabilities and kinetic instabilities.

When there is any kind of perturbation, the free energy available excites the waves and the plasma waves no longer remains in thermal equilibrium. Even though there exist an equilibrium because all the forces are balanced and there is no net force and it is possible to find a time independent solution to the wave. Perturbation makes the plasma waves unstable, which is always a motion that brings the plasma closer to true thermodynamic equilibrium by decreasing the free energy. Instabilities may be classified according to the type of free energy available to drive them. Few of them has been explained below.
