The Magnetic Dipole Moment — Question 2

Question 2

Quasistatic rotation measures the change in dipole potential energy

A rigid permanent dipole has fixed moment magnitude mu in uniform field B. An external agent slowly rotates it from theta=30.0 degrees to theta=120 degrees, where theta is measured from B. The dipole starts and finishes at rest; ignore dissipation.

1. Find the work done by the external agent and by the magnetic torque.

2. Use the energy curvature to classify the aligned and antialigned equilibria.

Question 2 - Solution

Quasistatic rotation measures the change in dipole potential energy

(a) Method and calculation

$$U(\theta)=-\mu B\cos\theta,\quad W_{\mathrm{ext}}=\Delta U=\mu B(\cos30^\circ-\cos120^\circ).$$$$W_{\mathrm{ext}}=\frac{\sqrt3+1}{2}\mu B,\quad W_B=-W_{\mathrm{ext}}.$$

(b) Interpretation and further calculation

$$\frac{dU}{d\theta}=\mu B\sin\theta=0\quad\text{at }0,\pi.$$$$U\prime\prime(0)=+\mu B>0\text{ (stable)},\quad U\prime\prime(\pi)=-\mu B<0\text{ (unstable)}.$$

Physical check

Zero torque identifies an equilibrium orientation; the energy curvature determines its stability.