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48 Chapter 3 Assuming that the electrons have essentially no inertia (their mass is small and so they react infinitely fast to changes in potential), the left-hand side of Eq. (3.5-7) goes to zero, and the net current in the system is also zero. Considering only electrons at a temperature Te , and using Eq. (3.2-7) for the electric field, gives qEz =e = kTe ne . (3.5-8) z ne z Integrating this equation and solving for the electron density produces the Boltzmann relationship for electrons: ne =ne(0)e(e /kTe), (3.5-9) where is the potential relative to the potential at the location of ne(0). Equation (3.5-9) is also sometimes known as the barometric law. This relationship simply states that the electrons will respond to electrostatic fields (potential changes) by varying their density to preserve the pressure in the system. This relationship is generally valid for motion along a magnetic field and tends to hold for motion across magnetic fields if the field is weak and the electron collisions are frequent. 3.5.2 Particle Conservation Conservation of particles and/or charges in the plasma is described by the continuity equation: n + n v = n s , ( 3 . 5 - 1 0 ) t where n s represents the time-dependent source or sink term for the species being considered. Continuity equations are sometimes called mass-conservation equations because they account for the sources and sinks of particles into and out of the plasma. Utilizing continuity equations coupled with momentum conservation and with Maxwell’s equations, it is possible to calculate the response rate and wave-like behavior of plasmas. For example, the rate at which a plasma responds to changes in potential is related to the plasma frequency of the electrons. Assume that there is no magnetic field in the plasma or that the electron motion is along the magnetic field in the z-direction. To simplify this derivation, also assume that the ions are fixed uniformly in space on the time scales of interest here due to their large mass, and that there is no thermal motion of the particles (T = 0).PDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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