Al fv n Wing Electric Fields Generated by a Conducting Object
Moving Through a Magnetized Plasma
by
Barry Arthur ,l inger
SUBMITTED IN PARTIAL FULFILLMENT
OF THE RE QUIREMENTS FOR THE
DEGREE OF
BACHELOR OF SCIENCE
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 1985
Massachusetts Institute of Technology 1985
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Department of Physics
June 3, 1985
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9ervisor
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dberg
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2
Alfven Wing Electric Fields Generated by A Conducting Object
Moving Through a Magnetized Plasma
by
Barry Arthur Klinger
Submitted to the Department of Physics
on May 20, 1985 in partial fulfillment of the
requirements for the Degree of Bachelor of Science
Abstract
A conducting body travelling through a plasma in the presence
of a magnetic field will oroduce electrical currents which
will in turn radiate electromagnetic waves in the olasma.
Becausp the object's velocity is greater than the velocity of
some components of the waves, a shocked wing, analogous to
Cerenkov radiation, is formed. We find the electromagnetic
fields from the low frequency contribution, for which
magnetohydrodynamic (MHD) aoproximations are valid, to the
soectrum of the waves.
A cold plasma model is used in order to solve Fourier
transformed electrodynamic equations for the electric field
in terms of induced current in the solid conductor.
The
field is then inverse transformed to get the field as a
function of nosition and time for specific case of a large
cylinder, with its princioal axis normal to the plane
containing the ambient magnetic field B and the velocity v of
the moving object.
To simplify mathematical calculations we
consider only the case where v is perpendicular to B.
3
Introduction
In this paper,
magnetized plasma,
we calculate the electric fields in a
generated by an electrically conducting
object moving perpendicular to the magnetic field lines.
The
interaction of the object with the magnetized plasma causes
the object to emit a whole spectrum of plasma waves.
Our
calculations are restricted to the low frequency range of
waves, for which magnetohydrodynamic (MHD) approximations are
acceptable.
An observer in the reference frame moving with the
conductor with the veloci ty V, detects an electric field
E=vxB/c p erpendicular to both the object's vel ocity and
the direc ti on of
the magn etic field.
result in a stat ic charge
to
In a vac uum this would
seoaration within th e conduct or.
For a conductor travel 1 ing through a plasma,
however, the
charges are abl e to leak off, and for certain physical
situations, a c urrent loo p is completed outide the cond uctor,
allowing a curr ent to flow inside the conducti ng body.
The
current, moving through the plasma with the ob ject, car ri es
with it
the ele ctromagnet ic
the form of ola sma waves.
disturbance that p ropagates in
Because a magnetize d plasma is an
anisotrooic med ium, the velocity of the waves are stron gly
dependent on th e directio n of their propogatio n.
This
leads
to the possibil ity that t he conductor can move at a hig her
velocity than t he phase velocity of the Alfv'en mode
propagating at large angl es with respect to the direction of
the magnetic field lines,
and "wings"
analogous to Cerenkov
4
radiation are oroduced as the conductor drags its olasma
waves behind it
(see page 16 for definition of the Alfven
mode).
This effect was examined by Drell, Foley, and Ruderman,
in their 1965 study (ref. 4) of the Echo weather satellite,
to account for Echo's anomalously high orbital decay rate in
terms of energy loss from MHD waves.
It is now expected that
any large conducting structures in low earth orbit will
radiate these plasma waves.
Proposed space shuttle missions
using tethered satellite systems will involve conducting
cable tens or hundreds of kilometers long, and will generate
potential difference on the order of kilovolts.
This will
correspond to significant radiation of energy through plasma
waves.
The Drell-Foley-Ruderman
expl anation for Ec ho did
not
gain wide acceotance until in situ observations of the
magnetic
field perturbation (Acuna
et al.,
ref.
1)
and later
of the plasma flow perturbation ( Belcher et al., ref.
3,
Barnett, ref. 2) generated by Io (an electrically conducting
moon of Jupit er)
in the Jovian magnetosphere.
These
observations confirmed the existence of the "Alfven wing"
oredicted by Drell et al.
generated by
Thus not only are these waves
man made objects,
speciman of t hem also exists.
but a dramatic natural
The interaction of metallic
asteroids wit h oulsar magnetospheres may be another
illustration of the effect.
The way in which the current circuit is closed outside
of the conducting body is interesting.
Charge can flow
freely only along the directi on of the magnetic field, which
in the cases desc ri be d above leads down into the atmosohere.
