Analytical study of the magnetic field generated by multipolar magnetic configuration

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Journl of Physics: Conference Series PAPE OPEN ACCESS Anlyticl study of the mgnetic field generted by multipolr mgnetic configurtion To cite this rticle: M T Murillo Acevedo et l 16 J. Phys.: Conf. Ser. 687 13 View the rticle online for updtes nd enhncements.

1 Journl of Physics: Conference Series PAPE OPEN ACCESS Anlyticl study of the mgnetic field generted by multipolr mgnetic configurtion To cite this rticle: M T Murillo Acevedo et l 16 J. Phys.: Conf. Ser View the rticle online for updtes nd enhncements. elted content - Opticl ner-field of multipolr plsmons of rod-shped gold nnoprticles A. Hohenu, J.. Krenn, G. Schider et l. - Microwve Plsm Source for Ion Sources with Multipolr nd Axil Mgnetic Fields Hitoshi Nihei, Junji Morikw nd Nobuyuki Inoue - The frustrted ferromgnetic S = 1/ Heisenberg chin in mgnetic field How multipolr spin correltions emerge from mgneticlly ordered sttes Andres M Läuchli, Julien Sudn nd Andres Lüscher This content ws downloded from IP ddress on 1/9/18 t 1:49

2 IMMPT15 Journl of Physics: Conference Series ) 13 doi:1.188/ /687/1/13 Anlyticl study of the mgnetic field generted by multipolr mgnetic configurtion M T Murillo Acevedo 1,, V D Dugr-Zhbon nd O Otero 1 Universidd Snto Tomás, Bucrmng, Colombi. Universidd Industril de Sntnder, Bucrmng, Colombi. E-mil: oswldoterolrte@gmil.com Abstrct. The mgneto-sttics field from prllelepiped mgnet which cn turn round n xis, is the first step to find the whole mgnetic field in multipolr configurtion. This configurtion is present in the ion sources, which re heted by electron cyclotron resonnce. We present the nlytic formuls to clculte this mgnetic field outside the volume of the mgnet. To model the mgnet, we considered constnt mgnetiztion vector inside of mgnet volume. Therefore, the mgnetic sclr potentil method cn be used. We present the results by hexpolr system. Their mgnetic field components re clculted on confinement region, severl grphics re shown with directions nd mgnitude s grdients of the mgnetic field to help understnd better the confinement system. Our results re confronted with experimentl ones. These formuls re very useful in reserch of plsm mgnetic confinement in ion sources through computtionl simultions. 1. Introduction To confine plsm in n electron cyclotron resonnce EC) ion source, trnsversl multicusp mgnetic field is used. This field helps to remove mgnetohydrodynmics instbilities due to the convex curvture of mirror s mgnetic field 1. It is common to use six or eight prllelepiped mgnets round cylinder dischrge cmer to crete cusp geometric form of mgnetic field, which chnges the mgnetic field curvture. The nlytic clcultion of mgnetic field is useful in plsm dynmics computtionl simultions. Although severl models use multipolr pproximtion, the interction between the mgneto-sttics field of the trp nd the microwves field exerts the biggest influence over plsm behviour; therefore is very importnt to get better model to clculte mgnetic field. An nlyticl result ws published in 4 for the modelling of prllelepipedic mgnets of vrious polristion directions. However, we present the cse, when the polristion vector stys constnt, but the whole mgnet cn turn round n xis nd we solve ech indetermintion present in the formuls. Therefore, our formuls re more useful for clculte multipolr mgnetic field. The multipolr mgnetic field hs been importnt in plsm studying from microelectronics fbriction to the fbriction flt pnel disply device. This system of mgnetic field is used in some configurtions of mgnetic confinement 3. It strts from mgnetic sclr potentil 5, but its equtions re not solved, insted the mgnetic fields components re clculted by using grdients, trnsforming those equtions until they trnsform into integrble form. The finl equtions re using to clculte the mgnetic field into cubic mesh. The mgnet is considered s mteril with constnt mgnetistion vector inside it, nd zero outside. The equtions re solved only in confinement Content from this work my be used under the terms of the Cretive Commons Attribution 3. licence. Any further distribution of this work must mintin ttribution to the uthors) nd the title of the work, journl cittion nd DOI. Published under licence by Ltd 1

