Electric Field/Plasma Oscillations

[JNBirDy]

Homework Statement

Suppose there is a "slab" of plasma in a gas, and let N be the density of free electrons/unit volume. If an external electric field is applied, all the electrons move upwards a distance of x, which produces a thin sheet of unbalanced negative charge -Nex per unit area at the top of the slab. This also leaves a sheet of unbalanced positive charge +Nex per unit are at the bottom.

i) Calculate the resulting electric field produced by the two sheet charges. In addition, calculate the restoring electric force it applies to a typical electron in the slab. Also show that after the external field is shut of the equation of motion of the electrons can be shown as:

d^2x/dt^2 = -(Ne^2)(x) / m$_{e}$ε₀

ii) Explain why that after the external electric field is turned off the electrons undergo simple harmonic motion, and determine its frequency.

Homework Equations

I believe the electric field produced by an sheet of charge is: E = σ/2ε₀

F = E / q

F = ma

The Attempt at a Solution

Not exactly sure how to start part i)...

If E = σ/2ε₀, would that mean that E = -Nex/2ε₀ for the top and E = Nex/2ε₀ for the bottom? With the electric forces being F = -Ne^2x/2ε₀ and F = Ne^2x/2ε₀?

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Any help is appreciated...

 

[GATES]

E = σ/ε₀ inbetween two sheets the fields are added because fields are directed in the same way between the sheets...

then the rest should work out...