Implementation of a planar coil of wires as a sinusgalvanometer. Analysis of the coil magnetic field

Transcription

1 mplmntation of a planar coil of wirs as a sinusgalvanomtr Analysis of th coil magntic fild Dimitar G Stoyanov Sofia Tchnical Univrsity, Slivn Enginring and Pdagogical Faculty, 59 Burgasko Shoss Blvd, 88 Slivn, Bulgaria dgstoyanov@abvbg Abstract Th papr prsnts a thortical analysis on th intraction btwn th Earth s magntic fild of a compass ndl and th magntic fild of a straight infinit currnt-carrying wir mplmntation of a planar horizontal coil of wirs has bn shown as a sinus galvanomtr Th magntic fild ovr th planar coil of wirs has bn xamind by xprimnt Th coil could b usd as a modl for straight infinit currnt wir in dmonstration st-ups or could b givn as an assignmnt in Physics laboratory workshops Kywords: magntic fild, Biot Savart law, sinus galvanomtr PACS : 15Pa, 7,55Dd, 755G 1 ntroduction n 18, Christian Orstd noticd that a compass ndl dflctd in th prsnc of a currntcarrying wir Howvr, th classical st-up mploys a sourc of stady powr supply and a currnt of 1- A, which maks th dmonstration vry difficult Hnc, our aim is to modify th Orstd xprimnt to mak it asy and low-cost and to us simpl voltag sourcs Th implmntation of a planar coil of wirs, which dflcts vigorously th compass magntic ndl (mor than 8 angular dgrs) whn a currnt of up to 1 A flows along it, has bn rportd (S [1]) Th planar coil crats a magntic fild in th middl of th bundl of wirs, which is vry clos to th magntic fild of a singl infinit straight currnt-carrying wir n [1] thortically prsnts th intraction btwn th magntic ndl of a compass and th magntic fild cratd by th coil As a rsult, th possibility for th coil and th compass to b usd as a tangnt galvanomtr has bn thortically and xprimntally provd Strong nonlinaritis in th functional dpndncs hav bn ascrtaind taking into account th inhomognity of th magntic fild around th currnt-carrying wir Thus, th nd to find anothr way to crat a galvanomtr using this horizontal coil mrgs Th aim of th following papr is to point out th possibility to us th coil and th compass as a sinus galvanomtr Thortical dscription of th intraction btwn a compass and a currnt-carrying wir 1 A magntic ndl in th magntic fild of a wir Th thortical analysis of th intraction btwn th magntic fild of a wir and th magntic ndl will b carrid out using th approach in [1] First, th intraction btwn a magntic ndl and th magntic fild of a straight infinit horizontal currnt-carrying wir will b xamind Lt a currnt-carrying wir li on th OY -axis of th Cartsian coordinat systm K (figur 1) A stady currnt flows along th wir with a magnitud of, having th sam positiv dirction as OY-

2 axis At a point on th OZ-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 th OZ-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 OY-axis (figur 1) 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 OX-axis is [1]: z = π ρ B (1) whr ρ = z + x () is th distanc from th givn point with coordinats of (x, y, z) to th wir axis Figur 1 Mutual disposition of a magntic ndl and a horizontal wir Du to th inhomognity of th magntic fild around 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 Th torqu of th whol ndl, M, is [1]: M z = mcos θ πz ρ dm = - mcosθf( δ) m πz (3) Th intgral on th right sid of (3) is a dimnsionlss function f( δ) of th paramtr δ (which is also dimnsionlss): sin θ δ = z (4)

3 Gnrally, th function of f( δ ) is vn and dcrass with an incras of δ du to th ndl symmtry which taks th pivot into account At zro, th function of f( δ ) has a valu of 1 A magntic ndl in th magntic fild of th Earth Th horizontal componnt of th induction of Earth s magntic fild is rprsntd by B r f vctor B r is at angl β with th OY axis of th coordinat systm, it has th following componnts in th sam coordinat systm r B = ( B sinβ, B cosβ, ) Th gomagntic fild is a homognous on Th intraction of th magntic ndl with it rsults in a torqu of M : M = mb sin( θ β) 3 A magntic ndl in th Earth s and th wir s magntic filds W assum th magntic ndl has a momnt of inrtia, J, towards its pivot Taking into considration (3) and (6), 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 [1]: (5) (6) d θ J dt = mb sin( θ β) m cosθf( δ) πz (7) Th abov quation has th following stationary solutions (i) At zro currnt along th wir and applying (7), w obtain: θ = β, and th ndl orints itslf along th Earth s magntic fild (ii) At θ = all parts of th magntic ndl ar at th sam distanc from th wir, and δ = Applying (7), w obtain: B sinβ = B = πz (8) Aftr transformation w obtain: = πzb sinβ = sinβ (9) whr is th quivalnt to th currnt of Earth s magntic fild whn z is givn: = πzb (1) Equation (9) can b usd as a mthod for masuring th currnt magnitud, which could b calld a sinus galvanomtr 3 Currnt coil in th mod of sinus galvanomtr

