Torque on an Electric Dipole in a Uniform Electric Field

Sections 1.9-1.11 considered the field produced by a dipole. This section instead asks what happens to a dipole when it is placed inside someone else's field -- specifically, a UNIFORM external electric field $\vec{E}$ (a field of the same magnitude and direction everywhere, such as the field between two large, closely spaced charged plates).

No net force, but a net torque. A dipole consists of a charge $+q$ and a charge $-q$. In a uniform field $\vec{E}$, the force on the positive charge is $\vec{F}_+=q\vec{E}$ and the force on the negative charge is $\vec{F}_-=-q\vec{E}$ -- equal in magnitude and exactly opposite in direction, so the net force on the dipole as a whole is zero:

A dipole in a uniform field therefore does not accelerate bodily in any direction. However, these two equal and opposite forces do NOT act along the same straight line -- they act at the two ends of the dipole, separated by the perpendicular distance between the two lines of action -- and a pair of equal, opposite, non-collinear forces is exactly what constitutes a couple, producing a net turning effect, or torque, even though the net force is zero.

Deriving the torque. Suppose the dipole moment $\vec{p}$ makes angle $\theta$ with the field $\vec{E}$. The perpendicular distance between the two parallel lines of action of $\vec{F}_+$ (through the $+q$ end) and $\vec{F}_-$ (through the $-q$ end) is $2a\sin\theta$ (the component of the separation $2a$ perpendicular to the field direction). The magnitude of the torque produced by a couple is (force) $\times$ (perpendicular distance between the two forces):

using the dipole moment $p=q(2a)$ from Section 1.8. In full vector form, since the torque is directed perpendicular to the plane containing both $\vec{p}$ and $\vec{E}$ (by the right-hand rule for a cross product),

$\vec{\tau} = \vec{p}\times\vec{E}$ …