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Hollow Cathodes 265 Fig. 6-12. Sheath potential and currents to the insert as a function of the resistive joule heating in the insert plasma. It is also possible to estimate the axial extent of the plasma in the insert region for Type A and some Type B cathodes with small orifices that again produce diffusion-dominated plasmas. This is useful in understanding the plasma “attachment” or “contact length” with the insert, which impacts where the electron emission can take place. As was shown in Chapter 3, the solution to the 2-D diffusion equation in cylindrical geometry is the product of a zero-order Bessel function radially times an exponential term in the axial direction: n(r,z)=n(0)Jo( C2 + 2 r)e z, (6.4-23) where is one over the e-folding distance of the plasma density from the reference location on axis at (0,0). This length can be found by considering the ion generation inside the insert. The ion current to the insert surface is the ion generation rate integrated over the volume inside the insert: RL Ii = 2 0 0 none iverdrdz . (6.4-24) Taking the axial integral in Eq. (6.4-24) to be approximately the e-folding distance ( L = 1 / ), Eq. (6.4-24) is simply R2 Ii = none ive . (6.4-25) The average plasma density is found from Eq. (6.4-5):PDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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