Physics 338 Advanced Physics Laboratory
Magnetic Torque
Report: Due Thursday, February 11, 2016
1
INTRODUCTION
In this experiment, you will investigate the behavior of a magnetic dipole in an external
magnetic field. Several of the topics studied here will play an important role in the study
of nuclear magnetic resonance.
2
OVERVIEW
Many of the important applications of magnetic dipoles involve atomic or sub-atomic systems, and hence are difficult to observe directly. In this experiment, you will study a
macroscopic classical magnetic dipole in an apparatus specifically designed to highlight
some of the basic properties of objects with both angular momentum and a magnetic
dipole moment.1
The magnetic dipole is a cue ball with a small cylindrical permanent magnet (0.375 inch
diameter, 0.25 inches thick) at the center. The cue ball has a small handle attached to it.2
The magnet acts as a magnetic dipole with its axis pointing along the the cue ball handle.
The handle will be used to hold a small rod for the static torque part of this experiment,
or to spin the cue ball for the dynamic parts of the experiment.
The magnetic field is supplied by a pair of coils wired in series. Each coil has 195 turns.
The equivalent radius of the coils is 0.109 m, and the equivalent center-to-center separation
between the coils is 0.138 m. Normally, the current passes in the same direction around
both coils, so that the magnetic fields from the two coils combine to give a relatively
uniform field at the center. If, however, you set the Field Gradient switch to On, then
the currents in the two coils go in opposing directions. More about this below.
The coil is designed to operate safely over the full range of currents supplied by the power
supply. However, the coils do tend to heat up if you keep the current on for too long, so
you should make sure to turn the current down to 0 when you are not actively making
a measurement. (As the coils heat up, their resistance increases. Hence the maximum
current (and hence maximum magnetic field) you can get decreases. Thus if you want to
cover the full range of available magnetic fields, try not to let the coils heat up much.)
The cue ball rides on a cushion of forced air to produce very low friction.
Lastly, there is a strobe light to be used to help measure the frequency of rotation of the
spinning cue ball in the third part of this experiment.
1
Much of this write-up is based on the instruction manual supplied by TeachSpin with the Mτ 1-A
apparatus.
2
There are two cue balls supplied with the unit. Either should work, but the one marked “2” appeared
to wobble slightly less during the precession experiments, so I would recommend using that one. In any
case, your lab report should specify which ball you used.
Physics 338
3
Magnetic Torque
Page 2
STATIC MAGNETIC TORQUE
3.1
Objective
The objective of this part of the experiment is to measure the magnetic moment µ of the
magnetic dipole (inside the cue ball). You will do this by a series of static measurements
balancing gravitational and magnetic torque.
3.2
Theory
A magnetic dipole µ in an external magnetic field B will tend to line up with the external
field. The torque is given by
~
τ~m = µ
~ ×B
(1)
Now suppose an additional gravitational torque is added. Specifically, suppose an additional mass m is added a distance r from the center of the ball. This will result in a
gravitational torque
τ~g = ~r × m~g
(2)
The system will be balanced when the sum of these two torques is zero.
3.3
Procedure
1. Measure all of the relevant constants: The size and mass of the cue ball, the length
of the handle of the cue ball, and the mass m of the small added weight.
2. Use the bubble level on the apparatus to make sure it is level. (The legs aren’t
adjustable – you have to put small slips of paper under the different corners to get it
level.)
3. Insert the aluminum rod into the handle of the cue ball. The rod has a steel tip at
one end and will stick to the ball easily. Slide the small plastic weight about halfway
onto the rod. Turn the air on and place the cue ball on the stand.
4. Make sure the field gradient and strobe light are off. Set the direction of the magnetic
field to “up.”
5. Slowly turn up the current. You should observe that for sufficiently large current the
magnetic torque balances the gravitational torque. Turn the current back down.
6. Your ultimate goal is to determine µ. The variables at your disposal are r and B.
The instruction manual supplied with the instrument suggests setting the current I
(and hence B) to a fixed value and then adjusting r until the cue ball is balanced.
I found it easier to pick a particular r and then adjust I until it was balanced. You
may choose whatever method you find easier.
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You will also have to define carefully what criteria you will use for the cue ball to
be balanced. (It tends to wander a bit due to the air flow and due to the earth’s
magnetic field.) I suggest trying to balance it at 90◦ from the vertical.
Measure I and r for at least 5 different currents running from approximately 2 A to
the maximum current you can get (about 4 A). Be reasonably careful, but don’t be
too picky. It’s better to get a number of pretty good trials than to spend all day
trying to get a single perfect one. Record your data in a neat table, including a rough
estimate of your uncertainties in your measurements.
Note that it may be difficult to measure r directly—you may wish to consider carefully
what to measure and how best to determine r. You might find it helpful to plan out
the analysis in the next section as you decide what, precisely, you need to measure.
Make sure you record clearly (probably with a sketch) what you actually did.
3.4
Analysis
1. Consult your introductory physics text to learn how to calculate B from the coil
measurements given above and the current I. Explain your final result.
2. Equate τm and τg to get an equation relating B and r. You should get a linear
equation involving µ. Use your data to determine your best estimate for µ (and your
uncertainty in µ).
3. What does your equation predict for the intercept of your graph? Does it agree with
your findings? What is the effect of the aluminum rod? Note that even without the
added mass m, the rod will supply a gravitational torque. (You can try it and see!)
How does that affect your analysis?
4
FORCE ON A MAGNETIC DIPOLE
4.1
Objective
In the previous experiment, you looked at the torque exerted on a magnetic dipole. In this
portion of the experiment, you will examine the net force exerted on a magnetic dipole.
