Volume 128, number 8 PHYSICS LETTERS A 18 April 1988

de M A G N E T I C F I E L D G E N E R A T E D BY AN ELECTRIC D I P O L E A N T E N N A IN AN A N I S O T R O P I C P L A S M A

V.I. K A R P M A N IZMIRAN, Troitsk, Academic City, 142092 Moscow Region, USSR

Received 20 January 1988; accepted for publication 15 February 1988 Communicated by V.M. Agranovich

dc magnetization generated by an ac electric dipole in a magnetized plasma is considered. The main result is the expression of the induced dc magnetic field in a system with axial symmetry. It has a general form valid for any ac source with the same symmetry.

1. Basic equations

An ac electric field can magnetize media; the cor- responding density o f the magnetic moment M is given by the expression [ 1 ]

M y = 1 0E.p . 16~ - ~ Ec, E~ , (1)

where a , fl, ~,= 1, 2, 3, E,~ is the dielectric tensor and B is the dc magnetic field. In its turn, the induced magnetic moment M is a source o f an additional dc magnetic field. In the present paper we investigate this effect produced by the ac electric field o f a di- pole antenna in a cold collisionless plasma. The problem under investigation is o f interest, e.g., for active experiments in space plasma.

The expression for the tensor e,~o in a cold mag- netized plasma can be written in the form

~.p=e( d ~ - B . B p / B 2) + q B . B ~ / B 2

- ig6o~yBr/B, (2)

where 5,a~ is a completely antisymmetric tensor with ~123-~ 1,

e = l + ~ 2 2 2 ~o~j/(co~,-~o ), ~ = 1 - y~. o~j/o~ ~ , j .,

g = _ ~ 2 2 og pjog ~j / og ( OJ cj - W 2 ) , (3) J

and o9 w, og~j are plasma and cyclotron frequencies o f

species j. In this paper we assume eq < 0 which is the condition o f the existence of the resonance cones where the ac field and, respectively, the effect under consideration may be large. Let B = Bo + B' , where Bo is the external magnetic field and B' is a dc addition due to the ac field produced by the antenna. In a small-fl plasma B' << Bo [2 ]. Then eq. ( 1 ) gives (for

Bolr~)

1 Mx - 8nBo Re{ ( q - e )E*xEz + igE*Ey } ,

1 My = ~ Re{ ( t / - E)E~Ez - igE*zEx} ,

1 Mz=

16rt

R f & × el~-~o(lExl2+lEyl2)-2ioO--~ffoE*Ey}.

Evidently,

O~ O~ Ogcj Og O~ ¢ncj OBo = ~ &ocj Bo ' OBo - ~ Oogcj Bo "

Below we assume that the antenna axis is parallel to Bo and thus the problem is axially symmetric. There- fore, we pass to cylindrical variables and the above expressions take the form

1 M r = 8~Bo R e { ( r l - ~ ) E * E z + i g E * E ~ } , (4a)

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Volume 128, number 8 PHYSICS LETTERS A 18 April 1988

1 M~- 8nBo - - - R e { ( q - ~ ) E ~ E z - i g E * E ~ } , (4b)

, {0, } M~=~6~xRe ~ o ( I E r +[E~,I2)--ZigE*E~, .

(4c)

In particular, for the ac field of the electron range frequencies (o9>> (O~ceWci) 1/2 with O9ce2 << O9p¢2 ) expressions (4) turn into the formulas of ref. [ 2 ] (where the signs in front o f g must be changed).

Consider now B'. In the axially symmetric case it satisfies the equations [ 2 ]

div B' - 1 0 OB'z (re'r)+ -bT =0, (5)

OO'z OO'r 4~(Omz Omr.~ Or Oz - \ Or Oz / ' (6)

OB~ = 4n OM~ Oz Oz " (7)

The solution to these equations with the condition B' ( R ) ~ 0 for R--,~ (R2=r2+z 2) was obtained in ref. [ 2 ]. After some transformations it can be writ- ten in the form

B'r=4nMr-Og2/Or, B'z=4~Mz-Og2/Oz, (8)

B~ = 4nM~ , (9)

g2(z, r) = - ~ r d~ - ~ Q - l / 2 ( x ) - - 0

O (pM~)), (10a) X(~(pMr)+ ~

where Q_ 1/2 (x) is the Legendre function of the sec- ond kind and

x = [ ( z - i f ) z + (r-p)2+2pr]/2pr. (10b)

Considering the integral in (10a), it is useful to take into account that

Q_l/2(x)~(2x) -1/2 (X>> 1) . (11)

2. The ac field of an extended electric dipole

As an important example, consider as a source of the ac field an electric dipole parallel to the z-axis with the charge density given by expressions

p(R, t)=Pqll (z)q± (r) cos cot, (12a)

