A Dipole in an Electric Field — Question 4

Question 4

Small angular oscillations and anti-aligned instability share one time scale

A freely rotating dipole with moment p and central moment of inertia I is placed in a constant uniform field E. Its angular equation is $I\ddot\theta=-pE\sin\theta$. No damping is present.

1. Linearize near alignment, derive the small-oscillation period and state how doubling E changes it.

2. Linearize near anti-alignment by writing $\theta=\pi+\phi$. Solve for a small displacement $\phi(0)=\phi_0$, $\dot\phi(0)=0$ while the linear approximation holds.

Question 4 - Solution

Small angular oscillations and anti-aligned instability share one time scale

(a) Method and calculation

$$|\theta|\ll1:\quad\ddot\theta+\Omega^2\theta=0,\quad\Omega=\sqrt{pE/I}.$$$$T=2\pi\sqrt{I/(pE)},\quad T(2E)=T(E)/\sqrt2.$$

(b) Interpretation and further calculation

$$\sin(\pi+\phi)\simeq-\phi\Rightarrow\ddot\phi-\Omega^2\phi=0.$$$$\phi(t)=\phi_0\cosh\Omega t.$$Alignment is stable under small angular perturbations; anti-alignment has zero torque exactly but amplifies any small displacement.

Physical check

The linear unstable solution ceases to be accurate once $|\phi|$ is no longer small.