Fundamentals of Plasma Physics

10 Chapter 1. Basic concepts

length is a microscopic length, then it is indeed an excellent assumption that plasmas re-
main extremely close to neutrality, while not being exactly neutral. It is found that the
electrostatic electric field associated with any reasonable configuration is easily produced
by having only a tiny deviation from perfect neutrality. This tendency to bequasi-neutral
occurs because a conventional plasma does not have sufficient internal energy to become
substantially non-neutral for distances greater than a Debye length (theredo exist non-
neutral plasmas which violate this concept, but these involve rotation ofplasma in a back-
ground magnetic field which effectively plays the neutralizing role of ionsin a conventional
plasma).
To prove the assertion that plasmas tend to be quasi-neutral, we consider an initially
neutral plasma with temperatureTand calculate the largest radius sphere that could spon-
taneously become depleted of electrons due to thermalfluctuations. Letrmaxbe the radius
of this presumed sphere. Complete depletion (i.e., maximum non-neutrality) would occur
if a random thermalfluctuation caused all the electrons originally in the sphere to vacate
the volume of the sphere and move to its surface. The electrons would have tocome to rest
on the surface of the presumed sphere because if they did not, they would still have avail-
able kinetic energy which could be used to move out to an even larger radius, violating the
assumption that the sphere was the largest radius sphere which could become fully depleted
of electrons. This situation is of course extremely artificial and likely to be so rare as to be
essentially negligible because it requires all the electrons to be moving radially relative to
some origin. In reality, the electrons would be moving in random directions.
When the electrons exit the sphere they leave behind an equal number of ions. The
remnant ions produce a radial electric field which pulls the electronsback towards the
center of the sphere. One way of calculating the energy stored in this system is to calculate
the work done by the electrons as they leave the sphere and collect on the surface, but a
simpler way is to calculate the energy stored in the electrostatic electric field produced by
the ions remaining in the sphere. This electrostatic energy did not exist when the electrons
were initially in the sphere and balanced the ion charge and so it must be equivalent to the
work done by the electrons on leaving the sphere.
The energy density of an electric field isε 0 E^2 / 2 and because of the spherical symmetry
assumed here the electric field produced by the remnant ions must be in the radial direction.
The ion charge in a sphere of radiusrisQ= 4πner^3 / 3 and so after all the electrons have
vacated the sphere, the electric field at radiusrisEr=Q/ 4 πε 0 r^2 = ner/ 3 ε 0 .Thus
the energy stored in the electrostatic field resulting from completelack of neutralization of
ions in a sphere of radiusrmaxis

W=

∫rmax

0

ε 0 E^2 r

2

4 πr^2 dr=πr^5 max

2 n^2 ee^2

45 ε 0

. (1.9)

Equating this potential energy to the initial electron thermal kinetic energyWkinetic
gives

πrmax^5

2 n^2 ee^2

45 ε 0

=

3

2

nκT ×

4

3

πr^3 max (1.10)

which may be solved to give

rmax^2 = 45

ε 0 κT

nee^2

(1.11)