so that E=E+uxB (in particular, for u =v the lorentz force is seen to be purely electrostatic; i F=qE). Most often the frame at u is chosen to be that moving at the mean mass velocity of the plasma Consider a plasma where there is a number density n, of the jon type of charged particle, which has a charge qj and moves at mean velocity Vi The net Lorentz force per unit volume is n, q(E+Vi X and since the plasma is neutral nai=0 =∑ n, q v,x B (4) But, by definition ∑nqv=j where j is the current density vector(A/m2). So, finally f=jxB (N/ Notice that vi in Equation(5)could be in any frame including the plasma frame ohm's Law In most cases, the dominant contribution to j(Equation(5))is from electrons, given their high mobility. In the plasma frame 1。=-en Notice that ve is the electron mean velocity vector, not to be confused with the mean thermal speed Ce. The picture of electron motions is that of a very rapid chaotic swarming of electrons back and forth going nowhere), except that the hole swarm "slowly"drifts at ve 16.522, Space P pessan Lecture 21 Prof. Manuel martinez Page 3 of 2116.522, Space Propulsion Lecture 21 Prof. Manuel Martinez-Sanchez Page 3 of 21 E + v x B = E' + v - u x B ( ) G JG JG JJG JGG JG so that E' = E + u x B JJG GGJG (2) (in particular, for u=v G JG the Lorentz force is seen to be purely electrostatic; i.e., F = qE' G JJG ). Most often the frame at u G is chosen to be that moving at the mean mass velocity of the plasma. Consider a plasma where there is a number density nj of the jth type of charged particle, which has a charge qj and moves at mean velocity vj JG . The net Lorentz force per unit volume is ( ) j j j j f = n q E + v x B ∑ G GJG JG (3) and since the plasma is neutral j j j ∑nq =0 : j j j j f = n q v x B ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ ∑ G JG JG (4) But, by definition, j j j j ∑nqv = j JG G (5) where j G is the current density vector (A/m2 ). So, finally, f = j x B G GJG (N/m3 ) (6) Notice that vj JG in Equation (5) could be in any frame, including the plasma frame. Ohm’s Law In most cases, the dominant contribution to j G (Equation (5)) is from electrons, given their high mobility. In the plasma frame, e e e j j = -en v GG JG  (7) Notice that ve JG is the electron mean velocity vector, not to be confused with the mean thermal speed ce . The picture of electron motions is that of a very rapid, chaotic swarming of electrons back and forth (“going nowhere”), except that the whole swarm “slowly” drifts at ve JG