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

• Maxwell’s equations
r
∂B
r
= −∇× E
∂t
r r
∇× B = µ 0
J
– ∇ ⋅ B r = 0 doesn’t help rbecause
∂(
∇ ⋅ B)
r
= −∇ ⋅∇× E = 0
∂t
r r r
– There are 14 unknowns in this set of equations - r
E , B,
J , u,
ρ,
p
– We have 11 equations.
• Ohm’s law
– Multiply the momentum equations for each individual species
by q s /m s and subtract
r
J
r
r r r 1 1 r r me
∂J
rr
= σ {( E + u × B)
+ ∇pe
− J × B − [ + ∇ ⋅(
Ju)]}
2
ne ne ne ∂t
r r
where J = ∑ qsnsusand σ is the electrical conductivity
s