Q ③R 22 The magnitude b of the magnetic field at a point p can be simply related to the amount of current i' which crosses the surface s; this surface leans on the ring that contains P and extends around the cathode tip as shown We have, from the integral Ampere's law, B(P)=Ho (24) 2πr where r is the radial coordinate of P. In particular, I'=I, the total current, for any point on the insulating backplate and I'is zero for points like Q, outside the cylinder. Approximate calculation of thrust There are two major contributions to the thrust of our coaxial accelerator. One of them is the familiar integral of the gas pressure over the back-facing surfaces. This is called the electrothermal"or aerodynamic"thrust, and would be the only one acting in a device where p/o dominates over(xB).u(or, for that matter, in a chemical rocket). The other component is the reaction to the Lorentz forces exerted on the plasma, and is physically applied as a magnetic force on some of the metalli conductors carrying current to the thruster. For example, looking at the figure, and assuming for simplicity that the conductors in the back are arranged symmetrically we see that at points like r the enclosed current for the loop shown is the total current I, and so B=ro 2r in the same direction as inside the engine. Across the radial wires, b goes to zero, but at least a part of each wire is subject to B, and since its current is radially outwards, the Lorentz force on it is to the left, i.e. a thrust At high efficiencies the elect etic thrust dominates over the electrothermal thrust. We can calculate it relatively easily with a few assumptions. First, we have exactly(by action and reaction) 顶xBdy=(x)xB 16.522, Space P pessan Lecture 21 Prof. Manuel martinez Page 12 of 2116.522, Space Propulsion Lecture 21 Prof. Manuel Martinez-Sanchez Page 12 of 21 The magnitude B of the magnetic field at a point P can be simply related to the amount of current I' which crosses the surface S; this surface leans on the ring that contains P and extends around the cathode tip as shown. We have, from the integral Ampère’s law, ( ) 0 I' BP = 2 r µ π (24) where r is the radial coordinate of P. In particular, I' = I , the total current, for any point on the insulating backplate, and I' is zero for points like Q, outside the cylinder. Approximate calculation of thrust There are two major contributions to the thrust of our coaxial accelerator. One of them is the familiar integral of the gas pressure over the back-facing surfaces. This is called the “electrothermal” or “aerodynamic” thrust, and would be the only one acting in a device where 2 j σ dominates over (j × B . u ) G JG G (or, for that matter, in a chemical rocket). The other component is the reaction to the Lorentz forces exerted on the plasma, and is physically applied as a magnetic force on some of the metallic conductors carrying current to the thruster. For example, looking at the figure, and assuming for simplicity that the conductors in the back are arranged symmetrically, we see that at points like R the enclosed current for the loop shown is the total current I, and so 0 I B = 2 r µ π , in the same direction as inside the engine. Across the radial wires, B goes to zero, but at least a part of each wire is subject to B, and since its current is radially outwards, the Lorentz force on it is to the left, i.e., a thrust. At high efficiencies, the electromagnetic thrust dominates over the electrothermal thrust. We can calculate it relatively easily with a few assumptions. First, we have exactly (by action and reaction) EM ( ) vol vol 0 1 F = j × B dV = × B × B dV ∇ µ ∫∫∫ ∫∫∫ G GJG JG JG (25)