The motion of charged particles was examined in the case of a homogeneous magnetic field together with an orthogonal electric field which has a gradient opposing voltage parallel to the electric field. Two regimes result: in one of these, the particles' rate of gyration is changed from the conventional gyrofrequency; in the other, acceleration of the particle takes place. Applied to a plasma, the theory predicts new electric currents orthogonal to magnetic fields

Cole, K. D.

English

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X-621-75-27
VE X B EFFECTS ON CHARGED PARTICLE MOTION
K. D. Cole*
February 1975
*On leave from La Trobe University, Bundoora Vic., Australia 3083
GODDARD SPACE FLIGHT CENTER
Greenbelt, Maryland
VE X B EFFECTS ON CHARGED PARTICLE MOTION
7-
K. D. Cole*
NASA/Goddard Space Flight Center
Greenbelt, Md. USA 20771
ABSTRACT
The motion of charged particles is examined in the case of
a homogeneous magnetic field B together with an orthogonal
electric field E, which has a gradient VE parallel to E. If
B 2 q 2  qVE
>0
m 2  m
the particles drift at right angles to E and B with a modified
gyrofrequency and produce a current in that direction. If
B 2 q 2  qVE
- <0
m 2  m
the particles not only drift in the direction of E X B but are
also accelerated in the direction of E, in which direction
they also produce a current.
*On leave from La Trobe University, Bundoora Vic., Australia 3083
iii
VE x B EFFECTS ON CHARGED PARTICLE MOTION
INTRODUCTION
The motion of a charged particle in orthogonal electric (E) and magnetic (B)
fields is considered in the case where VE is non-zero and parallel to E.
Two regimes of interest emerge. In one of these particles rate of gyration
is changed from the conventional gyrofrequency. In the other, acceleration of
the particle takes place. Applied to a plasma the theory predicts new electric
currents orthogonal to magnetic fields. The theory has wide application and
pertains to the outer magnetosphere of the earth, the solar atmosphere and to
many astrophysical situations of interest.
A MODEL
For a model calculation let the magnetic field be homogeneous and parallel to the
Ox axis of a Cartesian frame of reference (x, y, z). Let E and VE be parallel
to the Oz axis, and suppose
E = E + (VE)z (1)
where E o = constant and VE is assumed to be constant over the region of interest.
-o
Consider the motion of a charged particle of mass m and charge q in this field
configuration. The equation of motion is (in emu)
dv
m - = q[E o +(VE)z] +qvX B (2)
dt
In a frame of coordinates given by X = x, Y = y - E0 X Bt/B 2 , Z = z the equation
of motion yields
d 2 Y dZ
m - = q - B (3)
dt 2  dt
and
d2 Z dY
m -- q(VE)Z- q - B (4)
dt 2  dt
It follows that
d2Z VE
dt + (2 - - = -oC (5)
1
Wvhere
C= -\ dt
and
Bq
m
Case I
W2  qVE >0 (6)
m
Let
S22 = 2 qVE (7)
m
The solution of Equation 5 in the original (x, y, z) system becomes
u = U1  (8a)
E \ Ec 2 2  W
v = o + 1 W 1-- + - cos&2 +- w sing2t (8b)
B B 2 22 t
w - - B sin 2t + w' cos t (8c)w-
together with
x = u1 t (8d)
w2 E E W2 w2
t+ - sin + w(1 - cost) (8e)
wi
z =- sin 2t- - (1 - cos 2t) (8f)
where v = (u, v, w) and (ul , vi, w1 ) are the components of velocity at time t = 0
at the point (0, 0, 0). Equations 8 reduce to the well-known results when VE = 0
i.e., 2 = o.
2
It is seen that the rate of gyration of the particle is changed by the amount
qVE m -1 , being decreased from o to 92 when qVE is positive, and increased
from o to 92 when qVE is negative. This effect should change the characteristic
frequencies involved in propagation emission and absorption of waves in a plasma
in the magnetic field particularly at low frequencies. This effect will not be de-
veloped further here.
It is noted that Equations 8 imply perturbations of the ion-velocity distribution.
The original ion velocity distribution function f 1(u', v', w 1, 0) is modified to
become
)(ul, v1 , w 1 )
f(u, v, w, t) = fl (uv' v 1, w', O)
a(u, v, w)
In Case I,
)(ul, v', wl) 02) W2
=1--J cos O2t + -
a(u, v, w) 2 2+
Oscillations of the ion-veolcity distribution at the new frequency 92 will take
place. Also the usual E0 X B/B -2 drift is modified (see Eq. 8b) to yield the
mean drift of a particle"-along the Oy axis
o2 Eo 
2
v + V (9)2 B - 22
In the frame of reference moving with the drift velocity vD along the Oy axis,
the particle performs elliptical gyrations described by
Y* = v - sin Ut - - cos 2t
Z* v - 1 cos 92t + w' sin 92t
The sizes of the two axes of the ellipse are
v _)2 C,4 02 12
vI _ - ---+
B 6 24
and S22/C2 times this value. The ellipse is rotated about the Ox axis by an amount
tan-1 - wi 9
It follows from Equation 9 that for an assemblage of particles in a plasima, dif-
ferential drift occurs for particles of different w and different v1 . The effect of
v 1 could be expressed as a temperature effect in this case. In a uniform neutral
plasma consisting of one species of singly charged positive ions and electrons
this drift will cause an electric current in the y direction given by
yz=o =ne + - (ve) - 10)Jylz=0 B i2 s22 , 2
provided both 2i and 2 e > 0. In some situations, (v) and (ve) may be expected
