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An Electric Dipole in a Uniform Field

A uniform field pulls a dipole's two charges in opposite directions with equal force, so there is no net push, only a twist. The two forces make a couple whose torque is $\boldsymbol{\tau} = \mathbf{p}\times\mathbf{E}$, of size $pE\sin\theta$, always rotating the dipole toward alignment with the field. There is a stored orientation energy that goes with the angle, $U = -\mathbf{p}\cdot\mathbf{E} = -pE\cos\theta$: lowest when the dipole points along the field, highest when it points against it. Left alone, the dipole does not just snap into place; it swings past and comes back, librating about the field direction exactly like a pendulum about straight down, with small-oscillation period $T = 2\pi\sqrt{I/pE}$. Drag the dipole to any angle and release it, turn the field up to deepen the energy well and quicken the swing, and add a little damping to watch it spiral into alignment. The panel below is that energy well, with the total-energy line whose crossings with the curve are the turning points the dipole rocks between.

Figure 1. An electric dipole in a uniform field. Top: the field (teal) pulls the positive charge along it and the negative charge against it (a couple, gold), the torque rotating the dipole (purple p) toward alignment. Bottom: the orientation energy U = -pE cos(theta) with the total-energy line; the dipole librates between the turning points where they cross. Method: the rigid-pendulum equation I theta'' = -pE sin(theta), velocity-Verlet. Source: Griffiths, Introduction to Electrodynamics, 4th ed., Sec. 4.1.3.
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WHAT TO TRY

  • Drag the dipole to a large angle and release it: it swings toward the field, overshoots, and librates, the torque always pulling it back toward alignment.
  • Set damping to none: with no losses the dipole librates forever and the total-energy line stays put (energy is conserved).
  • Add damping: the total-energy line descends, the turning points close in, and the dipole spirals into alignment at the bottom of the well.
  • Turn the field up: the well deepens and the swing quickens, the period shrinking as $1/\sqrt{E}$.