Intermediate and Advanced Labs PHY3802L/PHY4822L

Torsional Oscillator and Torque Magnetometry

Labmanualandrelatedliterature

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Thetorsionaloscillatorandtorquemagnetometry1. PurposeStudythetorsionaloscillatorasanexampleofharmonicoscillator.Asanapplication,useitas

torquemagnetometertofindthemagneticmomentperNd2Fe14Bformula.

2. ApparatusTorsionalOscillatorapparatusfromTeachSpin,oscilloscope,multimeter,DCpowersupplyand

lowfrequencysourcegenerator.

3. DescriptionofexperimentTheapparatusallowstoapplyvarioustypesoftorques,suchasgravitationalormagnetic,static,

periodicorchaotic,torotateamasssuspendedonawirewithaknowntorsionconstant!.Inyour work, you will apply only magnetic torques by driving an electric current in a pair of

Helmholtz coils. The wire passes through the middle of the split coils and has a stack of 4

permanentmagnets(Nd2Fe14B)attachedtothewire.Themagneticfieldofthecoilswillinduce

atorqueonthemagnets(andthustothewire),generatingatorsionaldeformation.

Fig.1Sketchshowinghowthefield!ofthesplitcoilsgeneratesatorque!ontothemagneticmoment!ofthe

permanentmagnets,whichareattachedtothetorsionaloscillator(notshown).

Inthebackgroundpartofthelabreport,givetheentiretheoreticaldiscussionpresentedhere

aswellastheerrorpropagationstudy.

Dampedharmonicoscillator

In the first part of the experiment, you will apply a static magnetic torque and study the

dependencebetweentorqueandangulardisplacementduetotorsion.Whenyouwillsuddenly

turn off the torque, youwill observe oscillations back-and-forth, similar to themotion of an

harmonicoscillator.

The torsional oscillator has an angular degree of freedom, let’s call it θ, which follows theequationofmotionofanharmonicoscillator[1]:!" + !" + !" = 0where!ismass,! = !isthe angular acceleration, ! ≥ 0 is a friction constant, ! = ! and ! > 0 is a torsion (spring)constantdefiningtherestoring force.Withthenotations:dampingconstant! = !/(2!)andresonance frequency!! = !/! theequationofmotionbecomes! + 2!" + !!!! = 0withsolutions of the form !!". After substitution of ! and solving for !, one gets !± = −! ±

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!! − !!!and! ! = !!!!!! + !!!!!!with!±constantsdependingontheinitialconditions.Thesquarerootin!imposesthreesituations:

A.Overdampedoscillator! > !!, !± ∈ ℝWrite! ! and explain which of the !-values describes the exponential decay of! back toequilibrium.

B.Criticaldamping! = !!Write! ! .

C.Underdampedoscillator! > !!, !± ∈ ℂIn this case, !± = −! ± !" with ! = !!! − !! and ! ! = !!!" !! cos!" + !! sin!! =!!!!" cos !" − !! where!istheamplitudeoftheexponentiallydampedoscillationsand!!istheinitialphaseofthemotion.Itisessentialtonotethattheresonancefrequency!ofthetorsionaloscillatorisdecreasedbydampingeffects.If! ≪ !!,theTaylorexpansionof!gives! = !! 1− !! 2!!! .Asyouwillfindoutinyourwork,whentheeddycurrentsbrakesare

fully retracted, !! 2!!! is of the order of 10-6 - 10-7. Only in this case, one can consider

! ≈ !!andyouwillfind!!andtheso-calledqualityfactoroftheoscillator! = !! 2! .

Drivenharmonicoscillator

Inthesecondpartoftheexperiment,youwillapplyasmallperiodicmagnetictorque,usingan

alternatingcurrentsin!! ratherthanadccurrentasinthepreviousexperiment.Theequation

ofmotionbecomes:

! + 2!" + !!!! = ! sin !!!

where!! and!arethedrivefrequencyanditsamplitudeimposedbytheexternalsinusoidal

current generator. The solution ! ! in this case shows a transient regime exponentially

dampedinatimescale1 !followedbyasteady-stateregimewithasolution:

!! ! = !!!!

! !! !"# !!!!!

where! !! = !!!

!!!!!!!!! !!!!

!andtan! =!!!!!!!!!!!