3 IMMPT15 Journl of Physics: Conference Series ) 13 doi:1.188/ /687/1/13 volume, which is found outside mgnets inner region. Severl pictures show the curvture nd mgnitudes grdients.. Mgnet modelling The mgnet is modelling by considering constnt mgnetistion vector inside their volume, it is oriented rdilly from pole towrd the opposite pole, s is showed in Figure 1. Since the interest region is found outside mgnet, the mgnetic field cn be clculted through the method of grdient of sclr mgnetic potentil 5 since, not conduction density current exists in this Mr ) Mr ) ds plce: φm r) = s v r r dv.where, φm is the sclr mgnetic potentil, v) r r sv) is the mgnet boundry surfce, v is the mgnet volume. However, the second term is zero since the vector M is constnt inside. Then the potentil eqution cn be chnged by: Z z Z x1 Z z Z x dx dz M dx dz M 1) φm r) = sin θ r r sin θ r r z1 x1 z1 x Where x, x, x1 nd x1 re the limits on X xis, s is showed in Figure 1. The z1 nd z re the limits on z xis. The limits to x xis cn be found in this wy: x1 = 1 cos θ sin θ nd x1 = 1 cos θ + sin θ for inner pole s top. x = cos θ sin θ nd x = cos θ + sin θ for outer pole s top. Z z Z xr dx dz M We use the follow nottion φxl, xr, θ, )r) =, therefore: φm r) = xl sin θ r r z1 φx, x, θ, )r) φx1, x1, θ, 1 )r). The integrl respect to z cn be solved, so we get: φxl, xr, θ, )r) = f xl, xr, θ,, z )r) f xl, xr, θ,, z1 )r), where: f xl, xr, θ,, zb )r) = M xr p nd ρ θ,, x ) = x x ) + xl ln z zb ) + ρ θ,, x ) + z zb ) dx sin θ y + x cot θ. sin θ Figure 1. eference system Mgnet br cross-section). Bx component φm Since Bx = we need to solve: x f xl, xr, θ,, zb )r) M = x sin θ Z xr xl x x ) dx p p z zb ) + ρ θ,, x ) + z zb z zb ) + ρ θ,, x ) We cn solve this eqution through two substitutions nd using n integrl tble for one of integrls, 6. Then we get the following solutions for Bx component:

4 IMMPT15 Journl of Physics: Conference Series ) 13 doi:1.188/ /687/1/13 B x = p, θ) Sθ,, z, x )r) Sθ,, z, x )r) Sθ,, z 1, x )r) + Sθ,, z 1, x )r) + p 1, θ) Sθ, 1, z, x 1 )r) Sθ, 1, z, x 1 )r) Sθ, 1, z 1, x 1 )r) + Sθ, 1, z 1, x 1 )r) + sin θ Dx, θ,, z 1, z ) + Dx, θ,, z, z 1 ) + Dx 1, θ, 1, z, z 1 ) + Dx 1, θ, 1, z 1, z ) sin θ x + cos θy sin θ ) x sin θ) If y sin θ, p, θ) = x sin θ + y sin θ ) cos θy sin θ ) x sin θ) cos θ If y sin θ = Sθ,, z b, x uθ,, x ) + ) q )r) = M rctn θ,, z b ) + u θ,, x ) + z z b ) sin θ lθ, ) ) z zl M ln If sin θ = 1 y sin θ = x b x =, z z Dx b, θ,, z r, z l ) = r ) Dt x b, θ,, z r ) M ln Elsewhere D t x b, θ,, z l ) D t x b, θ,, z b ) = q θ,, z b ) + u θ,, x b ) + z z b ) sin θ uθ,, x ) = x + cos θy sin θ ) x sin θ ) ) ) h θ, ) = x + y cos θ y x sin θ Then: sin θ sin θ q θ,, z b ) = z z b ) + x sin θ + y sin θ ) cos θy sin θ ) x sin θ lθ, ) = sin θ hθ, ) = x sin θ + y sin θ ) cos θy sin θ ) x sin θ B y component For B y component, we hve: B y = j, θ) Sθ,, z, x )r) + Sθ,, z, x )r) + Sθ,, z 1, x )r) Sθ,, z 1, x )r) + j 1, θ) Sθ, 1, z, x 1 )r) Sθ, 1, z, x 1 )r) Sθ, 1, z 1, x 1 )r) + Sθ, 1, z 1, x 1 )r) + sin θ sin θ cos θ Dx 1, θ, 1, z, z 1 ) + Dx 1, θ, 1, z 1, z ) + Dx, θ,, z, z 1 ) + Dx, θ,, z 1, z ) j, θ) = sin θ ) y sin θ sin θ + x sin sin θ θ cos θy sin θ ) cos θ sin θ If y sin θ, x sin θ + y sin θ ) cos θy sin θ ) x sin θ) sin θ cos θ cos θ Otherwise 3

