A modification of Oersted experiment

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1 A modification of Orstd xprimnt Dimitar G STOYANOV Slivn Enginring and Pdagogical Faculty, Sofia Tchnical Univrsity 59 Burgasko Shoss Blvd, 88 Slivn, BULGARA dgstoyanov@abvbg Abstract Th papr dscribs a simpl stup of Orstd xprimnt A planar coil of wirs has bn usd to dflct vigorously th magntic ndl (mor than 8 angular dgrs) whn a currnt of up to 1 A flows along it Basd on thortical analysis th torqu on th magntic fild is analytically xprssd taking into account th inhomognity of th fild and th ndl shap What is mor, a procdur to masur th Earth s magntic componnt is implid and implmntd and its magnitud has bn stimatd following th sam stps PACS: 15 My, 755 Db, 843 Hh Ky words: magntic fild, Biot Savart law, Orstd xprimnt 1 ntroduction n 18 Christian Orstd noticd that a compass ndl dflctd its initially alignd northsouth dirction in th prsnc of a currnt-carrying wir That xprimnt has bn th first on to indicat that currnt-carrying wir producs a magntic fild Howvr, th classical stup mploys a sourc of stady powr supply and a currnt of 1 A, which maks th dmonstration vry difficult Hnc, w propos a low-cost typ of apparatus as a simpl solution and practical modification of Orstd xprimnt Thortical Analysis 1 A Magntic fild of a finit wir Lt a straight finit wir of lngth L lis on y-axis of th Cartsian coordinat systm (Fig 1) Th wir is orintd symmtrically to th origin of th coordinat systm A stady lctric currnt,, dirctd to th positiv y - axis flows along th wir Th currnt carrid by th wir crats a magntic fild around th wir

2 At any point on z - axis, at a distanc of z from th origin of th coordinat systm th vctor of th magntic induction, r B, is orintd to th positiv dirction of x - axis (in th stup prpndicularly in th drawing in figur 1, facing th papr) Th magnitud of th magntic induction vctor is obtaind by using Biot Savart law [1-3]: B x = 4 π + L / zdy ( y z ) L / + 3 / (1) Figur 1 A magntic fild of a finit currnt-carrying wir Th solution to th intgral is in [3]: B x = πz z L () Th first multiplir on th right of () is th fild of a straight infinit wir [1]: B ( ) = x πz (3) Th scond multiplir is a corrction for th (3) final lngth of th wir Th sam multiplir is a continuous function of L / z to th valu of 1 at L / z At L / z = 1, th multiplir has a valu of 986 whil at L / z =, it has a valu of 995 Hnc, at L / z, th fild cratd by th wir with an accuracy of 5 % is qual to that of an infinit wir For instanc, if z = 1 cm, w can us a wir lngth of cm

3 A magntic ndl in a magntic fild of wir W would lik to point out that th thortical analyss is totally basd on th intraction of th magntic ndl with th magntic fild of a straight infinit horizontal currnt-carrying wir Lt a straight infinit mtal currnt-carrying wir lis on y - axis of th Cartsian coordinat systm K (Fig ) A stady currnt flows along th wir with a magnitud of, having th sam positiv dirction as y-axis At a point on z-axis th suspnsion point lis at a distanc of z from th origin of th coordinat systm of a magntic ndl of a lngth Th pivot of th magntic ndl coincids with z-axis whil th magntic ndl lis and movs on а plan paralll to XOY plan Th actual position of th magntic ndl is charactrizd by th rotation angl of th ndl θ with rspct to th positiv dirction of y-axis Th magntic ndl has a magntic dipol momnt of m Th flowing currnt crats a magntic fild around th wir Th magnitud of th horizontal componnt B of th magntic induction vctor in th positiv dirction of x-axis can b obtaind by th Biot Savart law [1]: z = π ρ B (4) whr: ρ = z + x (5) s th distanc from th givn point with coordinats of (x, y, z) to th wir axis Figur Mutual disposition of a magntic ndl and a wir Th sparat parts of th magntic ndl at a rotation angl of θ ar at diffrnt distancs from th wir axis, so th magnitud of th magntic fild (4) xrtd on thm is

4 diffrnt Du to th inhomognity of th magntic fild round th wir th torqu of th magntic ndl M is obtaind by th intgration of th infinit small torqus cratd by th diffrnt infinit small parts of th magntic ndl Each infinit small part of th ndl of a lngth of dl is charactrizd by an infinit small dipol momnt of dm having th sam dirction as that of th ndl Th intraction of that part of th ndl with th magntic fild of th wir lads to a torqu of dm [1] zcosθ dm = dm π ρ (6) Th torqu of th whol ndl, M, is: M z = mcos θ πz ρ dm = - mcos θf( δ) m πz (7) Th intgral on th right sid of (7) is a dimnsionlss function f( δ) of th paramtr δ (which is also dimnsionlss) sin θ δ = z (8) Th thr-dimnsional configuration in Figur rprsnts th torqu having th sam dirction as that of th magntic fild of th wir Obviously, this can b don by adjusting th north-south dirction of th ndl to th positiv x-axis using th torqu Th functional dpndncy of f( δ) is dtrmind by th shap of th ndl Gnrally, th function of f( δ ) is vn and dcrasing with an incras of δ du to th ndl symmtry taking into account th pivot At zro, th function of f( δ ) has th valu of 1 For xampl, if th ndl has a rctangular shap, th solution to th intgral is: f( δ ) = arctg( δ) δ 1 - δ 3 4 δ δ (9) Th function of f( δ) is an important factor and it should b takn into considration for comparabl and z Howvr, thr is a grat varity of magntic ndl shaps manufacturd by diffrnt companis Thrfor, th dpndncy of f( δ ) cannot b considrd as univrsally valid and its thortical valus ar a rsult of complx mathmatical xprssions A tabulation of th function obtaind xprimntally should b usd in such cass

