94 Chapter 3. Motion of a single plasma particle
The total magnetic force isJtotal×B = (JM+J∇B+Jc+JP)×B=
−∇×
(
P⊥Bˆ
B
)
−P⊥
∇B×B
B^3
−P‖
Bˆ·∇Bˆ×B
B^2
+
ρ
B^2dE
dt
×B.
(3.133)
The gradBcurrent cancelspartof the magnetization current as follows:
∇×
(
P⊥Bˆ
B
)
+P⊥
∇B×B
B^3
=
[
∇
(
1
B
)
×P⊥Bˆ+
1
B
∇×
(
P⊥Bˆ
)]
+P⊥
∇B×B
B^3
=
1
B
∇×
(
P⊥Bˆ
)
=
P⊥
B
∇×Bˆ+
∇P⊥×Bˆ
B
(3.134)
so that
Jtotal×B=−[
P⊥∇×Bˆ+∇P⊥×Bˆ+P‖Bˆ·∇Bˆ×Bˆ−
ρ
BdE
dt]
×B.ˆ (3.135)
The first term can be recast using the vector identity
∇
(
Bˆ·Bˆ
2
)
= 0 =Bˆ·∇Bˆ+Bˆ×∇×Bˆ (3.136)
while the electric field can be replaced usingE=−U×Bto give
Jtotal×B=−(
P⊥−P‖
)
Bˆ·∇Bˆ+∇⊥P⊥−ρ
Bd(U×B)
dt×B.ˆ (3.137)
Here the relation
[
Bˆ·∇Bˆ
]
⊥=Bˆ·∇Bˆhas been used;this relation follows from Eq.(3.136). Finally, it is observed that
[
∇·
(
BˆBˆ
)]
⊥=
[(
∇·Bˆ
)
Bˆ+Bˆ·∇Bˆ
]
⊥=Bˆ·∇Bˆ (3.138)
so
(
P⊥−P‖
)
Bˆ·∇Bˆ=
(
P⊥−P‖
)[
∇·
(
BˆBˆ
)]
⊥=
[
∇·
{(
P⊥−P‖
)
BˆBˆ
}]
⊥.(3.139)
Furthermore,
ρ
B
d(U×B)
dt×Bˆ≃−
[
ρ
dU
dt]
⊥(3.140)
since it has been assumed that the magnetic field is time-independent. Inserting these last
two results in Eq.(3.137) gives
Jtotal×B=[
∇·
{(
P⊥−P‖
)
BˆBˆ
}]
⊥+∇⊥P⊥+
[
ρ
dU
dt]
⊥