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Class notes for EE5318/Phys5383 – Spring 2002
This document is for instructional use only and may not be copied or distributed outside of EE5318/Phys 5383
Single particle motion
Plasmas, by there very nature, exhibit collective behavior. Let us assume that we have a single
electron in a plasma which we can move at will. If that electron is not in the system, the rest of
the electrons and the ions will spread out in an approximately even distribution across the
system. (Note that this is even distribution has a lot of qualifications. For example walls greatly
disturb the charges… We will get to this more later.) Under this situation the distribution of
charges might look like:
Ions
Electrons
Now let us add our electron (noting that we could as easily have done this with an ion):
Ions
Electrons
Because our electron tends to push the other electrons away and draw the ions toward it we find
that we have rearranged our distributions. If our test charge now has some motion to it, we find
that all of the charged particles in a local area respond to this motion. This is a collective
behavior that is a requirement for our system to be in the plasma state. In general it is the
collective behavior that is most important to understanding how a plasma operates.
Unfortunately it also is fairly intensive mathematically. On the other hand, we can develop an
understanding of the plasma if we look at single particle motion in the plasma. (We just have to
remember that the rest of the charged particles move in response to the motion that we are
modeling and hence we change the E and B fields on our test charge.)
Lorentz force
The equation of motion for a single particle of charge q and mass m is given by the Lorentz force
dv
 q E  v  B .
law: m
dt
While this equation can be quite complex, for complex fields, it is often easiest to look at the
cases when the external fields are simple.
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Class notes for EE5318/Phys5383 – Spring 2002
This document is for instructional use only and may not be copied or distributed outside of EE5318/Phys 5383
Case 1: E  Ezzˆ; B  0
dv q
 E  v  B
dt m
q
 Ezzˆ
m
Integrating gives
x  x0  vx 0 t
y  y0  v y 0 t
Letting E  E 0 ; B  0




q
2 
Gives r = r0  v0 
Et


2m 0 

q
E t2
2m z
This is quite simple and not terribly informative.
This would imply that the charged particle accelerates to infinite velocities. In real systems the
electric field does not extend to infinity and collisions tend to slow the particle down. (Run away
charged particles do occur – eventually running into walls – resulting in the production of Deep
UV and x-ray production in some plasma systems.)
z  z0  vz0 t 
Cases 2: E  0; B  B0zˆ
dv q
 E  v  B
dt m
q
 v  B0 ˆz
m
Thus we find
dv x q
 v y B0
dt
m
dvy
q
  vx B0
dt
m
dvz
0
dt
To solve this set of equations, we must separate the components of the velocity. This is simple
to do by differentiating the equations again and substituting to give.
d 2 v x q dvy
q 2


B    B0  v x
dt 2
m dt 0
m
d 2 vy
q dvx
q 2


B    B0  vy
dt 2
m dt 0
m
2
d vz
0
dt 2
These second order equations are of course are easily solved as
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Class notes for EE5318/Phys5383 – Spring 2002
This document is for instructional use only and may not be copied or distributed outside of EE5318/Phys 5383
v x  v x0 e  ic t
v y  v y0 e i c t where  c 
q
B
m 0
vz  vz 0
Now taking into account the original coupled first order equations we find
v x  v 0 cos c t   
v y  v 0 sin  c t   
v z  v z0
Integrating a second time we find,
v
v
x  0 sin  c t     x 0  0 sin  
c
y
v 0
c
c
cos ct     y0 
v 0
c
cos 
z  vz 0 t  z0
rc 
v 0
c
It is easy to see that rc 
v 0
is the radius of a circular orbit around the magnetic field line; it is
c
known as the Larmor radius or the cyclotron radius. Further we note that the positively charged
particles orbit in a left-hand orbit while the negatively charged particles orbit in a right-hand
orbit.
Example
Of particular interest is the magnetic field required to give an electron a gyro-frequency of 2.45
GHz. This is of interest because it is required to understand Electron Cyclotron Resonance,
ECR, plasma sources. (Given time at the end of the semester, we will discuss these sources in
detail.)
fc 
c
2
q

B
2m 0
 2.8E6 B0 (Hz / Gauss)

B0  875G
Case 3: E  E zzˆ; B  B0 ˆz
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Class notes for EE5318/Phys5383 – Spring 2002
This document is for instructional use only and may not be copied or distributed outside of EE5318/Phys 5383
dv q
 E  v  B
dt m

xˆ yˆ zˆ 
q 
 Ezzˆ  vx v y v z 

m 
0 0 B0 


or
dv x q
 v y B0
dt
m
dvy
q
  vx B0
dt
m
dvz q
 E
dt m z
This is easy to solve as we have already done this as parts.
x  rc sin  ct     x0  rc sin  
y  rc cos ct     y 0  rc cos 
z  z0  vz 0 t 
rc 
q
Ez t 2
2m
v 0
c
Case 4: E  E x xˆ ; B  B0 zˆ: Note that E  Ey yˆ would also work here
dv q
 E  v  B
dt m

xˆ yˆ zˆ 
q 
 Ex xˆ  v x vy vz 
 or
m 
0 0 B0 


dvx q
 E   c vy 
?
dt m x
dv y
  c v x

?
dt
dvz
0
 vz  vz 0 ; z  v zt  z0
dt
The third equation is easy to solve while the first two are more difficult. Differentiating gives
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Class notes for EE5318/Phys5383 – Spring 2002
This document is for instructional use only and may not be copied or distributed outside of EE5318/Phys 5383
d 2v x
  c2 v x
2
dt

v x  vx 0 e  i c t
d 2v y
q

2   c  E x   cv y 
dt
m
qB E

  c  0 x   c vy 
 m B0

E

d 2 v y d 2 
Ex 
  c2  x  vy  but
 so
2 
2 vy 
B0

dt
dt 
B0 
vy 
Ex
 vy 0 e i c t
B0
Ex
B0
Plugging these into our initial equations gives

dv x q
E 
 Ex   c v y0 e  ic t  x 
dt
m

B0 
v y  vy 0 e i c t 
  c v y0 e  ic t
 i cv x 0 e ic t

v y 0  iv x 0
Let v   iv y 0  v x 0 giving
v x  ve  ic t  v cos ct   
 i c t
v y  iv e

