472 Chapter 16. Non-neutral plasmas
The surface chargeflows back and forth between the wall segments and ground. The charge
on the segment extending fromθ=−∆/ 2 to∆/ 2 is
Qs = Ls
∫∆/ 2
−∆/ 2
adθσ ̃leilθ
= Lsaσ ̃l
eil∆/^2 −e−il∆/^2
il
= −
2
l
Lsaε 0 Ssin
∆
2
. (16.67)
The electric currentI flowing through the resistor is the rate of change of this surface
charge, i.e., for thel=1mode this current is
I=2iωLsaε 0 Ssin
∆
2
. (16.68)
R
pure
electron
plasma
perfectly
conducting
wall
wall
pure segment
electron
plasma
Figure 16.2: Arrangement for resistive wall instability of a non-neutral plasma. Resistor
Ris connected between ground and an isolated segment of azimuthal extent∆and axial
extentLs.Remainder of wall is connected directly to ground.
The rate at which energy is dissipated in the resistor is
〈
I^2 R
〉
=
1
2
(
2 ωLsaε 0 Ssin
∆
2
) 2
R. (16.69)
If the mode frequency is allowed to have a small imaginary part then the rate of change of
electrostatic energy will be
P=2ωiδW. (16.70)