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MAGNETIC TOROUE z Experimenting with themagneticdipole Most of us have childhood memoriesof playing with magnetsand being fascinatedby their behavior. You were probably taught that this behavior could be explained by north and south "poles" that either attract or repel each other. However, since your introduction to classical electricity and magnetism,you have learnedthat a more fundamentalmodel hasbeen developedto explain magnetic interactions. This model involves the magnetic dipole, which itself can be modeled as a loop of current. By viewing permanent magnets as being composed of tiny magnetic dipoles, the magnetic field of permanent magnets can be predicted, as well as those magnets' behaviorin the presenceof external magneticfields. You have beentaught these conceptsfrom a theoretical standpoint. Now you need to test these theories in the laboratory. The Magnetic Torque instrument has been designedjust for this purpose,to cementyour knowledge of the magneticdipole through experiments. With Magnetic Torque, you'll be examiningthe behavior of a magnetic dipole in both uniform and non-uniform magnetic fields. You will also be taking data in order to calculate the dipole moment of your magnetic dipole. There are five different experiments that can be performed; each of these experimentsinvolves different combinations of mechanicsand E&M principles. Your instructor may pick and choosewhich experimentshe or she wishesyou to perforrn, but you might want to survey all five of the experimentsanyway. They'll help your understandingof physics, and may evencomein handvon a test! The Instrument ln orderto usethe MagneticTorqueinstrumentandlearnthe physicsprinciplesinvolved with it, it is first necessaryto understandthe instrumentitself. This sectionof the manual providesa descriptionof the variouscomponentpartsof the MagneticTorqueapparatus.Study it carefully beforeattendingyour laboratory session. A. TheMagnet What is referredto as "the magnet"is the componentof the instrumentthat housesthe fwo co-axialcoils, the air bearing,andthe strobelight. 1.-- Coils The coils are composedof copperwire that is wound on bobbins. Each coil has 195turns. The two coils are alwaysconnectedin seriesso that the samecurrentflows througheachturn. This currentis displayedon the analogammeter. It is importantto notethat the coils havesomeresistance,andthat resistanceis temperature-dependent. If currentis allowedto flow throughthe coils for a long time, or if the currentis high (-3-4 amps),the coils' temperaturebeginsto rise. You canfeel the increasein temperature.As the coils heatup, their resistancerises,and the currentsubsequently beginsto decrease sincethe power supplyis not current-regulated.It is thereforea good idea to turn the culrentdownto zero whenthe magnetis not beingused. Try to avoidusinghigh currents for any appreciablelength of time. The instrument is designedto sustain the full output power of the supply without any danger. However, since the maximum current will be decreasedas the temperature of the coils increases,you may not be able to obtain the highest fields if you allow the coils to get too hot. ln order to calculatethe magneticfield at the center of the apparatus,one needsto perform an integral, since each of the turns has a different radius and a different distance from the center of the pair. Such a calculation may be a bit tedious. Therefore, glven below are an equivalent radius and equivalent distance between the two coils, such that the 390 turns can be thought of as two separatecurrent loops. equivalent radius: 0.lo?m equivalent separation between thecoils: O,l3?m Using the equivalentradius and separationalong with the Biot-Savart Law, you should be able to calculate the magnetic field at the center, which is where the magnetic dipole will reside. It is only the magneticfield in a small region around the mi@oint of the coils' mis that is important in all of these experiments. The value for the magnetic field gradient at the center will be neededfor one of the experiments. The field gradient can be calculatedby differentiating the expressionof the magnetic field with respectto z, where z is the axial distancefrom the center of one of the coils. 