.Thismeansthattheoscillatorwillrotate

back-and-forthwiththesamefrequency!! astheexternaldrive.Twoessentialobservationsare to bemade here. First, while!! is fixed by the drive, the amplitude of the oscillations!!!!! !! strongly depends on the relationship between!! and!!: it is very small when

!! ≪ !" ≫ !! and maximum at the resonance condition !!,!"# = !!! − 2!! (you candemonstratethisbysolving!" !!! = 0).Notethedifferencebetween!!,!"#and!.Onlyatresonance,thetransferofenergybetweenthedrivingforceandtheoscillatorisoptimal,thus

generatingaperiodicaltorsionwiththelargestamplitude.Secondobservation:thereisaphase

shift!betweenthedriveandoscillator.In this second part of the experiment, youwill verify quantitatively the shape of the! !!

functionandidentifytheresonancefrequency.Youwillalsocheckqualitativelythevariationof

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!whengoingthroughtheresonancecondition.Torquemagnetometry

Inthethirdpartofthelab,youwillusethedataalreadyacquiredinpartone,tostudytheuse

oftheoscillatorasatorquemagnetometer.Asdetailedbelow,youwillfindthenumberofBohr

magnetonsperformulafortheNd2Fe14Bpermanentmagnetattachedtothewire.

4. MeasurementprocedureCaution:Followthe lab instructioncarefully.When indoubt,call theclass instructor.This

equipmentiswaytooexpensivetotakeanychances!!

Part1:Dampedharmonicoscillator:TorsionversustorqueandQfactor

1. Engagetheeddycurrentbrakesabout1/3orsoofthefullbrakingsetting.Thiswillquickly

stabilizetheangularpositionofthetorsionwirewhenchangingthemagnetictorque.

2. With the DC power supply turned off, connect it to the drive ports of the apparatus.

Connect the ports of the 1Ω resistor to a multimeter to read its voltage. Explain why the

readingofthisvoltageisactuallythecoilcurrentinamperes.

3. Readtheangularpositionwiththedcpowersupplyturnedoff,let'scallitθ0(thescaleisinradians).Makesuretoproperlyalignyoureyewiththetwoverticalmarksonthetransparent

plastic.UsetheZeroAdjustknoboftheapparatustozerotheindicationofthemultimeter.This

isyourfirstdatapoint.

4. Turnon thepowersupplywith thevoltageoutput set tominimum.Youwillnotea finite

current, 70-80 mA, which is normal. Read again the angular position. For these and all

subsequentangularmeasurements,youhavetosubtractθ0toobtainthetorsionangleθ.Haveacolumnwiththerawvalue,let'scallitθraw,andanotheroneforθ= θraw- θ0.Assumethatthe

readingerroris~halfofthesmallestdivisiononthescale.

5. Increasethevoltageuntilthenexttwodigitvalueoftheθraw ,suchas2.9,2.8,2.7or3.0,3.1,3.2,etc.Whenreachatwodigitvalue,makesurethattheangle isstableandrecordthe

coilcurrent inamps.Themaximumcurrentshouldnotbehigherthan~2.2Amps;duetocoil

warmingeffects,takedataquicklyandthenrampthecurrentdown.

6. FlipthewiresattheportsoftheDCpowersupplytoflipthedirectionofthemagneticfield

(seeFig.1).Repeatstep5aboveandbringthecurrenttominimum(~70-80mA).Intotal,you

shouldhavegatheredabout20datapoints.

7. Connect the oscilloscope: on channel 1 connect the voltage from the 1Ω resistor. Onchannel2,connecttheportthatmonitorstheangularpositionofthetorsionaloscillator.You

mayneedtoadjusttheparametersoftheoscilloscopeandrepeatsomeofthemeasurements.

Usualsettingforthetimebaseis~25sec/div:sincethedecayoftheoscillationsisquiteslow,

youneedquitea long time interval to record it. Set the trigger to channel1 in autoor scan

mode. This should allow you to chose the vertical scalewithmore ease. After that, set the

triggeron"normal"mode;inthisway,asuddenvariationincoilcurrentwillbeinterpretedby

theoscilloscopeasa"start"commandtobegindataacquisition.

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8. Slightly increase thecoil currentuntil youobserveadeviationθ= θraw- θ0ofonly0.1-0.15radians.Whentheangleisstable,gentlyfullyretracttheeddycurrentbrakesandwaitforthe

systemtostabilize.Don'ttouchthetable,anyvibrationwillbetransferredtotheoscillator.