Ex
E
 v sin  c t     x
B0
B0
v z  v z0 ; z  vzt  z0
This means that the particle travels along the as it would with just the magnetic field but it also
have a drift in the E B direction. We can calculate this in general. First the average force is
F  qE  v  B  0
Therefore
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Class notes for EE5318/Phys5383 – Spring 2002
This document is for instructional use only and may not be copied or distributed outside of EE5318/Phys 5383
F  B  qE  B   v  B  B  0

E  B    v  B  B
 B   v  B
 v B• B  B v • B
Now if there is no drift along B then we get
E B
v 
2
B
What we have described above is true in general. Assuming that we have any constant force that
is a right angle to the magnetic field.
Ftotal  B  0  F  B  qv  B  B

v drift 
F  B
qB 2
Case 5: E  0; B  B0  r • B  ... : Non - uniform magnetic field
Here we look at a magnetic field that is non-uniform in space. The Taylor series expansion of
such a field will be of the form
B  B0  r •  B  ...
This should be straight forward from
Bzy  Bz0  y y Bz ...
Now from Lorentz’s Force Law we have
dv
m
 q E  v  B  qv  B
dt
or in the y-direction
dvy
m
 qv x Bz
dt
 qv x Bz0  yy Bz  ...
(We can do the same thing in the x-direction.)
The force averaged over one gyration is
dv y
m
 q v x Bz0  yv x  y Bz  ...
dt
The first term is clearly zero. The second term is not as


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Class notes for EE5318/Phys5383 – Spring 2002
This document is for instructional use only and may not be copied or distributed outside of EE5318/Phys 5383
yv x  rc cos ct     y0  rc cos v 0 cos c t   
 rc v 0 cos  c t    - all of the other terms are zero
2
1
rc v 0
2
We can now plug this into our average force equation to get
dv y
1
m
  q rcv 0  y Bz  or in general
dt
2
1
  q rcv 0 Bz
2
1
1
 mv20 Bz
2
B
In the x-direction this becomes
dvx
m
 q vy Bz0  yv y  y Bz  ...
dt
but
=


yv y  rc cos ct     y0  rc cos v 0 sin  c t   
 rc v 0 cos c t   sin  c t   
=0
Thus, we have from above a drift velocity
F B
v drift   2
qB
1
B  Bz
rcv 0
2
2
B
1
1 B  Bz
 mv2 0
2
B B2
This leads into a topic known as magnetic mirrors. Magnetic mirrors are naturally occurring
phenomena that happen at the magnetic poles of planets and stars. In laboratory-based plasmas
magnetic mirror are used in some process systems to confine the plasma. (They were also used –
quite unsuccessfully - as a confinement mechanism for fusion plasmas.)

In a mirror the gradient of the magnetic field is parallel to the direction of the field lines. This
sort of arrangement is known as a cusp field and looks like the figure below. (This is the
geometry one finds with permanent magnets.)
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Class notes for EE5318/Phys5383 – Spring 2002
This document is for instructional use only and may not be copied or distributed outside of EE5318/Phys 5383
z

From Maxwell’s Equations we have
 •B  0
 - in cylindrical coordinates
r -1 r rBr   z Bz   0 - or
r -1 r rBr   z Bz 
Integrating over r and assuming  z Bz  is not a function r gives
  rB dr   r B dr
r
r
z
rBr  
z
r2
 B 
2 z z r 0
r
Br   z Bz 
2
r 0
Now from Lorentz,
F  qE  v  B

0


Fr  qv Bz  v z B   qv Bz


F  qv z Br  vr Bz 
 0

r
Fz  qvr B  v Br  qv Br  qv z Bz 


2
r0
Now
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Class notes for EE5318/Phys5383 – Spring 2002
This document is for instructional use only and may not be copied or distributed outside of EE5318/Phys 5383
v  v x  v y
2
2
 v
and
v
r  rc  
c
has to do with the direction of .
where  c 
q
B0
m
Thus
Fz  qv 
 q
rc
 zBz 
2
2
v
 z Bz 
2 c
  12 mv2
1
z Bz 
B
 z Bz  where   12 mv2
1
B
 is the magnetic moment of a particle gyrating around a point. This can be easily seen from
  IA - where I and A are the current and area
q
=  c rc2 
 2 


2
qv 
2 c
1
2
mv2
B
What does this mean?
As a particle moves into a region of increasing B, the Larmor radius shrinks but the magnetic
moment remains constant. (This is shown in a number of books, including Chen.) Since the B
field strength is increasing the particles tangential velocity must increase to keep  constant.
The total energy of the particle must also remain constant and thus the particle velocity parallel
to the magnetic field must decrease. This causes the particle to bounce off of the ‘magnetic
mirror’. (There are still ways for some of the particles to ‘leak’ through the mirror. This is one
of the major reasons that magnetic mirrors did not work in the fusion field.)
Additional drift motions
There are a number of additional drift motions that occur for single particles. We unfortunately
do not have time to cover these drifts. The other drifts include:
Curved B: Curvature drift
Non-uniform E
Curved Vacuum fields
Polarization Drift.
Most of these are covered in the main text or in Chen.
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