#7A&B Magnetic force between two conductors

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Rudolf Curtis
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1 #7A& Magnetic foce between two conductos This expeiment will be done duing two lab sessions. n the fist we become familia with the appaatus and make one set of measuements. n the second session, we complete the data-taking and analysis. The second week lab wok cannot be done unless the fist week has been completed, and the homewok fom the fist week is completed. Goals y doing this expeiment, you will 1. pefom an expeiment based on null measuements, and use an optical leve,. measue how the magnetic foce between two paallel conductos vaies with cuent, and with the sepaation of the conductos, 3. measue the fundamental constant, 4. lean about the magnetic field of the eath, and measue a component of that field. Reading The magnetic field and magnetic foces ae intoduced in Young and Feedman, Sec. 7., 7.6, and 7.7 in the 1 th ed. Souces of magnetic fields ae discussed in Sec. 8.1-, leading to the foce between paallel conductos in Sec in the 1 th ed. Theoy As discussed and studied in the pevious expeiment, a long, staight conducto caying cuent is suounded by concentic magnetic field lines. The magnitude of the magnetic field is /, whee is the adius of the cicle, i.e., the distance fom the conducto, and is a constant, equal to T m/a. (1T 1Tesla, the unit of magnetic field stength.) Figue 1 indicates the magnetic field due to the cuent diected to the left. The diection of is given by the ight-hand ule, i.e., when the thumb of the ight hand points in the diection of the cuent, the finges cul in the diection of. A conducto caying a cuent in a magnetic field expeiences a foce df d, whee d is the length of the conducto segment, and the vecto d is in the diection of the cuent in the conducto. The magnetic field poduced by the cuent in one conducto exets a foce on any neaby conducto caying a cuent. Fo two paallel conductos caying equal cuents in opposite diections, as in Figue 1, and as in ou expeiment, d and ae pependicula to each othe. at the uppe conducto is constant at any given value of, and the foce is epulsive. (Not ugly, but the conductos ae pushed apat.) The magnetic foce pe unit length on the uppe conducto is F. (q. 1) Thus the foce pe unit length between two paallel conductos one mete apat, each caying pecisely 1 A would be exactly 1 7 N/m. This is impotant because it is the official S definition of the Ampee. Although quation 1 is stictly valid only fo conductos of infinite length, it is a good appoximation if the sepaation is small compaed to the conducto length. Fo ou expeiment, / <.1 is acceptable. F Fig. 1. Paallel conductos caying cuents in opposite diections epel each othe. This diagam shows the magnetic field lines poduced by the bottom conducto and the esultant foce on the top conducto. f the cuents ae in the same diection the foce is attactive. -1-

2 Appaatus Take some time to appeciate the ceative engineeing that has gone into the design of the cuent balance, the pincipal piece of appaatus that you will use in this expeiment. This instument is as petty as any you will encounte in a lowe division physics lab! As indicated in Fig., the cuent balance has two paallel conductos, one suspended ove the othe. Obseve how the delicate balance of the top conducto is achieved. This conducto foms one side of a fame that ests on two knife-edge pivots. The equilibium position is vaied by moving the counteweight, but what eally clinches this is a magnetic dampe consisting of a metal paddle that dips into the field egion between two magnets. As its name indicates, the magnetic dampe seves to damp out ocking that would othewise esult afte the top conducto is displaced. Fig.. Cuent balance used fo measuing the foce between two paallel conductos. The bottom conducto is stationay, and the top conducto is one side of a delicately balanced fame. Small displacements of the top conducto ae detected by obseving appaent shifts in the position of a scale obseved by means of a mio mounted on the fame. The epulsive foce between the conductos can be measued by adding known masses to the mass pan. The schematic cicuit diagam does not accuately epesent the actual outing of the cuents. telescope scale knife-edge pivot mass pan mio magnetic dampe counteweight R A Note, in addition, how cuents ae bought pependiculaly into and out of the paallel conducto bas. (The simple cicuit schematic in Fig. does not epesent the actual cuent path.) The motivation fo this is to make the magnetic fields,, geneated fom the enty and exit cuents paallel o antipaallel to the cuents in the conducto bas, so that d. n this case, the enty and exit cuents poduce no foce on the conducto bas. Futhemoe, obseve how the enty and exit cuents ae outed though the knife-edge pivots, theeby minimizing any mechanical dag foces imposed by the equied use of some type of conducto to oute the cuents fom one ba to the othe. Also not indicated in this cicuit schematic is a evesing switch to change the diection of the cuent. Vetical displacements of the top conducto can be accuately measued using the telescope which views a scale via its eflected image in a mio attached to the conducto fame. Coss-hais within the telescope indicate the pecise point on the scale to which the telescope points. Figue 3 shows how this woks. Fo simplicity, it is assumed that the mio suface is initially in a vetical plane, and that the telescope is aligned hoizontally. Thus the telescope points to the position on the scale at the same height as the telescope. ut, as indicated in Fig. 3, should the top conducto move upwad, the mio attached to the conducto tilts accodingly, and the telescope then points to a highe position on the scale. --

