The Magnetic Dipole Moment — Question 4

Question 4

The dipole energy barrier separates oscillation from full rotation

A fixed-magnitude dipole starts aligned with uniform B and has initial rotational kinetic energy K0 about a frictionless planar pivot of inertia J. Its potential is $U=-\mu B\cos\theta$.

1. Find the critical initial speed separating bounded oscillations from continuous full rotations.

2. For K0=mu B/2, find the turning angle. Explain the special behavior exactly at the critical energy.

Question 4 - Solution

The dipole energy barrier separates oscillation from full rotation

(a) Method and calculation

$$\Delta U_{0\to\pi}=2\mu B,\quad\tfrac12J\omega_c^2=2\mu B.$$$$\omega_c=2\sqrt{\mu B/J};\quad |\omega_0|>\omega_c\text{ permits continuous rotation}.$$

(b) Interpretation and further calculation

$$K_0=\mu B(1-\cos\theta_{\max})=\mu B/2\Rightarrow\theta_{\max}=60.0^\circ.$$$$K_0=2\mu B:\quad |\dot\theta|=2\sqrt{\mu B/J}|\cos(\theta/2)|\to0\text{ as }\theta\to\pi.$$At this separatrix, the ideal dipole approaches the unstable top only asymptotically.

Physical check

Exactly reaching the barrier energy is not enough for finite-speed passage over the unstable orientation.