---
title: Electric Dipoles
module: Electric Fields
moduleNumber: 1
lessonNumber: 5
order: 105
summary: >
  Most neutral matter carries no net charge yet still responds to an electric field,
  because its positive and negative charge sit slightly apart. That separation is a
  dipole, moment $\vec p=q\vec d$ pointing from the negative to the positive charge,
  and it is the leading term in how any neutral distribution looks from far away. We
  derive the torque $\vec p\times\vec E$ and energy $-\vec p\cdot\vec E$ a uniform
  field imposes, the net force a field gradient adds, and the axial and equatorial
  $1/r^3$ fields the pair produces, then measure how far out the point-dipole
  approximation still holds.
topics: [Electric Fields]
draft: false
sources:
  - book: Tipler & Mosca
    ref: "Ch. 21 — The Electric Field I; §§21-5–21-6"
---

## Charge separation and the dipole moment

An electric dipole consists of charges $+q$ and $-q$ separated by a displacement
vector $\vec d$ directed from the negative charge to the positive charge. Its
electric dipole moment is

$$
\vec p=q\vec d.
$$

The direction is part of the definition. Reversing the arrow from positive to
negative reverses $\vec p$ and gives wrong signs in torque and potential-energy
calculations. The SI unit is coulomb metre:

$$
[\vec p]=\mathrm{C\,m}.
$$

An extended neutral charge distribution has dipole moment equal to the
charge-weighted sum of position vectors,

$$
\vec p=\sum_i q_i\vec r_i,
$$

or, in a continuous distribution,

$$
\vec p=\int\vec r'\,\d q.
$$

When the total charge is zero, this result is independent of the chosen origin.
Shifting the origin by $\vec a$ changes the general moment by
$-\vec a\sum_iq_i$, which vanishes for a neutral system. A nonneutral
distribution has an origin-dependent first moment; its monopole field dominates at
large distance.

$$
% caption: Electric dipole definition. The displacement vector and dipole moment
% both point from the negative charge to the positive charge, while the charge pair
% has zero total charge and a finite first spatial moment.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
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\draw[<->,black] (0.8,0.75)--(4.6,0.75);
\node[above] at (2.7,1.55) {moment direction};
\node[below] at (2.7,0.75) {separation};
\end{tikzpicture}
$$

The point-dipole model applies when the observation distance is much larger than
the charge separation. It keeps $p=qd$ finite while treating the two source
positions as nearly coincident. Near either charge, the separate point-charge fields
must be retained. The dipole approximation then fails because the two distances to
the observation point differ by a substantial fraction.

The term dipole refers to the leading spatial moment of a neutral source, not to a
material object with a fixed molecular shape. Two electrodes, a separated charge
cloud, and a polar molecule can share the same external dipole description at
distances large compared with their internal dimensions. Their internal responses,
mechanical constraints, and time dependence can differ.

## Forces and torque in a uniform electric field

Put the dipole in a uniform applied field $\vec E$. The positive charge has
force $+q\vec E$, and the negative charge has force $-q\vec E$. Their sum
is zero:

$$
\vec F_{\rm net}=q\vec E-q\vec E=\vec 0.
$$

The force lines act at distinct positions. They form a couple and produce torque
about the dipole centre. With $\vec\tau=\sum\vec r_i\times\vec F_i$,
the result is

$$
\vec\tau=\vec p\times\vec E,
\qquad
\tau=pE\sin\theta,
$$

where $\theta$ is the angle from $\vec p$ to $\vec E$. The torque turns
the dipole toward parallel alignment. It vanishes at parallel and antiparallel
orientations because the force arms are collinear with the applied field in both
cases.

$$
% caption: Uniform-field torque on a tilted electric dipole. The positive and
% negative charges receive equal opposite forces, giving zero net force but a couple
% whose lever arms rotate the dipole moment toward the applied-field direction.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
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$$

The sign of the torque is most safely obtained from the cross product. A drawing
that identifies only the torque magnitude loses the rotational sense. In a planar
view, use a chosen normal axis and test a small rotation: if the angle between
$\vec p$ and $\vec E$ decreases, the torque has the restoring sense for
that angle. The same procedure works for a negative rotation coordinate without
memorizing clockwise or counterclockwise rules from a page drawing.

