ESCAPING FROM EARNSHAW THEOREM

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1 ESCAPING FROM EARNSHAW THEOREM EMIL CAZAC, MIHAI MARICAR, ALEXANDR STĂNCILESC, MARILENA STĂNCLESC Key wods: Levitation, Stability aea, Diamagnetic mateials. A classical electodynamical esults known as Eanshaw theoem fobids the stable static levitation in stationay fields. Even though, pemanent magnets above supeconductos, the spinning top magnets and diamagnetic mateials do stable levitate. This pape is going to explain this loophole by showing how diamagnetic mateials escape fom theoem incidence by violating its tems. Fo bette analytical esults, the whole analysis is done ove a cylindical symmetic geomety. It is also estimated the tiny stability aea of the equilibium. A qualitative expeimental configuation emphasies the theoetical statements.. INTRODCTION Those who have studied levitation, chaged paticle taps, o magnetic field design fo focusing magnets have pobably un acoss Eanshaw s theoem [] and its consequences. Thee can be no puely electostatic levitato o paticle tap. If a magnetic field is focusing in one diection, it must be defocusing in some othogonal diection. Eanshaw s theoem applies to a test paticle, chaged and/o a magnet, located at some position in fee space with only divegence- and cul-fee fields. No combination of electostatic, magnetostatic, o static gavitational foces can ceate the thee-dimensional potential well necessay fo stable levitation in fee space. The theoem also applies to any aay of magnets o chages. At fist glance, any static magnetic levitation appeas to contadict Eanshaw s theoem. Thee must be some loopholes though, because magnets above supeconductos [], the spinning magnet top magnets [3], diamagnetic bodies [4] including oganic mateials, do stable levitate. This pape shows how the tems of this classical theoem can be violated by using mateials with special popeties which Eanshaw did not conside. A quantitative analye ove a cylindical symmetic geomety is also done and a qualitative expeimental configuation validates these theoetical esults. ) Polytechnic nivesity of uchaest, 33 Splaiul Independenţei, 64 uchaest, Romania, caacu@elth.pub.o, mm@elth.pub.o Rev. Roum. Sci. Techn. Électotechn. et Éneg., 49, 4, p., ucaest, 4

2 Emil Caacu, Mihai Maicau, Alexandu Stănciulescu, Mailena Stănculescu. TEORETICAL ISSES In ode to discuss the equilibium and stability of a igid body, we constuct its potential enegy and equie that the second spatial deivatives to be positive when the fist deivatives vanish [5]. The gavitational potential enegy is gav mg, taking the -axis as vetically upwads, while the inteaction enegy of a magnetic dipole m embedded in a magnetic field is [6]: mag m. So, the whole potential enegy of a magnetic mateial with volume V and magnetic susceptibility χ m is: ( χ V ). () gav mag mg m µ To balance the foce of gavity, we equie that: F ( χ µ ) ρgk m, () whee ρ is the mass density of the mateial to be levitated and k is the unit vecto in the vetical diection. This equilibium condition, while necessay, is not sufficient. Fo stability we must have positive cuvatue in the enegy suface in evey diection [5]. We can wite this condition as: x ; y ;. (3) A necessay condition fo stability can be obtained accoding to (3) equiing positive value fo the laplacian enegy: m. (4) ( χ V µ ) It is easy to demonstate that in a static magnetic field (iotational and divegenceless) the laplacian of flux density magnitude is always positive [7]. That means, as we can see fom (4), that mateials which have χ m <, can satisfy the stability conditions asked by (3). Diamagnetic mateials match this estiction and could stable levitate in a static magnetic field if the flux density gadient in stong enough to satisfy the equilibium condition (). Supeconductos, which ae pefect diamagnets (χ m ), fulfil this condition even bette than diamagnetic bodies and could also stable levitate in a static field. 3. EQILIRIM POINT AND ITS STAILITY AREA In this section, we ae going to estimate the equilibium point and its stability aea fo a vey small diamagnetic body placed on the symmety axis of a solenoid.

3 Escaping fom Eanshaw theoem 3 The equilibium points will be on -axis of the symmety. Then the condition that (, ) to be an equilibium point is:. k F m g χ µ ρ (5) The stability conditions in (, ) ask fo a minimum value of enegy, which means positive cuvatue function in evey diection: stability. adial ; vetical stability; ; (6) To complete the poblem, we expess the magnitude of the magnetic field in tems of its -component (, ) only. Taking into account that and, the following extension of (, ) aound (, ) deive: ( ) ( )( ). ) (, ; ) (, ; ) (, 4 '' ' (7) Accoding to the magnitude of the magnetic flux density extension fo the extenal magnetic field (7) and the expession fo potential enegy, vetical and hoiontal stability conditions given by (6) can be now ewitten:. ; '' ' '' ' D D h v (8) Fo achieving a stable static levitation of a diamagnetic body, the above condition, called disciminants of stability D v and D h, must be simultaneous satisfied. 4. QANTITATIVE AND QALITATE ANALYSIS As a model fo studying the stability aea of common magnetic field souce, we conside the field inside of a solenoid of length l and thickness bounded by

