# Electron Oscillations in a Plasma - trouble with electric fields

1. Dec 22, 2011

### knowlewj01

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:

$\rho_e = \rho_{e0} + \rho_{e1}$
$v_e = v_{e1}\hat{z}$

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 $-\alpha\rho_ev_e$

(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
$\frac{\partial \rho}{\partial t} + \nabla (\rho v) = 0$

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

where $f_{other}$ are force/vol due to other sources than pressure.

3. The attempt at a solution

a is straight forward:

$\frac{\partial \rho_{e1}}{\partial t} + \rho_{e0} \frac{\partial v_{e1}}{\partial z} = 0$

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

$f_e = -\frac{e\rho_e}{m_e}E$

$\rho_{e0}\frac{\partial v_{e1}}{\partial t} + \frac{e\rho_e}{m_e}E + \alpha\rho_ev_e = 0$

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

$\frac{\partial E}{\partial z} = -\frac{e\rho_{e1}}{\epsilon_0 m_e}$

does anyone know where to start to derive this relationship?

Thanks