# Potential

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8

6

4

2

0

0

1

2 r

3

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Electric field

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2

1

0

0

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2 r

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Figure 2: Comparison of unscreened Coulomb (upper curves) and screened Debye (lower curves) potential (left plot) and field (right plot). The r axis is in units of the Debye length λ_{D}; the units on the other axis are arbitrary.

We have here introduced a quantity called the Debye length

λ_{D }

=

_{0}KT 2n_{0}e 2

.

(23)

The solution of (22) is straightforward, and yields exponential solutions for rΦ. Rejecting the expo-

nentially growing solutions on physical grounds, we are left with

exp(−r/λ_{D}) Φ(r) = C r

.

(24)

To determine the constant C, we note that at small values of r, the exponential goes to 1. In this region, the potential should equal the normal Coulomb potential, since for small r, there is very little charge inside r that can do the screening. Hence, we must have C = 1/(4π _{0}), and so

Φ(r) =

1 4π _{0}r

exp(−r/λ_{D})

(25)

A comparison of the unscreened Coulomb potential in vacuum and the screened Debye potential in a plasma is seen in Figure 2. The very rapid decay of the Debye potential above r = λ_{D }(r = 1 in the figure) is clearly seen. This has important physical implications: on distances longer than λ_{D}, the influence of individual particles are unimportant. Only the collective fields created by the ”cooperation” of many charges are important. This is very fortunate, since it means that we do not have to keep track of what every particle is doing, but can concentrate on statistical properties like plasma density, velocity etc.

# 3.5.4 Equation of continuity

All phenomena of interest to us are deviations from perfect thermodynamic equilibrium, small or large. To study such phenomena in a plasma, one needs equations for the ion gas and the electron gas. The fundamental equation describing the behaviour of any gas is the continuity equation,

∂n_{α }+ ∇ · (n_{α}v_{α}) = Q − L. ∂t

(26)

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