Fundamentals of Plasma Physics

3.5 Drift equations 85

transverse to the initial direction of the magnetic field, i.e., a field in thexorydirections.

In a cylindrically symmetric system, this transverse field must be a radial field as indicated

by the vector decompositionB=Bzzˆ+Brrˆin Fig.3.7.

B

Bz

Br

r

z

field lines squeezed

together

Figure 3.7: Field lines squeezing together when B has a gradient.Bfield is stronger on the
right than on the left because density of field lines is larger on the right.

The magnetic field is assumed to be static so that∇×E= 0in which caseE=−∇φ

and Eq.(3.92) can be written as

m

dv‖

dt

=−q

∂φ

∂s

−μ

∂B

∂s

. (3.109)

Multiplying Eq.(3.109) byv‖gives

d

dt

[

mv^2 ‖

2

+qφ+μB

]

= 0, (3.110)

assuming that the electrostatic potential is also constant in time. Time integration gives

mv^2 ‖

2

+qφ(s) +μB(s) =const. (3.111)

Thus,μB(s)acts as an effective potential energy since it adds to the electrostatic potential

energyqφ(s).This property has the consequence that ifB(s)has a minimum with respect

tosas shown in Fig.3.8, thenμBacts as an effective potential well which can trap particles.

A magnetic trap of this sort can be produced by two axially separated coaxialcoils. On each

field lineB(s)has at locationss 1 ands 2 maxima near the coils, a minimum at locations 0