Fundamentals of Atmospheric, Ionospheric, and Magnetospheric Plasma Dynamics - The Earth's Ionosphere: Plasma Physics and Electrodynamics - page 59

and using (2.23a)

q ∂ dt · K g =− B / d a

Note that the vectors q B and d a are in opposite directions. Now, since B is
uniform in space, we can move it outside the integral and

q

r g /τ g K

⊥ /

t = K g /τ g = π (∂ B /∂ t ) = ( K

⊥ /

B )(∂ B /∂ t )

and finally

K

B 2 ∂μ/∂ t = ( K

⊥ /

B )(∂ B /∂ t )/ B − (∂ B /∂ t )

⊥ /

= 0

μ

and so is conserved.

For illustrative purposes, we can apply these findings to a “magnetic bottle”
geometry similar to the earth's magnetic field as illustrated in Fig. 2.10. Suppose
a particle starts on the symmetry axis at the place where the field strength is B 0
and the velocity vector of the particle makes an angle (the pitch angle) with

respect to the magnetic field line. At the magnetic equator, then,

α

V

|| =

V cos α , V

⊥ =

V sin α

carries the particle into a region of larger field strength, as
shown in Fig. 2.10b. However, if

The velocity V

||

increases MV 2

2 must
also increase. Since the particle energy must also be conserved, this can only

μ is conserved, as | B |

⊥ /

V

B

B

B 0

(a)

V II

V '

B

(into paper)

(b)

Figure 2.10 (a) Sketch of the magnetic field lines and the corresponding magnetic field
magnitude along the axis of a magnetic bottle. (b) The gyromotion associated with a
converging magnetic field.