There is a region in the uppe r atmosphere where there is a
high enough oarti cle density for some current to leak from
one field line to anoth er, though not yet enough neut ral gas
for charge carrie rs to recombine.
one side of the orbi tin g object,
to a lower altitu de,
Thus current can flow off
down a magnetic fiel d line
ho rizontally to another magnetic field
line, and up the fi el d line to the other side of the
conductor,
where it
Drell,
our oroblem.
et al
clo ses the circuit.
did
not evaluate the electric fiel ds for
The p uroo se of their paper was to show the
existence of Alfv en win gs and to derive an order of m agnitude
expression for the radi ated power.
In this paper, we find an
exoression for the elec tric field in the case of a
cylindrical condu ctor.
Like Drell,
et al.
we use a two component, singly
ionized cold plas na aporo ximation in our calculations.
cold plasma aparo
oressure gradient
tractable.
ximati
on
,
The
by neglecting the effects of, the
, makes the problem computationally
We re strict o urselves to the case of a
homogeneous, infi iite pla sma with a constant magnetic field
that is oerpendic ular to the (also constant) velocity of the
conductor.
In pri actice,
this is equivalent to assuming that
the scale length For chan ges in these parameters is much
larger than scale lengths for the conductor.
We also
b
restrict our study to the case of v << c A << c, where v is
the conductor soeed,
c is the speed of light,
and cA is the
Alfven velocity, a fundamental parameter of the plasma to be
defined below.
us,
Finally,
for the environments
such as the terrestrial
magnetosohere,
that interest
ionosphere or Jovian
we assume that the plasma particle density is
high enough so that the plasma frequency w
than the ion cyclotron frequency Q. (all
is much greater
of which are defined
bel ow).
Though many of the calculations will be applicable to
the whole soectral
range,
our final
valid for the very low frequency,
The plasma
W
=
results will
MHD domain (i.e., w << Q.).
frequency is defined as
4nee/me
(1)
n , e, and me are the number density, charge, and mass
where
of the electron,
respectively.
The cyclotron frequency is
Q = q P/(mc)
where
only be
(2)
B is the magnetic field and q and m are the charge and
mass of the species (either electron or ion).
Finally, the
Alfven soeed is defined as
cA
=
/ 4,tm n
where m is mass of an ion.
(3)
7
Fourier Transformed Electric Field
The derivation of the electric and magnetic fields for
the Alfv'en wings starts with the statement of the Fourier
transformed Maxwell equations, which lead to an equation for
the electric field in terms of a source current and
dielectric tensor.
These quantities are calculated,
utilizing the aonropriate MHD aoproximations,
and the
electric field, a function of frequency and wave number, is
Fourier inverse transformed to get the field as a function of
position and time.
The four dimensional Fourier transforms we use are
1
(
- c-)t) d3 x d t
F(x,t)e- i (k x
(4)
F(kdI)ei(k X--t)d 3kdo
(5)
When equations are transformed in this way
V operators are
F((,
)
-2
(2)
1
F(x,t) =
2
replaced with ik and 8/6t is replaced with -iw.
We combi ne
the Maxwell equations
VxE = -
-
VxB
=18E
=
(6)
t ++
4x<
(7)
in order to get
vx(?$xE)
1
1 2
2
=-g(6 /3t )E
-
(41T/c
2
Fourier transforming, this becomes
-. >
2-.
kx(kxE) + (w/c) E + (4tie/c 2 )j
with E,
j functions of (k,
(8)
=
0
) rather than o
(9)
current term may be divided up into a sourcce term
plasma term
j is
+
.
Equation (9)
The
(x,t).
becomes
js
and a
N
kx(kxE) + (w/c) 2 E + (4
i
)o/c2=
-(4 i/c)J
'p
s
We can define a dielectric tensor K such that
KE = E +
(10)
j(11)
To find an expression for K we will use the Fourier
analyzed Lorentz force equation to find the velocities (as a
function of k,w)
of the particles in the plasma,
we will derive is as a function of E.
from which
This information will
allow us to find K from its definition (for the following,
see Stix,
ref.
6).
The Lorentz force law for a particle is
q(E + vxB/c)
my
(12)
where v, m and q are the velocity, mass, and charge of the
I~
particle.