5 IMMPT15 Journl of Physics: Conference Series ) 13 doi:1.188/ /687/1/13 B z component For B z component, the integrl is more simple; first we integrte respect to x vrible doing the previous substitutions nd then integrte with z : ) kθ,, z, x )kθ,, z 1, x )kθ, 1, z 1, x 1 )kθ, 1, z, x 1 ) B z = M ln kθ,, z 1, x )kθ,, z, x )kθ, 1, z, x 1 )kθ, 1, z 1, x 1 ) Where, kθ,, z b, x b ) = q θ,, z b ) + u θ,, x ) + uθ,, x ). esults for θ = or θ = 18 Since these formuls do not work when θ = or θ = 18, becuse in this cses we hve: r r = z z ) + x cos θ) + y y ). z yr M dy dz We use the follow nottion, φ y l, y r, θ, )r) = z 1 y l z z ) + x cos θ) + y y ). therefore, we hve: φ y l, y r, θ, )r) = f y l, y r, θ,, z )r) f y l, y r, θ,, z 1 )r) f y l, y r, θ,, z b )r) = M yr B x component with θ = ) y l ) ln z z b ) + ρ θ,, y ) + z z b dy B x = S θ,, z ), r) + S θ,, z, ) r) + S θ,, z ) 1, r) S θ,, z 1, ) r) + S θ, 1, z ), r) S θ, 1, z, ) r) S θ, 1, z ) 1, r) + S θ, 1, z 1, ) r) If x cos θ =, S θ,, z b, y b )r) = x cos θ M x cos θ rctn y y b + q θ,, z b) + y y b ) + z z b ) Otherwise x cos θ B y component with θ = ) q θ,, z b ) = z z b ) + x cos θ) B y = D θ,, ), z 1, z r) + D θ,, ), z, z 1 r) + D θ, 1, ), z, z 1 r) + D θ, 1, ), z 1, z r) ) z zd M ln z z D θ,, y b, z n, z d )r) = n ) Dt θ,, y b, z n ) M ln D t θ,, y b, z d ) If x cos θ = y y b =, otherwise 4

IMMPT15 Journl of Physics: Conference Series 687 16) 13 doi:1.

6 IMMPT15 Journl of Physics: Conference Series ) 13 doi:1.188/ /687/1/13 B z component with θ = ) B z = M ln k θ,, ), z k θ,, ), z 1 k θ, 1, ), z k θ, 1, ), z 1 k θ,, ), z k θ,, ), z 1 k θ, 1, ), z k θ, 1, ), z 1 k θ,, y b, z b ) = q θ,, z b) + y y b ) + y y b We present picture of mgnetic field from cubic mgnet of 4 cm length, with mgnetiztion vector of G of mgnitude see Figure ). To confront this results, we tke dt from Dexter Mgnetic Tecnologies 7. We tke cubic mgnet with θ = 7, 1 = in, = in, z 1 = 1 in, z = 1 in, nd = in; esidul induction Br) G= 5, Mteril type, Nd -Fe -B. We use point x p, y p, z p ) to clibrte the mgnetic field mgnitude, for exmple we tke M = 1 nd clculte the mgnetic field mgnitude in this point, through our formuls, clled B f x p, y p, z p ). Then we get the experimentl result clled B e x p, y p, z p ), then M = B e x p, y p, z p )/B f x p, y p, z p ), this wy we cn get the correct mgnetiztion vector, in this cse is G. We put both experimentl nd simulted results in Tble 1. Therefore these results help us to show the vlidity of our formuls. Now we present the results for hexpole system. Tble 1. Confront of dt. Coordintes in) B G) Experimentl B G) Simulted x y z B x B y B z B x B y B z Tble. Physicl scheme hexpolr system). 3. Hexpolr system The physicl system consists in six prllelepiped br mgnets plced on the outside of cylinder, in lternting North/South polriztions, whose poles re trgeted rdilly; this system is showed in Figure. The cylinder chmber rdius is cm, with length of 5.97cm. Longitudinl size mgnet br is 1.cm, the inner pole rdius is 1 = 4.1cm in Figure ; the externl pole rdius is = 7.1cm, width =.5cm. For hexpole field simultion we tke the following ngles θ = 3, 9, 15, 1, 7 nd 33. The field grphic on XY section plne z = cm) is showed in Figure 3; where the cusp curvture is seen ner to mgnetic pole zone. In this figure we cn see grdient rdil from the colour scle of field mgnitude towrd Z xis; this property is necessry to press the plsm bck inside cmer dischrge. The field on z = 1.cm plne See Figure 4) is like one centrl configurtion since it is inside region covered by hexpole system. The field on z = 1.5cm plne Figure 5) is more longitudinl, specilly in regions ner mgnet poles; since this zone is outside of hexpole region. The longitudinl field view on x = cm plne, is given in Figure 6, the mgnetic field becomes more longitudinl, outside of region covered by mgnets, in the zone ner to the mgnets. 5

IMMPT15 Journl of Physics: Conference Series 687 16) 13 Figure. Physicl system. Figure 5. XY view plne z = 1.5cm) of the mgnetic field. doi:1.188/174-6596/687/1/13 Figure 3. XY view plne z =.cm) of the mgnetic field. Figure 4.

7 IMMPT15 Journl of Physics: Conference Series ) 13 Figure. Physicl system. Figure 5. XY view plne z = 1.5cm) of the mgnetic field. doi:1.188/ /687/1/13 Figure 3. XY view plne z =.cm) of the mgnetic field. Figure 4. XY view plne z = 1.cm) of the mgnetic field. Figure 6. YZ view plne x =.cm) of the mgnetic field. 4. Conclusion The nlytic clcultion from prllelepiped mgnet, which cn turn round n xis is very importnt to reserchers on plsm physics under multipolr mgnetic field, for exmple, in plsm mgnetic confinement in EC sources, since hexpole mgnetic field configurtions re n importnt term to improve the plsm dynmics simultions. Using n pproximte vlue of mgnetic field cn cuse mistkes. Although these errors cn be smll, the plsm simultions cycles need to be performed mny times to rech the stbility of the system. Therefore the errors cn increse when the number cycles grow. These clcultions serve to help reduce computtionl effects over the plsm behviour simulted. eferences 1 A Dinklge T Klinger G Mrx nd L Schweikhrd 5 Plsm physics confinement, trnsport nd collective effects 1st ed Berlin: Springer-Verlg) p 148 K Kim, Y Lee, S Kyong nd G Yeom 4 Surfce nd Cotings Technology H Tkekid nd K Nnbu 4 Phys D Appl Phys vud nd G Lemrqund 9 Progress in Electromgnetics eserch J Jckson 1999 Clssicl Electrodynmics 3rd ed United Sttes of Americ: John Wiley & Sons Inc) pp H Bristol 1957 Tbles of integrls nd other mthemticl dt 3rd ed New York: The mcmilln compny) p 93 7 Dexter Mgnetic Tecnologies 15 Field clcultions for rectngle 6

Jackson 2.26 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell

Jackson 2.26 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell Jckson 2.26 Homework Problem Solution Dr. Christopher S. Bird University of Msschusetts Lowell PROBLEM: The two-dimensionl region, ρ, φ β, is bounded by conducting surfces t φ =, ρ =, nd φ = β held t zero