Environmental Engineering Reference
In-Depth Information
Because the magnetic field is directed along the axis z , the force acting on a charged
particle is given by
e
c v τ H D v τ
@
H z
@
F z
D
.
2 c
z
Let us introduce the particle magnetic moment
μ
of a moving particle of charge e
in the usual way as
IS
c
μ D
,
(4.148)
where I is the particle current, S is the area enclosed by its trajectory, and c is the
velocity of light. Under the given conditions we have I
r L ,
D
e
ω
H /(2
π
)and S
D π
where r L is the Larmor radius of the particle. Hence, we have
H r L
2 c
e
ω
m
2
τ
2 H ,
v
μ D
D
where v τ D
H is the tangential component of the velocity of the charged parti-
cle, and the Larmor frequency is
r L
ω
ω
D
eH / mc .Neartheaxis,where H z
H ,
H
we obtain
m
2
τ
v
μ
H z
D
.
(4.149)
2
The force acting on the particle in the magnetic field direction is given by
D μ @
H z
@
F z
.
z
The minus sign means that the force is in the direction of decreasing magnetic
field.
Let us prove that the magnetic moment of the charged particle is an integral of
the motion, that is, it is a conserved quantity. One can analyze the particle motion
along a magnetic line of force when it is averaged over gyrations. The equation of
motion along the magnetic field gives md
z / dt
D
F z
D μ
dH z / dz , and since
v
dH z . From the energy con-
servation condition for the particle, we have accounting for (4.149)
D
dz / dt , it follows from this that d ( m
z /2)
D μ
v
v
z
m
2
τ 2 C
2
z
2
z
d
dt
m
d
dt
m
v
v
v
D
0
D
μ
H z
C
2
2
d
dt (
dH z
dt D
H z d
dt D
D
μ
H z )
μ
0.
This yields the equation for the magnetic momentum of the particle motion,
d
dt D
0 ,
(4.150)
so the magnetic moment is conserved during the motion of a charged particle in a
weakly nonuniform magnetic field. It should be noted that in deriving this formula,
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