Environmental Engineering Reference
In-Depth Information
Figure 4.25
Trajectory of a charged particle moving along the gradient of the magnetic field that
is directed perpendicular to the magnetic field.
charged particle rotates clockwise in the
xy
plane, so the upper Larmor radius of
particle rotation is larger than the lower one. As a result, a particle moves in the
x
direction,asshowninFigure4.25.
Let us rewrite (4.138) in the form
c
[
F
H
]
emH
2
w
D
,
where
F
is the force acting on a charged particle of charge
e
and mass
m
.Because
F
D
μ
r
2
/2
H
,weobtain
H
and
μ
D
m
v
2
[
r
H
H
]
H
2
w
D
v
.
(4.152)
2
ω
H
Hence, if the magnetic field is directed along the
z
-axis and it varies in the
y
di-
rection, a charged particle is moving in the
x
direction. From this it follows that
if the magnetic field strength has a maximum at the
z
-axis and decreases in the
direction, the particle moves along the
z
-axis with a helical trajectory.
4.5.5
Magnetic Traps
We can now analyze the motion of a charged particle in a magnetic trap. If the mag-
netic field has the form given in Figure 4.26 and the particle is moving from the
region of a weak magnetic field, the region of increasing magnetic field acts on the
charged particle like a magnetic mirror, and the particle reflects from this region
and returns. Indeed, as the magnetic field increases, some of the particle's kinetic
energy
m
2
τ
/2 that corresponds to the particle's motion in the direction perpendic-
ular to the magnetic field increases proportionally. If this value becomes as large as
the total initial kinetic energy, motion of the particle along the
z
-axis stops, and the
particle starts to move in the opposite direction. This is the principle of a magnetic
mirror in which a particle moves along a magnetic line of force between a region
with a weak magnetic field and a region with a strong magnetic field. We take
H
min
to be the minimum magnetic field and
H
max
to be the maximum magnetic field
along this magnetic line of force;
v
is the angle between the particle velocity and
magnetic field line at the field minimum. Then if this angle exceeds
θ
θ
m
,givenby
the relation
sin
2
θ
D
H
min
/
H
max
,
(4.153)
m
