Typically e<<c et us make a crude model of the motion of the electron swarm the net force on it per unit volume is fe=-eE+ ve X b)ne where E is used, since we are in the plasma frame. In steady state, this is balanced by the drag force opposing motion of electrons relative to the rest of the fluid, which we are assuming to be at rest and whose particles have, by comparison only a very slow thermal motion. To evaluate this drag let ve be the effective collision frequency per electron for momentum transfer. This frequency is defined such that in each collision with a particle of the rest of the fluid, the electron is, on average, scattered by 90, so that its forward momentum is completely lost. Then the mean drag force per unit volume is Equating the sum of (8)and(9)to zero, +VeX or, since j=eTe-e'-e-jxB Define the scalar conductivi (10) and the hall parameter B and we can write the generalized ohms law as 16.522, Space P pessan Lecture 21 Prof. Manuel martinez Page 4 of 2116.522, Space Propulsion Lecture 21 Prof. Manuel Martinez-Sanchez Page 4 of 21 Typically e v << ce JG . Let us make a crude model of the motion of the electron swarm. The net force on it per unit volume is e e ( ) e f = -e E' + v x B n G JJG JG JG (8) where E' JJG is used , since we are in the plasma frame. In steady state, this is balanced by the drag force opposing motion of electrons relative to the rest of the fluid, which we are assuming to be at rest, and whose particles have, by comparison, only a very slow thermal motion. To evaluate this drag, let e ν be the effective collision frequency per electron for momentum transfer. This frequency is defined such that in each collision with a particle of “the rest of the fluid,” the electron is, on average, scattered by 90D , so that its forward momentum is completely lost. Then the mean drag force per unit volume is e e e e ee e m f = -n m v = j e ν ν G JG G (9) Equating the sum of (8) and (9) to zero, ( ) 2 e e e e e n j = E' + v B m ν × G JJG JG JG or, since e e j v =- en G JG , 2 e ee ee e n e j = E' - j × B m m ν ν G JJG GJG Define the scalar conductivity 2 e e e e n = m σ ν (10) and the Hall parameter e e eB B = ; = m B ⎛ ⎞ β ββ ⎜ ⎟ ⎜ ⎟ ν ⎝ ⎠ JG G (11) and we can write the generalized Ohm’s law as