2. -- Asrbearing The air bearing is the sphericalhollow in the cylindrical brassrod that is supported on the bottom coil form. The bearing has a niurow opening that allows air to be pumped into the sphericalhollow. The ball sits in this hollow and floats on a cushion of air. This provides support without significant friction. The air pump is housed inside of the power supply. A vinyl hose attachesto the back of the power supply at one of its ends, and at the other end attachesby a threadedright-angle fitting to the under-side of the air bearing. Make zure not to restrict air flow by accidentally"kinking" the hose. 3 -- Strobe light The strobe light is located in an insulated housing on top of the upper coil. One can vary the frequency of the strobe flashesby a control located on the power supply front panel. The frequency of these flashes is automatically measured and read out to two significant figures on the power supply's front panel. This data is updated every l0 seconds. 4. -- Strobelight frequencydisplay Displaysthe frequencyof the strobe light in Hertz. Below the displayis the frequency-adjust.Turningthe knobclockwisewill increasethe frequency.efter adjusting the frequency,one needsto wait for the instrumentto countup to the actualfrequency. Next to the frequency-adjust knob is the on-offswitchfor the strobelieht. 5. - Air switch Allows you to turn offthe air pump when you are performingthe magneticforce experiments. 6. -- Pilot light Indicateswhenthe ac power is on for the entire systeminsidethe power supply case. B. The Accessories The accessories are thosecomponentsof the MagneticTorque instnrmentthat are not permanentlyattachedto the two main parts of the instrument. ThJy are usedto perform the variousexperiments.Lst's examinethem. l. - Cueballs Thesecue ballsare simplyaramithsnookerballswith a smallcylindricalpermanent magnetat their centersthat acts as if it were a magneticdipole. The magnitic dipole moinentpointsin the directionof the ball's handl". th" handli allowsyou to-spinthe LalL measureits rotationfrequency,and determinethe directionof the magneticmoment. The handleson the ballshavea smallaxialhole drilledin them. In this hole, a thin metalrod with an attachedweight is placedaspart of a staticmagnetictorqueexperiment. The aluminumrod hasa steeltip at one end that holdsfast to the magneticinsideof the ball. The movableweightis a smallclearplasticcylinderwith an O-ringinsidethat keeps the weight from involuntarily slipping on the rod. The weight is meantto be moved up anddown therod to varythe gravitationaltorque(Figurel). m16,vfr/lED STEL Dt.< T/P ,=--n/--- ;:?@=. .j.!------- / t?e /A/u&€ CU€ 8A/I F/6AI?E t 2. - Plastictow€f, The clearplastictube attachedto a cylindricalbasecan be placedon top of the air bearing. This apparatusis usedfor the magneticforce experiment.A nylon cap placedon the top of the nrbe holds the rod that zupportsthe sprhg. The other end of the spring is connectedto the msgnet. The position of the suspendedmagnetinsideof the tube canbe adjustedby movingthe rod insidethe cap. Thereis alsoa smallscrew-eyeattachedabove the magneticdipole that can be used to prevent the magnetfrom rotating on its gimbals (Figure2). Ball bearingsthat weigh one gram eachare providedfor calibrationof the spring. 3. - Rotatingmagneticfield This is an optional accessory,so it may not be included. The rotating magnetic field is simply a special configuration of permanentmagnetsand soft iron shims that providea horizontalmagneticfield. This uniformhorizontalmagneticfield (-1.0 mT) can be manuallyrotatedaroundthe air bearing. It has a hole in its basethat allows the air bearingto act as its rotationoris (Figure3). This magneticfield is usedto demonstrate nuclearmagneticresonance. '€JAa*Vt ./ 4. - Bulls-eyelevel If the air bearingis not level, an additionaltorque due to unequal air flow can result. Such a torque can produceerroneousdata.The bulls-eye level is simplya fluid-filledregionthat hasa bubblein it. Whenthe bubbleis within the circle, the apparatus is reasonablylevel. The leveling can easily b€ accomplished by placing shimsunderneaththe rubberfeet below the magnet. C. PowerSupply What we term the "power zupply" contains componentsother than the power supply. On the front panef starting from the left, is: g*?.e'etE l. - Analogarnmeter Reads the current passing through the coils (since the coils are connectedin series). The knob below the meter is usedto adjust the current. Turning the knob clockwise increases the currentthrough the coils. 2. -- Fielddirectionswitch Controls whetherthe magnetic field at the centeris eitherup or down. 2 3. - Fieldgradientswitch Controls wheth€r there is a magneticfield gredient at the centerof (gradient off). the coils,or a uniform magneticfield F/6utE On the backof the powersupplyarethe following: 1. -- on-offacswitchfor all of the components inside. 2. -- cordthat plugsinto the ac electricalsocket. 3. -- cinch Jonesconnectorthat connects the powersupplyto the magnet. 