Fundamentals of Plasma Physics

478 Chapter 16. Non-neutral plasmas

non-neutral

plasma

equilibrium

configuration

perturbed

configuration

x

y

z

−a 
a −a 
a

x x

perfectly

conducting

wall

0 0

Figure 16.3: Left: equilibrium configuration for non-neutral plasma of widthw, centered
between pair of perfectly conducting walls atx=±a.Right: perturbed configuration
where position ofx-midpoint of plasma is located at∆=∆cos ̄ ky.

(d) Since there is vacuum between the plasma and the wall andkd<< 1 ,what is

the limiting form of Poisson’s equation in the regions between the plasmaand

the walls?

(e) Assume thatwis small, so that the non-neutral plasma can be approximated

as being a thin sheet of charge. Being perfectly conducting, the walls must be

equipotentials, and so without loss of generality, this potential can be defined to

be zero. Show that the potential to the right of the charge sheet must be of the

form

φr=α(a−x) (16.91)

and the potential to the left of the charge sheet must be of the form

φl=β(a+x) (16.92)

whereαandβare coefficients to be determined by considering the jump in

electric field at the charge sheet.

(f) Taking into account the jump in the potential at the charge sheet and the continu-

ity in the potential at the charge sheet, solve forαandβ.Assume thatw<<a

consistent with the assumption that the non-neutral plasma can be considered as

a thin charge sheet. Which ofEy^2 ,Ex^2 is larger?