Environmental Engineering Reference
In-Depth Information
we assume
H
z
H
. Therefore, in analyzing the motion of a charged particle
in a weakly nonuniform magnetic field, for a given spatial region we choose the
magnetic field to be directed almost along the
z
-axis .
Let us analyze the motion of a charged particle in a weakly nonuniform magnetic
field along the magnetic field direction, which is taken as the
z
-axis. A particle is
moving along a helical trajectory, and the motion equation has the form
m
e
d
D
μ
@
H
z
@
z
D
ε
H
@
H
z
@
v
z
dt
D
F
z
,
z
where
ε
τ
is the transverse kinetic energy of the charged particle, and we use the
definition (4.149) of the longitudinal force and the value of the magnetic momen-
tum of the charged particle. Multiplying the motion equation by the particle velocity
along the magnetic field
z
,wereducethisequationtotheform
v
d
dt
D
ε
H
ε
dH
z
dt
z
.
On the basis of the conservation of the total electron kinetic energy
ε
C
ε
τ
D
const
z
of the charged particle, we obtain from this equation
ε
H
D
const .
(4.151)
Thus, the magnetic field gradient leads to exchange between the the kinetic en-
ergies of the charged particle in the longitudinal and transverse directions. If the
particle is moving in the direction of a magnetic field increase, the Larmor radius
increases in the course of this motion, as shown in Figure 4.24. Note that (4.151) is
analogous to (4.150). Indeed, the magnetic moment of an electron that is moving
along a circular trajectory is
c
D
ε
H
,
where the electric current is
I
IS
μ
D
D
e
ω
H
/(2
π
), and the area for this current is
S
D
r
L
,where
r
L
is the Larmor radius. Therefore, (4.151) corresponds to conservation
of the electron magnetic moment (4.149) in the course of exchange between the
transverse and longitudinal electron kinetic energies of the particle.
Another example of the behavior of a charged particle in a weakly nonuniform
magnetic field is its drift due to a magnetic field gradient. Indeed, let the magnetic
field be directed along the
z
-axis and be decreased weakly in the
y
direction. A
π
Figure 4.24
Trajectory of a charged particle moving in a weakly nonuniform magnetic field along
the magnetic field. Because of a variation of Larmor radius of the particle, its drift takes place in
accordance with (4.151).
