Teachspin - Magnetic Torque: Experiments with Electromagnetism, Torque, Simple Harmonic Motion, Precession and a classical analog of an NMR Spin Flip.

Magnetic Torque - "A New Classic"

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U of Chicago: Magnetic Torque & Spin Resonance (1.1 mb)
CalPoly SLO: A Novel Way to Spin the Sphere (194 kb)
Expanded experiments in PDF (111 kb)
Gravitation Torque Experiments (14 kb)
Demonstrating Magnetic Resonance (42 kb)

Magnetic Torque experiments include quantitative measurements of electromagnetism, magnetic and gravitational torque, simple harmonic motion, and precession along with an introduction to the concepts of gyromagnetic ratio, nuclear magnetic resonance and the NMR spin-flip.

Students develop an operational definition of magnetic moment and then measure the magnetic moment of a magntized disk four different ways. The first three measurements involve the same magnetic dipole, allowing more sophisticated students to evaluate the limitations of each type of measurement.

MAGNETIC TORQUE EQUALS GRAVITATIONAL TORQUE
This is a statics experiment. A gravitational torque is applied to the ball by a weight on an aluminum arm inserted in the ball's handle.

When the magnetic torque is adjusted to equal the gravitational torque:

µ × B = r × mg

Here, r is the radial distance of "m" from the center of the ball.

Since B and mg are measurable quantities, µ can be determined.

A graph of the gravitational torque, rmg, as a function of the magnetic field B yields a straight line whose slope is the magnetic moment µ. This experiment also provides an "operational definition" of magnetic moment; a one unit magnetic moment experiences a one N-m torque in a one tesla field. Simple algebra shows N-m/tesla=amp-m².

Notice that the line on the graph of Torque vs. Magnetic Field does not go through (0,0). This is because the magnetic torque must balance the combined torque of both the mass and the aluminum arm.

The intercept indicates the B field needed to balance just the torque of the aluminum rod but does not affect the slope from which µ is calculated.

HARMONIC OSCILLATION

The cue ball, supported in a uniform magnetic field by the air bearing, acts like a spherical physical pendulum.

Its oscillations are described by: - |µ × B| = I(d²q/dt²)
where I is the moment of inertia and q is the angular displacement.
For small q, the motion is simple harmonic with a period T given by:

A sample of one student's data is given to the right. The magnitude of the magnetic moment can be calculated from the slope of the line. Oscillation Data

PRECESSIONAL MOTION

Spinning the ball gives it angular momentum L, which can be determined by measuring its rotational frequency using the built-in strobe light and counter.

In a uniform magnetic field B, the precessional motion of the spinning ball with its "intrinsic" magnetic moment µ is described by: µ × B = dL/dt
The solution for the precessional frequency is:

W_p = µB/L One way to extract the magnetic moment from this phenomenon is to measure the precessional frequency as a function of magnetic field for a given angular momentum.

A sample of one student's data for this measurement is shown to the left. The linear behavior confirms the theory and the slope gives the third independent measure of µ.

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There are three objectives to the experiments which utilize the magnetized disk suspended from the calibrated spring:

To demonstrate that there is no net force on a magnetic dipole in a uniform magnetic field, only a net torque.
To recognize that there is a net force on a magnetic dipole in the presence of a magnetic field gradient.
To measure the magnetic moment using the relation:

F = µ_z(dB/dz)

Student data for magnetic force is shown on the left. The magnitude of this dipole moment will be close to those found for the dipole imbedded in the cue ball.

With the addition of a horizontal uniform rotating magnetic field, the Mt1-A becomes an instrument to demonstrate a classical analog for magnetic resonance.

Magnetic Torque with Rotating Field Attachment

Magnetic resonance is observed on atomic size particles that have both intrinsic magnetic moment and angular momentum.
The cue ball can be made to mimic the atomic particles by spinning it so that it has angular momentum as well as its "intrinsic" magnetic moment.

The horizontal magnetic field is manually rotated by the demonstrator at the Larmor precession frequency for the particular angular momentum "spun-up." A 90° or 180° nutation spin can easily be produced.

Given the approximate magnitude of the rotating field, students can even estimate the time necessary to nutate the ball by 90°.
Such demonstrations significantly enhance a student's understanding of the complexities of rotating coordinate systems, effective fields, and spin nutation that are fundamental to mastering the principles of magnetic resonance.

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