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20 Chapter 2 F = ion 0 dx Escreen 2 d dE(x) E 1 E(x)dx= accel EdE= E2 E2 . (2.2-8) o( accel screen) The net force on the grids, which is the thrust, is equal and opposite to the o T = F = 2 E2 E2 . (2.2-9) Therefore, the thrust in an ion engine is transferred by the electrostatic force between the ions and the two grids. In Hall thrusters, ions are generated in a plasma volume and accelerated by an electric field in the plasma. However, the presence of the transverse magnetic field responsible for the rotational Hall current modifies the force transfer mechanism. Assume for argument that the Hall thruster plasma is locally quasi- neutral ( qni qne ) in the acceleration region, where ne is the electron plasma density, and that in the acceleration zone the electric and magnetic fields are uniform. The geometry is shown schematically in Fig. 2-2. The ions are essentially unmagnetized and feel the force of the local electric field, so the o electric field forces on the ions between the grids: 1 ion o ( accel screen ) Acceleration Zone r E B z Fi Fe Fig. 2-2. Cross section of a Hall thruster showing electric and magnetic fields. force on the ions is Fion = 2 q ni E rdr dz (2.2-10) The electrons in the plasma feel an E B force and circulate in the system transverse to the electric and magnetic fields with the velocity ve = E B . (2.2-11) B2 The electrostatic force on the ions is the negative of the electrostatic force on the electrons due to their sign differences. The electrons are constrained not to move axially by the transverse magnetic field, so the force per unit area on the electrons (to the left) is balanced by the Lorentz force: Fe= 2 qneErdrdz 2 eneve Brdrdz=0 (2.2-12)PDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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