Figure 79: Charge density in a metalare also seen in plasma. Plasma is a gas of positive and negative charges. If you move the electrons
in the x-direction, the resulting force is the electric field that is caused by the protons. So Newton’s
law for a bunch of electrons in an electric field is
nmd^2 x
dt^2
=−neE (166)where the electric field E is determined by the following equation (out of Gauss law: ∇·E= ρ 0 ,
ρ=ne):
E=nex
0, (167)
withnas the electron density. Now we plug eqn. (167) in eqn. (166) and we get a differential equation
that just depends onx.
nmd^2 x
dt^2=−
n^2 e^2 x
0(168)
d^2 x
dt^2+ω^2 px= 0 (169)The solution of this differential equation is the same like for a harmonic oscillator. The frequencyωp
is called Plasma frequency and it’s defined by
ωp=√
ne^2
m 0