3 To be specific, if the conducto moves up distance a and it is distance b fom the knife-edge pivots, the mio tilts though angle φ a/b in the small angle assumption. Howeve, as shown in Fig. 3, due to the eflection, the appaent position of the coss-hais on the scale moves upwad though angle φ h/d. Thus the height change of the conducto is given by a h (b/d). The amplification of the displacement by the mio in this way has led to the effect being called an optical leve. φ mio knifeedge pivot b φ φ d φ top conducto scale DRAWNG NOT TO SCA a h final appaent position of coss-hais initial appaent position of coss-hais telescope Fig. 3. Pinciple of the optical leve. When the top conducto ba moves upwad though angle φ, the position on the scale to which the telescope points moves upwad though angle φ. Oveview of measuements and analysis Consideable cae is equied in this expeiment, because the masses ae small, typically 1 to milligams, and the cuents ae lage. The vey small epulsive magnetic foce between the conductos can be measued by knowing the mass on the mass pan that balances the magnetic foce. This is a fom of a null measuement technique. The initial efeence equilibium position of the top conducto (with zeo cuent) is detemined with the telescope. Then, a small mass m is placed on the pivoted top conducto, causing a deflection. The cuent is adjusted until the magnetic foce F balances the added gavitational foce mg, etuning the system to its initial position. Then, we have measued the magnetic foce to be F mg. y adjusting the counteweight, the efeence (equilibium) sepaation distance can be changed fom seveal mm to a few cm. Fo fou diffeent efeence sepaations, measue the cuent equied to balance vaious masses on the mass pan. (One efeence sepaation is to be completed in the fist lab class.) Change the cuent diection and ecod the equied cuent in both diections fo each sepaation and mass point. This povides a coection fo the additional small foce exeted on the top conducto by the eath s magnetic field, which has a magnitude of about.5 mt at Amhest. Only a component of exets a vetical foce, and you will also be able to extact, fom the data, a value fo this component of the eath s magnetic field. n the Appendix we show that an exact coection is obtained by eplacing with the poduct 1. The expession fo can be obtained by subtacting the fist pats of quations A3 and A4 fom each othe. With the coection fo the eath s magnetic field and quation 1, fom page 1, we expect that F/ mg/ 1 /, so we can wite ( 1 )/mg (/ )(/). At each sepaation, the cuents equied to balance seveal diffeent masses have been measued. Calculate the aveage of ( 1 )/mg at one value of the sepaation. Repeat this fo the othe sepaation values. A gaph of <( 1 )/mg> as a function of (/) is expected to be a staight line with slope /. The fundamental constant / can be calculated fom the slope of the gaph. Using the aveage values of ( 1 ) at each sepaation, you will also be able to calculate a value fo a component of the magnetic field of the eath. -3-

4 Set up The appaatus has aleady been wied but do not tun the cuent on just yet! Fist tace the path of the cuent though the cicuit, including the two switches. One switch meely connects and beaks the cuent, the othe eveses its diection. Designate one position of the evesing switch as the fowad diection, the othe as the evese diection. n ou labs the conductos ae set appoximately ast- West. With the evesing switch in the designated fowad diection, does the positive cuent go fom ast to West though the top conducto, o fom West to ast? The cuent is to be ead on the sepaate ammete. n seies with the ammete is a heostat a low-r vaiable esisto capable of handling lage cuents. The heostat can be used as a fine adjustment to the cuent; set it initially nea its mid-position. Measue the lengths of the paallel conducting ods. Fo the puposes of this expeiment, what is the elevant length to measue? Also conside the sepaation paamete that entes into the equations. What is this distance in the expeiment? Measuements of the diamete of the conductos in this lab wee all within.1 mm of 3.19 mm. Vaiations this small have no impact on the esults. You may use this value, o measue you appaatus with the micomete. Place a penny on the mass pan so that the two conductos gently touch. t is unlikely that they will make contact ove thei entie length. You may need to take the esultant gap into account. Use the feele gauges (.1,.,.3 mm) to estimate the maximum gap distance between the two conductos. As an appoximation, assume that when in contact, the aveage gap distance between the conductos is equal to half the maximum gap. Use this value to calculate c, the value of when the two conductos ae in contact. Adjust the telescope to obtain a shap image of the coss hais and scale. Measue the distances b and d indicated in Fig. 3. Recod the scale eading with the two conductos touching. Remove the penny and, if necessay, adjust the counteweight until the paallel conductos ae sepaated such that 6 mm. This will be the fist efeence sepaation. Recod the scale eading seen though the telescope. t is easy to make a mistake in setting 6 mm! What you ae looking fo is a change a in the height of the top conducto, which, when added to c, bings you up to 6 mm. Do not use the micomete to do this it is not designed fo this pupose. nstead use the telescope, emembeing that a change in the conducto sepaation of a coesponds to a change in the scale eading of h (d/b)a. Afte you make the adjustment, visually check that the cente-to-cente sepaation of the two conductos is indeed about 6 mm. -4-

