= Electric dipole-dipole interaction
The <electric dipole-dipole interaction> is the orientation-dependent <potential energy> of two <electric dipoles>. For fixed point <electric dipole moments> $\mathbf p_1,\mathbf p_2$ separated by $\mathbf r=r\widehat{\mathbf r}$ in <vacuum>,
$$
U(\mathbf r)=\frac{\mathbf p_1\cdot\mathbf p_2-3(\mathbf p_1\cdot\widehat{\mathbf r})(\mathbf p_2\cdot\widehat{\mathbf r})}{4\pi\epsilon_0r^3}.
$$
Here $\epsilon_0$ is the <vacuum permittivity>. For extended neutral objects this is the leading interaction when their separation is large compared with their sizes and their <electric dipole moments> are nonzero.
To derive the expression, evaluate the first <electric dipole>'s <electric field>,
$$
\mathbf E_1(\mathbf r)=\frac{3(\mathbf p_1\cdot\widehat{\mathbf r})\widehat{\mathbf r}-\mathbf p_1}{4\pi\epsilon_0r^3},
$$
and use $U=-\mathbf p_2\cdot\mathbf E_1$. The <potential energy> of a fixed <electric dipole> in an applied <electric field> follows by expanding the two constituent charges' electrostatic <potential energies>. <The Feynman Lectures on Physics, Volume II by Richard Feynman and others>[Section 11–3, equation 11.14, dipole energy]{ref}
For equal parallel <electric dipole moments> of magnitude $p$, a head-to-tail arrangement has $U=-2p^2/(4\pi\epsilon_0r^3)$ and is attractive along the separation direction. Placing the same parallel <electric dipoles> side by side gives $U=p^2/(4\pi\epsilon_0r^3)$ and a repulsive radial <force>. A freely rotating pair can also experience a <torque>; the orientation is part of the physical specification.
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