A
We orient the coordinate system so that B=B x 3
0
and the one vector equation (12) is equivalent to three
scalar equations
= (q/m)(E +v 2 P 0 /c)
(13a)
V2 =
(q/m) (E2 -v 1 P0 /c)
(13b)
v3 =
(q/m)E
(13c)
v
3
Fourier transforming,
v (,w)
= (iq/wm)(EI(k,r,)+v2 (kr)0/c)
v 2 ( kto) = (iq/wm)(E 2 (k ,W)-v1(kIto) B0/c)
= (iq/om)E
v3 kIo)
The first
3(
kw)
(14a)
(14b)
(14c)
two equations of this set can be rearranged to give
=
q -iwE1 + OE2
- 2 -
-
m
Q
-(15a)
-(
q QE1 + ioE
2
2 ~ ~ m
2
Q
-w
2
(15b)
9
Where, as defined above, Q=qP/mc.
j
The olasma current density
for the k t h soecies is given by
number density,
model
=
k k k , where nk
and qk = particle charge.
Note that for our
the velocity of the species is simply the velocity
already given in equations (13a)
to (15b); we are studying a
cold plasma in its prooer frame.
For a two component plasma,
jI,
(16)
nq ve + n iqiv ,
where the subscriots e and i refer
to electrons and ions,
resoectively.
Inserting (16)
and (15),
into (11),
and combining this with (14c)
we get
K=
S
iD
0
iD
S
0
0
0
P
(17)
with
2
W
S
1 +
2
(18)
2
e
P =1-
2
p
(19)
2
(20)
If
e i
eliminate
we
in (7) using (8), we see that (7) has the
form
AE
We want
=-(4-tioj/c2)
s
to solve equation (21)
of simol icity,
(21)
for
IEgiven j
.
For the sake
let k be entirely in the x -x3 olane,
with k,
10
=
2
ksin 0, k2 = 0,
k3 = kcos 9,
2S-k2cos 2 o
(
i(-) 2D
i( )D
A =
2
Then
k2sinOcosO
2-k
w )2 k 2CC
s inOcos 9
2
= ki + k2.
and kr
0
(22)
2p-k2 sin2e
0
C__
Thus we have
E = -(41tiw/c 2)A
(23)
s
where
bf
A~I
i (/c)
IA\
i(w/c) 2 Df
-bd
af-d 2
i(W/c) 2 Dd
2 Df
-bd
2 Dd
-i(w/c)
(24)
ab-(wD/c)2
and
a
with
(ro/c)
= det A = abf -
JA|
(w/c) 2 S
=
-
k2cos29, h
k2sin2n, and d = k 2 sin~coso.
define the
matrix M = IA IA
that k2 = 0.
To derive the
the case,
=
2f - bd2
(,/c)
(25)
S - k2
f = (W/C)2
_
Fo r conveni ence later, we
Origi nal ly we had assumed
elec tric fiel d when this is not
we use the rotation ma tri x
k
R
4f
= (1/k
-k?
0
1
0
k2
)
0
0
(26)
k
From now on denote the original A by A0 .
either E (or
js)
rotated frame,
_1 _1
RA0 R
in one frame by
we find that A-
Since multiplying
R gives us E (or J ) in the
in the rotated frame is A-
11
The Source Current
The soecific expression for js is dependent on the shape
of the conducting object.
There are some general attributes
of the source current that should be clear.
In the frame of
reference in which the conducting object is at rest (i.e.,
the plasma is flowing nast it),
stationary,
object.
the source current is a
time-independent current localized to within the
Transforming the electric fields from the olasma
frame to the (non-relativistic) object reference frame, and
denoting the object frame with a prime,
For a good conductor,
transformation x1 = x
E'=O.
we have E'
=
E-vBx/c.
We use the Galilean
- vt , x
= x2
x 3=x 3, and t'
=
t.
Figure 1 illustrates the orientation of the cylinder,
magnetic field, and coordinate system.
Fig. 1
--
Cylinder in a magnetic field
The specific system we will examine is a cylinder of
12
length 2a and radius b,
direction.
the x2
the axis of which is along the x
2
We assume that js is parallel
axis)
and is uniform.
jy
to vxB (i.e.,
along
Thus
< b
iSX')
(27)
0
> b
jy)
Fourier transforming Js, we get
1 2
(2'
n)
js (k,w)
(k-x-Wt) d 3xdt
is(*x t)e
(28)
5
changing variables from x to x' ,
52
e-i(Ik-x-[(-vk 1lt) d3 x'1d t
(2x)'
Because
fexp li(w-vk )tldt
is (k,W)
1
(2T)
=
2xn8(w-vk 1)9,
6(0-vk)
2
(29)
is
)eik x'd3x
(l'X
(30)
or
j
s(k,w) = 2n6(,-vkI)j'(k)
(31)
The integral is best done in a cylindrical coordinate system
in which r2
=
It
a
=
2 + x32
1
3t
k2
r
xb= tan~1(x3 /x 1 ), and
k /k1 ).
