NE528_HW4_Fall_2009_with_solutions - NE528 Fall 2009...

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NE528 Fall 2009 HW#4 ( Total 60 points ) Due Wednesday, October 21st by class time 1. (30 points) In obtaining the dispersion relation for electron and ion plasma waves, perturbations were assumed for velocity, density and E-field. Solution requires linearization of the equation of motion, continuity equation, and Poisson’s equation for the electrons but not for the ions. Now, we will allow temperatures to also include perturbation such that 1 o TTT = + and () 11 jk x t x TT e ω = . This means that the p n p n γ ∇∇ but p n KT KT n KT n p nKT KT n γγ + == ⎝⎠ + , which is applicable for both electrons and ions. Obtain the new dispersion relation for electron and ion plasma waves for this case, i.e. with addition of temperatures perturbations. Discuss the implication of temperature perturbations on the dispersion relations. Worked Solutions Solution for electron plasma waves Poisson’s equation o e . o oo e jkE n n E E x ρ εε ε ∇= = =− Continuity equation 1 10 1 . 0 o nn nu n tt jn n nk k u j n u ∂∂ +∇ = = 1 u x −= Equation of motion 2 3 33 3 3 n pN ee e pn N n n e nKT KT n n KT n KT p e e e e K T n K T n K T e e n KT e pK T n n K T K T n e e e e 3 x x ∇+ = + = + ∇ + ∇ +
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33 un KT ee mn enE KT n e x e e tx ∂∂ =− x 1 Linearizing 11 1 j m n u en E KT jkn n jkKT eo o e o ω −= Substituting 22 2 1 j 2 ) 1 () 2 3 ( 3 3 3 1 ( ) 1 3 1 1 e RMS e e e p o mnu en n jkKT o o kKT KT k mm u k o nk KT KT kv mu e jk u ωω =+ = + ++ e o u jk o n e m oe εω ε 2 3 3 1 1 e n kKT KT e o k m e e + u
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This note was uploaded on 12/08/2010 for the course PY 528 taught by Professor Bourham during the Fall '09 term at N.C. State.

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NE528_HW4_Fall_2009_with_solutions - NE528 Fall 2009...

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