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Electron Oscillations in a Plasma - trouble with electric fields

  1. Dec 22, 2011 #1
    1. The problem statement, all variables and given/known data

    In a cold plasma (neglecting thermal pressure) the background medium is motionless and uniform

    for the electrons:

    [itex]\rho_e = \rho_{e0} + \rho_{e1}[/itex]
    [itex]v_e = v_{e1}\hat{z}[/itex]

    where ρ is electron density and v is velocity. Subscript 0 denotes a constant value and 1 denotes a small pertubation due to the wave. Small quantities are zero to second order.
    all variables depend only on z and t

    the electrons experience a drag force/vol equal to [itex]-\alpha\rho_ev_e[/itex]

    (a) what is the linear conservation equation for mass?
    (b) what is the linear momentum equation for the electrons including a term responsible for force/volume due to electric fields
    (c) what equation relates the electric field to the prtubation in density?


    2. Relevant equations

    conservation of mass
    [itex]\frac{\partial \rho}{\partial t} + \nabla (\rho v) = 0[/itex]

    conservation of momentum
    [itex]\frac{\partial}{\partial t}\left[\rho v\right] + \nabla \left(\rho v^2\right) + \nabla P = f_{other}[/itex]

    where [itex]f_{other}[/itex] are force/vol due to other sources than pressure.



    3. The attempt at a solution

    a is straight forward:

    [itex]\frac{\partial \rho_{e1}}{\partial t} + \rho_{e0} \frac{\partial v_{e1}}{\partial z} = 0[/itex]

    i have the solution to b also, by taking the force per vol due to electric fields to be

    [itex]f_e = -\frac{e\rho_e}{m_e}E[/itex]

    [itex]\rho_{e0}\frac{\partial v_{e1}}{\partial t} + \frac{e\rho_e}{m_e}E + \alpha\rho_ev_e = 0[/itex]

    part c is where im stuck. i have the answer but im not sure how to get there, it's supposed to be:

    [itex]\frac{\partial E}{\partial z} = -\frac{e\rho_{e1}}{\epsilon_0 m_e}[/itex]

    does anyone know where to start to derive this relationship?

    Thanks
     
  2. jcsd
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