4. maleair hoseconnection-theair hosehasa femaleconnection that matesto it. EXPERIMENT 1: Magnetictorque equalsgravitationaltorque Objectives The main objectiveof this experimentis to measurethe magneticmoment(p) of the magneticdipole (which is the magnetinsidethe cue balls). You will also verify the functionalrelationship:pxB - rxmg. Equipment Magnet, power supply, air bearing,cue ball, aluminumrod with a steel end, weight,ruler,balance,andcalipers. Theory From your electricityand magnetismcourse,you shouldbe awarethat a loop of continuouscurrent is referred to as a magneticdipole. The neodyniumiron boron magnetized disk insidethe ball is not a loop of current. In fact it is a .375 inch diameter, .25 inchthick disk magnetized alongthe axisof the disk.But its magneticfield is suchthat it actsas if it werea magneticdipole. In a uniformmagneticfield (which is the caseat the centerof the two-coil configuration),a magneticdipole experiencesa magnetictorque that is givenby the expression: t-pxB Your magnet'sdipole momentis alignedparallelto the handleon the ball; the magnetic field producedby the coilscanbe eitherup or down alongthe coils' axis. If the magnetic field points up, and the magneticmoment is alignedat some angJe0 away from the directionof the magneticfield, the ball will experiencea torquethat will tend to rotateit so that the handleof the ball points upward. But if the aluminumrod is placedin the handleof the ball, thereis now anothertorque due to the earth'sgravitationalfield. The expression for this torqueis: T = txmB The gravitationaltorquetendsto causethe ball to rotateso that the ball's handlepoints downward(Figure4). Sincea net torquecausesa changein angularmomentum,theball {-,4'"tfarr{eege) F/aePE p) + will rotate if the gravitationaltorque is largerthan the magnetictorqug or vice versa. But whenthe magnetictorque is equalto the gravitationaltorque,the ball will not rotate, since the net torqueon the bail is zero. Thisconfigurationis mathematically represented by: pBsn9 = rmgsn9 Or: N=rmg So if we measurer for variousmagneticfields, the functionaldependence of r to I should be a straightline with the slopebeingan expressionthat containsp. But what's r, m, anrd B ? I is the magneticfield at the center of the coils, and can be calculatedfrom the known current. But there are two nr's and two r's that we needto consider. The first zl is the massof the weight, with its correspondingr being the distancefrom the center of the ball to the cetrterof massof the weight. The secondm is the massof the rod, with its correspondingr being the distancefrom the centerof the ball to the center-of-massof the rod. But rememberthat we're trying to measurep usingthe slopeof a line. It turns out that the massof the rd ad its center<f-massdistorce, r, cne combinedin a constantin the grqh of r vs.B, @d do rct affea ilre slope of the line. So the only r that one needs to measureis the r of the weight, andthe onlym that mustbe measuredis the massof the weight. This is the advantageof a slope measurementof p rather than a single point determinationwhere the massand center-of-massof the rod would be essential. So p is the unknown in this experiment. The independentvariable is the B-field at the center of the instrument, and the dependentvariable is r, the displacementof the center of massof the weieht from the center of the ball. Procedure a. First measureall ofthe constantsthat are involved in the experiment. Use a balanceto determinethe mass of the weight, calipers to measurethe diameter of the ball, and a ruler to measurethe length of the ball's handle. From these measurements,r of the weight can be determined. Make wre that you remember to keep all of your measurements in SI units, specifically keeping the magnetic field in Teslas, the length measurementsin meters, and the mass measTrrementsin kilograms. b. Next measurer of the weight for various magnetic fields. Turn on the power supply and the air. Keep both the field gradient and the strobe light off Set the direction of the magneticfield on'hp" so that the handle of the ball points upward when the ball rests on the air bearing. For small currentsthe gravitational torque is greater than the magnetictorque, so the current to start at is about 2.5 amps. With that current, adjust the position of the weight until the ball and the tip remain stationary at about 90 degreeswith respect to the vertical. You might have to steady the rod, weight, and ball with your hand, because the system tends to oscillate and drift due h part to the earth's magnetic field. When the gravitational torque equalsthe magnetictorque (rememberto continuously check the current to make sure that it remains at the set value), take the ball offof the air bearing and turn the current down to zero. Measure the length from the end of the handle to the center of mass of the weight. Add this value to the length from the center of the ball to the end of the handle,and the resulting value is the r of the weight. Repeat these steps for at least six different currents up to 4 amps. The magnetic field can be calculated from the currents. A data table with column headingssuch as the ones below is a good way to orgarize the measurements: 1(amps) B (teslas) r (meters) Reoort l. Composea datatable. 