4 Emil Caacu, Mihai Maicau, Alexandu Stănciulescu, Mailena Stănculescu 4 adius a espectively a Fig.. The field inside this solenoid on its symmetically -axis is [6]: ( l ) ( l ) µ NI ( ) ln ln. l a a a a l a a a a a a a a Fo the equilibium position is necessay that () to be satisfied. This can be easily achieved by changing the cuent though the solenoid windings, which scales the magnetic field stength () while peseving the geomety of the field lines. Theefoe the stable ones ae detemined by (8). It is impotant to undeline that the values fo cuent and tuns of the coil do not establish the stability aea, but only thei altenation can fulfil the equilibium condition (5) fo a specific diamagnetic mateial. It is easily to anticipate that levitation will occu in the end of the coil, in the most inhomogeneous field egion (the maximum field gadient), athe than in the cente of the solenoid, whee the field is almost unifom. (9) Fig. Geomety notation. Fig. The disciminants D v and D h. The vaiation of stability disciminants D v and D h ove -axis is pesented in Fig. fo a coil with l cm, a cm, a 8 cm and NI 5 Asp. Fom this vaiation we can pedict the stable egion, whee both disciminants ae positive. It is impotant to notice that thee ae two stability aeas whee the equilibium is stable. oth ones ae at the magins of the coil and have the same depth. The stable egion fo the above mentioned field souce is limited between.79 cm (D v () ) and.54 cm (D h () ) having a width of d coil.33 cm. Altenating the coil geometical data, this elative naow stability one can be enlaged. Fig. 3 and Fig. 4 show the altenation of stability aea due to the vaiation of adius a fom 4 cm to cm, and length fom cm to cm.

5 5 Escaping fom Eanshaw theoem Fig. 3 Stability aea via adius a. Fig. 4 Stability aea via length l. The stability aea can be inceased using a coil with a lage adius ove its length. nfotunately, this cannot be taken into account as a easonable solution fo enlaging the stability aea because, using this paticula geomety fo the magnetic field souce, the equilibium condition (5) is vey difficult to satisfy. The electic cuent needed fo obtaining levitation (even fo best diamagnetic mateials) would have vey high values that will damage (themal o dynamic) such magnetic field souces. A solution fo this poblem is the usage of a diffeent geomety fo the magnetic field souce that allows a lage stability aea fo levitation and also peseves the magnetic field values fo equilibium condition. 5. EXPERIMENTAL CONFIGRATION To show a the effect of diamagnetic levitation Fig. 5 pesents the suspension of a vey thin pyolytic gaphite (χ m 45-6 ) disc by a magnetic field souce of fou NdFe magnets (. T). This configuation is commecially a available at w.w.w.klangspiel.ch. Fig. 5 A disc of pyolityc gaphite suspended by a goup of NdFe pemanent magnets.

6 Emil Caacu, Mihai Maicau, Alexandu Stănciulescu, Mailena Stănculescu 6 6. CONCLSIONS The poblem of stable static levitation in stationay field is teated both qualitatively and quantitatively showing how the usage of special magnetic mateials (diamagnetic o supeconductos) can escape the configuation fom Eanshaw theoem. The equilibium and stability conditions ae analyed fo a specified configuation detemining the location of equilibium point and its stability aea. Fo the specific levitation aay a numeical simulation was also made pesenting the quantitative aspect fo these developed devices. Fom this examination we achieve excessive tiny values fo the stability aea that can be enlaged by using special magnetic field souces along with stong diamagnetic mateials. An expeimental model of diamagnetic levitation that qualitatively validates the theoetical esults is also exposed. The application of this pemanent magnet levitation can be found in vey high-sensitive gavity sensos o in designing fictionless suspension whose paametes (such igidity) can by contolled by adjusting the field pofile. Taking into account that most of oganic mateial ae diamagnetic, this kind of levitation can be inteesting fo sciences as biophysics, biology o biotechnology. Received on Octobe 5, 4 REFERENCES. M. F. Reusch, A poblem elated to Eanshaw s theoem, IEEE Tansactions on Magnetics, 3, 3, pp , May M. C. Maion-Pea, J. P. Yonnet, Axial beaings using supeconductos and pemanent magnets, IEEE Tansaction on Magnetics, 3, 3, pp. -4, M.D. Simon, L.O. Heflinge and S.L. Ridgway, Spin stabilied magnetic levitation, Ameican Jounal of Physics, 65, p. 86-9, E. Caacu, Stable magnetic levitation in stationay field using diamagnetic mateials Rev. Roum. Sci. Techn. Electotechn. et Eneg., 47, 3, pp. 7-77, uchaest,. 5. A. Temple, J. ickley, Static equilibium, Claenton Pess, Oxfod, A. Moau, aele electotehnicii Teoia câmpului electomagnetic, Editua Matix-Ro, ucueşti,. 7. M. D. Simon, A. K. Geim, Diamagnetic levitation: Flying fogs and floating magnets, Ameican Jounal of Physics, 87, 9, pp. 6 64, May.