x = x2 2 + r( 1sin4
k = k2 2 + kr (
I 'x'
k2 + k 2
1
2'
=
x 2 k2
sin4k
+ x 3 cos$ )
A
(32)
x$ k)
(34)
+X3coS k )
+ rkrcos$
(33)
In these coordinates,
js ( kw
6(W-vk)
(2
3/2
Integrating first over x 2'
e(
CJfAe
k 2 x 2 +krrcos0)dedrdrcx(35)
13
=
js (k , ) thi issimla (U)-v k ) sin( ak2)
toan2
ntera
ik rcos4dtrd
this is similar to an integral expression for the
(36)
Bessel
function
J 0 ( z)
=
i zcos9d
16
(Watson,
ref.
48,
).
js (k , )
(37)
CI
Inserting this into (36)
7).
-
i)sin( ak2 )
2
=
JJ
0
( rk r) rdr
we get
(38)
Luckily ,
S
x
n+1
Sn(cx)dx
-
(Gradshteyn, o 683, ref. 5),
js( kw)
so
6 (w-vkI)sin(ak2
)J(bk r)
(40)
k k
Let the total current be
j
(39)
n+1 (c)
I
= T7b
2
J.
Then our expression for
is
2I
js(
k ,o)
Thb
J 1 ( bk )
k2 k
sin(ak 2 )8(fo-vk )
1
(41)
14
Determinant of A
Rotations do not affect the determinant of a matrix, so
det A = det
A0 .
As we shall
later see,
the form of det Ao is
n a r tic u 1 a r 1y simole in the MHD regime, allowing for simple
factori zati on.
Without any aooroximation,
det A = (')2 {k 4(Pcos2O+Ssi
4 SP 2 O () S
nn 22 9)(
0 -(
+[S 2 -D 2 1sin
=
where k2
2
2
2 0+-
C
where k
k 3 + k-
2 2
2Oi
) 2 k 2 ((PSF1+cos
P [ 1
o SC0
4 PS 2-D 2 )l
=kkI+k2.
+ 2
(42)
The determinant is
in k 2 so
3,
()2 PP k 4 +([1-S/Plk2-2(-)2S)
2_
' 2 k + k2
det A =
quadratic
+( )(rD_
(S2-_D 2)) I
2 ]/P-S)k2+(
(43)
Written in factored form,
2
det A = ( )p(k
c
1
wi th
=
A
1
() 2 -k2S
P2
k2 _ ()
c
A
2
(44)
k 2+A
k3 2
A
k2 (1- S/ P) ('[-C
S+ 1k2 ( (1-S/P )(
2
- 1)
(45)
(46)
1+E -1)
and
4, 2 PD 2((,02 P-c2 k2 )
E
=
c4 k4 (S-P)
In low earth orbit or
02 >> 9
,
(47)
2
for the Jovian environment around Io,
and v2 << (
e c/W )2 <<
iep
C2
in which case
E
<< 1
so that
A
1
=
-(k
P
2 _
c ) 2p)
P
(48)
and
A
2
2 = k
-
(W/c)2
(49)
In the MHD regime, we have further simplification.
begin by finding a simpler expression for S.
Let us
From the
15
definition of w' and P, we have
Q./Q
= m e/Mi
<<
1
(50)
The MHD aporoximation is valid for very low frequencies:
<<
.
i'
Since
.
Thus,
S1 =+
2
p
p
2/(
p
)
(51)
. P , we have
i e'
o 2 >>
S =
.0
w
.Q
).
ie
Similarly, P = 1
-
(52)
(w /)
2 and w2 << W,2
00
so our exoression
for Al becomes
A
=
("/c)2S
(53)
If we define a new variable with dimensions of velocity,
c A = c/ -s
(54)
then
C2W2
det A =
2
( 2" - c k ))(W2
(55)
- CA 2k2)
AA3
What is the ohysical signific ance of c A
The equation
det Ao = 0 is the condition for a solution to the homogeneous
equation AoE=O.
orooagat ion of
0),
and det Ao
and
".
Physically, t his co rresoonds to the
plasma waves th rough a sourceless medium (i
=
0 gives the disp ers ion relation between k
In this disoersion r el ati on, we get two solutions,
9.