2lz qll ( 2 ) - - ~;(Z2_.l._12)2 ,

a 2

q± ( r ) = l~2(x2+a2)(y2+a2 ) , ( 1 2 b )

where 2l is an effective length, 2a a width and p the dipole moment. It is assumed that

l>>a . (13)

I f l- ,0, we have qll(z)--,-O'(z), ql(r)--, O(x)O(y), and the field E is infinite on the reso- nance cone

7lz[ = r , ~2-~- - E / q > 0 . (14)

At finite l and a, the resonance cone turns into a conic layer of a width l (cf. ref. [ 2 ] ). We assume that l is small enough. Then E, being large inside the layer, sharply decreases outside it. The field E is large also in the near zone of the antenna. Thus, one should expect that the induced dc magnetic field is large in the regions of the resonance cone and the near zone and, therefore, we consider here these two regions. Fortunately, one can use a comparatively simple quasi-static approximation there.

Now, compute E for the charge distribution (12) in the quasi-static approximation. At l= 0 we have (see, e.g., refs. [2,3])

E o = - P O v Q - ' ( z , r ) , (15)

Q(z, r)= (9,2z2-r2) l/2 , (16a)

arg Q = 0 (9'1z[ > r ) ,

a r g Q = - ( g / 2 ) signq ( 9 ' l z l < r ) . (16b)

Note that eqs. (16) define Q(z, r) only for real z. In the following, it is also convenient to introduce the function S(z, r) defined in the whole complex plane of z as the analytical continuation of Q(z, r) from the region z> r/9' with the cut at

- r / 7 < z < r / y . (17)

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Then, outside region (17) for real z we have

S(z, r) =Q(z, r) (z>r/y) ,

S(z, r ) = - Q ( z , r) ( z < - r / ~ , ) . (18)

In region (17), the function Q(z, r) is the limit of S(z, r) taken from the upper (lower) half-plane for r/<0 (~/> 0).

Now, the field at l•0, a ~ 0 can be written as follows

( E ( z , r ) = - p l O v f d r ' q l ( r ' ) l ( z , l r - r ' , ) ) nrl~, Oz

(19a)

where

l(z, r)= i dg ~-i--~ Q - l ( z - ~ , r) . - o o

(19b)

From eqs. (19) one concludes, in particular, that

E± (z, r) = - E ± ( - z , r),

Ez(z, r ) = E z ( - Z , r) . (20)

Therefore, it is sufficient to investigate the case z> 0. To do this, we rewrite (19b) in the form

d~ S-~ (~, r) I(z, r)= (z_~)2+l 2 C

- - r l y

+2 f d~ (z_~)2 +l 2 Q- ' (~ , r) , (21)

where S((, r) is defined in (18) and contour C is shown in fig. 1. It rounds the cut in the upper (lower) half-plane of ( for q < 0 (r/> 0). Now we close the in- tegration path in the first integral of (21 ) by the half- circle with R ~ in the upper (lower) half-plane for r/< 0 (r/> 0 ) , to obtain

I(z, r) = (n/l) [72 (z - i l sign r ] ) 2 - - r 2 ] - 1 / 2 - - r / ~

+2 f d~ ( z_~)2+l 2 Q- ' (~ , r ) . (22) --ocz

Substituting (22) into (19a) and assuming in the first integral a- ,0 , we have

E= (Tp/~l)V{z[y2( z - i l sign rl)2-r z ] -3/2)+BE,

(23)

f Fig. 1. Integration contour Cin expression (21) for q<0.

where

~E(z ,r )= 4pl v ( ~ dr' ql (r ')

× Q - l ( z - ¢ , I r - r ' l ) ) ,

i d¢¢ (~2+/2)2 ~min

(24)

and ( m i n = Z - { - I r - r ' l / 7 . One can see that both terms in (23) are finite at all real z and r, and at

R>>I (25)

the term fiE is small in comparison with the first term in expression (23). In the following we restrict our- selves by condition (25) and, thus, omit fiE in (23). Now, taking into account relations (20), we have for zX0 Er ~ (3p?/q)zrO. -5 , (26a)

Ez ~ - (PT/?l) (2y2gZq-r2)0-5 , (26b)

E~ = 0 , (26c)

O_ = [yZ(z_il I )2_r2 ] ,/2, (26d)

l~ = l sign (r/z) . ( 2 6 e )

Therefore, under condition (25), the expression for the electric field of the extended dipole, parallel to the z-axis, may be obtained, approximately, from that

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of the infinitely small dipole (eq. ( 15 ) ) by means of the substitution z - , z - i l~ in Q(z, r). A similar results has been obtained for the ring antennas in refs. [4,5 ]. Far off the resonance cone, i.e. at ]y2zZ- r2 [ >>l 2, expressions (26) approximately coincide with ( 15 ). The region l y2z 2 - r21 < l 2 may be considered as the resonance conic layer for the extended dipole. The field in the resonance layer decreases as R -3/2 which is slower than in the near zone outside the layer, where E ~ R - 3.

