Fundamentals of Plasma Physics

1.7 Quasi-neutrality 9

neutrality means that

∑

σ=i,enσ^0 qσ= 0causing the terms independent ofφto cancel in
Eq.(1.4) which thus reduces to

∇^2 φ−

1

λD^2

φ=−

qT

ε 0

δ(r) (1.5)

where the effective Debye length is defined by

1

λ^2 D

=

∑

σ

1

λ^2 σ

(1.6)

and the species Debye lengthλσis

λ^2 σ=

ε 0 κTσ

n 0 σq^2 σ

. (1.7)

The second term on the left hand side of Eq.(1.5) is just the negative of the shielding
cloud charge density. The summation in Eq.(1.6) is over all species that participate in the
shielding. Since ions cannot move fast enough to keep up with an electron test charge
which would be moving at the nominal electron thermal velocity, the shielding of electrons
is only by other electrons, whereas the shielding of ions is by both ions and electrons.
Equation (1.5) can be solved using standard mathematical techniques (cf. assignments)
to give

φ(r) =

qT

4 πǫ 0 r

e−r/λD. (1.8)

Forr << λDthe potentialφ(r)is identical to the potential of a test particle in vacuum
whereas forr >> λDthe test charge is completely screened by its surrounding shielding
cloud. The nominal radius of the shielding cloud isλD.Because the test particle is com-
pletely screened forr >>λD,the total shielding cloud charge is equal in magnitude to the
charge on the test particle and opposite in sign. This test-particle/shielding-cloud analy-
sis makes sense only if there is a macroscopically large number of plasma particles in the
shielding cloud;i.e., the analysis makes sense only if 4 πn 0 λ^3 D/ 3 >> 1 .This will be seen
later to be the condition for the plasma to be nearly collisionless and so validate assumption
#1 in Sec.1.6.
In order for shielding to be a relevant issue, the Debye length must be small compared
to the overall dimensions of the plasma, because otherwise no point in the plasma could be
outside the shielding cloud.Finally, it should be realized thatanyparticle could have been
construed as being ‘the’ test particle and so we conclude that the time-averaged effective
potential of any selected particle in the plasma is given by Eq. (1.8) (from a statistical point
of view, selecting a particle means that it no longer is assumed to have a random thermal
velocity and its effective potential is due to its own charge and to the time average of the
random motions of the other particles).

1.7 Quasi-neutrality

The Debye shielding analysis above assumed that the plasma was initiallyneutral, i.e., that
the initial electron and ion densities were equal. We now demonstratethat if the Debye