Fundamentals of Atmospheric, Ionospheric, and Magnetospheric Plasma Dynamics - The Earth's Ionosphere: Plasma Physics and Electrodynamics - page 57

A gyrating particle has a magnetic moment since it carries a current I and
surrounds an area r g , where r g is the gyroradius, so π

r g

μ = IA = I π

The current equals the charge divided by the gyro period, τ g , and using

τ g = 2

and r g = MV

qB π r g / V

⊥ / ⊥

yields the magnetic moment

MV 2

2 B

μ =

⊥ /

= K

⊥ /

B

where K is the perpendicular kinetic energy of the particle. Thus, the force due

to the magnetic field gradient is

⊥

F =− ( K

⊥ /

B ) ∇ B

and the corresponding “gradient drift” is

qB 2

MV 2

2 B B qB 2 W D =− μ ∇ B × B / =

⊥ /

×∇ B /

Notice that the gradient and curvature drifts are proportional to the particle
perpendicular and parallel energies, respectively, and are in the same direction in
a dipole field. The gradient-driven motion can be visualized easily with reference
to Fig. 2.9. If B is downward, as in the figure, a gyrating particle will have a

slightly smaller radius of curvature in one portion of its cycle than in the other.
A net drift results to the right for positively charged particles and to the left for
negatively charged particles.

∇

B

B 3 B

B

Ion drift

Big radius

Small radius

Electron drift

Figure 2.9 Gyromotion in a magnetic field with a gradient pointing downward.