29:28 Physics II
Spring 2006
Laboratory Supplement

Simple Harmonic Motion of a Magnetic Dipole in a Magnetic Field

This lab will utilize the equipment from Experiment E9. The purpose will be to demonstrate an interesting aspect of magnetic dipoles in uniform magnetic fields, and also provide an independent means of computing the dipole moment of the magnet used in Experiment E9.

There is to be no write-up with this experiment. Fill in data in the table provided in this note, and attach additional calculations, observations, and conclusions on an accompanying sheet of paper.

I. Introduction
A magnetic dipole in a magnetic field feels a torque $\tau$ given by

$$\vec{\tau}= \vec{\mu} \times \vec{B}$$ (1)

The angular momentum of a rotating object is

$$\vec{L} = I \vec{\omega}$$ (2)

where $I$ is the moment of inertia and $\vec{\omega}$ is the angular velocity.

We can combine these equations for the equation of motion of a magnetic dipole, which also possesses a moment of inertia $I$, in a magnetic field,

$$\frac{d(I\vec{\omega})}{dt} = \vec{\tau}= \vec{\mu} \times \vec{B}$$ (3)

Since the vectors point in the same direction, and the angular speed is the time derivative of the angle of rotation, the magnitudes of the vectors obey the following equation

$$I \frac{d^2 \theta}{dt^2} + \mu B \sin \theta = 0$$ (4)

If we assume small oscillations with $\sin \theta \simeq \theta$, this equation is that of a simple harmonic oscillator, with motion

$$\theta(t) = \Theta_0 \cos (\Omega t)$$ (5)

where

$$\Omega = \frac{2 \pi }{T} = \sqrt \frac{\mu B}{I}$$ (6)

II. Procedure
Set up the magnetic dipole in the Helmholtz coils as last week. Turn on the air flow to minimize friction between the dipole (encased in the opaque plastic sphere). Set the current in the Helmholtz coil to 1 Amperes.

Rotate the dipole through an angle of about $20^{\circ}$ and release it. Notice that it undergoes simple harmonic (oscillatory) motion. Measure the period as precisely as possible using a stopwatch to measure the time it takes to swing through many (say 20) oscillations, and derive the period. Make several such measurements and take the average. Note also the error in the measurement, which is roughly the difference between a typical individual measurement and the mean. Record your data in the table below.

Repeat this procedure for coil currents of 2 and 4 amperes. Use the calibration formula from Exercise E9 to determine the strength of the magnetic field at the position of the magnetic dipole. Record the data for the magnetic field as well. From equation (6) above, we can see that the magnetic moment of the dipole is given by

$$\mu = \frac{4 \pi^2 I}{T^2 B}$$ (7)

The result for $\mu$ from the three different coil current settings should be the same. Remember that the moment of inertial for a uniform sphere is

$$I = \frac{2}{5}M R^2$$ (8)

where $M$ is the mass of the sphere and $R$ is the radius. Measure these properties of the sphere with instruments in the laboratory, and calculate the moment of inertia of the sphere.

Current (Amp)	B (Tesla)	Period $T$ and Error (s)	Magnetic Moment $\mu = \frac{4 \pi^2 I}{T^2 B}$

III. Questions and Results
Calculate the mean of your values for $\mu$, and give the fractional departure of the most discrepant value. Compare your result for $\mu$ obtained here with what you obtained last week. Look at the variables which went into calculating $\mu$ and think about which one might cause the greatest error.
IV. Precession
You know from your studies of physics that an object with angular momentum, subject to a torque, will precess. The magnetic dipole will precess if it is given angular momentum. Rotate the dipole to an angle with respect to the magnetic field and spin it about its axis. Observe its precession. Repeat this observation with different values of the magnetic field. Comment on your observations on this form.