Consider the following basic equation for the equilibrium of a plasma column with cylindrical geometry (see problem 12.9 in Chapter 12)

(a) Verify that, for the theta pinch, this equation reduces to

whereas, for the longitudinal pinch, it becomes

(b) For the cylindrical screw pinch, in which both B_θ and B_z are nonzero, assume that the longitudinal current density and the kinetic pressure are given, respectively, by

Verify that

Show that B_z(r) satisfies the equation

From this equation determine B_z(r) and make a plot showing p(r), J_z(r), B_θ(r), and B_z(r) as a function of r.

Data from problem 12.9.

Consider a cylindrically symmetric plasma column (∂/∂z = 0, ∂/∂θ = 0) under equilibrium conditions, confined by a magnetic field. Verify that in cylindrical coordinates the radial component of (5.1) becomes

Using Maxwell equation (5.2), show that

From these results, obtain the following basic equation for the equilibrium of a plasma column with cylindrical symmetry

Give a physical interpretion for the various terms in this equation.

Equations

Transcribed Image Text:

d dr B B²/ p+ + 2μο | 2 μο = 1 B μο γι