But w/k = v ., the nhase
CA and w/k = ±cA cos
velocity of the wave; these so 1 uti on s of the dispersion
/k
=
±1
equation imoly
which v
p
other
two modes of hr op agat ion of the wave: one for
is ind eoendent of the
angl e of propagation,
for which v, varies with cos e
the
The former mode is
known as the "fast mode," the latter is the "Alfven mode,"
16
and c A is the "Alfven velocity."
calculate the group
It is instructive to
velocity of the Alfven mode waves,
for
this measures the velocity of signal or energy propagation
due to Alfven waves.
Grouo velocity is given by v
and w =
cA X3.
k3,
so v g
= Vk
'
Though the phase velocity of
Alfven waves can have components in any direction,
energy only orooaoate along the maqnetic
signal and
field lines,
with
velocity cA*
This leads to a compelling model
of Alfven waves as
disturbances of the magnetic field lines themselves,
an a1 )gous to waves oro oogating al ong
strings.
Indeed, in
magn etohyd rodynamics, the magneti c f ield lines can be said to
be "1frozen into"
the o lasma,
and the Alfven mode is ourely
tra n ;verse (no comores sion) like the transverse disolacements
on a stri n
This "string" model for Alfv6n waves is further
ill us trated by a look at the definition of the Alfv'n
vel oc ity.
Using our aoproximation for S, we have
= =A jc2 Q
/w2
For a singly ionized plasma,
(56)
ne = ni,
and since n m
= P
,
the i on density of the plasma (and approximately the total
densi ty,
2
cA
since the electrons contribute negligible mass),
B2 /4n
0
(57)
2
Bo/4'r is twice the magnetic energy density, or magnetic
tensi on
of the field lines,
while P is the mass density.
This is completely analogous to the formula for the wave
17
speed in a wire,
mass/length.
v2 = T/X , where T is tension and X is
Integrals for E(x,t)
Knowing E(k ,w) (at least in principle),
we can calculate
E(x,t) from the inverse Fourier transform (5).
( ,t) =
t)d3kd
i (k
A i
(58)
In the reference frame of a conducting object moving with
velocity V, the current density vector is not a function of
frequency.
density is
In the oroper frame of the plasma, the current
js (I,)
moving in the"x
1
=
2n5(
-v - k) j' (k).
direction, so js k ,w)
The conductor is
2n6(w-vki)3
(k)
,
h.e nc e
47Ti vk
E(,t) =
)2
S() e ('x-k
1 A-
1vt)d3k
2-n) 2c2
E(x,t) =
RMR1
4dik
v
1
(59)
( )ei(k-x-klvt)d3k (60)
c2detA
(2n)2
In the MHD range, as we saw, the poles of the denominator o f
our integrand have a oarticularly simple form.
Inserting the
MHD exoression for det AO into the Fourier integral
(60)
,
and
remembering that due to the delta function w = vk1 , we get
.,4ni
E(x,t)
c
-
The integral
2 v k RM~
2
(61)
k space for which k
R-
2
2
s k )vt)
2d
3k
(61)
for E(x,t) has poles at the points i n
=
Ak
3 /v
or k1 = +cAk/v.
oole is ourely imaginary for real k's because k
The latter
= k sin e,
so for the latter pole to exist sin 0 = cA/v, which is larger
than one.
19
The other poles are re levent, and their existence
suggests the possibility of
evaluating the k 3 integration by
means of a closed contour i ntegral
in the complex k 3 plane,
using the residue theorem t o find the integral on a path
around the ooles.
Considerations involvi ng minute corrections due to
dissipative processes
(e.g. , collisions) allow us to
determine which of the pole s is to be included in the
integration path.
We find that the nole at k 3 = vki/cA is
actually at k3 = vk1/cA + i E,
where
E is some small,
The other pole is at k 3 = -vkl/c A
n.umber.
nositive
-
Our k3 integral is
E (k)eik3
(62)
3dk3
From thi s we ca n complete a closed loon with a semi-circular
oath in either
k3 ol ane.
For
the uoner or lower half-plane of the complex
integration over this semi -ci rc le,
where we
exn(ia),
take the limit of
from either 0 to aT (a
E(k) is of order
> 0)
R +
,
le t k 3 = R
and whe re a goe s
or from 0 to - -n (a
< 0)
Since
1/k2 or lower, as 1ong as the exp ix 3 k3)
term is bounded, the contribution of the semic irc 1 e to the
total closed cont our integral is 0.
x 3 > 0, e xp(ix
If
for R
+
c
3k 3
) =
exn(-Rx 3 cOs
for the upoer semicircle.
we would choose t he lower semicircle .