9. Makesurethattheoscilloscopetriggerindicates"Ready".SuddenlyturnofftheDCpower

supply, which should trigger the data acquisition by the oscilloscope, and do not produce

vibrations (touching the table ormoving chairs) during that time. At the end, an oscillatory

signalshouldbeplotted,withavisibledecayallowingtofind!! !, thetimeduringwhichthe

oscillationshalveinsize.Savethescreenimage(andshowitinyourreport)usingtheUSBport

oraphonecamera.

10. Use theoscilloscope functions toestimate theperiodof theoscillations!! = !!!! (zoom in

thetimedomain).Useverticalandhorizontalcursorsto findthisperiodaswellas!! !.NowyoucandisconnecttheDCpowersupplyandthemultimeter.

Part2:Drivenharmonicoscillator:Amplitudeversusdrivefrequencyandresonance

11. Withthelowfrequencysignalgeneratordisconnected,turnitonandadjustitsparameters:

mostimportantly,theamplitudeat0.1V(don'triskhugeforcesonthetorsionoscillator!!!).Set

ittodeliversinewavesandstartwithafrequencyof0.8Hzor800mHz.

12. Slightlyengageoneofthetwobrakes.Youwanttoreachthesteadystate!! ! quicklybutyoudon'twanttodecrease

significantly!!,!"# and! !!,!"# . Themagnet of thebrake

shouldbarelycomeclosetotheCudisc,likeinthisphotoon

theleftbrake.

13. Nowyoucanconnectthesignalgeneratortothedriveportsofthecoils(liketheDCpowersupplypreviously).YoushouldseesmalloscillationsoftheCudisc.Ontheoscilloscope,setthe

triggeronchannel2andadjustitsleveltohaveasteadydataacquisitiongoingon.Adjustthe

the time base to see one oscillation (faster acquisition). Use the automatic measurement

function of the oscilloscope, tomeasure peak-to-peak amplitude of the torsional oscillations

(thatis,channel2!!!).14. Withastepof5mHz,record!!! from0.82to0.92Hz.Adjusttheverticalscaletomaximize

thesizeoftheoscillationsontheoscilloscopescreen.Thereisquiteadifferencebetween!!! atresonanceandoff-resonance.Recordthefluctuationsof!!! asitsuncertainty!!.15. Take 3 screen captures: before, at and after resonance, showing the signals from both

channels. As in Fig. 4, average 128 traces, place the vertical cursors andmeasure∆! beforesavingthephoto(∆!isthetimeintervalbetweentwominimaondifferentchannels).

Part3:Torquemagnetometry:Magnetictorqueanalysis

16. ThedataanalysiswillbedescribedintheAnalysissectionofyourreport.

5. Analysis

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Part1

Read carefully [2] to get acquainted with the following reasoning. To get to the correlation

between applied torque and torsion angle, let's start with the static equilibrium condition:

!!"#$%&'( = !!"#$%"&orinotherwords!" cos! = !".Sincethefield!isdirectlyproportionaltothecurrent!imposedbytheDCpowersupply,onecanwrite! = !".Discusswhy! = 0.058Nm/radand! = 3.234mT/A.Assumethatthesevalueshavenouncertainty.

Justify why a plot !/ cos! vs ! is more

appropriatetoourstudy,andnot!vs!.Givethevalue of θ0. Show the data in a table withcolumns !,!!"# ,!,! cos! ,!!"# where !!"# isthe uncertainty of !/ cos! and is evaluatedfollowing theerrorpropagationtheoryyougave

inthebackgroundsection.

Execute the plot !/ cos! vs ! with the

appropriateerrorbars(likeinFig.2)andperform

a linear interpolationtoobtain theslopeand its

uncertainty.Calculate!anditsuncertainty!!.

Fig.2Data(dots)andfit(line)of!/ cos !vs!.To calculate the Q-factor, show the screen capture of the decaying oscillations in the case

where the brakeswere fully retracted. Explain how you got the values for!! and!! !; givethemwithyourestimateduncertainties.

The Q-factor is defined as ! = !!!! and has the meaning of a ratio "energy stored/energy

dissipatedbycycletimes2π".Inthecasestudiedhere,youwillsubstitute!with!! !.Solvefor!fromtheequation!!!!! ! = 1 2andshowthattheQ-factorcanbeexpressedas! = !

