2.7 Sheath physics and Langmuir probe theory 53and the equation of motion isρDU
Dt=J×B−∇P. (2.100)
(b) Double adiabatic regime, the collision frequency is insufficient to equilibrate
perpendicular and parallel temperatures so thatP‖B^2
ρ^3=const.,P⊥
ρB=const. (2.101)and the equation of motion isρDU
Dt=J×B−∇·
[
P⊥
←→
I + (P‖−P⊥)B̂B̂
]
. (2.102)
- Faraday’s Law
∇×E=−
∂B
∂t. (2.103)
- Ampere’s Law
∇×B=μ 0 J. (2.104) - Ohm’s Law
E+U×B=ηJ. (2.105)
These equations provide a self-consistent description of phenomena that satisfy all the
various assumptions we have made, namely:
(i) The plasma is charge-neutral since characteristic lengths are much longer than a
Debye length;
(ii) The characteristic velocity of the phenomenon under consideration isslow com-
pared to the speed of light;
(iii) The pressure and density gradients are parallel, so there is no electrothermal EMF;
(iv) The time scale is long compared to both the electron and ion cyclotron periods.
Even thought these assumptions are self-consistent, they may not accurately portray a
real plasma and so MHD models, while intuitively appealing, must be used with caution.
2.7 Sheath physics and Langmuir probe theory
Let us now turn attention back to Vlasov theory and discuss an immediatepractical ap-
plication of this theory. The properties of collisionless Vlasov equilibria can be combined
with Poisson’s equation to develop a model for the potential in the steady-state transition
region between a plasma and a conducting wall;this region is known as a sheath and is
important in many situations. The sheath is non-neutral and its width is of the order of a
Debye length. The exact sheath potential profile must be solved numerically because of
the transcendental nature of the relevant equations, but a useful approximate solution can
be obtained by a simple analytic argument which will now be discussed. Sheath physics is
of particular importance for interpreting the behavior of Langmuir probes whichare small