A physical dipole can be rigidly mounted, freely rotating, or coupled to a torsion
spring. A rigid mount experiences reaction torque from its support, so the observed
orientation remains fixed despite nonzero electromagnetic torque. A free dipole
accelerates rotationally according to

$$
I_{\rm rot}\alpha=\tau_{\rm net},
$$

where $I_{\rm rot}$ is the moment of inertia about the rotation axis. A torsion
support adds a restoring torque that determines the equilibrium angle through a
torque balance.

**Rotational potential energy and stability.**

During quasistatic rotation in a uniform field, external work equals the change
in electric potential energy. Integrating the opposing external torque gives

$$
U(\theta)=-\vec p\cdot\vec E
=-pE\cos\theta+C.
$$

Choosing zero energy at $\theta=\pi/2$ sets $C=0$. Parallel alignment has
$U=-pE$, and antiparallel alignment has $U=+pE$. The energy difference between
those orientations is $2pE$.

$$
% caption: Dipole potential energy as a function of orientation. The minimum at
% parallel alignment has positive curvature and is stable against small rotations;
% the maximum at antiparallel alignment has negative curvature and is unstable.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
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$$

Equilibrium requires $\d U/\d\theta=0$, giving the two torque-free orientations.
Stability comes from the second derivative:

$$
\frac{\d^2 U}{\d \theta^2}=pE\cos\theta.
$$

At parallel alignment this derivative is positive. A small angular displacement
raises energy and produces torque toward the minimum. At antiparallel alignment the
derivative is negative; a small displacement lowers energy and torque drives the
dipole farther from that orientation. Zero torque alone therefore does not identify
a stable configuration.

For small displacement $\varphi$ about the stable orientation,
$\sin\varphi\simeq\varphi$, and the torque is approximately
$\tau\simeq-pE\varphi$. A freely rotating rigid dipole then undergoes small
angular oscillations with

$$
\omega_0=\sqrt{\frac{pE}{I_{\rm rot}}}.
$$

The approximation requires $|\varphi|\ll1$ in radians and a field uniform across
the dipole. Damping from a support, a fluid, or internal molecular processes reduces
the amplitude and changes the observed response without changing the static energy
curve.

## Exact field on the dipole axis

Place the dipole on the $x$ axis with $-q$ at $-d/2$ and $+q$ at
$+d/2$. At an axial observation point $x>d/2$, both point-charge fields lie
along the axis and their magnitudes subtract:

$$
E_{\rm axis}
=\frac{1}{4\pi\varepsilon_0}q
\left[
\frac{1}{(x-d/2)^2}
-\frac{1}{(x+d/2)^2}
\right].
$$

The positive charge is nearer and gives the larger outward contribution. The net
field therefore points along $\vec p$ on the positive side of the dipole.
On the negative side, both directions reverse together and the field again points
along $\vec p$. Field direction on the axis is the same on both sides even
though the electric potential changes sign across the dipole centre.

$$
% caption: Axial electric field of a dipole. At a point beyond the positive charge,
% the nearby positive-source contribution exceeds the more distant negative-source
% contribution, leaving a net field along the dipole moment direction.
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\node[acc,above] at (4.85,2.05) {near};
\node[black,below] at (4.85,0.78) {far};
\end{tikzpicture}
$$

At distances $x\gg d$, expand the two denominators to first order in $d/x$:

$$
\frac{1}{(x-d/2)^2}-\frac{1}{(x+d/2)^2}
\simeq\frac{2d}{x^3}.
$$

The axial far field is

$$
\vec E_{\rm axis}
\simeq\frac{1}{4\pi\varepsilon_0}
\frac{2p}{x^3}\,\hat x.
$$

The $x^{-3}$ dependence is shorter range than the $x^{-2}$ field of a net
charge. Equal positive and negative charge cancel the monopole term; their
separation leaves the dipole term as the leading far contribution.

