The Magnetic Dipole Moment — Question 8

Question 8

A gyroscopic magnetic moment precesses without changing its angle to a uniform field

A rapidly spinning dipole obeys the supplied relation $\vec\mu=\gamma\vec L$, with constant signed gyromagnetic ratio gamma. Its only torque is $\vec\mu\times\vec B$ in uniform B. Treat L as classical angular momentum.

1. Write the angular-momentum equation as a precession law and find the angular-velocity vector.

2. Show that the magnitude of L and its angle to B remain constant. Give the precession period.

Question 8 - Solution

A gyroscopic magnetic moment precesses without changing its angle to a uniform field

(a) Method and calculation

$$\frac{d\vec L}{dt}=\gamma\vec L\times\vec B=(-\gamma\vec B)\times\vec L.$$$$\vec\Omega=-\gamma\vec B.$$

(b) Interpretation and further calculation

$$\frac d{dt}L^2=2\vec L\cdot\dot{\vec L}=0,\quad\frac d{dt}(\vec L\cdot\vec B)=0.$$$$T=\frac{2\pi}{|\gamma|B},\quad\cos\theta=\frac{\vec L\cdot\vec B}{LB}=\text{constant}.$$

Physical check

The torque changes the direction of a gyroscopic angular momentum, whereas damping would be needed to drive alignment.