First, you will examine the net force in a uniform magnetic field. Next, you will examine
the net force in a magnetic field with a strong gradient.
4.2
Procedure
1. Replace the cue ball with the plastic tower. Inside the tower, suspended from a spring,
is a small permanent magnet similar to the one inside the cue ball. This magnet is
mounted on an axle so it is free to rotate and align with an external magnetic field.
The magnet is also suspended from a spring so that you can determine the net force
acting on the magnet.
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2. Make sure the field gradient is off. Turn the current off and note the position of the
hanging magnet. If necessary, adjust the position so that the magnet is approximately
at 0 on the scale on the tower.
3. Turn on the current to 2 A. What is the position of the hanging magnet now? Try
it with 4 A. Reverse the direction of the magnetic field and repeat. Summarize your
results. (No graphs or calculations are required.)
4. Now turn the field gradient on. Repeat the measurements at 0, 2, and 4 A. Summarize
your results. (No graphs or calculations are required.)
5
5.1
PRECESSIONAL MOTION OF A SPINNING DIPOLE
Objective
In this experiment, you will observe the effect of an applied magnetic field on a magnetic
dipole that also has angular momentum. Elementary particles such as electrons, protons,
and neutrons, have an intrinsic angular momentum. In this experiment, you will give the
cue ball a “spin” angular momentum by spinning it rapidly around its handle. You should
find that the spinning cue ball precesses about the direction of the magnetic field. By
measuring the rate of precession, you will make a second determination of the magnetic
moment µ of the cue ball.
5.2
Theory
The motion of a spinning top subjected to an external torque is discussed in most mechanics
texts. Only a brief outline will be given here.
As discussed above, the torque τ on a magnetic dipole µ in an external magnetic field B
is given by
~ .
~τ = µ
~ ×B
(3)
But it is also true that
~τ =
~
dL
.
dt
(4)
Hence we can write:
~
dL
~ .
=µ
~ ×B
dt
(5)
Ignoring the small precessional motion, µ is in the same direction as L, so we can rewrite
~
µ
~ = µ L/L
and then rewrite the right-hand-side of the torque equation to get
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~
~
~
~
dL
~ = µL × B
~ =L
~ × µB = − µB × L
~ .
=µ
~ ×B
dt
L
L
L
(6)
This is of the general form
~
dL
~ ×L
~ .
=Ω
dt
(7)
This final equation describes a spinning top that precesses with angular frequency
Ω=
µB
L
(8)
~
around the vertical axis defined by B.
In this experiment, you will set L to a fixed value and then measure Ω, for a range of
applied magnetic fields B. From that data, you will determine µ.
5.3
Procedure
First, you need to develop the technique to set L to a known constant value. You will use
the equation L = Iω, where I is the rotational inertia of a sphere, and ω is the angular
velocity of the sphere.
1. Place the cue ball on the air bearing and turn on the air. Practice spinning the ball
by the handle. You don’t have to spin it too quickly, but you do want it to spin
steadily at a reasonable rate. You may find it convenient to tap the handle gently
with your fingernail or with a plastic pen to try to stop it from wobbling. Don’t
strive for perfection—the ball will wobble no matter what you do—but try to get it
reasonably steady.
2. Turn on the strobe light and set it to a frequency somewhere between about 4.5 and
6 Hz. Reocrd the frequency. Note that the apparatus displays the linear frequency
f , not the angular frequency ω.
3. Now spin the ball and steady it so that the strobe light illuminates the handle. The
ball will gradually slow down. When its rotational frequency is the same as the strobe
light frequency, the white dot on the cue ball handle will appear to stand still.
Note that the ball will slow down during the course of the experiment. That’s ok.
The frequency range suggested is such that the rotation won’t slow down too much
during the time it takes the ball to precess around once.
4. Now you are ready to take measurements. Turn the current off, set the field gradient
off, and the field direction up. Spin the cue ball. When the cue ball’s rotation rate
matches the strobe light, turn the current quickly up to 1 A and use a stop watch to
record the time it takes the cue ball to precess around once. You may find you have
to try several times to get a decent measurement.
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Again, don’t strive for perfection. The ball will always wobble some. If in doubt,
take several trials and record them all.
5. Repeat your measurements for currents up to the maximum of 4 A.
5.4
Analysis
1. Determine L, the angular momentum of your spinning cue ball.
2. Use your data and Eq. 1 to determine your best estimate for µ (and your uncertainty
in µ).
3. Compare this to your earlier result for µ.
4. What does your equation predict for the intercept of your graph? Does it agree with
your findings?
6
DEMONSTRATING MAGNETIC RESONANCE
Lastly, you will observe the classical “spin flip” when the precessing magnetic dipole is
subjected to a rotating magnetic field perpendicular to the externally applied static field.
6.1
Theory
See class notes.
6.2
Procedure
Place the “rotating” magnetic field apparatus over the air bearing. This will supply a small
(∼ 1 mT) horizontal magnetic field. Spin the cue ball, steady it, and then turn the current
on to a moderate value so that the cue ball precesses as in the previous part.
~ rot at the same frequency as the ball is precessNow, rotate the additional magnetic field B
~ rot perpendicular to µ. Observe how the ball “flips”.
ing. Be sure to always keep B
~ rot at a different frequency or in the other direction. You
For variations, try rotating B
should find that such conditions do not lead to a spin flip.
6.3
Analysis
No analysis is required. Just give a qualitative statement of your observations.
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