A Dipole in an Electric Field — Question 2

Question 2

Quasistatic rotation distinguishes external work from field work

A permanent rigid dipole of magnitude p rotates in a uniform field of magnitude E. Let $\theta$ be the angle between the moment and field, and choose $U(90^\circ)=0$. An external actuator slowly rotates it from $30^\circ$ to $120^\circ$ without changing its kinetic energy.

1. Derive U from the field torque and calculate the external work.

2. Find the work done by the field and the largest opposing torque magnitude encountered during this rotation.

Question 2 - Solution

Quasistatic rotation distinguishes external work from field work

(a) Method and calculation

$$\tau_\theta=-pE\sin\theta,\quad\frac{dU}{d\theta}=-\tau_\theta\Rightarrow U=-pE\cos\theta.$$$$W_{\mathrm{ext}}=\Delta U=pE(\cos30^\circ-\cos120^\circ)=\frac{\sqrt3+1}{2}pE.$$

(b) Interpretation and further calculation

$$W_{\mathrm{field}}=-\Delta U=-\frac{\sqrt3+1}{2}pE.$$$$\max_{30^\circ\le\theta\le120^\circ}|\tau_\theta|=pE\quad\text{at }\theta=90^\circ.$$Quasistatic balance requires the actuator torque to oppose the field torque at each angle, not merely on average.

Physical check

The potential energy rises during the entire prescribed rotation since $\sin\theta>0$ along the path.