5 3 A magntic ndl in th magntic fild of th Earth Th Earth has its constant magntic fild f th ndl was placd only in th Earth s magntic fild, it would orint to th magntic mridian of th gographical point, whr th compass lis That dirction is to b nominatd as north Th gomagntic fild is a homognous on Th horizontal componnt of th induction of Earth s magntic fild orints th positiv dirction of y-axis (Fig ) and has a magnitud of B Th intraction of th ndl with it rsults in a torqu of M M = mb sin θ (1) 4 A magntic ndl in Earth s magntic filds and th wir W assum th magntic ndl has a momnt of inrtia, J, towards its pivot Taking onto considration (7) and (1), th simultanous action of both th Earth s magntic fild and th on cratd by th wir can b xprssd by following quation of twisting motion: d θ J = mb sin θ m cosθf( δ) dt πz Th abov quation has th following stationary solutions: A) at zro currnt along th wir and applying (11), w obtain: θ = ; and th ndl orints th Earth s magntic fild B) at non-zro currnt along th wir and applying (11), w obtain: (11) tgθ = f ( δ) πzb = (1) whr is th quivalnt to th currnt of Earth s magntic fild whn z is givn: = πzb (13) Equation (1) xprsss th dpndncy of th dflction angl of th ndl on th wir currnt Hnc, this dpndncy can b usd as a mthod for masuring th currnt magnitud along a wir t could b calld tangnt-galvanomtr and that is how th vrtical currnt coils ar calld [4, 5] Th minus (-) sign in (1) mans that whn th currnt flows along th positiv dirction of OY axis th ndl dviats on th right 3 Exprimntal Stup 31 A planar coil of wirs Our nxt stp was to dvis and mak a planar coil of wirs with ovrall dimnsions of cm long and 3 cm wid to wind th wir in th middl sction as many tims as possibl and crat a bundl of windings thr

6 Figur 3а A planar coil of unvn windings Figur 3b A planar coil of vn windings Th magntic fild cratd by th sid wirs in th middl of th bundl of wirs is compnsatd bcaus aras AB and GH, BC and FG, CD and EF mutually nutraliz thir filds du to th symmtrical location of th aras and to th fact that th currnt flows along thm symmtrically Thus, th fild cratd by th bundl of wirs in th middl of th coil is what should b rgistrd Figur 4 Framwork of th planar coil of wirs usd in th stup

7 Figur 3a shows an unvn numbr of windings in th bundl whil in figur 3b th numbr of windings in th bundl is vn n th stup w prfrrd th option with th vn numbr of windings taking into considration that th currnt input and output wr locatd on on sid of th plan framwork W usd a dilctric sht (plastics sht in our stup) and 8-mm hols wr drilld into it Th location of th hols is prsntd in Figur 4 Th wir was wound onto th sht whil th hols wr usd to fastn th framwork Th bundl of windings was locatd undr th dilctric sht Whn at work, th planar coil was placd horizontally and th compass lay on th dilctric sht in its gomtrical cntr t was th sht gaug that dtrmind th minimum distanc btwn th compass and th bundl windings 3 Th horizontal fram as a sourc of a magntic fild n th stup w usd th planar coil with vn numbr of windings Th total numbr of windings in th bundl was 4 A stady currnt was supplid by a DC powr supply rctifir with an output voltag up to 15 V and a max currnt of А Rsistanc of 15 Ω / powr of W/ was connctd in srial into th powr supply and th planar coil as to b usd as a ballast Figur 5 Th bunch windings orintd th north-south dirction by th compass

8 At work th planar coil was put on horizontal tabl and th compass lay on th dilctric sht in its gomtrical cntr (Figur 5) Th bunch windings orintd th north-south dirction dtrmind by th compass as th currnt was switchd off along th fram Th compass scal was st to zro whn th coil was orintd that way and th currnt was switchd off, th magntic ndl pointd north, i θ = Thus, th magntic fild cratd by th bunch of windings B had a dirction prpndicular to th dirction of th horizontal componnt of Earth s magntic fild, B Whn th currnt was switchd on along th coil, th magntic ndl dflctd lft or right at th angl of θ with rgards to th currnt dirction (S Figur 4) Th angl of dflction on th lft has a positiv sign Currnt of flowd along ach winding Th rang of chang of was from -87 А to 87 А at a maximum dflction of θ = 84 Figur 6 rprsnts th masurmnts of th dflction angl of θ as a multiplication of th currnt along ach wir by th numbr of windings, N 9 θ, o , -16, -8,, 8, 16, 4, N, A Figur 6 Dflction angls of th magntic ndl as a multiplying function of th currnt by th numbr of wirs N

