Appendix B ELECTRON DIFFUSION IN A MAGNETIC FIELD 1) No B Field In electron momentum balance, main forces are pressure gradient and collisional retardation(no inertia ) -nme ve Also P =n. kT. VP =kT Vn Solve for flux In, v. =m. -Vr This is Ficks law of diffusion n Ve =-D. Vn, with a diffusivity (v。=∑ncQ, collision frequency) 2)WthB(⊥tovP) Add magnetic force neme Vev- en. ve×B (4) To solve for n ve form VPe xB=-mevene Ve XB-ene ve XB xB, ×xB=(.6,B Eliminate ve xb between these two equations, simplify 16.522, Space P pessan Lecture Prof. Manuel martinez Page 8 of16.522, Space Propulsion Lecture 15 Prof. Manuel Martinez-Sanchez Page 8 of 11 Appendix B ELECTRON DIFFUSION IN A MAGNETIC FIELD 1) No B JG Field In electron momentum balance, main forces are pressure gradient and collisional retardation (no inertia): e ∇ ν P -n m v e ee e JG  (1) Also P = n kT e ee , ∇ ∇ P kT n . e ee  Solve for flux: e e e e e e kT n v =- n m ∇ ν JG (2) This is Fick’s law of diffusion e n v = -D n e ee ∇ JG , with a diffusivity e e e e kT D = m ν (3) ( e je e j ν = ncQ ∑ , collision frequency) 2) With B to P ( ) ⊥ ∇ e JG Add magnetic force: e e ∇ ν P = -n m v - en v ×B e ee e e JG JG JG (4) To solve for e n ve JG , form ∇ ν P ×B = -m n v ×B - en v ×B ×B e eee e e e ( ) JG JG JG JG JG JG , and use 0 ( ) () 2 v ×B ×B = B v . B - B v . e ee JG JG JG JG JG JG JG Eliminate ( ) v ×B e JG JG between these two equations, simplify: