In this experiment, a permanent magnet in the shape of a thin cylindrical rod is suspended from a rigid support by a cotton thread. The cylindrical axis of the magnet is in the horizontal plane, and thus the plane of oscillation is also the horizontal plane. The equilibrium direction of the magnet is determined by the horizontal component of the Earth's magnetic field; that is, the magnet acts as a compass and aligns itself north-south.

When the magnet is displaced from its equilibrium direction, it oscillates in simple harmonic motion. The period of the oscillation depends on the magnetic dipole moment of the magnet and on the strength of the magnetic field; thus, if an additional magnetic field is applied the period of oscillation can be changed. By measuring the period as a function of the applied field and plotting a graph of the results, one obtains a straight line from which the magnetic moment of the magnet and the horizontal component of the Earth's magnetic field can be calculated.

The restoring torque due to the fiber is very small compared to the the restoring torque due to the magnetic field

The magnitude of the magnetic torque vector is given by the familiar rule for the cross product

The angular velocity is the time rate of change of the angle

The equation of motion is now

If the applied magnetic field is aligned parallel to the Earth's magnetic field, then

The equation for the square of the frequency of the magnet's oscillation is

- Use MKS units throughout this lab. That is, convert all length
measurements to meters, all mass measurements to kilograms, etc.
If MKS units are used for the inputs to calculations, the results will
automatically come out in MKS units as well. The units of magnetic dipole
moment are A.m
^{2}(amp meter^{2}) and the units of magnetic field are T (tesla).

- Use a ruler to measure the average outer radius of the coils. Each
coil should be measured several times in different directions and the
results for both coils averaged together.

- Next, we need to calculate the inner radius of the coils. This is
where the wood stops and the copper wire begins. The total number of
turns of wire on both coils together is N=120. There are 60 turns on each
coil. Measure the diameter of the copper wire several times using a micrometer under the platform where
the wire is easily accessible.
**DO NOT PULL THE WIRE OUT OF THE COILS!**Count the number of turns of wire visible in the top layer. You can now calculate how many layers deep the copper is wound, and knowing the diameter of the wire you can find the inner radius of the coil.

- Find the average radius (R) of the coils by averaging the inner and
outer radii. Use this average radius to calculate C.

- Use a triple beam balance to
measure the mass (M) and a Vernier
caliper to measure the length (L) of the cylindrical magnet. Record
these data for use in calculating the moment of inertia (I) of the magnet.

- Suspend the magnet by thread so that it hangs in the central region of
the coils. Adjust the thread so that the magnet hangs in a horizontal
level position.

- When the freely suspended magnet becomes stationary, it will point in a
magnetic north-south direction (by definition). Verify this direction
with the compass held far away from the metal tables and coils.

- Connect the coils, power supply, and
ammeter in a series circuit. This
allows the coil current (i) to be read as the supply voltage is varied.

- Supply about 0.30 amps to the coils. Pull the magnet out of the way
while holding the compass between the coils. Note the direction of the
applied magnetic field. Rotate the coils so that the applied magnetic field
aligns with the Earth's magnetic field. Replace the cylindrical magnet
and make sure that it is level.

- Supply 0.20 amps to the coils. Wait about thirty seconds for the coils
to reach thermal equilibrium. Readjust the power supply if necessary.
Set the magnet into oscillation about an axis along the thread with amplitude
no more than about 20
^{o}. Time 20 oscillations. Calculate the frequency (f).

- Repeat for currents 0.30 A, 0.40A, ..., 1.20A.

- Plot f
^{2}vs. i and fit the best straight line though the data.

- Find the magnetic dipole moment of the magnet from the slope of the plot.
No error estimate is required.

- Use the magnetic dipole moment of the magnet and the y-intercept of the
plot to calculate the horizontal component of the Earth's magnetic field.
No error estimate is required.

- Find C numerically with units and an error estimate.

- Find I numerically with units and an error estimate.

- Why was it desirable to limit the coil current to 1.2 A? List several
reasons.

- Identify two sources of random error and two sources of systematic error.

- How would your plot differ if the Earth's magnetic field B
_{E}and the applied magnetic field B_{A}had been in opposite directions rather than in the same direction? (Hint: Would the slope change? Would the y-intercept change?)

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