Download BACKGROUND: Maxwell`s Equations (mks)

BACKGROUND:
Maxwell’s Equations (mks)

E 

  B  0,
B
E
,
t
1 E
c2  t
 B j 
neglect if v <<c
E
Since
2

c t
We have used
Bl
2 2
c t
l 2 B
  
t  lc 2

2
v B
lc
H  B ,D  E
2

v
c
2
2
B
to eliminate
H (the magnetic field) and D (the electric displacement)
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Star Formation & Plasma Astrophysics
Definitions and Units
The magnetic permeability
The permittivity of free space
   0  4   10 7 Hm -1
   0  8.854  10 12 Fm -1
c2  1  0  0
Electric field
Charge density
Current density
E Vm-1 
 Cm

-2
j Am 

-3
Observers often measure magnetic induction in Gauss
(rather than Tesla) and energy in erg (rather than Joules)
4
1G  10 T
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-7
1erg = 10 J
Star Formation & Plasma Astrophysics
THE MAGNETOHYDRODYNAMIC
(MHD) EQUATIONS: DERIVATION
• These equations describe how a magnetic field
interacts with a plasma (ionized gas).
• What happens when you put a magnetic field
into a plasma?
• Force (F) on a particle of charge (q) moving in
an electric field (E) and a magnetic field (B):
F  qEr,t   rÝ Br,t  mrÝ
Ý
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Star Formation & Plasma Astrophysics
Simple case
z
• No electric field E = 0
• Uniform magnetic field
B = (0,0,B)
B
y
x
Force equation
mÝ
rÝ qrÝ zˆ B
has components
AS 4002
mxÝ
Ý qByÝ 

myÝ
Ý qBxÝ
Star Formation & Plasma Astrophysics
Resultant particle motion
The solution describes
helical motion
xÝ v cos t   

yÝ v sin t   

zÝ const
where the gyro frequency
(or Larmor frequency)
  qB m
and the perpendicular velocity
is constant
2
v
2
2
 xÝ  yÝ
B
The gyroradius
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r L  v 
Star Formation & Plasma Astrophysics
Consequences
• Charged particles are tied to fieldlines (electrons
and protons gyrate in opposite directions).
• Even when the plasma is effectively collisionless
( mean free path >> typical lengthscales ) the
magnetic field causes the plasma to behave
collectively.
• The gyroradius defines the lengthscale on which
particle motions are organised.
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Star Formation & Plasma Astrophysics
• Both conductivity and viscosity across
magnetic field lines are much less than that
along field lines.
• Applying a uniform electric field perpendicular
to B causes the centre of the helix to drift in the
direction of E, while applying an electric field
along B causes particles to be accelerated.
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Star Formation & Plasma Astrophysics
The Equation of Motion
• The equation of motion for charged particles is
where
dv
nm  nF  p  P
dt
n is the particle number density,
m is the particle mass,
F is a combination of the Lorentz force and gravity,
F  q  E  v  B  m g
p is the pressure (assumed to be a scalar), and
P describes the momentum transfer by collisions.
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Star Formation & Plasma Astrophysics
Write this equation separately for electrons and ions:
dv e
ne me
 ne q  E  v e  B  ne me g  pe  P ei
dt
dv i
ni mi
 Zni q E  v i  B  ni mi g  pi  P ie
dt
Add these using
Zni  n e
  ni mi  ne me  ni mi
j  qZni v i  n e v e 
ne me v e  ni mi v i
v
ne m e  ni mi
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Star Formation & Plasma Astrophysics
dv e
dv i
ne me
 ni mi
 ne qv i  v e   B
dt
dt
p  ni mi  ne me g
to get
dv
 
dt
j  B  p   g
Lorentz force
This is just the same as the normal equation of
motion for a fluid, except for the Lorentz force
which describes the interaction between the
magnetic field and the plasma.
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Star Formation & Plasma Astrophysics