Experiment MF Magnetic Force

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1 Expeiment MF Magnetic Foce Intoduction The magnetic foce on a cuent-caying conducto is basic to evey electic moto -- tuning the hands of electic watches and clocks, tanspoting tape in Walkmans, stating cas, opeating efigeato compessos, etc. In this expeiment, you will investigate the magnetic foce between two cuent caying wie coils. One coil has 10 tuns and the othe has 38 tuns. The 10-tun coil will be taped to one end of a pivoted balance beam. The beam pivots on two pins that also make electical contact, allowing cuent to flow onto the beam and though the 10-tun coil (figue 1). Figue 1: Top view of balance with 10-tun coil taped to undeside of foam coe The 38-tun coil will be positioned on the table diectly below the 10-tun coil. A cuent taveling though both coils will poduce a magnetic foce between the coils. You will measue the magnitude of this foce, by noting when the magnetic foce between the coils is balanced by the weight of squaes of aluminum foil that ae placed on the othe end of the balance beam. Figue : Undeneath side of balance beam with 10-tun coil 1

2 Deivation of the foce equation: Conside the case when the cuent in the two coils ae flowing in the same diection. Calculate the foce by fist calculating the magnetic field, B 1, of the lowe coil, (coil #1) at the uppe coil. Then we use the magnetic foce law, d F = Id B 1, to calculate the foce, df, on a small cuent element, Id, in the uppe coil # (figue 3). We then sum up the total foce on all n tuns in the uppe coil, F = n coil # d F. In ou expeiment set-up, the two coils ae a sepaated by a distance d that is much smalle than the adius of eithe coil. As a fist appoximation, we can teat the two coils as if they wee paallel wies sepaated by a distance d. Then we can use Ampèe s Law to calculate the magnetic field, B 1, fom the lowe coil and compute the foce on the uppe coil. Figue 3: Foce diagam on coil # fo attactive foce Note that in ou appoximation when we calculate the foce, df, on a small cuent element, Id, in the uppe coil, we ae neglecting the contibution to the magnetic field, B v, 1 fom pats of the lowe coil that ae not diectly below the small cuent element in the uppe coil. (This will ove-estimate the foce somewhat (can you see qualitatively why?) but the eo is not moe than about 10% with you aangement). Each tun in coil #1 has a cuent I which poduces a tangential magnetic field B tun at d (a distance d away), whose magnitude is then appoximately the field of a single long staight wie, µ 0 I B tun =. πd

3 The constant µ = 4π T m A and is called the magnetic pemeability of space. The magnitude of the total field B 1at d is equal to times the field due to a single tun B 1 = µ 0 I. πd In figue 3, the diection of the magnetic field at d points towads the cente of the cicle and is pependicula to the cuent element Id, in the uppe coil. The magnetic foce df exeted on d is given by: df = Id B 1. Since the cuents ae flowing in the same diection in the two coils, the diection of this foce at d points downwads towads coil #1. Thus the foce is attactive between the coils when the cuents flow in the same diection. The magnitude of this foce is given by df = Id µ 0 I πd Evey element d of coil # expeiences this foce, in the same diection -- vetically down, (if the cuent diections ae such as to poduce attaction). The total length of wie in coil # is n π, whee is the adius of both coil #1 and coil #. Thus the magnitude of the total foce of attaction on coil # is given by:. πµ 0 I F = n df = n Id µ 0 I = n πd = µ πd 0 n I d whee is the coil adius, d the sepaation between the cente of the two coils (see the figue above), and and n the numbe of tuns in each of the coils. By Newton s Thid Law of Motion, the total foce on coil #1 is equal and opposite to the foce on coil #. The magnetic foce is balanced by aluminum foil weights that ae placed on the othe side of the pivot, at an equal distance fom the pivot as the cente of the uppe coil. If the weights ae all the same-- cm x cm of foil, thei weight will be: F = nmg = nρatg. gav 3 whee g = 9.8m/ s, A = 4.0cm, ρ = kg / m, t = m, and n is the numbe of foils. The balance just begins to tip when the toque fom the magnetic foce equals the toque fom the aluminum weight. Since the moment ams ae equal, the foces must also be equal, Thus we have F = F. gav 3