4 So far, th thortical analysis to us th compass magntic ndl as a sinus galvanomtr has bn prsntd To implmnt it in rality would rquir gratr accuracy than as a tangnt galvanomtr Morovr, th masurd currnt valus must b limitd to th valu of (taking into account (9) ) 31 A masuring procdur for a sinus galvanomtr Th procdur to masur th currnt along th wir is as follows: 1 Th noncurrnt-carrying wir orints itslf along th north-south dirction by th compass (figur a) Th compass scal is st to zro Th wir and th compass ar stadily fixd, rspctivly A currnt flows along th wir with a magnitud which is of intrst to us (figur b) Th magntic ndl dflcts at a crtain angl 3 Th currnt-carrying wir and th compass ar rotatd simultanously as far as th compass ndl sts to zro (figur c) 4 Th currnt along th wir is switchd off and th dflctd angl of β is dtrmind by th compass scal (figur d) Th currnt magnitud is stimatd applying (9) Figur а nitial position Figur b Currnt is on Figur c Rotation of th wir Figur d Currnt is off

5 3 Dscription of th st-up and considration of rsults Th abov-mntiond st-up was carrid out using a planar horizontal coil of wirs dscribd in [1] and shown in figur 3 Th total numbr of windings in th bunch was 4 (N=4) Whn a currnt of flows along ach currnt-carrying wir, th total currnt along th bunch windings is N A stady currnt sourc was providd by a dc powr supply rctifir with an output voltag of up to 15 V and a max currnt of А A rsistanc of 15 Ω with a disprsing powr of W was connctd in srial as a ballast into th planar coil n th st-up, th planar coil was placd horizontally and th compass lay in th middl of th bunch windings (S figur 3) Figur 3 Framwork of th planar coil of wirs usd in th st-up First, th bunch windings wr orintd along th north-south dirction by th compass as th currnt was switchd off along th fram Th compass scal was st to zro Thus, th magntic fild cratd by th bunch windings, B, had a dirction, which was prpndicular to th dirction of th horizontal componnt of th Earth s magntic fild, B Thn, th currnt was switchd on along th fram and th magntic ndl dflctd at an angl of θ Th angl of dflction on th lft has a positiv sign (S figur b) Nxt, th fram and th compass and th compass on it wr rotatd so as to rst th compass ndl to zro (figur c) Finally, th currnt was switchd off and th dflction angl, β, was dtrmind by th compass scal Taking th xprimnt into account and applying (9), w obtain πb N = zsinβ (11)

6 o o o Th rang of th dflction angl, β, was from 75 to + 75 at an intrval of 15 Figur 4 rprsnts th masurmnts obtaind from th multiplication of th currnt,, by th numbr of windings, N, as a function of th dflction angl, β, of th compass ndl acting as a sinus galvanomtr Th graph also displays th linar dpndnc of th currnt on sin β with a high cofficint of corrlation N, A 15 y = 1646x + 31 R = Figur 4 Masurmnts taking into account (9) applying sin β As a rsult, th dflctd angl of th compass ndl masurd as prcisly as possibl maks us bliv that th horizontal coil of wirs could b usd as a sinus galvanomtr ithr for dmonstrations or could b givn as an assignmnt in Physics laboratory workshops For instanc, th obtaind avrag valu of = A in th masurmnts (figur 4) and at z = 11 9 mm th magnitud of th horizontal componnt of Earth s magntic fild has bn stimatd at 6 B = ( 7 69 ± 5 ) 1 T at Sofia Tchnical Univrsity, Slivn Enginring Pdagogical Faculty 4 Analysis of th planar coil magntic fild Th linarity of th dpndnc (11) and th simpl calculations urgd us to us th planar coil as a sinus galvanomtr in stimating th magntic fild cratd in th ara of th bunch windings of th coil W tak th Earth s magntic fild as a standard and for th magntic fild cratd by th currnt applying (11), w obtain sin β B B = sinβ = N B πz (1)

