PHYSICS 334 - ADVANCED LABORATORY I
Spring 2000
Purposes: Examine the phenomena associated with the dynamics of a magnetic moment. Measure a magnetic moment and verify the dynamical expression =
µ × Β.
Background: Results from introductory magnetostatics (torque on dipole, magnetic dipole moment) and rigid body motion (angular momentum, precession).
Additional information is found in the TeachSpin manual.
Protocol: You will use two methods to determine the magnetic moment of the
‘snooker’ ball containing a small (0.375 in diameter, 0.25 in thick) permanent cylindrical magnet at its center. The magnet will act as a magnetic dipole for the purposes of this experiment. The ball will be immersed in the effectively uniform magnetic field produced by two coils.
The effective magnetic field at the center of the coils is given by
B = {1.36(3)
×
10
-3
T/A}
×
I where I is the current supplied to the coils. The direction of the field is determined by the direction in which current is supplied to the coils.
Follow the instructions and record all measurements in your laboratory notebook. Remember that all numbers should have an indicated uncertainty associated with them. While one student may serve as recorder for the group, all students should transfer that student’s records to their own lab notebook.
The power supply for the coils is NOT current-regulated. As the coils heat up, the current will vary. You should make your measurements as quickly as possible and record the current when you have established the desired conditions in each part. Avoid high currents for long periods of time. Turn the power supply off when not in use.
For extra credit, perform the precessional motion experiment (“C”) given in the TeachSpin manual.
Part I: Determination of magnetic moment by equalizing gravitational and magnetic torque.
From your introductory electricity and magnetism study, you know that in a static uniform magnetic field, a magnetic dipole experiences a torque given by
τ
=
µ × Β.
The magnet at the center of the snooker ball in this apparatus experiences just such a torque when current is supplied to the coils. With a net torque on the ball, the ball will rotate and the aluminum handle will be seen to move.

For this apparatus, a gravitational torque counteracting this magnetic torque can be supplied by a test mass placed on the aluminum handle of the snooker ball. The torque provided by this test mass m when it is a distance R from the center of the ball is given by
τ
= R
× mg
.
When these two torques are equal, the ball will not rotate in either direction.
By sketching a free body diagram of the situation in your lab book, you should show that the various quantities mentioned above, in the absence of all other torques, are related by the expression
R
= µ
B mg
Thus, if we plot a graph of R versus B for various magnetic fields when R is adjusted to balance the gravitational and magnetic torques, a straight line should result where the slope of the line is given by (
µ
/mg).
This line would have an intercept of zero in the absence of all other torques. However, the aluminum rod also supplies a torque to the system. Derive the contribution of the aluminum rod to the total gravitational torque on the ball. Show in your lab book that this contribution does not affect the slope of the line of R versus B for various magnetic fields, but instead determines the intercept of the line.
Note the cautions mentioned above in the protocol. Further, for all the measurements in this part of the laboratory, keep the strobe light and field gradient off.
1.
Measure all quantities and dimensions needed, recording them and their associated uncertainties in your lab book. The key items needed include, for instance, the diameter of the ball, the length of the aluminum handle, and the mass of the test mass.
2.
Set the field direction ‘up’. Turn on the air supply to the air bearing. For small currents, the gravitational torque overwhelms the magnetic torque, so start with a current of 2.5 A to the magnetic coils.
3.
Adjust the position R of the test mass so that the rod remains in the horizontal plane when you release the ball. Since the air bearing is nearly frictionless, you will need to observe the released system carefully since it will precess in the earth’s magnetic field. When the gravitational torque equals the magnetic torque, turn the air supply to the air bearing off, remove the ball and determine R.
4.
Repeat step 2 for four more current settings up to a maximum current of 3.5 A.
5.
Determine the magnetic moment of the ball from the data obtained in steps 2 and
3 by performing a weighted least squares fit to the data. Compare the value of the intercept with the expected value based on the mass and length of the aluminum handle.

Part II. Magnetic moment determination from the oscillation of the ball as a spherical pendulum.
In the absence of other torques, small oscillations of the ball and magnet within the uniform magnetic field are described by those of a spherical pendulum. This is discussed in the TeachSpin manual on pages 10-11. The period of oscillation T for this motion is given by
T
2
2
= π
B
K where K is the moment of inertia of the ball, given by
K
=
2 mr
5
2 where m is the mass of the ball and r is its radius. Thus, by determining the period of the oscillations of the ball for different magnetic fields, the magnitude of the magnetic moment
µ
can be determined. A graph of T slope 4
π 2
K/
µ
.
2
versus B
-1
should yield a straight line with
Note the cautions mentioned above in the protocol. Further, for all the measurements in this part of the laboratory, keep the strobe light and field gradient off. Because there are no gravitational torques involved in this part of the lab, only small currents need be used here.
1.
Turn the air on to the air bearing. Put the ball on the air bearing such that the rod is vertical.
2.
Turn the current on to the coils at a value of 1.0 A.
3.
Give the rod a small angular displacement from the vertical. Determine the time for 20 oscillations. Use that time to determine the period and its uncertainty.
4.
Repeat step 3 for ten more settings of the coil current up to a maximum of 3.5 A.
5.
From the data in steps 3 and 4, use a weighted least squares fit to determine the magnetic moment of the magnet within the snooker ball and the uncertainty associated with that measurement.
Summing up:
Compare the results of the measurements in Parts I and II. If you perform the precessional motion experiment, compare those results as well. Discuss sources of error and how your measurements might be improved. What are the conclusions of your analyses?