4 nρatg = µ 0 n I. d This equation shows us that the cuent squaed depends linealy on the numbe of foils pesent, i.e. I ρatgd = n. µ 0 n This is why you wee asked to plot I vs. n. The slope of the I vs. n plot is given by slope = ρatgd. µ 0 n Then fom the best staight line fit of you data, you can calculate the magnetic pemeability of space using 1 µ = ( ρatgd 0 ). slope n In this expeiment you depended on the multimete calibation in ampees when you measued cuent. It is this vey sot of expeimental appaatus that was long used to define the cuent unit, the ampee, in tems of geomety and mechanical foces, setting the constant µ 0 in Ampee s law at exactly µ = 4π 10 7 N A. 0 You can ead you esult backwads, so to speak. Find out what value you own measuements, using the ampee as the unit of cuent, yield fo the constant µ 0. Appaatus The wie coils ae made by winding #6 enameled wie aound a soda can. Befoe winding, pepae the soda can by taping a sheet of pape aound it -- this pape will help the wie coil slide off the can. Then wind the wie aound the can leaving about 8 of wie at each end of the coil. Afte making the coil, caefully slide it off the can. Then use a couple of pieces of tape to hold them in thei ound shape. Figues 1 and above show the completed appaatus. Figues 4 though 6 povide a geneal idea of how the balance is constucted. Begin by dawing 3 lines acoss the beam, one at the cente of the beam and the othes at positions 1-1/ fom each end of the beam. Stick the T- pins into the foam-coe to make a pivot. Place and align each pin on the cente line. Using a line you maked on the beam as a guide, tape the lighte, ten-tun coil in place. Cut the leads to the 10-tun coil to convenient lengths that will allow thei ends to be wapped aound and soldeed to the pins. Remove about 1/ of the enamel insulation at the ends of the leads to the two coils. The enamel insulation on those wies is tenacious; folding a piece of 4

5 emey cloth in half and dawing the wie though it will emove the insulation. Altenatively stip off the enamel with you wie stippe. You will see a bight coppe colo when the insulation has gone, but check with you ohmmete to be sue. Wind the bae ends of the wie a couple of tuns aound the pins. Solde the wies to the pins. Figue 4: wies soldeed to pins Figue 5: Spacing between the 10-tun and 38-tun coils Figue 6: Lay out of coils on foam coe and base Use double-sided sticky tape to hold the cone baces and the 38-tun coil to you desk. Aange things so the coils ae paallel with the beam in its hoizontal balanced position. The coil end of the beam is much heavie than the othe end. You can balance it by taping one o two pennies, appopiately located, on the light end. As you tape the pennies into position, be caeful not to bea down on the beam while it s esting on the beaings: you ll distot the pin positions. Remembe that the beam will stat to swing slowly as it neas balance, and that sensitivity to a few squae centimetes of aluminum foil is quite good enough. (A squae centimete of the aluminum foil povided has a mass of about 4.3 mg.) You can use a piece of aluminum foil o a slive of tape fo the final adjustment of balance. Late, it may help to make a stop out of wadded foil placed on the desk at the cente of the bottom coil. Pess down gently until the 5

6 beam is whee you want it. In the expeiment it will lift off when the unbalanced foce of gavity (weight) just exceeds the magnetic foce. If the electical path though the touching sufaces of pin and beaing appeas eatic o intemittent, use emey cloth to clean the sufaces. Expeiment Depending upon the winding diection of the coils, the cuent though them will geneate magnetic foces such that the coils will eithe attact o epel each othe causing them to move. Exchanging the ends of one coil in the cicuit will evese the diection of the foce (although evesing the diection of the cuent in both coils will not). Connect the following components in a seies cicuit as shown in the figue 7: The coil on the balance beam. (Connect it into you cicuit by clipping alligato clips to the cone baces. The othe coil of the balance. The 8W filament of the 1157 bulb. You LVPS The piece of #16 alloy 800 esistance wie fom you ed box, a stiff shiny wie about 1.3 mm in diamete with a esistance of1.0 Ω m. Connect you MMM with the clips 0. m apat along the #16 esistance wie. You will measue the cuent flowing though the cicuit by measuing the voltage acoss the esistance wie. Figue 7: Wiing diagam 6

