A Brief Introduction to Space Plasma Physics.pdf - Institute of ...

– On time scales much shorter than
r
∂B
∂t
r r
= ∇× ( u × B)
– The electric field vanishes in the frame moving with the fluid.
– Consider the rate of change of magnetic flux
dΦ
dt
=
d
dt
r
∂B
∂t
∫ B ⋅ndA
= ∫ ⋅ ndA + ∫ B ⋅(
u ×
A
r
ˆ
– The first term on the right is caused by the temporal changes
in B
– The second term is caused by motion of the boundary
– The term is the area swept out per unit time
A
r r
u × dl
0
– Use Stoke’s theorem
Φ r
d ⎛ ∂
⎞
= ∫ B r r
⎜ ( ) ⎟
− ∇× u × B
⋅ nˆ
=
dt ⎝ ∂
dA
A
t
⎠
– If the fluid is initially on surface s as it moves through the
system the flux through the surface will remain constant
even though the location and shape of the surface change.
ˆ
τ D
C
r
r
r
dl )