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Hall Thrusters 369 where the wall collision frequency w is neglected in the plume. 7.4.1.2 Transport Along the Magnetic Field. In the direction along the magnetic field lines, the J B cross product in equation is zero and Eq. (7.4-1) becomes Je = E + p eiJi . en The electric field along the field line is then E= p+ Je eiJi = . en the electron momentum (7.4-14) (7.4-15) With the standard assumptions used along magnetic fields in many plasmas of zero net current ( je ji ) and uniform electron temperature, Eq. (7.4-15) can be solved for the potential along the field line to give = +Tln n . (7.4-16) o e no This equation was derived in Section 3.5-1 and represents the simple Boltzman relationship for plasmas with Maxwellian electron distribution functions. It is often called the barometric law in ion thruster literature and the thermalized potential in Hall thruster literature. Thus, the transport along the magnetic field lines is usually considered to be classical. It is commonly assumed that the density gradient along the magnetic field line is relatively small, so the potential change along a magnetic field line from Eq. (7.4-16) is essentially zero. Therefore, within about kTe /e , the magnetic field lines represent equipotential lines in the plasma. The simplifying assumptions leading to this conclusion (zero net current, Maxwellian electrons, and small density gradient along the magnetic field lines) are often used and may introduce significant errors in some cases. Nevertheless, the thermalized potential has been used for many years [3] in the design of Hall thrusters to relate the magnetic field shape to the electric field in the plasma [11]. 7.4.1.3 Continuity and Energy. Continuity for the neutrals in the thruster can be expressed as no + vo no = neno ive , (7.4-17) tPDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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