3. dc magnetic field generated by ac electric dipole

Substituting expressions ( 26 ) into (4), one obtains

M r ~ 3( l+y2)TZp 2 zr (2y2zZ+r2) (27a) - 8rcBoq I QI 1o

Mz~- 972p2 de z2r 2 16nl/20Bo 1011o, (27b)

3~2gp 2 lr M ~ 87tBo~ 2 1(21 ~o [2~2(22+12) - r 2 ] sign(qz) ,

(27c)

10~12~ [ (y2z2_r2)2 + 2~z12(72z2 +r2) ]1/2 (27d)

These formulas are valid under condition (25) for z,~0. From them it is seen that the components of M reach their maximum values on the resonance cone and sharply decrease with the distance from it. At ~[ z l = r one obtains

9( 1 +72)7p2 sign (qz) M ~ - 8 n B o ~ / (2fl)Sr , (28a)

9p 2 0E 1 Mz~ 16nq 2 OBo (2yl)Sr ' (28b)

M~,~ 3~'gP2 sign lqz) (28c) 16nBoq 2 (2y/)4r 2"

Note, also, that according to general relations (4) and (20)

M r ( z , r ) = M r ( - Z , r ) ,

Mz(z, r ) = M ~ ( - z , r) ,

M~(z, r) = - M ~ ( - z , r) . (29)

Now, to compute the dc magnetic field, it is ne-

cesary to calculate the potential £2 (z, r) from ( 10a ). We restrict ourself to large distances from the source. Since M(( , p) rapidly decreases at (2q_p2>> l 2, the main contribution to f2(z, r), in the region (25), comes from the domain (2 q_p2 << R 2. Therefore, the variable x in (10) may be considered as large and one can use the asymptotics ( 11 ) which gives

Q-1/2 (x) ~Tt(pr)1/2( 1 + z( /R2) /R .

Taking into account (29), we obtain

$'2,.~#z/R 3 , (30)

# = 2 ~ d~ dppMz(~ ,p) . (31) --c '~ 0

Expression (30) is the potential of a dc magnetic di- pole with the moment #, parallel to the z-axis.

Thus, the dc magnetic field B' (z, r), generated by an ac dipole, consists of two parts. The first one, 4~M(z, r), is the contribution from the local mag- netization of the plasma. The second part has the form of the field produced by a dc magnetic dipole with the moment (31 ). Evidently, this result is gen- eral: it is true not only for the ac electric dipole, but for any local ac source oriented along the z-axis, pro- vided M(z , r) satisfies symmetry conditions (29). In particular, this theorem is valid for the ac mag- netic dipole as well as for the ring antennas, perpen- dicular to the external magnetic field Bo. Generalizations of these results to arbitrary orien- tation of the ac source will be considered elsewhere.

In the cases of the ac electric dipole, the value of the induced magnetic moment # may be estimated by means of expressions (27). The result is

# ~ (9pZ/r12~ 3 ) (Oe/OBo) (27l)-3 (32)

Thus, the dipole part of B' increases with decreasing of the source length l as 1-3. Comparing eqs. (30) and (32) with (28), we conclude that the term 4rim is the largest one inside the resonance cone layer. Outside it, the largest is the dipole term, i.e. -Vg2.

Finally, we remark that if the amplitude of the ac source is slowly modulated with a frequency o9' (o9' << o9), then # is also modulated and the system ma be considered as a magnetic dipole antenna gen- erating a electromagnetic field with the frequency o9'. For to' << ogci, i.e. for frequencies o9' in the MHD

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range, the field o f such magne t i c d ipoles has been f o u n d in ref. [6 ]. O f course, in a d d i t i o n to this field one should take in to accoun t a field p r o d u c e d by the t e rm 4riM. Th i s will be cons ide red in a separate

paper.

References

[1] L.P. Pitaevski, Zh. Eksp. Teor. Fiz. 39 (1960) 1450 [Sov. Phys. JETP 12 (1961) 1008].

[ 2 ] V.I. Karpman, Zh. Eksp. Teor. Fiz. 89 ( 1985 ) 71; 90 ( 1986 ) 1136 (E) [Sov. Phys. JETP 62 (1985) 40; 63 (1986) 662 (E)].

[3] A.A. Andronov and Yu.V, Chugunov, Usp. Fiz. Nauk_116 (1975) 79 [Sov. Phys. Usp. 18 (1975) 343].

[4] V.I. Karpman, Fiz. Plasmy 12 (1986) 836 [Sov. J. Plasma Phys. 12 (1986) 480].

[5] V.I. Karpman, Phys. Lett. A 117 (1986) 73. [6] V.I. Karpman and E.M. Maslov, Zh. Eksp. Teor. Fiz. 93

(1987) 1696.

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