Fundamentals of Plasma Physics

98 Chapter 3. Motion of a single plasma particle

a Poisson-like equation. Since no current loops can exist at infinity, the field prescribed
by Eq.(3.146) must be produced by a set of coaxial coils having various finite radiirand
various finite axial positionsz.
The axial magnetic field is
Bz=

1

2 πr

∂ψ

∂r

. (3.153)

Nearr= 0,ψcan always be Taylor expanded as

ψ(r,z) = 0 +r

∂ψ(r= 0,z)

∂r

+

r^2

2

∂^2 ψ(r= 0,z)

∂r^2

+... (3.154)

Suppose that∂ψ/∂ris non-zero atr= 0, i.e.,ψ∼rnearr= 0.If this were the case,
then the first term in the right hand side of the last line of Eq.(3.152) would become infinite
and so lead to an infinite current density atr= 0. Such a result is non-physical and so we
require that the first non-zero term in the Taylor expansion ofψaboutr= 0to be ther^2
term.
Every field line that loops through the inside of a current loop also loops back in the
reverse direction on the outside, so there is no net magneticflux at infinity. This means
thatψmust vanish at infinity and so asrincreases,ψincreases from its value of zero at
r= 0to some maximum valueψmaxatr=rmax, and then slowly decreases back to zero
asr→∞.As seen from Eq.(3.153) this behavior corresponds toBzbeing positive for
r rmax.A contour plot of theψ(r,z)flux surfaces and a plot
ofψ(r,z= 0)versusris shown in Fig.3.13.

r, 0 

r

r

B

r,z
const.

z

Figure 3.13: Contour plot offlux surfaces