**Equatorial field and far-field geometry.**

At a point on the perpendicular bisector of the charge pair, each source is the
same distance away. The vertical components of the two point-charge fields cancel,
while their horizontal components add opposite to $\vec p$. For perpendicular
distance $y$, the exact magnitude is

$$
E_{\rm eq}
=\frac{1}{4\pi\varepsilon_0}
\frac{p}{\left(y^2+d^2/4\right)^{3/2}}.
$$

The direction is $-\hat p$. Far from the pair,

$$
\vec E_{\rm eq}
\simeq-\frac{1}{4\pi\varepsilon_0}
\frac{p}{y^3}\,\hat p.
$$

The axial far field has twice the equatorial magnitude at equal distance. This
geometric ratio checks a dipole-field calculation.

> **Worked example (axial and equatorial dipole field).** A dipole of moment
> $p=8.0\times10^{-11}\ \mathrm{C\,m}$ is observed at $r=0.50\ \mathrm m$, far compared
> with its charge separation. On the axis,
>
> $$
> E_{\rm ax}=\frac{2kp}{r^3}=\frac{2(8.988\times10^9)(8.0\times10^{-11})}{(0.50)^3}
> =11.5\ \mathrm{N/C},
> $$
>
> pointing along $\vec p$. On the perpendicular bisector at the same distance,
>
> $$
> E_{\rm eq}=\frac{kp}{r^3}=\tfrac12 E_{\rm ax}=5.75\ \mathrm{N/C},
> $$
>
> pointing opposite $\vec p$. The factor of two between axial and equatorial fields is
> the signature of the $1/r^3$ dipole pattern.

$$
% caption: Equatorial dipole field. The vertical components from the positive and
% negative charges cancel at the perpendicular-bisector sample point, while their
% horizontal components add opposite to the dipole moment direction.
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\draw[->,acc,very thick] (2.9,2.75)--(1.75,2.75);
\node[acc,above] at (1.95,2.80) {net E};
\end{tikzpicture}
$$

The far-field vector at arbitrary angle $\theta$ to $\vec p$ is

$$
\vec E(\vec r)
\simeq\frac{1}{4\pi\varepsilon_0r^3}
\left[3(\vec p\cdot\hat r)\hat r-\vec p\right].
$$

It contains a radial component proportional to $2p\cos\theta$ and a polar
component proportional to $p\sin\theta$. The vector expression should be used
when observation points are neither axial nor equatorial; assigning an axial
formula to an off-axis point misses a transverse component.

## Nonuniform fields and translational force

Uniform fields produce torque but no net force on an ideal fixed dipole. In a
nonuniform field, the two charges sample different field magnitudes and directions.
A sufficiently small fixed dipole has net force

$$
\vec F=\nabla(\vec p\cdot\vec E).
$$

The derivative summarizes the difference between the force on the positive charge
and the force on the negative charge. It does not replace those two forces when the
field changes significantly across the separation or when the dipole itself changes
orientation while moving.

$$
% caption: Translational force on a dipole in a nonuniform electric field. The
% nearer charge samples a stronger local field than the farther charge, so the two
% opposite charge forces no longer cancel and the pair has a net force in addition
% to its possible torque.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
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$$

An induced dipole has $\vec p$ dependent on the applied field. A common
linear model is $\vec p=\alpha\vec E$, where $\alpha$ is polarizability.
The energy and force must then account for the work required to induce the dipole;
the familiar fixed-dipole expression cannot be inserted unchanged. Dielectric
polarization treats this material response in detail.

The force direction also depends on constraints. A free permanent dipole can rotate
toward the local field before translating, changing the value of
$\vec p\cdot\vec E$. A dipole held at fixed angle by a support
can experience a different translational force and a support torque. State whether
the moment is fixed in the laboratory frame, aligned with the local field, or
determined by material polarization before using a compact gradient formula.


