Fundamentals of Plasma Physics

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)