2. Includea samplecalculationof your determinationof the magneticfield from the measured current. t, Graphr vs..8. 4 . Calculatethe magnitudeof your magneticmoment,p. 5 . Extrapolateyour graphin order to determineits y-intercept. From this value, determinethe location of the rod's centerof mass. Comparethis calculated valuewith a measured locationof the rod's centerof mass. EXPERIMENT 2: Harmonicoscillationof a sphericalpendulum Objectives The primaryobjectivesof this experimentare to determinethe dipole momentof the magnetinsideof the ball, and to studythe behaviorof a physicalpendulum'ssmall amplitudeoscillation. Equipment Magnet,powersupply,air bearing,cueball, stopwatctr,calipers,andbalance. Theory This experimentinvolvesdynamlbsprinciples. From classicalmechanics, you are familiarthat a net torqueon an objectcausesa changein that object'sangularmomentum, givenby the expression: dIr sr ) r' = - /-/ dt For our particular systern, if the cue ball is placed in the air bearing with a uniform magnetic field present, and if the "intrinsic" dipole moment of the ball is displaced some angle away from the direction of the magnetic field, the ball will experiencea net torque and will changeits angular momentum. However, it's important to note the direction of the magneticmoment relative to the magnetic field. If the magneticmoment in the ball is displacedan angle d from the axis of the coils (the direction of the field), it experiencesa restoring torque that acts against the angular displacementof p. Thus, the differential equationthat describesthe motion of the ball havingmomentof inertia1is. pxB=I dze ar, Where d is the angulardisplacement from the directionof B. The minussign indicates that the torqueis restoringin nature. In scalarform we have: - PBsin0=, d20 Or, But for smallangledisplacements, sinde d, so -pB9=I d20 ar, We'lI guess the solution of this equation to be 0(t7= Acosail, where a and A are constants. Substitutinginto the differential equationwe have: - pBAcov)t ---Maz coy)t For this equationto be true for all times t, ,'=lu where aris the angularfrequencyof oscillation. If Z is the period of oscillation, 2n T --0) Thus,the finalexpression is: Remember that this is only applicable for a small qngle displacement. l can be well approximatedas the moment of inertia of a uniform solid sphere,namely: ,, mr' = I 5 wherez is the massof the ball and r is the ball's radius. B is againthe independent variable,and Zcan be measuredusinga stopwatch.A graphof Tz w. * shouldyield a straightline whoseslopeincludesthe magneticmoment. Procedure a. First, determinethe momentof inertia of the cue ball using its massand its radius. The masscan be determinedusing a balance,and the radius can be determinedusingthe calipers. b. For this experiment, the field gradientshouldbe off; the strobelight, off; the air on; andthe field'trp". Becausethe magnetictorqueis the only torqueinvolvedin this experiment,the experimentcanbe performedat low currents(smallB). Place the cue ball on the air bearingand set the currentat or near one amp. Give the handleof the ball a smallangulardisplacement from the vertical. Releasethe ball from rest, and it will oscillate. With a stopwatclqmeasurethe amountof time it takesthe ball to completetwenty full cyclesof motion. Make sure you include the counting of the first full cycle. This measuredtime dividedby twenty will be the period of oscillationfor the ball at that particular applied magneticfield. Repeatthis for currentsup to 4 amps. Becauseof the 1/B independent variable that will be graphed,it's a good ideato obtaindatafor quite a few differentlower currentsin order to obtain an even distributionof data points on the graph of Tz vs. I I B . Plot your datain a tablewith columnheadingssuchas the ones below: 1 (A ) B (T) I (20) (s) T: tl20 (s) 'T' (s') l/B (T-I) Reoort l. Composea datatable. Showsamplecalculations. 2. Graphthe functionalrelationshipofl vs. 1/ B . 