> **Worked example (torque and energy on a dipole).** Two charges of magnitude
> $2.0\ \mathrm{nC}$ are separated by $4.0\ \mathrm{cm}$, so the moment is
>
> $$
> p=qd=(2.0\times10^{-9})(0.040)=8.0\times10^{-11}\ \mathrm{C\,m}.
> $$
>
> Place the rigid dipole in a uniform field $E=3.0\times10^4\ \mathrm{N/C}$ at
> $60^\circ$ from the moment. The torque magnitude is
>
> $$
> \tau=pE\sin60^\circ=(8.0\times10^{-11})(3.0\times10^4)(0.866)=2.1\times10^{-6}\ \mathrm{N\,m},
> $$
>
> directed along $\vec p\times\vec E$. The potential energies at $60^\circ$ and at
> alignment are
>
> $$
> U_{60}=-pE\cos60^\circ=-1.2\times10^{-6}\ \mathrm J,\qquad
> U_0=-pE=-2.4\times10^{-6}\ \mathrm J,
> $$
>
> so turning from $60^\circ$ to alignment releases
> $\Delta U=U_0-U_{60}=-1.2\times10^{-6}\ \mathrm J$ — energy the field delivers as
> rotational kinetic energy of a freely pivoting dipole.

The sine factor keeps the field component perpendicular to the moment, which is what
produces the turning effect; omitting it gives the right $pE$ only at $90^\circ$. A
viscous medium dissipates part of the released energy and a torsion support stores
part of it. Two checks keep the signs right: the torque vanishes at $0^\circ$ and
$180^\circ$ while the energy sits at an extremum, and the parallel-to-antiparallel
energy gap is $2pE$.

A common sign error draws the moment from positive to negative charge; the checks
above catch it.

## Permanent and induced dipoles

A permanent dipole has a charge separation before an applied field is introduced.
Its moment may rotate or be constrained, but its magnitude is fixed in the stated
model. Its interaction energy at fixed moment is

$$
U_{\rm perm}=-\vec p_0\cdot\vec E.
$$

An induced dipole is produced by the applied field. For an isotropic linear response,

$$
\vec p_{\rm ind}=\alpha\vec E.
$$

Reducing the field to zero removes the induced moment. In this linear model, the
moment follows the applied-field amplitude throughout the response. Its energy follows
from the response built gradually as the field is raised. Direct substitution of
$\alpha\vec E$ into the fixed-moment expression omits that assembly work:

$$
U_{\rm ind}=-\frac12\alpha E^2.
$$

The one-half factor accounts for the work required to polarize the object. Omitting it
overstates the magnitude of the induced-dipole energy and gives an incorrect gradient
force. The distinction matters even when a permanent and induced moment happen to
point in the same direction.

A material can have both contributions. The total moment then depends on what is
held fixed, what is allowed to rotate, and how rapidly the field is changed.
Orientational averaging, nonlinear response, and saturation require models beyond
the simple scalar polarizability. The two expressions above apply only after those
physical conditions have been made explicit.

A linear isotropic induced dipole has force in a slowly varying field

$$
\vec F_{\rm ind}=\frac12\alpha\nabla(E^2).
$$

Reversing the field direction leaves $E^2$ unchanged and does not reverse this ideal
force. A fixed permanent moment has a different reversal signature. This difference
can separate the two responses in a controlled experiment.

**Gradient-force calculation and measurement.**

A permanent dipole constrained to a fixed orientation in a one-dimensional field map
has

$$
F_x=\frac{\d}{\d x}(\vec p\cdot\vec E).
$$

Suppose the moment is held parallel to an axial field
$E_x(x)=E_0+Gx$. The force is

$$
F_x=pG.
$$

Positive $G$ means that the aligned field magnitude grows toward positive $x$, so an
aligned permanent moment is pulled in that direction. An antiparallel moment has the
opposite force.

> **Worked example (translational force in a field gradient).** A dipole of moment
> $p=8.0\times10^{-11}\ \mathrm{C\,m}$ is held parallel to an axial field with gradient
> $G=2.0\times10^{5}\ \mathrm{N/(C\,m)}$. The net force is
>
> $$
> F_x=pG=(8.0\times10^{-11})(2.0\times10^{5})=1.6\times10^{-5}\ \mathrm N,
> $$
>
> toward increasing field. A uniform field ($G=0$) gives zero net force, confirming
> that only the gradient — not the field itself — translates a dipole. Reversing the
> moment reverses the force, which is how a measurement separates a permanent dipole
> from an induced one that always seeks the stronger field.

The calculation requires the field to vary little across the charge separation and the
support to prevent rotation. A free dipole can turn before translating, changing the
relevant dot product.

