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Basic Plasma Physics 85 and the ion energy conservation gives The charge density in Eq. (3.2-5) can be written 12 M v i 2 = e ( s – ) 2e 1/2 (3.7-44) (3.7-45) (3.7-46) (3.7-47) v i = M ( s – ) . = i+ e = Ji – Je = Ji M – Je m . vi ve s– 2e 2e Poisson’s equation can then be written in one dimension as dE = = Ji M – Je m . dx o o s– 2e o 2e Integrating once gives o E2 =2J M –( – )1/2 –2J m 1/2. 2 i2e ss e2e For space-charge-limited current flow, the electric field at the right-hand boundary (the edge of the plasma) is zero and the potential is = s . Putting that into Eq. (3.7-47) and solving for the current density gives Je= MmJi. (3.7-48) If the area of the two plasmas in contact with each other is the same, the electron current crossing the double layer is the square root of the mass ratio times the ion current crossing the layer. This situation is called the Langmuir condition (1929) and describes the space-charge-limited flow of ions and electrons between two plasmas or between a plasma and an electron emitter. For finite initial velocities, Eq. (3.7-48) was corrected by Andrews and Allen [13] to give J e = Mm J i , ( 3 . 7 - 4 9 )PDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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