3. Calculatethe magnitudeof your magneticmomentfrom the slope of this glaph. EXPERIMENT 3: Precessional motion of a spinningsphere Objectives The mainobjectiveis to measurethe dipole momentof a permanentmagnetinside the cue ball. A secondaryobjectiveis to observeand quantifythe motion of a spinning spheresubjectto an externaltorque. Equipment Magnet,power supply,air bearing,cue ball, strobelight, stopwatclr,calipers,and balance. Theory When the magneticmomentis displacedsomeangle from the direction of the magneticfield, the magneticdipole (and subsequently a torque that the ball) experiences causesa changein the ball's angularmomentumin the directionof the torque. This is the centralprincipleof this experiment.The ball is displacedfrom the vertical positionand spurqwith its spin-axisthe u<isthat runs through the handleof the ball. This createsa large spin angularmomentum. The spin axis will remainin a fixed position until the uniformmagneticfield is turnedon. Whenthe magneticfield is turnedon, the magnetic dipolewill experience a torque. This torquewill causea changeof angularmomentumrn the direction of the torque, but becausethe ball already has a large spin angular momenturqit will precess.This motionis similarto that of the spinninggyroscopein the earth's gravitationalfield. You may be familiar with gyroscopicmotion from your mechanics course. The differentialequationfor the motionof the ball is: dL PXB= dt A sideview of the anzularmomentumvectorsis shownbelow: Insind If we look from abovethe spinningba[ at the changeof the angularmomentumfor a short time Al, the picturelookslike the onebelow: L.sind/r Sinces: r0, we can show that: AI "- A,0L"sn?/ LL L0 =V Lsin4' N as At-+O, dL E = QoL "ine' whereQ is theprecessional angularvelocity.Sincep is in thesamedirectionasL, -dL = FB sin9' So pBsnd: and pB: QrI' o-dsn?l The precessionalfrequency(in radianVsecond)is the dependentvariable. It can be determinedby meazuringthe time neededfor the handleof the ball to precessthrouglt 2tr radians,andthendividingthat time by 2n. The magneticfield is the independent variable. The magnitudeof the angularmomentumI is a constantthat can be measured usingthe strobelight. The handleof the ball hasa white dot on its top. As the ball spins the strobelight reflectsoffof this white dot. Whenthe strobelight is flashingat the same frequencyat which the white dot is spinning,the dot will appearstationary. Thus, from the displayedstrobefrequencyand the measurement of the momentof inertia,the spin angularmomentumof the ball at that time canbe calculated.The graphof Oo vs. I will yielda straightline ifZ is heldconstantthroughoutthe experiment.Fromthe slopeof this line,p canbe determined. Procedure ' a. First measurethe constants.The momentof inertiaof the ball is assumed to be that of a solid sphere.The massof the ball needsto be measured with the balance; its radius,with the calipers. The other constantis the spin angularmomentumof the ball. A constantvalue of the spin angularmomentumof the ball can be accomplished by fixing the frequencyof the strobelight. We recommendchoosing a frequencysomewherebetween4.5 and 6 Hertz. Set the strobe light at a frequencyin that rangeandcheckit throughoutthe experimentto makesurethat it remainsconstant. In order for the strobelight to illuminatethe white dot, the room shouldbe darkened.It is not necessary to makethe room completelydark in orderto usethe strobelight effectively. You canmaintainamplelight for writing andworking with the instrument. b. Oncethe strobelight has beenset to someparticularfrequency,record that number. Turn the air on, leavethe field gradientofl and set the field direction 'tlp". Have a stopwatch ready,but don't furn on the currentquite yet. Before you beginmakingmeasurements, practicespinningthe ball. A good techniqueis to spinthe ball (giveit a good,hardspin!)andthenusethe tip of your fingernailto directit to spinaboutthe handle'saxiswith the handlein the strobelight. Notice that the frequencyof the ball's spin does change with time. If you graph this change,it closelyapproximatesan exponentialdecay. It's becauseof this decay that the rangeof frequencybetween4.5 and 6 henz is advised. In this range,the rotationalfrequencydoesnot changesignificantlyduringthe time it takesthe ball to precessthroughoneperiod. Whenyou masterthe spin technique,begn the measurements. Leavethe currentofl spinthe ball up, adjustit so that it's bathedin the strobelight, andthen watchthe white dot. You'll seeit all aroundthe handleat first, but as the ball slowsdowq the white dot will begrnto havea regularrotationof its own. This rotationwill slow down until the white dot stops. As soon as it stops,turn the currentup to I amp,and time a period of the ball's precession.Recordthis time for that current, turn the current o4 and spin the ball up again. Do the same procedure,but this time use 1.5 amps,and continueon in 0.5 amp intervalsuntil you reach4 amps. This shouldgive you enoughdata. Recordyour datain a table with columnssuchasthe onesbelow. 