Since three conditions were imposed, integration of the equations(1)to(3)will yield the voltage profile and also the current density j. the result is 4 ana aso 4 Va x d Equation(8)in particular shows that the field is zero(as imposed )at x=0, and 3 d at x=d(the accelerator grid). This allows us to calculate the net electrical force per unit area on the ions in the gap as the difference of the electric pressures on both faces of the slab A=2(3d丿=9na and this must be also the rocket thrust(assuming there is no force on ions in other regions, i. e. a flat potential past the accelerator). It is interesting to obtain the same result from the classical rocket thrust equation The mass flow rate is m and the ion exit velocity is F mc=e A Using Child-Langmuir's law for j(Equation 6), this reduces indeed to Equation(9) For a given propellant(mi) and specific impulse(c/g, the voltage to apply to the accelerator is fixed 16.522, Space P pessan Lecture 13-14 Prof. Manuel martinez Page 5 of 2516.522, Space Propulsion Lecture 13-14 Prof. Manuel Martinez-Sanchez Page 5 of 25 Since three conditions were imposed, integration of the equations (1) to (3) will yield the voltage profile and also the current density j. The result is 1 2 3 2 0 2 i 4 2 e Va j = 9 m d ⎛ ⎞ ε ⎜ ⎟ ⎝ ⎠ (6) and also ( ) 4 3 x x = -Va d ⎛ ⎞ φ ⎜ ⎟ ⎝ ⎠ (7) ( ) 1 3 4 Va x Ε x =- 3d d ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ (8) Equation (8) in particular shows that the field is zero (as imposed) at x=0, and is 4 Va - 3 d at x=d (the accelerator grid). This allows us to calculate the net electrical force per unit area on the ions in the gap as the difference of the electric pressures on both faces of the “slab”: 2 2 2 F 1 4 Va 8 Va = A 2 3d 9 d 0 0 ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ ε ε (9) and this must be also the rocket thrust (assuming there is no force on ions in other regions, i.e., a flat potential past the accelerator). It is interesting to obtain the same result from the classical rocket thrust equation. The mass flow rate is m = j mi A e i , and the ion exit velocity is i 2eVa c = m , giving i i F m 2eVa m c= j AA e m = i Using Child-Langmuir’s law for j (Equation 6), this reduces indeed to Equation (9). For a given propellant (mi) and specific impulse (c/g), the voltage to apply to the accelerator is fixed: 2 m ci Va = 2e (10)