7 Figue 8: Expeimental set-up Plug in the wall tansfome of the LVPS; the lamp will tell you if cuent is flowing. Adjust the pot of the LVPS; the MMM should ead on the 50µ(50mV) scale between 0 mv and 100 mv coesponding to cuents fom 0.1 A to 0.5 A. When you have things woking, convince youself that the balance can show the magnetic effect, exchanging leads to see both attactive and epulsive foces. The idea is to balance the magnetic foce with the foce of known masses of foil acting on the balance. Seveal aangements of the expeiment s paametes ae possible. The balance can be set to attact o to epel, the cuent can be aised o loweed, and the weights can exet thei foce at eithe end of the beam. Think though how these setups vay and what might be the advantages of one ove the othe. Having the coils close togethe is a good idea; the magnetic foce, whethe attacting o epelling, is stongest in that position. Also the coil spacing is easie to measue, and moe epoducible, unde those conditions. But should the coils be wied to epel o attact? Both ways wok. But the attaction choice is pefeable because once the coils stat to sepaate the attactive foces ae educed in two ways simultaneously: by the loweed cuent and by the inceased sepaation of the coils. Taking Data So tun the cuent up high. Set the coils to attact, and put weights, squaes of aluminum foil, on the line at the fa end of the balance, but not enough to ovecome the magnetic attaction. Then, binging the cuent down slowly and as smoothly as you can, find the cuent whee the educed attaction lets the balance tip. Suitable weights ae cm cm squaes of foil. Cut out thee weights of that size. Gentle tapping of the table may help ovecome any tendency to stick. Make seveal measuements of the cuent at which the balance tips with all thee weights. Then, emove one weight at a time and again measue the citical cuent. Repeat the measuements to make sue they ae ok and tabulate the data, numbe of weights vesus cuent. 7

8 Plot the data with the numbe of weights, n, hoizontally and cuent squaed, I measued in [A ], vetically. If no weights wee on you balance, and it balanced at zeo cuent, then the oigin (0,0) is also a point. Fit a best staight line by eye. Estimate the slope and the eo on the slope. This can be done by taking the extemal (maximum and minimum) values of the slope and computing the diffeence with the best fit slope. Analyzing the Data Fom the best staight line fit of you data, you can calculate the magnetic pemeability of space using Eo Analysis 1 ρ At g d µ 0 = ( ). slope n Conside the uncetainties in the quantities that appea in the slope: g, ρ, A, t, d, and. Which eo contibutes the most? Which eos can you safely ignoe? Pats List Red Box 1' 800 wie 1 lamp socket (#1157) aluminum foil LVPST Test Kit 1 #1157 tail lamp Expeiment MF Pats Kit 40 ft #6 enameled coppe wie 1 by 10 piece of foam coe T-pins 1 cone baces 1 piece of double sided tape 1 ft # ed insulated standed wie 1 ft # black insulated standed wie 8

9 Expeiment MF (Magnetic Foce): Calculating the magnetic pemeability In you expeiment, depending upon the winding diection of the coils, the cuent though them will geneate magnetic foces such that the coils will eithe attact o epel each othe causing them to move. a) Is you appaatus set up to epel o attact? Biefly explain how you intend to measue the cuent that flows though the coil that will poduce a magnetic foce that will just balance the weight of the foil. b) What is the adius of you coils? What is the distance d between you coils? c) You will measue the cuent that will poduce a magnetic foce that will just balance one, two, and thee squaes of the foil. Suitable weights ae cm by cm squaes of foil. Fo each numbe of weights, n, make seveal measuements of the cuent that balances the weight. Aveage you values. d) Make a table of you data with columns fo weight (in tems of the numbe of weights, n), the cuent, I, in [A], and the cuent squaed, I, in [A ]. e) Make a plot of the cuent squaed I, in [A ], vetically vs. the weight (in tems of the numbe of weights, n) hoizontally. If no weights wee on you balance, and it balanced at zeo cuent, then the oigin is also a point. f) Use a linea egession to find the slope and intecept of a staight-line fit of you plot. (You may also find the best fit staight-line by eye.) g) Estimate the eo on the slope. This can be done by taking the extemal (maximum and minimum) values of the slope and computing the diffeence with the best fit slope. h) Then fom the best staight-line fit of you data, calculate the magnetic pemeability of space using 1 µ = ( ρ Atgd 0 ), slope n whee g = 9. 8m / s, A is the aea of you foil, ρ = 7 aluminum, t = m is the thickness of the aluminum, = 38, n = 10, is the adius of you coils, and d is the sepaation between you coils kg / m is the density of Conside the uncetainties in the quantities that appea in the slope: g, ρ, A, t, d, and. Which eo contibutes the most? Which eos can you safely ignoe? 9

10 Poblem 1: Expeiment Magnetic Foce In the magnetic foce expeiment, a cuent I = 0.5A is passed in seies though a 38 tun coil taped to a table and a 10 tun coil which is taped to a balance diectly above the 38 tun coil. The distance between the 10 tun coil and the 38 tun coil is d = 0.5cm. Each coil has a adius of = 6.0cm. a) Calculate the magnitude of the magnetic field oiginating fom the 38 tun coil at any point on the 10 tun coil. Clealy indicate any appoximations you make. b) Calculate the magnitude of the magnetic field oiginating fom both coils along the cental axis at a height z = 0.5cm above the plane of the 38 tun coil. You may take this point as the cente of the 10 tun coil. 10

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