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50 Chapter 3 where the linearized convective derivative has been neglected. Linearizing the continuity Eq. (3.5-12) gives dn1 =–no v1 zˆ, (3.5-19) dt where the quadratic terms, such as n1v1 , etc., have been neglected as small. In the linear regime, the oscillating quantities will behave sinusoidally: E1 = E1 ei(kz– t) zˆ v 1 = v 1 e i ( k z – t ) zˆ n1 = n1 ei(kz– t) . (3.5-20) ( 3 . 5 - 2 1 ) (3.5-22) This means that the time derivates in momentum and continuity equations can be replaced by i t , and the gradient in Eq. (3.5-17) can be replaced by ik in the zˆ direction. Combining Eqs. (3.5-17), (3.5-18), and (3.5-19), using the time and spatial derivatives of the oscillating quantities, and solving for the frequency of the oscillation gives nee2 1/2 p = o m , ( 3 . 5 - 2 3 ) where p is the electron plasma frequency. A useful numerical formula for the electron plasma frequency is fp= p 9 ne, (3.5-24) 2 where the plasma density is in m–3. This frequency is one of the fundamental parameters of a plasma, and the inverse of this value is approximately the minimum time required for the plasma to react to changes in its boundaries or in the applied potentials. For example, if the plasma density is 1018 m–3, the electron plasma frequency is 9 GHz, and the electron plasma will respond to perturbations in less than a nanosecond. In a similar manner, if the ion temperature is assumed to be negligible and the gross response of the plasma is dominated by ion motions, the ion plasma frequency can be found to bePDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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