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B
2
so this term cancels the
ˆ
s
component of
∇
/
2
μ
0
and we have finally
B
2
μ
0
R
∂
B
2
μ
0
n
B
2
μ
0
b
b
f
B
=
/
2
n
ˆ
−
/
2
∂
n
ˆ
−
∂
/
2
∂
B
forces are therefore equivalent to a pressure
B
2
μ
0
acting
isotropically in the plane perpendicular to
B
plus a force
T
B
also normal to
B
,
which acts in the plane of curvature of the magnetic field. The sense of
T
B
is
similar to that of forces due to tension in a stretched string, which are parallel to
ˆ
The magnetic
J
×
/
2
n
and proportional to 1
R
(see Fig. 2.13). As noted above, for a dipole field the
tension and pressure forces in the
/
n
direction exactly cancel (as they must when
ˆ
∇×
.
The case of an equilibrium requires
d
V
B
=
0
)
/
dt
=
0, which in turn implies
from (2.57)
0
=−∇
p
+
J
×
B
which is equivalent to
p
μ
0
B
2
∇
+
/
2
=
(
B
·∇
)
B
/μ
0
The magnetic pressure (
B
2
enters just like the particle pressure, while
the curvature (tension) term appears on the right-hand side of this force balance
equation. These concepts are quite useful in discussing magnetospheric dynamics
and equilibria.
Equation (2.60d) also has an interesting and useful interpretation. Assuming
first that
/
2
μ
0
)
σ
is infinite,
∂
B
/∂
t
−∇×
(
V
×
B
)
=
0
Consider the magnetic flux
φ
across an arbitrary surface
moving with
velocity
V
:
B
·
V
d
l
d
φ/
dt
=
(∂
B
/∂
t
)
·
d
a
+
×
S
The first integral yields the change in
due to the time variation of
B
and the
second integral yields the change due to the motion of the surface. Working on
this second term,
B
φ
·
V
d
l
=−
×
(
V
×
B
)
·
d
l
=−
∇×
(
V
×
B
)
·
d
a
S
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