= Electric dipole
{wiki}
A finite <electric dipole> consists of equal and opposite <electric charges> separated by a nonzero <displacement>. It has <charge neutrality> but a nonzero <electric dipole moment> $\mathbf p=q\mathbf d$. An ideal point <electric dipole> is the mathematical limit $d=|\mathbf d|\to0$ with $\mathbf p$ fixed, so $q$ must increase as $1/d$. Shrinking the separation while holding $q$ fixed instead makes the <electric dipole moment> vanish. The ideal <electric dipole> describes an exterior <electric field>; it does not reproduce the two near-charge singularities or the internal structure of a <molecule>. https://farside.ph.utexas.edu/teaching/jk1/lectures/node20.html[Richard Fitzpatrick, Electric Dipoles: the fixed-moment point limit]{ref}.
Place the pair's midpoint at the origin and write $k_e=1/(4\pi\epsilon_0)$, where $\epsilon_0$ is <vacuum permittivity>. By <superposition principle> and <Coulomb's law>, its exact <electric potential> away from the two <point charges> is
$$
\Phi_{\rm pair}(\mathbf r)=k_eq\left(\frac1{|\mathbf r-\mathbf d/2|}-\frac1{|\mathbf r+\mathbf d/2|}\right).
$$
For $r=|\mathbf r|\gg d$, retain the first nonzero term of a <Taylor series>. The <electric potential> and its negative <gradient> become
$$
\Phi_{\rm dip}(\mathbf r)=k_e\frac{\mathbf p\cdot\widehat{\mathbf r}}{r^2},
\qquad
\mathbf E_{\rm dip}(\mathbf r)=\frac{k_e}{r^3}\left[3(\mathbf p\cdot\widehat{\mathbf r})\widehat{\mathbf r}-\mathbf p\right],\qquad r>0.
$$
Here $\widehat{\mathbf r}=\mathbf r/r$ and the products in parentheses are <dot products>. These are exact exterior formulas for the ideal <electric dipole> and distant approximations for a finite pair. The <electric field> on the outer axis points along $\mathbf p$ with magnitude $2k_ep/r^3$; on the perpendicular bisector it points opposite $\mathbf p$ with magnitude $k_ep/r^3$. This directional structure is what a scalar total <electric charge> misses. https://www.feynmanlectures.caltech.edu/II_06.html#Ch6-S2[Feynman, Leighton and Sands, II.6.2: exact pair potential and dipole expansion]{ref}.
In a prescribed uniform external <electric field> $\mathbf E_{\rm ext}$, a rigid pair has opposite <forces> $\mathbf F_\pm=\pm q\mathbf E_{\rm ext}$. Its net <force> is zero, but summing their moments about the midpoint gives the <torque>
$$
\boldsymbol\tau=\frac{\mathbf d}{2}\times q\mathbf E_{\rm ext}+\left(-\frac{\mathbf d}{2}\right)\times\left(-q\mathbf E_{\rm ext}\right)=\mathbf p\times\mathbf E_{\rm ext}.
$$
The <cross product> makes the <torque> perpendicular to both the <electric dipole moment> and the external <electric field>. If $\theta$ is the angle from $\mathbf E_{\rm ext}$ to $\mathbf p$ in their plane, the signed turning <torque> is $\tau_\theta=-pE_{\rm ext}\sin\theta$. It tends to reduce a small positive $\theta$. https://openstax.org/books/university-physics-volume-2/pages/5-7-electric-dipoles[Ling, Moebs and Sanny, University Physics 2, 5.7: forces and torque on a dipole]{ref}.
The external interaction <potential energy>, with zero chosen at $\theta=\pi/2$, is
$$
U=-\mathbf p\cdot\mathbf E_{\rm ext}=-pE_{\rm ext}\cos\theta,\qquad \tau_\theta=-\frac{dU}{d\theta}.
$$
Parallel alignment minimizes this <potential energy>; antiparallel alignment maximizes it even though the <torque> is zero in both positions. This omits the pair's constant internal binding <potential energy> and assumes a fixed permanent <electric dipole moment>. https://farside.ph.utexas.edu/teaching/jk1/Electromagnetism.pdf[Richard Fitzpatrick, Classical Electromagnetism, exercise 2.10: dipole torque and energy]{ref}.