9 4 Considration of Rsults 41 Th planar coil as a practical modification of Orstd xprimnt Th planar coil framwork shown in Figur 4 is on of th many xampls which could b usd whn w do not want to us mtal fastning lmnts or glu Crtainly, thr ar many othr options as wll Th planar coil in th stup maks th ndl dflct vigorously (mor than 8 angular dgrs) whn th currnt flow is up to 1 A Th magntic ndl can considrably dflct if a small battry of 15 A is dirctly connctd to th coil as a voltag sourc (without any ballast rsistanc) Thus, Orstd xprimnt can b conductd in an asy and simpl stup f th planar coil of wirs has L / z < 1, th sam rsults ar obtaind: th currnt carrying wir crats a magntic fild and th fild dirction is dpndnt on th currnt dirction Howvr, prcis quantitativ masurmnts will show srious dviations from Biot Savart law ranging from 1 to 15 % Figur 4 rprsnts a planar coil of wirs of L / z > Th magntic fild cratd by th bundl of windings and of accuracy bttr than 1% adjustd th magntic fild of a singl straight infinit wir carrying a currnt of N Consquntly, it can b usd as a modification of Orstd xprimnt 4 Th planar coil of wirs as a tangnt-galvanomtr Th prsnc of a sourc of magntic fild adjusting th magntic fild of a singl straight infinit rctilinar currnt carrying wir can b usd and b implmntd in many othr stups with practical application Th abovmntiond thortical assumption (1) to us th planar coil of wirs as a tangnt-galvanomtr could bcom rality if an accurat graduatd curv was to b drawn to tak into account nonlinaritis 43 Masurmnt of th horizontal componnt of Earth s magntic fild W would lik to point out to th following procdur for masuring th horizontal componnt of Earth s magntic fild t is basd on th abovmntiond thortical analysis and on th possibility for planar coil of wirs to b usd as a tangnt-galvanomtr Firstly, w transform (1) by using (8) into: tgθ N = 1 Lsinθ f z (14) Thn, th xprimntal rsults shown in Figur 6 ar procssd by MS Excl Figur 7 rprsnts th lft sid of quation (14) as a function of sin θ, i th dynamics of argumnt δ (8) Th inhomognity of th magntic fild of th wir lads to th dynamic charactr of f( δ ) Nxt, th graphic dpndncy of th obtaind xprimntal data is automatically approximatd by using Trndlin MS Excl function which conducts a polynomial approximation of th xprimntal points

10 Trndlin MS Excl automatically prforms th following substitution (abscissa and ordinat in th graph): tgθ y =, N (15) x = sin θ (16) -tanθ/n, 1/A y =,795x 4 -,148x 3 -,519x +,1x +,694,7,6,5,4,3,,1, -1, -,5,,5 1, Figur 7 Th dpndncy of th lft sid in (14) on sin θ sinθ At sin θ =, th valu of th approximating function (shown in Figur 7) is as follows: y () = 694 Taking into account f () = 1and comparing (14) and (15), w obtain:

11 1/ = 63 ± A 1 (17) Th variations of ach xprimntal points around th approximating function (valud at ± А -1 ) shown in Figur 6 ar takn into account in (17) Ths variations ar dpndnt on th masurd accuracy of th dflction angl of th magntic ndl, θ n th stup w masurd th distanc z, from th axis of th bundl of windings to th magntic ndl and it was: z = 119 mm Th obtaind z, th rsults from (17) togthr with (13) ld to: B = (67 ± 8)1 6 T (18) Hnc, th magnitud of th horizontal componnt of Earth s magntic fild in th ara of Slivn Enginring and Pdagogical Faculty was stimatd 4 Conclusion Firstly, th proposd typ of apparatus for Orstd xprimnt is vry simpl Th planar coil of wirs maks th magntic fild dflct vigorously (mor than 8 angular dgrs) whn currnt up to 1 A flows along it Furthrmor, a considrabl dflction can b obsrvd by using a small battry of 15 V as a powr supply, connctd with planar coil of wirs without ballast Scondly, th thortical analysis points out to dpndncy of th magntic ndl torqu on th shap of th magntic ndl taking into account th inhomognity of th magntic fild around th currnt carrying wir Furthrmor, it outlins that th planar coil of wirs can b usd as a tangnt-galvanomtr Finally, a masuring procdur of th horizontal componnt of Earth s magntic fild is implid and implmntd and its magnitud is stimatd Rfrncs [1] J D Jackson 1975 Classical Elctrodynamics (Nw York: Wily) pp [] J A Miranda Magntic fild calculation for arbitrarily shapd planar wirs Am J Phys [3] T Charitat, F Granr 3 About th magntic fild of a finit wir Eur J Phys [4] [5] C Young 1989 Th Pnguin Dictionary of Elctronics: Scond Edition, Editor: Valri llingworth (London: Pnguin Books Ltd) p564