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240 Chapter 5 Homework Problems 1. For an ion accelerator that is described by the Child–Langmuir law, derive the dependence of the minimum Isp on the beam voltage for a given thrust level. 2. A 1-kV ion accelerator has a grid spacing of 1 mm and a screen aperture diameter of 1 mm. a. What is the space-charge-limited beamlet current density for Xe+ assuming a very thin screen grid and a planar sheath? b. If the screen grid is 0.25 mm thick, what is the maximum beamlet current density for a non-planar sheath? How does this compare to the classic planar Child–Langmuir result? 3. An ion thruster with a grid diameter of 20 cm has a beam current density that varies with the radius as kr2 , where k is a constant. a. If the peak current density on axis is J p and the current density at the edge of the grid is Jp /10 , find the expression for J(r). b. If the peak current density is 5 mA/cm2, what is the total beam current? c. Whatistheflatnessparameter? d. What is the percent reduction in the beam current compared to the case of a uniform beam current density of the peak value (the flatness is 1)? 4. An ion thruster has a beam plasma potential of 20 V and an electron temperature in the beam of 5 eV. a. For a plasma potential at the screen grid sheath edge of 1000 V, what potential must be established in the accel grid aperture to keep the electron backstreaming current to 1% of the beam current? b. Neglecting space charge in the beamlet, what voltage must be applied to the accel grid to achieve the minimum potential in (a) for the case of a 3-mm screen grid diameter, 0.25-mm screen grid thickness, 2-mm accel grid diameter, and 0.5-mm accel grid thickness with a 1-mm grid gap? c. If the beamlet current is 0.2 mA and the beamlet has a diameter in the accel grid aperture of 1 mm, what must the accel grid voltage be to maintain the 1% backstreaming current specification?PDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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