\Image[/electric-dipoles-pair-torque.png]
{title=Neutral charge pair and orientation energy in a uniform electric field}
{description=Original analytical illustration for a rigid <electric dipole>. Equal opposite <forces> cancel in translation but give a clockwise <torque> at the depicted positive angle. The <potential energy> curve is the dimensionless model $U/(pE_{\rm ext})=-\cos\theta$, not measured data.}
A nonuniform external <electric field> can exert a net <force> on a neutral <electric dipole>. Before taking the small-pair approximation, the exact external <force> is $\mathbf F=q[\mathbf E_{\rm ext}(\mathbf R+\mathbf d/2)-\mathbf E_{\rm ext}(\mathbf R-\mathbf d/2)]$, where $\mathbf R$ is the midpoint. A first-order <Taylor series> gives, for a fixed <electric dipole moment> and an external <electric field> varying slowly across $d$,
$$
\mathbf F\simeq(\mathbf p\cdot\nabla)\mathbf E_{\rm ext}=\nabla(\mathbf p\cdot\mathbf E_{\rm ext})
$$
in <electrostatics>. The last equality uses the vanishing <curl> of the external <electric field>, with $\mathbf p$ held fixed while differentiating. Thus uniform external <electric fields> turn a misaligned <electric dipole>, whereas gradients can also move its midpoint. For example, the local source-free <electric field> $\mathbf E_{\rm ext}=Gx\widehat{\mathbf x}+(E_0-Gy)\widehat{\mathbf y}$ gives $\mathbf F=(Gp_x,-Gp_y,0)$, so reversing the <electric dipole moment> reverses the net <force>. This stipulated local model has zero <divergence> and zero <curl>.
A motivating experiment is Filsinger and colleagues' study of electrostatic deflection of <polar molecules>. Benzonitrile ($\mathrm{C_7H_5N}$) in <helium> expanded from a pulsed valve into <vacuum>; two 1-mm skimmers narrowed the beam before a 15-cm deflector. A grounded trough and a high-voltage rod produced a vertical <electric field> gradient. Scanning a focused <laser> vertically downstream and recording benzonitrile ions from <photoionization> measured the beam's relative density. With voltage off, the profile spanned about 2 mm; applying voltage shifted and broadened it upwards. At 10 kV, many <molecules> lay outside the original profile. https://pure.mpg.de/pubman/item/item_736019_2/component/file_736018/433451%28filsinger%29.pdf[Filsinger et al., Quantum state selection, alignment, and orientation of large molecules using static electric and laser fields, sections II and III A, figure 3]{ref}.
Interpretation uses a <quantum state>'s <Stark effect> <energy> $W(E)$: $\mu_{\rm eff}=-dW/dE$ gives $F_y=\mu_{\rm eff}\,\partial_yE$. States whose <energy> falls with increasing <electric field> seek stronger fields. The voltage-off profile controls the mechanical aperture. Inferring state populations needs calculated trajectories, <speed> distributions and a rotational <temperature> model; profile displacement alone does not measure a permanent <electric dipole moment> uniquely. This is an electric <Stark effect> experiment, with <photoionization> occurring after deflection. https://pure.mpg.de/pubman/item/item_736019_2/component/file_736018/433451%28filsinger%29.pdf[Filsinger et al., section III A: Stark slopes, trajectory modelling and state selection]{ref}.
For the symmetric finite pair, the first omitted <electric multipole expansion> term after the <electric dipole> is third order, so typical relative errors scale as $(d/r)^2$. This stronger error estimate need not hold for an arbitrary neutral <molecule> with a nonzero <electric quadrupole moment>. A concrete check for a pair on the $z$ axis compares the exact and ideal <electric fields> at $r>d/2$ on the positive axis and at $r>0$ on the perpendicular bisector:
$$
\frac{E_{z,\rm pair}(0,0,r)}{E_{z,\rm dip}(0,0,r)}=\left(1-\frac{d^2}{4r^2}\right)^{-2},
\qquad
\frac{E_{z,\rm pair}(r,0,0)}{E_{z,\rm dip}(r,0,0)}=\left(1+\frac{d^2}{4r^2}\right)^{-3/2}.
$$
Both perpendicular-bisector components are negative. At $r=5d$, the ratios are approximately $1.0203$ and $0.9852$: errors of $+2.03\%$ and $-1.48\%$ relative to the ideal <electric field>. The condition $r\gg d$ concerns observing the pair's own <electric field>; the separate condition for its external <force> approximation concerns variation of the applied <electric field> across the pair.
\Image[/electric-dipoles-finite-field.png]
{title=Finite charge pair approaching its ideal electric dipole field}
{description=Original geometry and analytical field-ratio curves for two <point charges> $\pm q$ separated by $d$. The outer axial <electric field> points along the <electric dipole moment>; the perpendicular-bisector <electric field> points against it. The curves compare exact <Coulomb's law> sums with the ideal <electric dipole> prediction and are models, not experimental measurements.}
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