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42 Chapter 3 v = E B vE , (3.3-16) B2 which is the “E cross B” drift velocity. In this case, the drift is in the direction perpendicular to both E and B, and arises from the cycloidal electron motion in the magnetic field being accelerated in the direction of –E and decelerated in the direction of E. This elongates the orbit on one-half cycle and shrinks the orbit on the opposite half cycle, which causes the net motion of the particle in the E B direction. The units of the E B velocity are vE = E [V/m] (m/s). (3.3-17) B [tesla] Finally, consider the situation of a particle gyrating in a magnetic field that is changing in magnitude along the magnetic field direction zˆ. This is commonly found in electric propulsion thrusters relatively close to permanent magnets or electromagnetic poles-pieces that produce fields used to confine the electrons. Since the divergence of B is zero, Eq. (3.2-3), the magnetic field in cylindrical coordinates is described by 1 (rBr)+ Bz =0. (3.3-18) r r z Assuming that the axial component of the field does not vary significantly with r and integrating yields the radial component of the magnetic field with respect to r, Br r Bz . (3.3-19) 2 z The Lorentz force on a charged particle has a component along zˆ given by Fz qv Br, (3.3-20) where the azimuthal particle velocity averaged over a Larmor-radius ( r = rL ) gyration is v = v . The average force on the particle is then F z 1 m v 2 B z . ( 3 . 3 - 2 1 ) 2 B z The magnetic moment of the gyrating particle is defined asPDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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