1(A) B (T) Z' G) Qn: Qn)lI (r-t) Report l. Composea datatable. Showsamplecalculations. 2. Graphtherelationship Q vs.B. 3. Calculatethe magnitudeof the your magneticmomentfrom the slopeof this graph. EXPERIMENT 4: Net force in a magneticfietd gradient Objectives Therearethreemainobjectivesto this experiment.The first is to demonstrate that in a uniformmagneticfield, thereis no netforce on a magneticdipole, only a net torque. Secondly,this experimentdemonstrates that there is a netforce on a magneticdipole whenit is in thepresenceof a magneticfield gradient. The third objectiveis to measure the dipole moment,usingthe fact that the net force on a magneticdipole in a magnetic field gradientis proportionalto the magneticmoment. Equipment Magnet,power supply,plastictower apparatus,calibratedspringthat is supported insidethe plastictower, permanentmagnetdisk mountedin a gimbat,ball bearingsthat serveasweights,ruler, andbalance. Theory The principalplayerin this experimentis the magneticdipole. If we modelthe magnetizeddisk insidethe cue ball as a currentloop, and placeit in a uniform magnetic field that is directedalongthe axis of the loop, the force on any infinitesimalsectional of the loop is givenby: dF: idlxB where i is the loop current. The same amount of current passesthrough each infinitesimalsectionof the current loop. Using the right-handrule one can seethat every dF adds to another alF of equal magnitudethat points in the opposite direction. Therefore,there is no net force on a current loop in a uniform magneticfield. If this modelrepresentsour magnetizeddisk then there shouldbe no net force on the disk in a uniform magneticfield. However,if the disk is placedin a non-uniformmagneticfield, that is, a spatial magneticfield gradient,thenthereis a net force on it. To showthis, let's assumethat the directionof the magneticfield is a directionparallelto the z-a>ris. This magneticfield changesonly n the z-directioq where it changesits magnitudewith increasingz. But Mar<well'ssecondequatiorg =0 f"B.nda saysthat the flux of the magneticfield througha closedsurfaceS must equalzero. If a closedsurfacewas eonstructed aroundthe field alongthez-axis,therewould be a net flux of the magneticfield throughthe surface,whichwould violateMaxwell'ssecondequation. We can thus concludethat if thereis a magneticfield that is changingin space,it must changein more than one direction. It follows that in our particularsituation,the field linesare no longerparallelto the aris of the magneticdipole,but are bowed,and curve awayfrom the z-axis.Usingthe expression for the force on an infinitesimalsectionof the currentloop, it canbe shownthat theremustbe a net translationalforce on the magnetic dipole' . It turnsout that the expression for the magnitudeofthis forceis: dBF- = u--dz We can measurethis force. Using the plastic tower apparatusand applyinga field gradient,the suspendedmagnetexperiencesa translationalmagneticforce. For our apparatus,this force is nearlyconstantover a fairly wide range of z-values,sincethe magneticfield gradientis reasonablyconstant. However,there is anotherforce on the magnet,the forcethat the springapplies. The force due to the springis given suspended by Hooke'sLaw, F: kz whereft is the springconstant,andz is the displacement of the magnetfrom its equilibrium position (the position where the magnetresides in the absenceof a magneticfield gradient). The fact that therearetwo forcesactingon the magnetmeansthat the magnet will displaceeitherupwardor downwarduntil the springforce equalsthe magneticforce. At that point the magnetwill ceaseits acceleratiorU and if one stopsits motion, it will z,the following expressioncanbe written: cometo rest. At this displacement F"etr,g : Fficld gradicat or pr= dBF_dz The magneticfield gradientcan be calculated.First calculatethe on-axismagneticfield produced by two coils with currentsin the opposite directions. Then differentiatethat expressionwith respectto z, with the axis of the coils being the z-axis. The spring constantf canbe measuredusingthe ball bearingsas weightsto calibratethe spring. The z is then the dependentvariabl ^a d.isplacement ", ff is the independentvariable, See'Electricity and Magnetism",EdwardM. Purcell,McGraw-Hill, page411, ISBN 0474049084 . proportional to the current. The graph of z vs. be plotted, and from