A Dipole in an Electric Field — Question 9

Question 9

A dipole can follow a rotating field only below a torque threshold

A uniform field of magnitude E rotates counterclockwise in a plane at constant rate $\omega>0$. A permanent dipole of magnitude p rotates about a fixed center with viscous resisting torque $-b\dot\theta$, where $b>0$. In the overdamped model inertia is neglected. Let $\delta=\omega t-\theta$ be the field lead angle.

1. Derive the constant phase-lag condition and the maximum drive rate for synchronous rotation.

2. For $b\omega/(pE)=0.600$, find the stable lag and dissipated power. Explain which of the two lag solutions is stable.

Question 9 - Solution

A dipole can follow a rotating field only below a torque threshold

(a) Method and calculation

$$b\dot\theta=pE\sin\delta,\quad\dot\delta=\omega-\frac{pE}{b}\sin\delta.$$$$\dot\delta=0\Rightarrow\sin\delta=\frac{b\omega}{pE},\quad\omega\le\omega_c=\frac{pE}{b}.$$

(b) Interpretation and further calculation

$$\delta_s=\sin^{-1}(.600)\approx36.8699^\circ,\quad P=b\dot\theta^2=b\omega^2.$$$$\delta=\delta_s+\eta:\quad\dot\eta\simeq-\frac{pE}{b}\cos\delta_s\,\eta.$$The branch with $\cos\delta_s>0$ is stable; the supplementary branch is unstable. At the threshold the two branches meet and linear restoring relaxation vanishes.

Physical check

Although the angle between moment and field is constant in synchronous motion, the external field drive continually supplies dissipated mechanical energy.