Potential Energy of an Electric Dipole in an External Field

When a dipole (dipole moment $\vec{p}$) is placed in a UNIFORM external electric field $\vec{E}$, the two equal and opposite forces $qE$ on its two charges are equal in magnitude and opposite in direction, so the NET force on the dipole is exactly zero -- but, unless $\vec{p}$ happens to be exactly parallel or antiparallel to $\vec{E}$, the two forces act at different points and so exert a net torque, $\tau = pE\sin\theta$, where $\theta$ is the angle between $\vec{p}$ and $\vec{E}$. This torque always acts to rotate the dipole toward alignment with the field (decreasing $\theta$ toward $0$), so a dipole placed in an external field acquires a potential energy that depends on its ORIENTATION $\theta$ -- a genuinely new feature, since a single point charge's potential energy in an external field never involved any orientation at all.

Deriving $U(\theta)$. Take the reference orientation to be $\theta = 90^\circ$ (dipole perpendicular to the field), and assign it $U = 0$. The work an EXTERNAL agent must do, against the restoring torque, to rotate the dipole quasi-statically from $\theta = 90^\circ$ to some general angle $\theta$ is

Since $\cos 90^\circ = 0$, this simplifies directly to

$U(\theta) = -pE\cos\theta = -\vec{p}\cdot\vec{E}$ …