(12) where as given in Equation (2),E=E+ux B Remembering that the gyro frequency(the angular frequency of motion of an electron orbiting about a perpendicular magnetic field B)is o (13) i. e, it represents the ratio of gyro frequency to collision frequency it can be expected to be high at low pressures and densities where collisions are rare, and also at high magnetic field, where the gyro frequency is high. In many plasmas of interest in MHD or MPD, B-1 Electromagnetic Work The rate at which the external fields do work on the charged particles can be calculated (per unit volume)as W=∑q(E+vxB) where we used(Vx B).V,=0. We see here that the magnetic field does not directly contribute to the total work, since the magnetic force is orthogonal to the particle velocity it does, however, modify Eor j(depending on boundary conditions) and through them it does affect W This total work goes partly into heating the plasma(dissipation) and partly into bodily pushing it(mechanical work). To see this, notice that E-uxB)j=Ej+j 问uxB)j=×B) Also, using Ohms law j=(6+j×p where we used (jxB).j=0 16.522, Space P pessan Lecture 21 Prof. Manuel martinez Page 5 of 2116.522, Space Propulsion Lecture 21 Prof. Manuel Martinez-Sanchez Page 5 of 21 σ β E' = j+ j × JJG GGG (12) where, as given in Equation (2), E' = E + u x B JJG GGJG . Remembering that the gyro frequency (the angular frequency of motion of an electron orbiting about a perpendicular magnetic field B JG ) is e = eB m ω , e = ω β ν (13) i.e. , it represents the ratio of gyro frequency to collision frequency; it can be expected to be high at low pressures and densities, where collisions are rare, and also at high magnetic field, where the gyro frequency is high. In many plasmas of interest in MHD or MPD, β ∼ 1. Electromagnetic Work The rate at which the external fields do work on the charged particles can be calculated (per unit volume) as ( ) j j j j j W = q n E + v × B . v ∑ G JG JJG JG j j j j = E . q n v ∑G JG or W = E . j G G (14) where we used ( ) v × B . v 0 j j ≡ JG JJG JG . We see here that the magnetic field does not directly contribute to the total work, since the magnetic force is orthogonal to the particle velocity; it does, however, modify E G or j G (depending on boundary conditions), and through them it does affect W. This total work goes partly into heating the plasma (dissipation) and partly into bodily pushing it (mechanical work). To see this, notice that W = E . j = E' - u × B . j = E' . j+ j × B . u ( ) ( ) G G JJGG JG G JJGG G JG G (using ( ) () u × B . j = - j × B . u G JGG GJG G ). Also, using Ohm’s law ( ) 2 1 j E' . j = j + j × . β j = σ σ JJGG G G G G where we used ( ) j × . j = 0 β GGG