!"!!! !!!

.

Calculate!anditsuncertainty,followingtheerrorpropagationtheory.Part2

Showthedatainatablewithcolumns!! ,!!!,!! where !! = !!

!! is the drive frequency. Plot it

similarly to the example in Fig.3, with error bars

(yourplotwillhavemoredatapoints).Theredfit

is optional, and you can use a data analysis

softwareforsuchapurpose.Otherwise,useyour

datatoestimate!!andtheoscillatorbandwidth!". The bandwidth is defined as the frequencyseparation between points where the amplitude

drops by a factor of 1 2 (and not by 1/2 !!explain why!). The fit would give you a better

resolutionof!!and! = !" 2.

Fig.3Data(dots)andfit(line)of!!!vs!!.

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In Fig. 3, the amplitude shows a quality factor a little bit reduced by the effect of the eddy

currents brake, to about ~600-700. The Q-factor is given by ! = !!!". Calculate Q and its

uncertainty.

Showthe3screenshotsanddiscussthephaseshift!betweenthetwochannels.Anexample

isshowninFig.4for!!=0.83,0.863(resonance)and0.91Hz.Yourfrequencies,especiallytheresonance value,may be different. Use the vertical cursors to showminima (ormaxima) on

differentchannels,tobeabletocalculatethephaseshift.Knowingthatthetwosignalsaresine

waveswiththesamefrequency,calculatethephaseshift! = !!∆!forallthreesituations.What is the range over which ! should vary when going from 0 to infinity, through the

resonance(seetheequationoftan!)?Withinthisrange,whereis!atresonancelocated?Isyour data qualitatively in agreement with the theoretical expectations? Can you imagine a

reason that couldcreateanadditional constant shiftbetween thecoil current (orB)and the

signalmonitoringtheangularposition!?

Fig.4ExampleofscreenshotsusingtheUSBport(data

isexportedasatableaswell).Theverticalcursorsshow

successive minima on different channels to show the

phase difference. Note the values of ∆! for !!=0.83,0.863and0.91Hz,startingfromtopleft,right,bottom.

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Part3

Usethetotalmagneticmoment! ± !!calculatedinPart1,tofindthedensityofmagneticmoment

!,asexplainedin[2].Assumethatthevolumeof

the 4 magnets ! is known precisely (no

uncertainty). Finding! is already an example of

torque magnetometry and in the following youwill use crystallographic information to estimate

themagneticmomentperformulaofNd2Fe14B.As

described in [3], thiscrystalhasa tetragonalunit

cell (shown in Fig. 5)withdimensions! =8.78Åand ! =12.21 Å. The unit cells contains ! = 4formula,thatis17x4=68atomsperunitcell.

Sinceyouknowthedensityofmagneticmoment

!, find the total magnetic moment per formula

and itsuncertainty.Compareyourvaluewiththe

value measured in [3] at 293K using neutron

diffraction.UseTable II in[3]anddetailhowyou

calculatetheirtotalmomentperformula.

Fig.5UnitcellofNd2Fe14B,from[4].

6. Additionalquestions6.1 Both ! and !" are quantities which are specific properties of the oscillator in itsenvironment, independentofanydrive.Actually, in thecaseof freelydecayingoscillations in

Part1(thatis,nodrive!! )whatisthemeaningofafinitebandwidth;itistheeffectofwhatphenomena?

6.2Optionally,asafollow-upto6.1,discussalsothemeaningofacurvelookingliketheonein

Fig.3but in the caseofnoexternaldrive.Moreprecisely,howcanadecayingoscillationby

relatedtosuchacurve?

7. References[1]J.R.Taylor,ClassicalMechanics,chapter5,UniversityScienceBooks,USA(2005).[2]TorsionalOscillator-InstructorGuidebySpinTeach,sections2.2-2.5,availableonline(PDF).

[3]J.F.Herbst,J.J.Croat,W.B.Yelon,"StructuralandmagneticpropertiesofNd2Fe14B",JournalofAppliedPhysics57,4086(1985).[4]J.F.Herbst,J.J.Croat,F.E.Pinkerton,"RelationshipsBetweenCrystalStructureandMagnetic

PropertiesinNd2Fe14B",PhysicalReviewB29,4176(R)(1984).