Fundamentals of Plasma Physics

86 Chapter 3. Motion of a single plasma particle

between the coils, andB(s)tends to zero ass→±∞.To focus attention on magnetic
trapping, suppose now that no electrostatic potential exists so Eq.(3.111) reduces to

mv‖^2

2

+μB(s) =const. (3.112)

Now consider a particle with parallel velocityv‖ 0 located at the well minimums 0 at time
t= 0.Evaluating Eq.(3.112) ats= 0,t= 0and then again when the particle is at some
arbitrary positionsgives

mv^2 ‖(s)

2

+μB(s) =

mv^2 ‖ 0

2

+μB(s 0 ) =

m

(

v‖^20 +v^2 ⊥ 0

)

2

=W 0 (3.113)

whereW 0 is the particle’s total kinetic energy att= 0.Solving Eq.(3.113) forv‖(s)gives

v‖(s) =±

√

2

m

[W 0 −μB(s)]. (3.114)

IfμB(s) =W 0 at some positions,thenv‖(s)must vanish at this position in which case
the particle must reverse its direction of motion just like a pendulum reversing direction
when its velocity goes through zero. This velocity reversal corresponds to a reflection of
the particle and so this configuration is called a magneticmirror. A particle can be trapped
between two magnetic mirrors;such a configuration is called a magnetic trap or a magnetic
well.

B

s 0

s 1 s 2

field

lines

“ potential”

hill

“ potential”

hill

“p otential”

valley

z

Figure 3.8: Magnetic mirror

IfW 0 > μBmaxwhereBmaxis the magnitude ats 1 , 2 then the velocity does not go to
zero at the maximum amplitude of the mirror field. In this case the particle does not reflect,
but instead escapes over the peak of theμB(s)potential hill and travels out to infinity.
Thus, there are two classes of particles: