A modifid xprimnt of Orstd Dimitar G STOYANOV Faculty of Enginring and Pdagogy in Slivn, Tchnical Univrsity of Sofia 59, Bourgasko Shauss Blvd, 88 Slivn, BULGARA E-mail: dgstoyanov@abvbg Abstract Simpl quipmnt for dmonstration of Orstd s xprimnt is suggstd in this articl Th offrd horizontal planar coil of wirs allows to b obsrvd a vigorous dflction of th compass ndl (mor than 8 dgrs) during transmission of currnt up to 1 A As a rsult from th thortical analysis an analytical dpndnc of th xprssion for th torqu on th magntic ndl is obtaind This dpndnc rads th fild non-homognity and th magntic ndl form PACS: 15My, 755 Db, 843 Hh Ky words: magntic fild, Biot-Savart law, xprimnt of Orstd 1 ntroduction n 18 Christian Orstd noticd a compass ndl was dflcting from its normal north-south dirction if it was nighbouring on a wir through which an lctric currnt was flowing That was th first proof that an lctric currnt producd a magntic fild as it flw through a wir Th dmonstration of this fact nds th us of a DC powr supply guaranting th flowing of currnt of 1- A This is th rason th dmonstration of this classic ffct to b ralizd rlativly difficulty Thrfor w st ourslvs th task finding a way for a modification of Orstd s xprimnt in ordr its ralization to b asir using mor accssibl mans and powr supply Modification of Orstd s xprimnt 1 Magntic fild of finit wir W considr a straight finit mtal wir with a lngth L lying on th y-axis of th Cartsian coordinat systm (Figur 1) Th wir is situatd symmtrically in rspct to th origin of th coordinat systm Through th wir a constant lctric currnt is flowing possssing magnitud and dirction coinciding with th positiv dirction of y-axis Th flowing currnt is producing a magntic fild in th wir surrounding n a point, lying on z-axis at a distanc r z from th origin of th coordinat systm, th vctor of magntic flux dnsity (magntic induction) B is dirctd in th positiv dirction of x-axis (in this
cas it mans prpndicularly to th drawing in figur 1 with a dirction from us to th list) Th vctor magnitud of th magntic flux dnsity is drivd (obtaind) from Biot-Savart s law [1-3]: Figur 1 A magntic fild of finit wir with flowing currnt B x = 4 π + L / zdy ( y z ) L / + 3 / (1) Th solution of th intgral is givn as B x = πz 1 + 1 z L () Th first multiplir in th right sid of quation () is th fild of straight infinitly long wir [1]: B ( ) = x πz (3) Th scond multiplir is a corrction of th final lngth of th wir This multiplir is incrasing function of L/z, which strivs asymptotically for th magnitud 1 whn L / z At L / z = 1 and L / z = th multiplir has valus 986 and 995, rspctivly Thrfor w may maintain at L / z th fild producd by th finit wir is qual to th sam producd by th infinitly long wir at accuracy of,5 % For xampl, if wir with lngth cm z = 1 cm w can us a
Horizontal plan coil Th nxt stag of th concpt ralization is th cration of a horizontal planar coil of wirs with ovrall dimnsions cm and 3 cm This coil crats conditions for rpatdly passing of th wir through th invstigatd spac, which in this cas is th middl of th middl bundl of wirs This is similar to a coil with a particular winding configuration Figur a Unvn coil Figur b Evn coil f th minimum configuration shown in Figur a is usd, thn it is possibl an unvn numbr of wirs passing through th middl bundl of wirs to b ralizd But if th shown in Figur b configuration is usd an vn numbr of wirs passing through th middl bundl of wirs is ralizd Th facility of working with vn coil is that th currnt input and output ar situatd on th on sid of th coil Th shown windings ar charactrizd with th fact that th magntic filds which th sid-lin wirs produc in th middl of th middl bundl of wirs mutually compnsat bcaus of thir symmtry, i th sctions AB and GH, BC and FG, and CD and EF nutralizd thir filds ach othr Th horizontal coil was mad as in a list of dilctric matrial (a list of plastic matrial in this cas) hols with diamtr 8 mm wr drilld Th position of hols is prsntd in Figur 3 Th wir was projctd on th list as th hols playd th rol of supporting and partitioning lmnts Th middl bundl of wirs was situatd undr th list of dilctric matrial whil th compass was situatd abov th dilctric list in its gomtrical cntr Thus th list dtrmind th minimum distanc btwn th compass and th wirs 3 Dscription of th xprimntal stand and xprimntal data n th particular cas w prfrrd an vn winding as a variant For this purpos a coil consisting of 3 sctions ach containing 4 turns was mad Thus th numbr of wirs through th middl bundl was N = 4 A rctifir a DC powr supply with input voltag up to 15 V and maximum currnt A was usd Rsistor of 15 Ohm with ratd output powr W was srial connctd as ballast to th coil
Figur 3 Th ralizd horizontal planar coil of wirs During work th planar coil was put on horizontal plan as th middl bundl of wirs was orintd to th dirction north-south dtrmind by th compass, and th currnt through th coil was turnd off Th magntic fild producd by th middl bundl of wirs B at th pointd abov coil orintation had a dirction prpndicular to th horizontal componnt of Earth s magntic fild B Whn th currnt through th coil was turnd off th compass ndl pointd north, i θ = Whn th currnt through th coil was turnd on th compass ndl dflctd to th lft or right with angl θ accordingly to th currnt dirction Th dflction angl to th lft had a positiv sign Th masurmnts wr don for an lctric currnt flw through ach wir of th bundl Th rang of currnt chang was from -87 А up to 87 А, in which th maximum compass ndl dflction was θ = 84 Th rsults from th masurmnts of th angl of compass ndl dflction as a function of th multiplication of th currnt by th numbr of wirs N ar rprsntd in Figur 4 This figur shows th offrd horizontal coil of wirs allows during th flow of currnt up to 1 A, a strong dflction (mor than 8 dgrs) of th compass ndl to b obsrvd
Considrabl dflctions of th compass ndl can b carrid out using a small battry of 15 V as a sourc of voltag st on dirctly to th coil Th prsnc of such sourc of magntic fild which wll modls th magntic fild of a singl straight infinit horizontal wir is possibl and is ncssary for th dvlopmnt of xprimntal problms mploying such quipmnt 9 θ, o 75 6 45 3 15-15 -3-45 -6-75 -9-4, -16, -8,, 8, 16, 4, N, A Figur 4 Dflction angls of compass ndl as a function from th multiplication of currnt by th numbr of wirs N 3 Thortical dscription of th intraction btwn th compass and th magntic fild 31 A compass ndl in th magntic fild of wir For simplicity in our thortical considrations w will invstigat th intraction btwn compass ndl and magntic fild of straight infinit horizontal wir through which a currnt is flowing
Lt a straight infinit horizontal wir through which a currnt is flowing ly on y-axis of Cartsian coordinat systm K (Figur 5) A dirct currnt with magnitud and dirction coinciding with th positiv dirction of y-axis is flowing through th wir n a point lying on z-axis at distanc z from th origin of th coordinat systm is situatd th hanging point of compass ndl with lngth Th axis of rotation of compass ndl coincids with z-axis, and th compass ndl is lying and moving in a plan paralll to th plan XOY Th currnt position of compass ndl is charactrizd by th angl of rotation θ of th ndl in rspct to th positiv dirction of y-axis Th compass ndl has a magntic dipol momnt m Figur 5 Mutual disposition of compass ndl and wir Th flowing currnt producs a magntic fild in th wir surroundings Th magnitud of th horizontal componnt B of th vctor of magnt flux dnsity dirctd to th positiv dirction of x- axis is obtaind from Biot-Savart s law [1]: z = π ρ B (4) whr ρ = z + x (5) is th distanc from th crtain point with coordinats (x, y, z) to th wir axis Bcaus of th non-homognity of th wir magntic fild th torqu on th compass ndl M has to b obtaind by intgration of th invisibl torqus cratd by th singl indivisibl parts of compass ndl Each singl indivisibl part of th ndl with lngth dl is charactrizd with
indivisibl dipol momnt dm dirctd along th ndl dirction As a rsult of th intraction btwn this part of th compass ndl and th wir magntic fild a torqu dm will appar zcosθ dm = dm π ρ (6) Upon th ntir compass ndl th torqu M will b M z = mcos θ πz ρ dm = - mcos θf( m πz (7) Th intgral on th right sid of quation (7) is a dimnsionlss function f( of th paramtr δ, which is also dimnsionlss sin θ δ = z (8) n th spcial configuration prsntd in Figur 5 th torqu is with such dirction, so that it must dirct th north pol of compass ndl along th positiv dirction of x-axis, i along th dirction of th wir magntic fild Th functional dpndnc f( is dtrmind by th ndl form For a ndl with rctangular form th solution of intgral givs f( δ ) = arctg( δ 1 - δ 3 4 δ + 5 6 δ - 7 + (9) For a ndl with triangl form w obtain arctg( ln(1 + δ f( δ ) = - δ δ ) δ 1-6 4 6 δ δ + - + 15 8 (1) n gnral for th function f( δ ) w can say that indpndntly from th ndl form bcaus of th ndl symmtry in rspct to th rotation axis th function is vn and dcrasing with incrasing of δ On th zro δ = th function f( δ ) has magnitud 1 For comparabl and z th function f( δ ) is a significant factor and has to b takn into account Unfortunatly a larg varity of compass ndl forms manufacturd by diffrnt producrs is availabl This is th rason that th function f( cannot b unifid, and for its thortical obtaining
ar ncssary complx mathmatical xprssions An xprimntal tabulation of th function is th way out of this situation 3 A compass ndl in Earth s magntic fild Th plant Earth posssss own constant magntic fild f a compass ndl is put only in Earth s magntic fild it orintats along th dirction of th magntic mridian of th gographical point in which th compass is This cardinal point w will call north Earth s magntic fild is homognous W will suppos that th horizontal componnt of Earth s magntic flux dnsity is pointd along th positiv dirction of y-axis (Figur 5) and has magnitud B As a rsult of th intraction btwn th compass ndl and arth s magntic fild a torqu M appars M = mb sin θ (11) 33 A compass ndl in th magntic filds of Earth and wir Lt th compass ndl hav an inrtia momnt J in rspct to its axis of rotation During th simultanous action of Earth s magntic fild and th magntic fild producd by wir and taking onto account quation (7) and (11) w can writ d θ J dt = mb sin θ m cosθf( πz (1) This quation has th following pculiar stationary solutions: A) f no currnt is flowing through th wir from (1) w obtain θ =, i th compass ndl is orintd along th dirction of Earth s magntic fild Hr with B) f a currnt is flowing through th wir from (1) w obtain tgθ = f ( πzb = w sign th currnt quivalnt to Earth s magntic fild at a givn distanc z btwn th wir and th hanging point of compass ndl πzb = (13) (14) Th quation (13) givs th dpndnc btwn th dflction angl of compass ndl and th currnt flowing through th wir This dpndnc can b a basis for th cration of mthod for masuring th magnitud of a currnt flowing through a wir as th rspctiv dvic can b calld a tangnt-galvanomtr So it is calld [4, 5] in th cas whn windings of currnt ar vrtical
Th sign minus in (13) mans that during th currnt flowing through th wir in th positiv dirction of y-axis th compass ndl will dflct right 34 Discussion of rsults Th thortical possibility for th us of a horizontal fram of wirs as a tangnt-galvanomtr is shown in th considrations abov Probably for th us of th horizontal coil of wirs as a tangntgalvanomtr th drawing of prcis graduation curv will b nough for th rading of th occurrd non-linaritis Hr as an illustration for th possibilitis of work w will tak advantag of th fact that th coil of wirs was usd as a tangnt-galvanomtr abov Using (8) w can rwrit (13) in th form tgθ N = 1 Lsinθ f z (15) Som of th rsults prsntd in Figur 4 ar usd now Th procssing of xprimntal data is don using MS Excl n Figur 6 th lft sid of quation (15) is shown as a function of sin θ, which is th dynamic part of th argumnt δ of f( from (8) Th significant dynamics of f( gnratd by th non-homognity of th magntic fild producd by th wir can b sn in this figur For th approximation of th graphic dpndnc th function Trndlin of MS Excl is usd For th ralization of this function a polynomial approximation of th last squars mthod can b don using som programms From th valu of th approximating function, shown in Figur 6, for sin θ = and taking into account that f () = 1 w obtain 1/ 1 = 63 ± A This allows th magnitud of th horizontal componnt of Earth s magntic fild on th trritory of Faculty of Enginring and Pdagogy in Slivn to b dtrmind For our quipmnt th distanc btwn th middl bundl of wirs and th compass ndl was z = 119 mm, from which w obtain B = (67 ± 8)1 6 T (16) 4 Conclusion As a conclusion w can stat that simpl quipmnt for th dmonstration of Orstd s xprimnt is suggstd Th prsntd horizontal planar coil of wirs allows by flowing a currnt of 1 A through a wir a grat dflction (mor than 8 dgrs) of compass magntic ndl to b obsrvd Significant dflctions of th compass ndl can b ralizd using a small battry of 1,5 V as a sourc of voltag
,7 -tanθ/n, 1/A y =,795x 4 -,148x 3 -,519x +,1x +,694,6,5,4,3,,1, -1, -,5,,5 1, Figur 6 Dpndnc of th lft sid of quation (15) vs sin θ As a rsult of th thortical analysis an analytical dpndnc of th rotation momnt upon th compass ndl is obtaind This dpndnc rads th non-homognity of th magntic fild producd by th flowing through th wir currnt and th influnc of th ndl form Th possibility for th us of a horizontal coil of wirs as a tangnt-galvanomtr is also shown sinθ Rfrncs [1] J D Jackson 1975 Classical Elctrodynamics (Nw York: Wily) pp 169 73 [] J A Miranda Magntic fild calculation for arbitrarily shapd planar wirs Am J Phys 68 54 8 [3] T Charitat, F Granr 3 About th magntic fild of a finit wir Eur J Phys 4 67-7 [4] http://nwikipdiaorg/wiki/galvanomtr [5] C Young 1989 Th Pnguin Dictionary of Elctronics: Scond Edition, Editor: Valri llingworth (London: Pnguin Books Ltd) p564