= Solution
An <electric dipole> is a localized pair of opposite <electric charges>, or more generally a neutral <electric charge> distribution with nonzero first electric moment. For charges $-q$ and $+q$ separated by <displacement> $\mathbf d$ directed from negative to positive, the <electric dipole moment> is
$$
\boxed{\mathbf p=q\mathbf d,\qquad [p]=\mathrm{C\,m}.}
$$
A neutral <molecule> can have a permanent <electric dipole moment> when its positive and negative charge centres do not coincide. An applied <electric field> can also produce an induced <electric dipole moment> by distorting an initially nonpolar <electric charge> distribution.
At distance $r\gg|\mathbf d|$ in <vacuum>, expand the difference of the two point-charge potentials to first order in the separation. The leading <electric potential> is
$$
\Phi(\mathbf r)=\frac{\mathbf p\cdot\widehat{\mathbf r}}{4\pi\varepsilon_0r^2}.
$$
The monopole term cancels because the net <electric charge> is zero. Taking $\mathbf E=-\nabla\Phi$ gives the <electric dipole field>
$$
\boxed{\mathbf E(\mathbf r)=\frac{3(\mathbf p\cdot\widehat{\mathbf r})\widehat{\mathbf r}-\mathbf p}
{4\pi\varepsilon_0r^3}.}
$$
On the dipole axis its magnitude is $2p/(4\pi\varepsilon_0r^3)$ and its direction is along $\mathbf p$; in the equatorial plane the magnitude is $p/(4\pi\varepsilon_0r^3)$ and the field points opposite to $\mathbf p$. The $r^{-3}$ decay is faster than the $r^{-2}$ field of an isolated <point charge>.
In a uniform external <electric field>, the opposite forces on the two charges cancel in their resultant but form a <couple>. Its <torque> is
$$
\boldsymbol\tau=\mathbf p\times\mathbf E,\qquad
|\boldsymbol\tau|=pE\sin\theta.
$$
For a fixed permanent <electric dipole moment>, the interaction <potential energy> is
$$
\boxed{U=-\mathbf p\cdot\mathbf E=-pE\cos\theta.}
$$
Differentiating with respect to orientation recovers the torque. Parallel alignment minimizes $U$ and is stable against a small rotation; antiparallel alignment maximizes $U$ and is unstable. Rotating a fixed dipole slowly from parallel to antiparallel requires external <work> $2pE$.
A nonuniform electrostatic <electric field> can also exert a net <force>. Expanding the two charge forces gives $\mathbf F\simeq(\mathbf p\cdot\nabla)\mathbf E$, which equals $\nabla(\mathbf p\cdot\mathbf E)$ for a fixed dipole in a curl-free electrostatic field. Thus a suitably aligned dipole is attracted toward stronger field. An induced <electric dipole moment> requires care in computing energy: when $\mathbf p=\alpha\mathbf E$ for constant isotropic <electric polarizability>, building the dipole costs energy and the effective interaction is $U=-\alpha E^2/2$, rather than substituting directly into the fixed-dipole formula. These field, torque and energy effects explain molecular orientation and <electric polarization>.
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