7 Hnc, th magntic fild of th coil grows in linarity with th currnt incras of N at on and th sam distanc of th compass from th wir z Morovr, th inclination of th straight lin is disproportional to z Th distanc from th wir to th compass z could vary by placing svral pads undr th compass To mak th analysis of th dpndnc, B, asy a transformation with th sam linar connction along z (at constant β ) could b applid: N sinβ = πb z (13) 41 Dscription of th st-up and considration of rsults n th st-up, th fram was placd horizontally on a plan and th compass lay in th middl of th bunch windings (S figur 5) A numbr of dilctric plats, n =,1,, 3, 4, with on and th sam width of d = 86 mm wr put btwn th coil plan and th compass Figur 5 Location of th compass and pads Th dilctric plats st th compass ndl from th wir at a distanc of z = z + nd (14) Hr z = 119 mm is th minimum distanc btwn th wir and th compass magntic ndl, which is obtaind whn th compass dirctly lis on th plastic sht of th planar coil Th xprimnt was carrid out as follows:

8 First, th bundl windings wr orintd along th north-south dirction by th compass on an off-currnt coil of wirs and th compass scal was st at zro Thn, th currnt,, was switchd on along th fram and th magntic ndl dflctd at an angl of θ Th angl of dflction on th lft has a positiv sign (S figur b) Nxt, th fram and th compass on it wr rotatd so as to rst th compass ndl to zro (figur c) Finally, th currnt was switchd off and th dflction angl, β, was dtrmind by th compass scal (figur d) Howvr, th dpndncs of (1) and (13) could b dtrmind asily if th angl, β, was givn to dtrmin th currnt which sts th compass ndl to zro, th valus rang of th dflction angl, β, wr from o o o to + 75 at an intrval of 15 1 B /B d 1d d 3d 4d N, A Figur 6 Variations of th coil magntic fild as a function of th currnt Figur 6 shows th masurmnts of B / B as a function of th multiplication of th currnt,, by th numbr of windings, N, using diffrnt numbr of pads undr th compass Th graph also displays a linar dpndnc on th currnt along th coil taking into considration th masuring accuracy of th angl, β Morovr, it could also srv as an xprimnt chck-up of (1) confirming th linar dpndnc on th currnt Finally, th graph shows that th inclination of th straight lin dcrass with th incras of n, which lads to th ris of z (as a consqunc of (14)) 4 Considration of rsults Th sam xprimntal rsults can b obtaind using (13) Figur 7 shows th masurmnts of th multiplication of th currnt,, by th numbr of windings, N, as a function of th pads numbr placd undr th compass at on and th sam magntic ndl s dflction angl, β A linar dpndnc of th currnt on n can b obsrvd as all lins cross at on and th sam point of th X axis qual to / d Th obtaind graph vrifis (13) z

9 6 N, A 5 4 β=75 β=6 β=45 β=3 β= n Figur 7 Currnt dpndnc on th numbr of pads at on and th sam angl of β Th statistics procssing of th xprimntal rsults provids th following avrag curv: N sin β = n, A (15) Th obtaind avrag valu of (z ) = 1545 A diffrs with 6% from th on w hav in paragraph 3 Th variation could b xplaind by th scal divisions of th compass usd in th stup Each on rprsnts 3 which maks it possibl for accidntal rrors of up to 5 to occur in th angl Figur 8 shows th thortical dpndnc and th xprimntal data of th rlation btwn th magntic fild B(h) at th hight of h ovr th planar coil of wirs B() as a hight function ovr it Th graph displays that th planar coil of wirs magntic fild is clos to th thortical curv (8) at th distanc of up to 4 cm ovr it Data points corrspond to h = n d

10 1 B(h)/B() 8 Thory Data h, cm Fig 8 Comparison btwn th thortical dpndnc and th xprimntal data 5 Conclusion To sum up, a thortical analysis of th intraction btwn th compass magntic ndl and th currnt-carrying wir s magntic fild has bn don Th possibility to us a horizontal coil of wirs as a sinus-galvanomtr has bn shown and tstd by xprimnt Thrfor, w ascrtain th magntic fild of th planar coil of wirs is clos to its thortical valu at th distanc of up to 4 cm ovr it, taking into account th masuring accuracy of th ndl s dflction angl Hnc, th planar coil of wirs can b usd as a modl for a straight infinit currntcarrying wir in dmonstrations or can b givn as an assignmnt to studnts in Physics laboratory workshops Rfrncs [1] Dimitar G Stoyanov 9 A modification of th Orstd xprimnt Eur J Phys 3 641