# 3.5.2 Fluid in equilibrium

In thermodynamic equilibrium, the plasma follows the Maxwell-Boltzmann distribution (11). The potential energy of a charge in an electrostatic field is U = qΦ, where Φ is the electrostatic potential (SI unit: V). Thus, (11) and (12) implies

m n_{i}(r) = 2πKT

3/2

exp

−

1 2

mv^{2 }+ eΦ(r) KT

d^{3}v = n_{0 }exp

−

eΦ(r) KT

(17)

and, in the same way,

n_{e}(r) = n_{0 }exp

eΦ(r) KT

.

(18)

These equations are known as the Boltzmann relations for ions and electrons. Their basic content is very simple: where the potential energy is low, there will be a lot of particles; where it is high, the particles are scarce. A familiar example where the potential energy is due to gravitation rather than electric effects is the ordinary air around us. The higher the potential energy, i.e. the height (U = m g h in this case), the lower the density.

# 3.5.3 Electrostatic (Debye) shielding

A charge +Q will affect the motion of other charges in the plasma. If a particle is negative, its orbit will bend towards Q, if it is positive, the orbit will be bent away from Q. The net result is that negative particles will spend more time near Q and positive particles will spend more time away from Q, thus creating a net negative charge density in space near Q. In this way, the positive charge Q is shielded by a cloud of negative particles. In the same way, a negative particle is shielded by positive particles. This is known as Debye shielding, and is mathematically treated as below.

For an equilibrium plasma, the ions and electrons follows the Boltzmann relations (17) and (18).

# The charge density (14) in the plasma is then

ρ(r) = e(n_{i }− n_{e}) = n_{0 }

exp

−

eΦ(r) KT

− exp

eΦ(r) KT

.

(19)

The electrostatic potential is determined by the charge density through E = −∇Φ and Gauss’ law for the electric field (3). Hence, we have

# ∇^{2}Φ = −∇ · E = −

0

=

n 0 e 0

exp

eΦ KT

− exp

−

eΦ KT

.

(20)

# This is a nonlinear ordinary differential equation, solvable only by numerical methods. However,

for the case eΦ/KT quantity to get

1, we can expand in powers of eΦ/KT and neglect higher terms in this

∇^{2}Φ =

n 0 e 0

1+

eΦ KT

+

...

−

1−

eΦ KT

+

...

≈

n 0 e 2 2 0 K T

Φ.

(21)

To solve this equation around a charge Q in the plasma, we may assume spherical symmetry and introduce spherical coordinates:

∇^{2}Φ =

1 d^{2 }r dr^{2 }

(rΦ) ≈

n 0 e 2 2 0 K T

Φ=

1 λ 2 D

Φ.

(22)

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