3.5 Drift equations 79To simplify the algebra for the averaging, a local Cartesian coordinate system is used with
xaxis in the direction of the gyrovelocity att= 0andzaxis in the direction of the
magnetic fieldat the gyrocenter.Thus, the Larmor orbit velocity has the form
vL(t) =vL 0 [ˆxcosωct−yˆsinωct] (3.79)where
ωc=qB
m(3.80)
is called the cyclotron frequency and the Larmor orbit position has the form
rL(t) =vL 0
ωc[ˆxsinωct+ ˆycosωct]. (3.81)Inserting the above two expressions in Eq.(3.78) gives
F∇B =qv^2 L 0
ωc〈[ˆxcosωct−ˆysinωct]×([ˆxsinωct+ ˆycosωct]·∇)B〉. (3.82)Noting that
〈
sin^2 ωct〉
=
〈
cos^2 ωct〉
= 1/ 2 while〈sin(ωct)cos(ωct)〉= 0,this reduces
to
F∇B =qvL^20
2 ωc[
xˆ×∂B
∂y−ˆy×∂B
∂x]
=
mvL^20
2 B[
xˆ×
∂(Byyˆ+Bzˆz)
∂y−yˆ×
∂(Bxxˆ+Bzzˆ)
∂x]
=
mvL^20
2 B[
zˆ(
∂By
∂y+
∂Bx
∂x)
−yˆ∂Bz
∂y−xˆ∂Bz
∂x]
. (3.83)
But from∇·B= 0, it is seen that
∂By
∂y+
∂Bx
∂x=−
∂Bz
∂zso the ‘gradB’ force isF∇B=−
mv^2 L 0
2 B∇B (3.84)
where the approximationBz≃Bhas been used since the magnetic field direction is mainly
in theˆzdirection.
Let us now define
Fc=−mv^2 ‖gcB̂·∇B̂ (3.85)
and consider this force. Suppose that the magnetic field lines have curvature and consider
a particular point on a specific field line. Define, as shown in Fig.3.5, a two-dimensional
cylindrical coordinate system(R,φ)with origin at the field line center of curvature for this
specific point and lying in the plane of the field line at this point. Then, the radial position
of the chosen point in this cylindrical coordinate system is the local radius of curvature of
the field line and, sinceφˆ=B,ˆ it is seen thatB̂·∇B̂=φˆ·∇ˆφ=−R/R.ˆ Thus, the force
associated with curvature of a field line
Fc=mv^2 ‖gcRˆ
R(3.86)
is just the centrifugal force resulting from the motion along the curve ofthe particle’s
guiding center.