Map the field at the actual sample positions before force measurements. Calibrate the
balance mechanically, record a zero with the field source inactive, and reverse the
source polarity while holding sample orientation and position fixed. The permanent
dipole force reverses when the field reverses, while a stable weight offset does not.
Half the difference of the two balance readings isolates the field-odd component.

Repeat the measurement over several positions. A plot of reversal-isolated force
against the independently measured gradient tests the proportionality to $pG$. A
nonzero intercept can indicate balance offset, unaccounted sample charge, or a
gradient referenced to the wrong position. A changing slope can indicate dipole
rotation, a nonlocal field variation, or an induced response added to the assumed
permanent moment.

An induced dipole requires the half-energy model instead. Its force depends on the
gradient of $E^2$, not generally on the signed gradient of one field component.
A polarity reversal leaves $E^2$ unchanged, so an ideal induced force does not
reverse. This field-reversal behavior provides a practical diagnostic under the
stated isotropic linear response when the apparatus controls charge leakage,
orientation, and field-source drift.

For matched source polarities, separate the measured force into odd and even parts:

$$
F_{\rm odd}=\frac{F(+E)-F(-E)}{2},
\qquad
F_{\rm even}=\frac{F(+E)+F(-E)}{2}.
$$

The leading signatures apply to a sample held at the same position and orientation
for both readings.

| response model | leading force along $x$ | reversal signature | control requirement |
| --- | --- | --- | --- |
| permanent dipole | $p\,\partial E_x/\partial x$ | odd | hold $\vec p$ fixed |
| linear induced dipole | $(\alpha/2)\,\partial(E^2)/\partial x$ | even | keep $\alpha$ and geometry unchanged |
| residual free charge | $qE_x$ | odd | measure or bound $q$ independently |

A permanent-moment assignment requires a residual-charge test, since residual charge
has the same odd reversal parity. A uniform-field measurement leaves the charge force
while removing the permanent-dipole force, separating the two contributions.


## Model audit and uncertainty

The worked torque calculation has independent uncertainty from charge, separation,
field magnitude, and angle. For the magnitude
$\tau=pE\sin\theta$, a first-order estimate is

$$
\left(\frac{u(\tau)}{\tau}\right)^2=
\left(\frac{u(p)}{p}\right)^2+
\left(\frac{u(E)}{E}\right)^2+
\left(\cot\theta\,u(\theta)\right)^2.
$$

The angular term becomes large near parallel or antiparallel alignment because torque
is then small. Report an angular uncertainty in radians and avoid estimating a
moment from points where the torque is comparable with the torsion-balance zero.
Energy differences use the same measured $pE$ scale, but their sensitivity to angle
is different because they contain cosine differences rather than a sine.

A reversal matrix separates response types. For a permanent dipole held in a fixed
orientation, reversing the applied field reverses the torque and the fixed-orientation
gradient force. For a linear induced dipole, the leading energy and force depend on
$E^2$ and do not reverse under ideal field reversal. Run both polarities at matched
field magnitude, retain the sample position, and record any residual odd and even
components. A charge on the sample can produce a force linear in the field and may
therefore imitate a permanent-dipole reversal signature unless it is independently
controlled.

The gradient experiment also requires a scale audit. Map the field at positions
bracketing the dipole, fit the local derivative, and compare scans taken in both
stage directions. A stable current or voltage source is not enough if the sample
moves under force into a different part of the field. Repeat one midrange location
after each scan. A changed reading can reveal balance drift, charge leakage, lead
motion, or a sample orientation that was not actually constrained.

Test the limiting cases before interpreting a fitted moment. With the field
source inactive, reversal-isolated torque and force must vanish within uncertainty.
With the gradient reduced while field orientation remains fixed, the constrained
permanent-dipole force must approach zero. With the field made uniform over the
sample region, the net force must vanish even though a tilted permanent dipole can
retain torque. These distinct limits keep the torque, energy, and translation models
from being mixed into one unsupported force law.


**Dipole field lines and equipotential geometry.**

Field lines and equipotential curves encode complementary local information. At any
regular point, an electric-field vector is perpendicular to the equipotential curve
through that point and points toward lower potential. This statement concerns local
direction. It does not turn the spacing of arbitrary drawn equipotentials into a
numerical field scale unless successive curves represent a stated equal potential
increment.