a);
this go es to 0
Simi larly, for
X3
<
0,
In either case, the
residue theorem a nd Cauchy's theorem tel 1 us that if g( z) is
analytic,
2.0
g(z)
dz
(63)
= 20ig(z 0 )
z-zo
For
x3 >
0,
the role at vkl/cA + ie is inside the loop;
only
for X3 < 0,
only the pole at -vki/cA
the limit as
+
F
-'
iE
inside.
is
Taking
0,
E(klk2,k3) e ik 3x3dk3
2 v5 k5 RM~1
R
4ni
cA
We 2
c )
= 2,n i
s (k19k2 , k 1 sgn(x3)
C
2vk
gn
V2k2-c k2 )sg n (x3)
3
(64)
or
Ek,t)
1k x1 2
Cjs e
s
wk3
lk2 (1-S/P)P ok3
=
where C is the a matrix derived from RM~ 1R~
x1-vt+(v/cA
(65)
dk dk2
and x 1
x3'
As shown above,
only the x 2 comoonent of the vector
j
is nonzero for the case we are studying,, so only (C 1)12,
(C~
) 2 2 , and
frequencies,
C2
1 =
C2 2
=
C23
=
(C
1
) 3 2 contribute to the
integrand.
For all
the expressions for these are
2
(1-S/P)(k
21c
_
(1-S/P)(k2_
2c
2...
K2K3
2
P)k 1 k2
2P)k2
(66)
(67)
(68)
21
Evaluation for E(x,t)
The expressions in the MHD regime for the electric
fields due to our source are
E
1
s 1 2 e(
k2 +k 2
:
c2 A
=
01
1
2
-GO
2cf?
A_
E2
c
E3 =----v
As
sk 2
k 2+k 2
jsk 1
shown before,
2
i71k1+x2k
2Is
(70)
dk 1dk2
(71)
2
the source factor for a cylinder of length
F1+(v/c A)
~k1
1(bkI)sin(ak 2)
,Fb
(72)
k1k2
three definite integrals
All
2dk dk 2
(i1 k1 +x 2 k
2a and radius b is (kr= k +k 2
is
(69)
k 1+x2 k2)dk dk
can be calculated
Combining (69) and (72) , we have
analytically.
(bk
21 scA
bc
)sin(ak2 i (~
2 e
12
1 +k~
ik
+x k
k 1112
2
)
dkIdk
2
(73)
2
1
Seoarating the double integral and only taking the terms that
symmetric
with resoect to k
PI sc A
1~~~~~~~
dk
E 1 = nbc2
Using a table of integrals,
sin( ak 2 )sin( x2 k2 )
-
(Gradshteyn & Ryzhik,
bc
2
1 (bk
0
1
2'
( sin(ak 2 )sin( x 2 k2)
k
2
k1 2 +k 2
2
k1
n 715,
)
(74)
we find
-eS-
I (a-x 2 ) k 1-eI -
1(a+x 2 )k i)
1 4
k2 +k2
2=
IscA
dk =
and k
(75)
ref. 5) which gives us
-a-x 2 Ik1ea+x21k1)sin(~ 1kI)dk
(76)
22
Again turning to the integral tabl e(Gradshteyn & Ryzhik,
763, ref.
p.
5) ,
x-
21 CA
(1
Lb
= bc
(77)
b
j
or
E=
(2IscA/b 2 c 2 )R 1 (r2- r1 )
-2
X
1
b
(78)
where
1-r
2
Solvinq for ri
2
1 1-r
r 2
2
r2
)[-\,A +p -A
-
AZ+BS-A
where
co
=
21s cA/(bc)2
A
=
(x2 !a) 2 + b 2
B
=
2xl(x
(79)
2
and r2 , we find that
C sgn(x
E
(a+x )2
2 b2
-2
(a-x2 )
2
_
I
(80)
(81)
(82)
-x
(83)
a).
Similarly, for E 2 , we have
E 2=~ ~
21 cA k? S
2 J
bc
ki
bk ) sin(ak
k1
(
k +x k )dk
+ k2
1 dk2
(84)
Seoarating the k1 and k 2 integrations,
41 sCA(
E 2~
2
dk-J (bk
_bc k 1
)e
dk 2
1k
We first evaluate the integral in k2'
k2 sin(ck 2 )
dk
2
2
2
k1 +k 2
(Gradshteyn
= 2e
E
2-
t bc
-0
( b
(86)
1 sgn(c)
)e
1
using
- Ic k 1
& Ryzhik p. 406,
21c A
k2 si n(ak2 )cos( x 2 k
2
2
85)
k +k 2
ref. 5) leaving
1k
e- I(a-x
2
2 )k1
1
sgn(a-x 2
23
+e-
(87)
a + x 2) k 1I sgn( a+x2 ) dk
1
Since the integrand is symmetric, only the cosine part
exp( iikj
) contributes to the integral.
