Download Magnetic Torque

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).
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STEL
Dt.<
T/P
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;:?@=.
.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