dc Magnetic field generated by a local source of the rf electromagnetic field in a collisionless...

Volume 131,number1 PHYSICSLETTERSA 1 August 1988

dc MAGNETIC FIELD GENERATED BY A LOCAL SOURCEOF THE rf ELECTROMAGNETIC FIELD IN A COLLISIONLESS PLASMA

V.1. KARPMANIZMIRAN,AcademicCity, Troitsk, MoscowRegion142092, USSR

Received20 April 1988; acceptedfor publication 18 May 1988Communicatedby V.M, Agranovich

It is shownthat a local rf electromagneticfield sourceproducesa demagneticfield asamagneticdipole. Its magneticmomentis found.If therf field is slowly modulated,thesourceradiatesadditionalwaveswith themodulationfrequency.In a magnetizedplasmathismayprovidea methodof MHD wavesgeneration.

Thegenerationof a dc magneticfield by alocalrf rot H=O, H=B—.4itM, divB=O. (2)electromagneticfield source (antenna)in a colli- Assumethat the if source is local. Thenfrom (2)sionlessplasmais of principal interestas an impor- ~tant nonlinear effect available for experimental 0 0W5

observation.It isof specialinterestinconnectionwith B = 4ItM—V ‘P+B°, (3)activeexperimentsinspaceplasma.In thispaperwe L~!P=47t divM, ~P(R)—~O (R—~O). (4)showthatat largedistancesfrom thesource,thegen-erateddc magneticfield hasa dipole character,i.e. From eq. (4) we havetheantennamaybe consideredalsoasa dc magneticdipole. Generalexpressionof its magneticmoment yJ(R)= — J divM(R ~dR’. (5)is found asanintegralof quadraticcombinationsof IR~Rthe if electric field producedby the antenna.If the The point R= R’ = 0 is supposedto be inside theif field is modulated,the antennain a collisionless source.Then,sinceM(R’) rapidlydecreasesatR’ >1,magnetizedplasmaactsas a sourceof MHD waves whereI is the effective size of the source,andusingwith frequencyequalto the modulationfrequency, the expansion

Thedensityof the dc magneticmomentproducedby the if electric field R—R’ I ‘=R~(1 +R~R’/R2+...), R’ <<R,

E(R,t)r=Re{E(R)exp(_iat)} we have (6)

is given by the expression[1] ~P(R)~mR/R3, (7)

(1) m=JdR’M(R’). (8)

where i, ji, y= 1, 2, 3, ~ap(W, B°) is the dielectric Thus, in the lowest orderof 1/R, the function ~P(R)tensorandB°is the externaldc magneticfield. Let is thefield potentialof adcmagneticdipolewith theBB°+B’ (for simplicity we considerB°=const) momentm definedby expression(8). Formulas(1),whereB’ is the dc field due to the media magneti- (3), (7) and(8) give a generalsolutionof theprob-zation describedby eq. (1). Neglectingthe dissi- lem of the dc magneticfield generationby an an-pation,we have~afi = E~a. tennaat largedistancesincomparisonwith thesource

The basic equationsfor the dc magneticfield are size. The next terms in expansion(6) lead to the

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Volume 131, number 1 PHYSICSLETTERSA I August 19855

momentsofhigherorder. It is worth pointingoutthat sphere),one canshow that B’ ~ B°[4]. Theneqs.theseresultsarevalid notonly for a plasma,butalso (10) and(11) maybelinearizedwith respectto B’.for any gyrotropic media.The presenceof the ex- In this approximation the last term in expressionternalmagneticfield is not necessary(if B°—s0,the (11) canbealsoneglectedandprojectionof eq. (10)quantityM definedby (1) may be finite). on the z-axisgives

If the antennahasa rotationalsymmetrywith re-spectto the external magneticfield B°which is as- p=p~+ —~— (�,~— ~)E~EJ~. (12)sumedto be directedalong the z-axis (e.g., electric 1 it

rf dipole, ring antenna,etc., orientedparallelto ~), Projectionof eq. (10) on thexy-planewith accountthen of eq. (12) and the equation

M:(r, z) =M(r, —z), M~(r, z)=—M1 (r, —z), div rotB’ =0

wherer=R1. From that and (8) we have m~=O, again leadsto expression(3). Therefore,relations(3) and (12)constitutethe necessaryandsufficientcondition of magneto-hydrostaticplasma equilib-

C , , rium in the rf antennafield with accountof pon-m=2ic I dz dr rM~(r,z ). (9).5 .5 - deromotiveeffects.

Finally, it is evidentthat if theamplitudeof the rfThus, in caseof rotationalsymmetrywith respectto field is modulatedwith frequencyQ~zw, thenm Os-

the z-axis, the dc magneticmomentis parallelto ~ cillates with the samefrequencyQ and the systemand its valueis definedby expression(9). For the maybeconsideredasa magneticantenna,generatingelectricdipoleantennathisresult waspreviouslyob- a electromagneticfield with thefrequencyQ. Thisistamedin ref. [21, where the r.h.s.of (9) wasalso of interest for active experimentsin the magneto-calculated, sphere. At Q<<w~,(w~~is the ion cyclotron fre-

It is useful to note that the expressionsobtained quency),i.e. for the MHD frequencyrange,the ra-above also follow from the equationof magneto- diation field of a magneticdipole hasbeenfoundinhydrostaticplasmaequilibrium with accountof the ref. [5]. The contributionfrom the term 4itM andponderomotiveforce producedby the rf antenna alsothe radiationdue to the striction effectswill befield. Thementionedequilibrium equationis of the consideredin a separatepaper.form

Vp+ ~— (BxrotB)=f, (10) References

[1] L.P. Pitaevski Zh. Eksp. Teor. Fiz. 39 (1960) 1450 lSov.wherep is kinetic pressureand f is the pondero- Phys.JETP 12 (1961) 10081.

motive force densityhaving the form [1,3] [2] VI. Karpman, Phys. Lett. A 128 (1988) 439

[3] H. Washimi and VI. Karpman, Zh. Eksp. Teor. FIz. 7!((aflóap)V(E~1~fl)+BXrothu4’+MaVBa. (1976) 1011 [Sov.Phys.JETP44 (1976) 528].l6it [4]V.1.Karpman,Zh.Eksp.Teor.Fiz.89(1985)71;90(1986)

(11) 1136 (E) lSov. Phys. JETP 62 (1985) 40:63 (1986) 662(E)l.

Considering,for simplicity, the low jiplasma(which [5] V.1. Karpmanand E.M. Maslov, Zh. Eksp. Teor. Fiz. 93is the case, in particular, in the earth magneto- (1987) 1696.

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