16.6 Assignments 477
uniformity, the same radial electron density profilene(r), and are in equilibrium:
(a) a cylindrical pure electron plasma immersed in a strong magnetic fieldB=Bˆz
whereω^2 pe<<ω^2 ce
(b) anunmagnetizedquasi-neutral electron-ion plasma which has infinitely massive
ions.
What is the ion density in the quasi-neutral plasma and what electric field is produced
by this ion density? How does the force associated with the electric field produced by
the ions in the quasi-neutral plasma compare to the magnetic force associated with
electron rotation (finiteueθ) in the pure electron plasma? What is the net equilibrium
force on the electrons in the two cases? Assuming that the electrons have zero mass,
which plasma has more free energy?
- Free energy associated with sheared velocity profile in slab approximation:Suppose
a non-neutral plasma does not rotate as a rigid body and instead has a sheared angular
velocity. An observer rotating with the plasma at some radiusrobswould conclude that
the plasma has positive angular velocity forr>robsand a negative angular velocity
forr <robs.Since the rotational velocity is due to anE×Bdrift, this means that
the effective radial electric field measured in the observer frame must change sign at
r=robs.
(a) Show that this means that there is an effective charge sheet atr=robs.
(b) Consider the limiting situation where the above situation is modeled using slab
geometry so thatr→xandθ→y,B=Bzˆand the non-neutral plasma initially
centered aboutx=0, has widthwand a uniform densitynfor|x|<w/ 2 .This
equilibrium is sketched in Fig.16.3. There is vacuum in the region betweenthe
plasma and perfectly conducting walls atx=±awherea>w/ 2 .What are the
electric field and the plasmaflow velocity forx> 0 and forx<0?
(c) Now suppose that the location of the non-neutral plasma charge sheet is per-
turbed so that instead of being centered atx= 0,the location of the center
becomes a function ofyand is given byx=∆(y)=∆cos ̄ kywherekd<< 1
so that the wavelength in theydirection is very long. What is the boundary
condition onEyand hence onφat the walls?