5 Measuements Pecautions The set up of the appaatus is delicate. Take geat cae not to bump the table o appaatus, fo once you stated a seies of measuements, any distubance may equie you to begin again. 1. At a efeence sepaation of 6 mm, stat with mg on the pan. The stating masses fo othe sepaations ae suggested on the lab epot sheet.. Check that the voltage knob on the powe supply is fully counte clockwise, and then tun on the powe supply. Slowly tun the knob to aise the cuent indicated on the ammete until the conducto is nea the efeence position. (The cuent should be kept below about 14 A, and at no time should the cuent exceed 15 A.) 3. Tweak the cuent slightly to estoe the conducto to the efeence position. Recod the value of the cuent 1 to the neaest.1 A o bette. Revese the diection of the cuent and adjust it as necessay to etun to the efeence position. Recod the cuent. 4. Continue these measuements (items 1 though 3 above) with diffeent masses, inceasing the mass in 4 o 5 steps as suggested on the lab epot sheet. At each mass, adjust the cuent to etun to the efeence position. f moe than about 14 A is equied, educe the mass by mg. Do not exceed 15 A unde any cicumstances. Remembe to take measuements with the cuent in both diections. 5. As you go along, calculate the poduct 1 and diffeence 1 at each point. 6. When you have eached the maximum mass and cuent, emove all masses fom the top conducto, and etake the efeence and contact scale eadings to ensue that nothing has changed. Notify the instucto if the eadings diffe fom you stating values. 7. Repeat the entie sequence of measuements fo additional efeence sepaations. Use, 6, 8, and 1 mm. f thee is time, do a 4 th one at 18 mm. Analysis With the coection fo the eath s magnetic field and quation 1, fom page 1, we can wite: F mg 1 1 mg Fo each equilibium sepaation, the quantity 1 /mg should emain constant. (The Appendix shows that analyzing the esults in this way eliminates the effect of the eath s magnetic field.) Calculate the aveage and standad deviation of ( 1 )/mg fo each value of the sepaation. Recall that the sample standad deviation of a quantity x measued N times is defined as: 1 N σ x ( x i x ) whee x 1 N x i, N 1 i1 N i1 You now have the aveages <( 1 )/mg> fo each value of the atio (/). Plot a gaph of the aveages <( 1 )/mg> as a function of (/). Plot these values with eo bas. The uncetainty in esults in an eo ba along the hoizontal axis fo each point, and the standad deviation of ( 1 )/mg is the eo ba fo each point in the diection of the vetical axis. The gaph is expected to be a staight line with slope /, and the ecipocal of the slope is expected to be / 1 7 T m/a. Detemine the value of the fundamental constant /4. As a bypoduct, the data also povide an appoximate detemination of the component of the eath s magnetic field that causes the top conducto to deflect in the vetical diection. Study the Appendix and deive an equation that expesses only in tems of measued quantities: mg (the mass on the mass pan), 1,, and (the length of the top conducto). -5-

