NE528_HW1_with_worked_solutions_Fall_2009 - NE528 Fall 2009...

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NE528 Fall 2009 HW#1 (total 90 points) Due Wednesday, September 9 by class time 1. (10 points) Derive the expression of the Debye length for a pure hydrogen plasma in a steady-state situation, where electron and ion temperatures are not equal, i.e. e TT i , and both ions and electrons follow the Boltzmann relation, i.e. , ee kT ei eo io nn e e Φ Φ == 2. (20 points) When solving Poisson’s equation using Boltzmann’s distribution for both electrons and ions, as per Problem # 1, a solution for the Debye length for 2-temperayture plasma was obtained. The Boltzmann’s distribution has the potential Φ in the exponent in which no fluctuations appear on the potential. N ow we have a situation in which the potential has fluctuations in the form ( ) Φ+Φ ± , which will affect the Boltzmann’s relation for both electrons and ions and hence the Boltzmann’s relation for electrons and ions will be () exp e kT e = ± and ( ) exp i e kT =− ± , respectively. Thus, for fully ionized pure hydrogen plasma in which electrons and ions are at different temperatures, what is the effect of this potential fluctuation on the Debye length? Work out the solution starting from Poisson’s equation. 3. (20 points) It was shown that the Boltzmann distribution could be derived from argument of particle dynamics, which gives a distribution for electrons and ions in presence of bulk motion, and hence the electron and ion densities can be expressed by 2 1 2 exp e mv e kT −Φ and 2 1 2 exp ii i e kT , respectively. Derive the expression of the Debye length in presence of bulk motion. How this Debye length differs from that in absence of bulk motion? 4. (20 points) The Boltzmann relation in presence of bulk motion, as derived from argument of particle dynamics, shows that the time-independent solution gives a Boltzmann distribution for electrons, with “v” is the velocity of the bulk of electrons, as: Φ = e o e kT e m n n 2 e v 2 1 exp In obtaining this relation, the force equation assumed the right hand side (RHS) composed of 2 terms, electric force term and pressure term
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e e e e e e e e e e p x e n p eE n p qE x v v t v m n Φ = = = + . The pressure gradient term was taken as the gradient in the density BUT assumed no gradient in the temperature ( p e = n e kT e ). Re-work the solution with the assumption that both electron density and temperature are spatially dependent and hence the electron number density can be expressed by: 2 0 11 () 2 x ee e e mV e kT x dx kT x x x eo nn e φ ⎡⎤ ⎛⎞ ⎢⎥ −− + ⎜⎟ ⎝⎠ ⎣⎦ = 5. (20 points) Plasma has several frequencies that characterize the nature of its electrostatic local oscillations and the oscillations that rises as a result of having plasma under the influence of a magnetic field.
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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_HW1_with_worked_solutions_Fall_2009 - NE528 Fall 2009...

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