to be zero. This would be the case if the velocity distributions were Maxwellian
before time t = 0 at which time the electric field might be assumed to be "switched
on. "
Some interesting consequences of Equation 10 follow by considering the direction
of VE with respect to that of E. If VE is positive i. e., in the direction of Oz,
then
Gi < o i and &2e > We
Consequently the current given by Equation 10 flows in the direction of Oy in re-
gions of positive E and the opposite direction in regions of negative E. On the
other hand if VE is in the opposite direction to Oz the current flows in the nega-
tive y direction in regions of positive E o and the opposite direction in regions of
negative Eo . These consequences can be stated as follows: that the current flows
in the direction +VE X B according as VE is in the same or opposite direction to
E. The current may appropriately be referred to as the VE X B current.
If boundaries impede the flow of this current a secondary electrostatic field
would be set up which would cause a component of plasma drift in the Oz for
similarly directed E0 and VE and in the negative z direction for oppositely di-
rected E and VE.
Case II
In the case of extremely large values of VE we may have
2  VE <0 (11)m
Let
2 - qVE _ 2
m
4
The solution of Equation 5 in the original system (x, y, z) is
u = uI  (12a)
v =  , + v+ - -+2 cosh t + w1 - sinh t (12b)
B B2 o2
w =-- v - ) sinh Ot+ w cosh Ot (12c)
B
together with
x = ult (12d)
[v 2 E E\ 02
Y- v - sinh t + - wl(cosh Ot - 1) (12e)
Wo
z - sinh t - vi - (cosh t - 1) (12f)
where again (ul, v 1, w1) are the components of velocity at time t = 0 at the point
(0, 0, 0). Again Equations 12 reduce to the well known results when VE = 0 i.e.,
0= LW.
It is observed that the particle does not gyrate but is continually accelerated in
the electric field. The original velocity distribution f(u 1, vw, wi, 0) of a group
of similar particles would be modified to become
f(u, v, w, t) = f(u1, v1, w , 0) 1 + - cosh Ot - -
o2 02
A group of similar particles will contribute the following amounts to electric
current in the Oy and Oz directions respectively.
n-eE w? 2 2
B coshq51
JBi(x 2, zt) - i 2
(13a)
+ (w!) sinh Oit
nieEo oi (i
i(x, t) - - sinh it + (v1) sinh it + (w) cosh Oit  (13b)B i i
5
Where Eo vi and w i are defined at time t = 0 at the point (0, 0, 0) and x y z aire
the mean coordinates
S= (x) = (u1)t (13c)
W 2 CO 2 t O 2 E °  W
-=(y) = (V1 ) + -2 -2 sinh 0 t - - t - (w ) (cosh Ot - 1) (13d)
( 2 o3 t 2 B o2
(w') o E o w
z = (z) = sinh Ot - (vl) (cosh Ot - 1) + - (cosh Ot - 1) (13e)
o~2 B o2
Suppose that the plasma is uniform, singly ionized, collisionless and neutral
and that both 0i and g2e are real. This will be shown in a later section to be a
useful special case. Further suppose that (v1) and (w') are zero for both ions
and electrons up till time t = 0 at which time the electric field is "switched on."
It follows from 13 and 9 that the total current after time t = 0 at the point (x, 3,
z, t) is given by
neeEo  02
j(x, , z, t) e (cosh ¢it - 1) - (14a)y B [0 2  22
neeE
o oijz(xo, , , t) = - sinh ¢it  (14b)
B 0i
In the rare event that both 0, and Oe are real the current is given by
n eE 2 2)
_o _i e (cosh et - 1 (15a)jy(R, y, z, t) = B - (cosh Oit - 1) (cosh et - 1 5a)
e
neeE° i °e (j(xz [- sinh it - -- sinh Oet (15b)
B i e
In order then to calculate the distribution of current at some time t the position
of the origin of the particles of the current can be found indirectly from Equations
13d and 13e. This gives the value of Eo which must be used to calculate jy and
jz in Equations 14a and 14b. It is assumed in this calculation that the original
electric field configuration is maintained. In practice this may be caused by the
flow of charged particles along the magnetic field to and from some reservoir
of charge outside the region of interest. The currents jy and Jz may be appro-
priately referred to as the VE X B and V E currents respectively.
Suppose that, to a sufficient approximation, the magnetic field lines are equipo-
tentials of the electric field (E) and suppose a constant gradient of E exists over
6
a dimension L. The total potential change is then
S V EL 2  (16)
2
The values of L which satisfy the condition for acceleration to take place are such
that
L < B- 1 12m AD/q|'A (17)
An alternative way of expressing the condition is given by
L < (E)B - 1 m (18)B -
B q
where E is a characteristic value of the field within the system. In words, the
inequality 18 states that acceleration becomes possible when the scale of the
system becomes less than the gyroradius that one would associate with a particle
of speed equal to the well known drift speed (E/B). A general condition for the
operation of the acceleration mechanism is
q 2 B2
qVE > - (19)
m
The applications of this theory to structures in the magnetosphere and plasmas
in other planetary and astrophysical settings are being investigated by the author.
7


Effects of opposing voltage and magnetic fields on charged particle motion

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