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106 Chapter 4 Electrons that are incident in this area will be either reflected by the magnetic mirror, electrostatically repelled by the sheath potential, or lost directly to the anode. Electrons that are incident between the cusps encounter a transverse magnetic field and are reflected from the boundary. The contours of constant magnetic field shown on the right in Fig. 4-10 illustrate that the total field is essentially constant across the boundary at a distance sufficiently above the magnets, although the component of the field is changing from purely perpendicular at the cusp to purely parallel between the cusps. An analysis of the magnetic field strength for various multipole boundaries was published by Forrester [23] and discussed by Lieberman [24]. Since the divergence of the magnetic field is zero, the field satisfies Laplace’s equation, and the solution for the lowest-order mode at a distance from the magnets greater than the magnet separation can be expressed by a Fourier series. This gives a magnetic field strength above the magnet array described by B (x,y)= wBocos x e y d, (4.3-1) y 2 d d where Bo is the magnetic field at the surface of the magnet, d is the distance between the magnet centers, w is the magnet width, and the y-direction is perpendicular to the wall in Fig. 4-10. Due to localized magnet positions, the field has the periodic cosine behavior along the surface of the wall illustrated in the figure. In addition, the magnetic field decreases exponentially away from the wall all along the boundary. Atthecusp,thefieldactuallydecreasesas1/d2 duetothedipolenatureofthe permanent magnet. This rapid decrease in the field moving away from the magnet illustrates the importance of placing the magnets as close to the plasma Field Lines Contours of Constant Total B d Alternating-Polarity Magnet Rings w Fig. 4-10. Cross section (side) view of a six-ring-cusp magnetic multipole boundary showing the magnetic field lines and examples of contours of constant magnetic field.PDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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