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and the top

The type of precession we all have experience with is angular momentum precession. This is exemplified by the top or by the bicycle wheel.

Suppose we hang a bicycle wheel by a string from its axle. If we tie the string at a short distance $\mbox{\it\bf d}$ from the wheel centre, the force of gravity $M\mbox{\it\bf g}$ will apply a torque to the wheel,

$(1) \qquad\qquad \tau = \mbox{\it\bf d}\time M\mbox{\it\bf g}$

Therefore the wheel will not stand upright, it will flop over and fall.

If we instead spin the wheel, it will acquire an angular momentum $\mbox{\it\bf L}$ about its axis and Newton's law of motion impose that the torque determines the rate of change of $\mbox{\it\bf L}$ . The consequence is that the wheel does not flop over any more, instead it precesses about the direction of $\mbox {\it\bf g}$ .

The frequency of this precessional motion is called Larmor frequency.

The sketch on the left is that of a top which spins about its axis and precesses about the vertical direction, although not at right angles with it.

The very same thing happens if we place an elementary magnetic moment $\mbox{\it\bf m}$ (a suitable elementary particle) in a uniform magnetic field $\mbox{\it\bf B}$ . This is because the torque has a similar geometric appearence to that of Eq. (1)

$(2) \qquad\qquad \tau = \mbox{\it\bf m}\time \mbox{\it\bf B}$

and also because an elementary magnetic moment $\mbox{\it\bf m}$ is due to an Amperian current, i.e. an elementary charge orbiting in an atomic loop, or spinning, in some more intrinsic sense. In summary there is an angular momentum $\mbox{\it\bf L}=\hbar \mbox{\it\bf I}$ such that:

$(3) \qquad\qquad \mbox{\it\bf m} = \gamma \hbar \mbox{\it\bf I} = \gamma \mbox{\it\bf L}$

where $\gamma$ is the magnetogyric ratio of the moment and $\mbox{\it\bf I}$ is, for instance, a (quantum) spin vector (we have specialized already on spin, the intrinsic angular momentum of certain nuclei).

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