form of cosine,
exoonential
fe-c k
(bk
)(1/k 1 )dk1
1
1
of
Using the
and knowing that
(1/b)(
=
c2 +b' -c)
is true for c > 0, we can integrate (87)
2 2.
to get
2
1
.i,
(88)
1
+4lfa-x
2
2
2 +b2 -21 a-x a-) +
+ii1 )+22
-2 Ia+x
a+x2I+iR 1 ) 2 +b 2
22I
+
i
Ha
sgn(a+x 2)(
(89)
2 1 )]
We can rewrite (P9)
in terms of
the A's and P's defined
Since
above.
+
A+i P
the final
2
A-iP
+B2 +A
(90)
expression for E2 is
sgn(a-x 2 )
~t0
A +B +A-
+ sgn(a+x
2)
+ +
The final expression for E3
-4a/ ]2
(91)
has particularly simple form.
We
must evaluate the integral
fT
rT'N)sr
4--T5~ v
-)7brc
i)z
0
C).J
(9
a-)
which can be written
21s
E3=nb
cO
dk
d(bk )e
1k 1(2i)
f dk2
(e1
2+a)k
2
-e -i(x2 +a)k2 ).
(93)
Using the defi ni ti on of the delta function, we get
21 v
E-
+ib-)kx
1(2,)[6(x 1 (bk )ei
+a) l(2i)f
1
1 k dki (94)
24
or
161sv
E
3
=-
(x
f
+a)-6(x2-a)
j (bk I)sin(~x 1 k i )dk 1
Using the table of integrals,
(ref. 5, o. 730).
s in ( ns in~
nf
~b
I
we see that
( 1/b))
-2
b2
(J (bk1 )sin(i 1 k1 )dk
(95)
(96)
ncos(nt/2)
+
1(7
-b2)
n
1>b
We restate the other two components of E along with the
x3
comoonent:
E=
Cosgn(x
1
) 4
C0rsgn(a-x 2
E2 =
+ sgn(a+x 2)
-A_ -
\A
4f+
+ B_+A
-4a/-F
A ++B +A+
x
161 v
a( x2 +a) -a( x 2 -a )]
b
(98)
1
b 2 -i
E3
x, -
1
(99)
(
21 CA/(bc) 2
vt + (v/cA )x3
(I x 1 <b)
2
0
where Co=
(97)
-A+
.
A =(x
2.ra)
2 +b 2
-x 2
1I)>b)
B =2x,(x 2 ra)
and
xi=
Results
The first
thing we can discover
confirmation that a "wing"
from our results is the
is indeed an appropriate
description for the electric fields due to olasma waves from
the conducting object,
(ref.
4).
The
function of
x2
as first
described by Drell,
et al.
wing stems from the fact that E must be a
and
constant along the
This means that the fields are
xi
"V" shape of x -vt+(v/cA)x 3 /=constant, or
in the object' s reference frame,
along x +(v/cA)1x
3
I
=constant.
This make s the (x
which to plot
the electric field lines.
cylinders with a/b =
Note that for
circular; this
is ohysically
,x2 ) olane the most convenient one in
We do so for
1, 10/3, and 10/1 (see figures 2 t0
large distances, the field lines are nea rly
is the field due to a line dipole.
Thi s li mit
reasonable because the distant field is c aus ed
by the charge associated with the two currents induced in the
olasma
by the conducting object.
These currents lead to a
line dipole.
Figures (2 to
assumotion that js
) also illustrate the degree to which our
s
2 is valid.
Since
j
'
is in the same
direction as E' inside the conductor, we see that for a/b=1
or 10/3,
E and js are reasonably well
(but not exactly)
aoproximated by vectors in the x2 direction, while for a/b=10
the aooroximation is better.
E1,
E2,
and E (=
the E3 component)
E+E 2
,
Figures (
i.e.,
to
) are graphs of
ignoring the 8-function from
as functions of x 2 , olotted for our three
26
values of a/b and for
c=1.
x2
various x1 .
For
these olots we set
As can be seen from the analytic expressions for E, as
0.
oE I and E2
endc ao 0 f the cylinder
There is also a discontinuity at each
reflecting the' discontinuity in js
Suggestions for Further Study
The formulas we have derived for E detail the field due
to low frequency waves
in a cold pl asma.
