Fundamentals of Plasma Physics

16.6 Assignments 479

(g) Suppose that the length of the plasma in theydirection isLwhereLis an

integral multiple of 2 π/k,the wavelength in theydirection. By calculatingEx

to the right of the charge sheet and also to the left of the charge sheet, show that

the energy stored in the electrostatic electric field is approximately

W =

ε 0

2

h

∫L/ 2

−L/ 2

dy

∫a

−a

dxE^2 x

wherehis the length in thezdirection. How doesWchange when∆ ̄is in-

creased? What does this suggest about the stability of sheared velocity profiles

with respect to perturbations as prescribed in (c)?

3. Image charge in cylindrical geometry:Two lines chargesλ 1 andλ 2 are aligned along
   thezaxis and located atr 1 =x 1 xˆandr 2 =x 2 ˆxas shown in Fig.16.4. What values
   ofx 2 andλ 2 are required in order for the cylindrical surface

√

x^2 +y^2 =ato be

an equipotential as would be required in order to have a perfectly conductingwall at

r=a? The following hints should be useful:

(a) Define a cylindrical coordinate systemr,θ,zso that the cylindrical surface is

given byr=aˆr.Determine the potential on the cylindrical surfacer=a.

(b) Show that|ˆr−ˆxx 1 /a|=|ra/xˆ 2 −ˆx|ifa/x 2 =x 1 /a.

(c) Show thatλ 2 =−λ 1 is required in order for the cylindrical surfacer=ato be

an equipotential.

perfectly

conducting

w all

x 1

(^1)
2
x 2
a
Figure 16.4: Line chargeλ 1 is displaced a distancex 1 from axis of perfectly conducting
cylindrical wall of radiusa.An image chargeλ 2 is located atx 2.

4. Diocotron mode energy using image charge method:A cylindrical non-neutral plasma
   is translationally invariant in thezdirection and is bounded by a perfectly conducting