6 -6- Appendix gnoing the magnetic field of the eath, the magnetic foce exeted by the bottom conducto on the top conducto is given by (A1) whee is the length of the top conducto. n the unavoidable pesence of a magnetic field fom the eath, the total vetical magnetic foce on the top conducto becomes Thee ae a few things to note about this quation : n geneal, the magnetic foce exeted by the eath on the top conducto has both vetical and hoizontal components. Ou expeiment is sensitive only to the component of the eath s field that gives a foce in the vetical diection. The vetical magnetic foce exeted by the eath on the top conducto can be upwad o downwad, depending on the diection of the cuent. This accounts fo the ± signs in q.. The magnetic foce due to the eath is independent of the conducto sepaation ; it depends only on the cuent in the top conducto. f, no magnetic foces act on the top conducto. n ou expeimental pocedue we fix the foce on the top conducto by choosing a paticula mass m, and then adjusting the cuent to exactly offset the added weight mg. Diffeent cuents will be obtained coesponding to diffeent situations. f we have q. (A1). f the magnetic foce exeted by the eath is upwad: (A3) When, as obtained by simply evesing the cuent diection, a magnetic foce of equal magnitude is obtained in the downwad diection: (A4) quating qs. (A3) and (A4): (A5) Hence, the simple substitution of the poduct fo in q. (A1) coects fo the eath s magnetic field. The coection is exact and theefoe applies at all distances, even when is so lage that the foces the conductos exet on each othe ae much weake than the magnetic foce imposed by the eath. mg F mg F ( ) ( ) F F ± ± ±

7 #7 aboatoy Repot Magnetic foce between two conductos - Name: Patne: ab Section: Date: Refeence sepaation #: (~ 8 mm) m (mg) 1 1 / mg (A /N) Aveage. Standad Deviation. 1 S Con Scale eading at contact mm S Ref nitial ef. scale eading mm S Ref S Con Diffeence D scale mm Conducto motion D scale /M mm Contact sepaation c mm c D scale /M nitial sep. mm Recheck (with m ): Final scale eading at contact mm Final efeence scale eading mm Final sepaation mm Refeence sepaation #3: (~ 1 mm) m (mg) 1 1 / mg (A /N) Aveage. Standad Deviation. 1 S Con Scale eading at contact mm S Ref nitial ef. scale eading mm S Ref S Con Diffeence D scale mm Conducto motion D scale /M mm Contact sepaation c mm c D scale /M nitial sep. mm Recheck (with m ): Final scale eading at contact mm Final efeence scale eading mm Final sepaation mm 1

8 Refeence sepaation #4: (~ 18 mm) m (mg) 1 1 / mg (A /N) Summay data Aveage. Standad Deviation. 1 S Con Scale eading at contact mm S Ref nitial ef. scale eading mm S Ref S Con Diffeence D scale mm Conducto motion D scale /M mm Contact sepaation c mm c D scale /M nitial sep. mm Recheck (with m ): Final scale eading at contact mm Final efeence scale eading mm Final sepaation mm (mm) / < 1 / mg> AVG (A /N) 1 AVG. Ref. Sep. #1 Ref. Sep. # Ref. Sep. #3 Ref. Sep. #4 Scatchpad Taget Refeence Sepaation o Contact sepaation c Subtact: conducto motion equied Multiply by M Requied change in scale eading Scale eading at contact S Con Add & Round Taget S Ref nitial Ref. Scale Reading

9 Detemining the value of /: We saw in the laboatoy desciption that the gaph of ( 1 /mg) as a function of (/) is expected to have a slope equal to (/ ). Plot this gaph including uncetainties as eo bas, and detemine the slope. The ecipocal of the slope should be equal to / 1-7 T m/a. f ( 1 /mg) is plotted in units of A /N, thee will be a missing facto of 1-6, and the ecipocal of the slope should be equal to. T m/a. Attach you gaph to this woksheet. (Note on units: The expession fo the foce on a cuent in a magnetic field tells us that 1 N 1 T m A.) Slope of gaph ± (units) Final calculated value of / T m/a; stimate uncetainty in / T m/a Compae you esult with the expected value, and comment on the diffeence elative to the uncetainties. Detemining the magnetic field of the ath: Fom quations A3 and A4 in the Appendix, we can find an expession fo containing only diectly measued values. A simple analysis follows if we just subtact one equation fom the othe to obtain OR Δ Δ Using the aveage value of Δ at each efeence sepaation, plot the gaph of Δ as a function of, and detemine the slope. Fom this slope and you own measued value of /, calculate as slope /. This is the component of the magnetic field of the eath esponsible fo the vetical foce on the uppe conducto. Attach you gaph to this woksheet. Final calculated value of component T Uncetainty in value of component ± T Which component of is this? What is the expected value of this component, and how does it compae with you measuement? 3

Current Balance Warm Up

Current Balance Warm Up PHYSICS EXPERIMENTS 133 Cuent Balance-1 Cuent Balance Wam Up 1. Foce between cuent-caying wies Wie 1 has a length L (whee L is "long") and caies a cuent I 0. What is the magnitude of the magnetic field