In order t o get
complete expression for the electri c field, we woul d have
a
to
extend our calculations to include the comolexity o f high er
frequency waves.
the total
The imoortance of the MHD contrib uti on to
electromaqnetic field is related to the si ze an d
soeed of the object.
As noted earl ier, for MHD, w
Pecause the current is stationary i n the reference
the conductor,
the fields will
w=vk,
scale length of the object.
frame of
The major contr ibuti o n to
so k1 < 0 /v.
be in this range
05.
if
L > v/Q., where L is the
For low earth orbit, L is on the
order of 20 m.
Another useful study would be the calculation of E due
to the Alfven wings of a sohere.
Also,
one would like to
correct the expressions for E in order to take into account
the fact that the cold plasma model is only an aoproximation
to actual warm nlasmas.
An interesting exoeriment would be to measure the Alfven
wings oroduced by the space telescooe (to be launched in
1986) or due to a tethered satellite system.
The MHD
contributions calculated here could be compared to the
27
measured electric fields.
dw
I
I
I
I
I
I
A
I
I
I
I
£
I
T
-- I~g
4C
--
FF be2
04
( a= 10, bzlo)
6c)
g
I
a
I
I
__________________________________Xa~
40 -
1C)
i~-e
Fig
)
-V elj
(lu
30
F
1(X)
II
To
-100
I
-50
1
L1
F I I
I
I
Veld
-ie o r
i
I
I
U
.II
I
I
50
-
FV 4-
V
-
(a
1004.
=lo, b r 3
I
~U
I
~
I
I
I
I
XI
x
40-4-
30 +
,
I..
10
. . .. . . . .
.. II --3c
-3c)
FIJ.
K -
KFte
a
,
---
4Q
a
=
1
/O
.
0
0/b
-
0.5
0/b
*
0.2
0/r,
.
21b
107,
75
2
T
4
s
t 0
-40
,10
30
0/ b
4
R/b
5
40
50
100
j-o
-
-30
-10Is
to
-lot
0/b
E0 (x)
N/b -
/i/b
/b
2
0.75
I
.
R/b- 3
-.-
4
5
R,/b
R/b
-4o
L
-30
9.0
-e1I
I
30
4o
IL 0
u
to
-
t
40/
0
-
- o
S/b
-
4
0/b
0/b
3
I
-5
-1
6
-
,
(
E
fur-
.
0,
50
10r i5 v
R/b
R/b
R/b
-
1
F,(xx\
R/b
0.667
=
-
2
5/b
. 2.5
6/b
3
0.333
*
IL)
o) C 30 40
3o
40
10 -10
-40 -30
6c/b
O
0.333
*
4-= 0.667
6/b
R/b
6/b
6/b
R/b
=
=
1.5
2
= 2.5
I
*
-4
2/h
*
6/b
-
6
36/b - 3
-4~~30
j
j
1
- i
7-
+
E
-fio
y
4
.-
40
-30
to
aC)
~
40
5./= I
Ro2
+
R/b
3
a 0.75
R/b a
~
0.5
R/b
-
R/b
R/b
=
2
-
4
)
RibE
R/b
0
.
R/b a
R/b
0.5
. 0.75
R/b
R/b
-
1
R,/b
*
0
- 3
R/b-2
X/b - 4
V/b -
R/b
R/b
*
*
0.5
0.75
5
R/b -
R,b.
-
to
I
3
4'-
5'
Tdr-
JO
G~~/O,
-110
bz;
-IOU'7
IOl -'
1-0
35
References
1. Acuna, M.H., and F.M. Ness. Standing Alfven Wave Current System at Io:
Voyager I Observations. JGR:85]3-8522 30 September 1981
2. Barnett, A.S. In Situ Measurements of the P'lasma Bulk Velocity Near the
Io Flux Tube, submitted for publication 1984
3. Belcher,J.W., C.K. Goertz, J.D. Sullivan, M.H. Acuna. Plasma Observations
of the Alfven Wave Generated by Io. JGR 86:8508-8512 1981
4. Drell, S.D., H.M. Foley, M.A. Ruderman. Drag and Propulsion in the Ionosphere: An Alfven Propulsion Engine in Space. JGR 70:3131 1965
5. Gradshteyn, I.S., and I.M. Ryzhik.
Products. Academic Press, 1965
6.
Stix, T.H.
Table of Integrals, Series, and
The Theory of Plasma Waves.
McGraw Hill 1962
7. Watson, G.N. A Treatise on the Theory of Bessel Functions.
University Press, 1962
Cambridge