flow at uof the plasma in the total magnetic field B, while any external electrodes are short-circuited, so that E=0 in the laboratory frame. Then E=E+uxB=uxB In order of magnitude e=uB. Neglecting the Hall effect, then ,j-ouB The induced field obeys separately its own Amperes law V j, where is the es). Thus, in order of magnitude s wir ma current density this is because V x Bext =0 in the plasma(outside the coil where I is the characteristic distance for variation bd i e. the plasma size. Altogether, B lu B (20) B=Bo +Bind This indicates that the field created by the plasma currents becomes comparable to the external field when the so-called"magnetic Reynolds numbe en=μolou becomes of order unity. For a high power Argon MPD accelerator, a=1000mho/m,u-10,000m/sec,l=0.1m, Rn-106×102×103×104=1 and so, operation with self-induced magnetic field becomes possible. This simplifies considerably the design, since no heavy and power-consuming external coils are needed The order of magnitude also indicates however, that external field augmentation may be desirable under some conditions; the issue of self-field vs. applied field devices is not yet fully resolved A Simple plasma Accelerator Consider a rectangular channel with two conducting and two insulating walls, as shown 16.522, Space P pessan Lecture 21 Prof. Manuel martinez Page 8 of 2116.522, Space Propulsion Lecture 21 Prof. Manuel Martinez-Sanchez Page 8 of 21 flow at u G of the plasma in the total magnetic field B JG , while any external electrodes are short-circuited, so that E=0 G in the laboratory frame. Then E' = E + u × B = u × B JJG GGJG GJG . In order of magnitude, E' u B  . Neglecting the Hall effect, then, j u B ∼ σ . The induced field obeys separately its own Ampère’s law ind 0 B ∇ × =j µ JG G , where j G is the plasma current density; this is because ∇ × B = 0 ext JG in the plasma (outside the coil wires). Thus, in order of magnitude B j l ind 0 ∼ µ where l is the characteristic distance for variation Bind ,i.e., the plasma “size.” Altogether, B l u B ind 0 ∼ µ σ (20) B=B +B 0 ind JG JG JG , so ind 0 ind 0 B l u B +B ∼ µ σ This indicates that the field created by the plasma currents becomes comparable to the external field when the so-called “magnetic Reynolds number” R = l u em 0 µ σ (21) becomes of order unity. For a high power Argon MPD accelerator, σ ∼  1000 mho/m , u 10, 000 m/sec, l 0.1 m , so -6 -1 3 4 R 10 × 10 × 10 × 10 = 1 em ∼ and so, operation with self-induced magnetic field becomes possible. This simplifies considerably the design, since no heavy and power-consuming external coils are needed. The order of magnitude also indicates, however, that external field augmentation may be desirable under some conditions; the issue of self-field vs. applied field devices is not yet fully resolved. A Simple Plasma Accelerator Consider a rectangular channel with two conducting and two insulating walls, as shown: