NE528_Fall_09_Mid_Term_Takehome_Exam_worked_solutions

NE528_Fall_09_Mid_Term_Takehome_Exam_worked_solutions - NE...

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NE 528 Fall 2009 Mid Term Take Home Exam (Due on Friday November 6 on or before 5:00pm) 20% of total course grade 1) ( 20 points ) Derive the dispersion relation for electron plasma wave in presence of a steady-state uniform magnetic field where ˆ o B zB = and ˆ E xE = . What is the phase velocity and discuss its meaning Worked Solutions o B = and E = Equation of motion () 11 1 1 1 1 1 hence e ee e x e e o e e e o e eo x o o y o e oxo o eo y o x o x eo e ce y u mn enE en u B P t un KT tx x equation j m n u en E en u B jkKT n en u B eB y equation j m n u en u B j u jmn m uj γ ωγ ω =− × −∇ ∂∂ × −⇒ −= = −− = = 1 x u From linearization of Poisson’s equation: 1 1 oo jkE n E j n k ε = From linearization of the continuity equation: o x nk nu = We can now substitute in the linearized x-equation: 1 o e e j mnu enE enu B jKT j kKTn 1 γγ 1 1 1 1 o x o o o e o c o x e x jmnu e n e ju nB j k K u T j u k e 2 o e 2 2 2 o e c o e e e T mm k K n + + = i.e. 2 2 2 2 ce e e pe k KT m =+ +
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22 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 33 () 3 ee pe pe ce pe ce RMS pe ce RMS pe ce RMS pe c ce eR M S KT KT kk v k mm v k v k for = 3 in a 1- D in x - direction Hence v k γ γγ ωω ω ωγ =+ + + + + + + + + This is the same dispersion relation previously obtained for electron plasma waves EXCEPT the addition of the electron cyclotron frequency; it is also the same dispersion equation for electrostatic electron oscillations perpendicular to B “the upper hybrid frequency) EXCEPT the addition of the thermal motion. 2 2 2 OR pe c 2 RMS ce Upper Hybrid F pe requency Electron Plasma Wave RMS e v vk k = + + ±²³ ² ´± ² ³ ² ´ 2 + The phase velocity is then given by 2 2 2 OR pe c RMS ce Upper Hybrid F pe requency Electron Plasma Wave RMS e v k = + + ² ³ ² ´ + The phase velocity is then given by 2 2 ce + pe RMS φ + = vv 2) ( 80 points: 40 for part a, 20 for part b and 20 for part c ) The single fluid equation does not recognize the individual particles nor are their charge, electron and ion fluid equations summed up to reduce the individual equations to one equation. When studying an experimental transient plasma system, several unusual behaviors were observed: - The plasma is NOT quasi-neutral. - The rate of electron-ion collision is faster than that of ion-electron collision ie e i ei R m m R = . - The plasma fluid velocity varies along the discharge axis [ ( ) 0 u u ]. - The radial density profile is ( ) a / r cos n n 0 = . - The magnetic field varies radially as o B BCos r π = , and thus ( ) 0 B B - The electron and ion kinetic temperatures are ( ) cos / o TT r a = and 4 e i T T a = .
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Starting from the two fluid equations, one for electrons and one for ions: a) Obtain a single fluid equation for this transient nonquasi-neutral plasma , which has a higher collisional rate of electrons over ions. Show all solution steps . You may leave integrals as is without integration.
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NE528_Fall_09_Mid_Term_Takehome_Exam_worked_solutions - NE...

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