Transcript Lecture_6 - University of Massachusetts Lowell

Lecture_6
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3.5 Nonuniform B Field
Gradient-B drift
Assume B  B has a gradient  B. Assume :
dB
B  x  B  x z
B 
y (Fig.3.5)
dx
Since rL  mv eB, the Lamor radius is smaller where B is larger,
etc. (Fig. 3.6). This leads to the grad-B drift velocity
1
B  B
v B   v rL
where B   B
2
2
B
sign for ions, -sign for electrons  current!
Dimensional analysis of 3.41 ....
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 3.41
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3.5.2 Centrifugal Force:
Curvature drift
Assume a charged particle moves along a curved field line.
2
 v// 
Centrifugal acceleration is  r =   r.
 r 
mv//2
Centrifugal force (Figure 3.7): Fc  2 R c
Rc
2
See Fig. 3.7 for radius of curvature R c . Starting with (3.24):
Fc  B
v cB 
qB 2
mv//2 R c  B v//2 Rc B rc  b v//2 1
v cB  2
 2

rc  b
2
2
Rc qB
Rc reinisch_85.511
qB m
Rc qB m
3
v//2
 v cB  
rc  b
Rc 
 3.43
The total drift velocity in a non-uniform B field is:
v B  v B  v cB
1

v//2
B  B
v B B    v rL

rc  b 
2
B
Rc 
2

rc and b are unit vectors.
Evaluate B in cylindrical coordinates:
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1

v//2
B  B
v B    v rL

rc  b 
2
B
Rc 
2

In cylindrical coordinates:
B 
B
1 B
B
r
φ
z where R is the radius of curvature.
R
R 
z
If we assume a cylindrical field, B =B  R  φ, then
B
B
B
 0 and
 0  B 
rc . Then

z
Rc
B


B

r
c
1

Rc
v//2
v B    v rL

rc  b 
2
B
Rc 
2





1

1
v//2
v//2
B b  rc
B 1
   v rL

rc  b     v rL

R B
Rc 
R B Rc 
2

2
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
 rc  b

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Ampere's law:   B  0
1   RB 
B
B
 B B 
B 
z
 z  0 

R R
R
R
 R R 
 1
v//2 
B 1
Substitute into v cB     v rL

 rc  b
R B Rc  
 2
2
1
v// 
B
1
2  rc  b
v B    v rL

.
 rc  b    v rL   v// 
Rc B Rc  
2
 Rc 
2
v
But rL 
or rL   v . Therefore

1 2
2  rc  b
v B    v  v// 
2
 Rc 
 3.44 
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3.6 Adiabatic Invariants
Hamiltonian mechanics , working with a generalized coordinate
q and its conjugate momentum p , shows for periodic motions that
the action remains invariant for slow changes (adiabatic)
in the system!!!! The action is defined as the integral over one or
several periods of the motion: J 
 pdq
For our gyromotion, a good coordinate is the azimuthal angle  ,
and the conjugate momentum is the angular momentum l  mv rL . Then
J   pdq 
2
 mv r d  2 mv r
 L
 L
 3.46 
0
1 2
mv
2 mv v
4 m W 4 m
2 mv rL 
 4 2


m


q B
q
  m  const First adiabatic invariant
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 3.53
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3.7 Magnetic mirrors
Let's look at a B field that converges in space as shown in Fig. 3.8.
Within a neighborhood r >> rL , the field can be considered
cylindrical around the central axis in direction z. Then
B  Br  z  r  Bz  z  z with Br  Bz .
From Maxwell's equation   B  0, and in cylindrical coordinates
1 d
dBz
B 
0
 rBr  
r dr
dz
dBz
rBr    r
dr
dz
1 dBz
Br   r
2 dz
 3.57 
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Assume a particle moves with velocity v // in the z direction, i.e.
parallel to the magnetic field. The magnetic moment  m remains
constant when the particles moves into larger B fields, from
B0 to B:
W0 W
v2 0 v2
B 2
2

, or

 v 
v0
B0
B
B0 B
B0
 3.58 
v increases proportional to B.
Can v increase indefinitely?? No. The total energy of the particle
1
1
m  v2 0  v//2 0   m  v2  v//2  .
2
2
increases, v//2 decreases until v//2  0  Mirror reflection!
is conserved: W 
When v2
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The reflected particle will go back to the point with B=B0 and
onward. If the field becomes stronger again, as in Figure 3.9, v //
decreases again until it reflects again: magnetic bottle.
v
The pitch angle  is defined as tan  
or
v//
2
v
W
2

sin 
 sin   2

where W = const.
2
2
2
v  v// W
v  v//
v
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B 2
sin 2
v2
B
From  3.58  : v  v0 . Therefore
 2 
2
B0
sin  0 v0 B0
2

 3.62 
Here  0 is the initial pitch angle at z  z0 . At reflection sin 2  1, or
B0
1
v2
B
2



sin


0
sin 2 0 v2 0 B0
B
If the max field strength is Bmax , then all pitch angles  0 for which
sin 0 
B0
are reflected (confined in the bottle).
Bmax
 3.63
Loss cone: discuss Fig. 3.10.
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Magnetic bottle bounce period
A charged particle in a magnetic bottle bounces back between
the mirror points. The time to move from the minimum at z 0 to the
reflection point z max is T 
zmax

z0
Tb  4
zmax

z0
Tb  4
B z 2
dz
2
and v//  v cos   v 1  sin   v 1 
sin  0
v//
B0
zmax

z0
dz
. The total bounce period is then:
v//
dz
v 1
Bz 2
sin  0
B0
See Table 3.2
 3.64 
Second adiabatic invariant:
JL 
zmax
 mv dz  4mv 
//
z0
B z 2
1
sin  0 dz
B0
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The Magnetic Dipole Field
Earth's magnetic field is approximately that of a magnetic dipole
(Fig. 3.11) with a magnetic moment M 'E (recall M' = I  A):


M E'
ˆ , i.e.,
ˆ
B  R,     0

2sin

r

cos

λ
4 r 3
ME
ME
BR   3 2sin 
B  3 cos  , and
r
r
ME
2
2
B  BR  B  3 4sin 2   cos 2 
r
M
B  3E 1  3sin 2 
r
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 3.67 
 3.68
13
Equation of magnetic field line
From  3.67  BR  
ME
2sin 
3
r
B 
ME
cos 
3
r
BR
 2 tan . Use stream line approach:
B
R

BR
dR
dR
dR


 2 tan .  
 2  tan  d 
B Rd 
Rd 
R
Re
0
ln
R
 2 ln cos   R  Re cos 2 
Re
Re is the radius to the field line at the equator. Set Re  L RE :
R
 L cos 2 
RE
 3.69 
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Variation of B along field line
ME
B  3 1  3sin 2 
r
R
 L cos 2 
RE
 3.68 
 3.69 
From  3.68  and using  3.69  :
ME
ME
2
2
B  R,    3 1  3sin   3 3
1

3sin

6
R
RE L cos 
ME
Set B  RE , 0   BE  3 , then:
RE
BE 1  3sin 2 
B  L,    3
L
cos 6 
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 3.72 
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Path element along fiel line
ds 2  dR 2  R 2 d  2
2
2
ds
ds
 dR 
2
2  dR 

 
 1 R 
 R ,
 , R  LRE
d
dR
 d 
 d 
dR
2
From R  LRE cos  
  LRE 2 cos  sin 
d
ds

 LRE cos  1  3sin 2 
 3.71
d
ds
1  3sin 2 

dR
2sin 
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