3) A Z-pinch is a cylindrically symmetric plasma column with current in the z direction
only. Consider a static ideal MHD Z-pinch equilibrium with
where cl is a constant.
(a)Calculate Be(r) and p(r). Express your answers in terms of I, the total current. Sketch
the fields and the currents.
(b) (Extra credit) Since J(r) vanishes for large r the Z pinch is apparently confined by its
own current. Doesn't this violate the virial theorem(a plasma can not be confined by its
own currents)? Explain.
(From Problem 5.2, Ideal Magnetohydrodynamics,J.P. Freidberg)
0
= a'trc,
,[
&)=
I
~(l+*]~
a2nc,
I
2 - -If!Ga
I
B,o,
i
-
=I(.
2
r
r r-
Icr)
C
b
+ I ]
2
- I+ -I
1
2
a-
\
I
5
-4
=hi
-
-
=u k ,
tgr=&)
9
,
.& v -
3 a - cr&t
LL
@
"%'lcOJ"
1,
I-
(I+ X i L ) S
r/a-
cr
-
ail-
m
&
B
- I
-W
+ Y ~ =( + p
T
-J- L
2, ) & = P