2.8 Assignments 61
(i) By taking the second moment of the Vlasov equation for each species (i.e., use
v^2 / 2 ) and summing over species show that
N
2
DP
Dt
+
N+ 2
2
P∇·U=−∇·q+J·(E+U×B)
Hints:
(a) Prove thatU·
(
mσ
∑∫
v′v′fσdNv
)
=PUassuming thatfσis isotropic.
(b) What happens to
∑
σmσ
∫
v^2 CσαfσdNv?
(c) Prove using the momentum and continuity equations that
∂
∂t
(
ρU^2
2
)
+∇·
(
ρU^2
2
U
)
=−U·∇P+U·(J×B).
(ii) Using the continuity equation and Ohm’s law show that
N
2
DP
DT
−
N+ 2
2
P
ρ
Dρ
Dt
=−∇·q+ηJ^2
Show that if both the heatflux term−∇·qand the Ohmic heating termηJ^2 can be
ignored, then the pressure and density are related by the adiabatic conditionP ∼ργ
whereγ= (N+ 2)/N.By assuming thatD/Dt∼ωand that∇∼kshow that the
dropping of these two right hand terms is equivalent to assuming thatω >> νeiand
thatω/k >>vT.Explain why the phenomenon should be isothermal ifω/k <<vT.
- Sketch the current collected by a Langmuir probe as a function of the bias voltage
and indicate the ion saturation current, the exponentially changing electron current,
thefloating potential, and the plasma potential. Calculate the ion saturation current
collected by a 1 cm long, 0.25 mm diameter probe immersed in a 5 eV argon plasma
which has a densityn= 10^16 m−^3. Calculate the electron saturation current also (i.e.,
the current when the probe is at the plasma potential). What is the offset of thefloating
potential relative to the plasma potential?