the slope of # " the expectedline, the magneticmoment can be determined. Procedure a. For this experiment,you'll need both the power supply and the magnet coils, but the air bearing is not needed,so the air should be turned off. Place the clear plastic tower on top of the air bearing. Take the cap for the tower, and insert the rod with the spring and suspendedmagnet into the hole in the cap. There is a screw in the cap that can be used to hold the rod in place. Placethe cap on the top of the tower. Now adjust the length of the rod so that the suspendedmagnet is in the center of the coils, that is, where the center of the ball was when it rested on the air bearing. For the first part of the experiment, loosen the screw on the suspendedmagnet so the magnet is free to rotate about a horizontal axis. Set the field gradient off, and turn up a current. Now keep your eye on the magnet, and changethe field direction. Observewhat happensand write it down. b. For the next portion of the experiment,the sameequipmentwill be used, except that now the suspendedmagnet should be screwed down to prevent it from rotating. Tighten the screw eye so that it is in a position where the flat sidesof the magnetare facing up and down. Determine the spring constant *. To do this the magnetic field must be turned off. Hang the magnet down as far as possible, so that only a small portion of the rod is showing above its support. Mark the position of the magnet on the side of the plastic tower with some masking tape. Measure the length of the rod above the plastic holder. Next, take the cap-rod-spring device out of the tower, and add one ball bearingto the magnet. Then place the cap-rod-springdevice onto the tower and adjust the rod so that the magnetis again at the tape mark. Measure the length of the rod above the holder. Subtract from this length the length of the rod when the magnet was in equilibriurn, and the resulting length will be the displacementof the magnet from equilibrium due to one balVweight. (The balls have a massvery close to one gftrm. You can check that with a balance.) Repeat this for two, three, and four balVweights hangttg from the suspendedmagnet. Then graph the force on the magnei (zg) vs. displacement. The slope of that line will be fr. c. Now that the springconstanthasbeendetermined,you can proceedwith the field gradientexperiment.First, use the set screwto lock the dipole in placeto preventit from rotating. With all of the weightsoffof the magneticdipole,adjust the position of the rod so that the magneticdipole is at the centerof the coils. Agaiq placea pieceof maskingtapeon the sideof the plastictube so that you will know the locationof this center. Measurethe lengthof the rod whenthe magnet 'bn" position, is at this center position. With the field gradientswitch in the slowly turn the currentknob to 0.5 amps. The dipolewill displacesomeamount up or dowrqdependingon which directionthe field is relativeto p. It is probably bestto switchthe field directionso that the magnetdisplacesdownward. When the magnetis at the centerof the coils,there'smorerod insidethe tubethan there is outside. Oncethe magneticdipole hasdisplaced,returnit to the centerof the coils by adjustingthe supportrod. Thenmeasurethe lengthof the rod showingabove the tower. Thedifferencebetweenthis lengthandthe lengthof rod showingwhen thereis no magneticfield gradientis the displacement of the magneticdipole due to the magneticfield gradient. Record this displacement z for the current L Proceedfrom an initial current of 0.5 ampsto 4 amps,in 0.5 amp increments. Again, continuouslycheckthe ammeterat high currentsto make sure that it is stayrngat the valuethat you needfor your particularmeasurement. The displacement z is the dependent variablefor the experiment,*tl" $ is the independent variablethat varieswith diffFerent currents. The spring "onr,fi, dB ,t is a constant,so 1r can be obtainedfrom the slopeof the graph of z vs. . * Recordthe datain a tablewith columnssuchasthe onesbelow: 1(A) dB/dz (T/m) z (m) Report l . Write down your observationsfrom part a. of the procedure. Offer an explanationfor theseobservations. 2 . Composea datatablefor the secondpart of the experiment,showingsample calculations. .dB J. u ra p nzvs. &. 4. Calculatethe magnitudeof the magneticdipolemomentfrom the slopeof this graph. The of this experimentis to determinethe magnetMipol6froment of the of the dipolein the cue [--!he secondobjectiveis to veri&-*rcfft dependence distanton-a:rismagneticfield Materials probe,cueball, roll oftapE Gaussmeter will do rulerthat canbe zupportedandheldin anupri of cylindricalsupport itionanda3x5