Dipole field lines leave the positive charge and enter the negative charge. The
pattern is not radial except extremely close to either charge. Between the charges,
the field direction is mainly from positive to negative. On the perpendicular
bisector, the field points opposite the dipole moment. Equipotential curves are
symmetric under reflection across the dipole axis and across the perpendicular
bisector with the potential sign reversed. The zero-potential curve includes the
perpendicular bisector for an equal charge pair, but a zero-potential curve is not a
zero-field curve: its normal field component can be nonzero.

$$
% caption: Dipole field-line and equipotential geometry. Field arrows run from positive to negative charge and cross equipotential curves at right angles; the perpendicular bisector is a zero-potential curve but not generally a zero-field curve.
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  \draw[->,acc,thick] (1.50,1.38).. controls (2.30,0.30) and (3.50,0.30)..(4.30,1.38);
  \draw[dashed,black] (2.9,0.25)--(2.9,2.95);
  \node[black,above] at (2.9,2.95) {zero potential};
  \node[acc,below] at (2.9,0.22) {E lines};
\end{tikzpicture}
$$

The field-line topology distinguishes axial and equatorial observations without
memorizing a sign rule. Along the dipole axis, the field points in the direction of
the moment on both far sides of the pair. Along the perpendicular bisector, it points
opposite the moment. Equipotential curves provide the same check: their normals must
agree with the vector direction obtained by adding the two point-charge fields.

Near either charge, the single-charge term dominates and the local pattern looks
radial. At distances comparable with the separation, neither the point-charge picture
nor the far dipole picture can be discarded. Use the exact two-charge field or a
numerical vector map there. A diagram that smoothly connects the near and far regions
must have tangent directions checked against the summed field rather than sketched as
arcs joining plus to minus.

**Far-field ordering and the point-dipole limit.**

The external field of a localized charge distribution can be organized by its spatial
moments. The total charge gives the monopole term, which falls as inverse distance
squared in the field. If total charge is zero, that term cancels. A nonzero dipole
moment then gives the leading field, which falls as inverse distance cubed. If both
the net charge and dipole moment vanish, the next nonzero multipole moment controls
the far field and falls more rapidly still.

The ordering applies at far distance and compares source size with observation
distance. The exact two-charge distribution remains the source description at every
radius. For a dipole with separation $d$, the small parameter is $d/r$. The
point-dipole model requires this ratio to be much less than one. The axial and
equatorial expressions then agree with the leading terms of the exact two-charge
field. When $d/r$ is not small, the neglected terms can change magnitude
substantially and can also change the direction predicted by an oversimplified
far-field sketch.

$$
% caption: Far-field decay ordered by the leading nonzero moment. A net charge falls
% off slowest as inverse distance squared, while a neutral dipole falls faster as
% inverse distance cubed; a source with both lower moments cancelled decays faster still.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->,black] (0.3,0.4)--(6.1,0.4) node[right] {distance};
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$$

The point-dipole model has two distinct limits. The observation point must be far
from the charge separation, and the applied field used in a force calculation must
vary little across that separation. The first is a source-field approximation; the
second concerns an external field. Satisfying one does not automatically satisfy the
other. A small dipole can be observed far away while placed in an external field with
a short variation scale, or a large observation distance can coexist with a poorly
constrained dipole orientation.

A controlled audit compares exact and approximate values at several values of
$d/r$. The relative error should decrease as the observation point is moved farther
away. If an experiment fits a cubic falloff only over points close to the source, the
fitted coefficient can absorb near-field corrections and no longer represent the
dipole moment alone.

**Constrained field-gradient force calculation.**

A rigorous point-dipole force calculation begins by stating the constraint. Let a
permanent moment be held fixed along the positive $x$ direction. Let the applied
field on that axis be

$$
\vec E(x)=\bigl[E_0+Gx+Hx^2/2\bigr]\hat x.
$$

The point-dipole result is

$$
F_x=\frac{\d}{\d x}(\vec p\cdot\vec E)
=p(G+Hx).
$$

At a specified position $x_s$, the calculation uses the local derivative
$G+Hx_s$, not a gradient measured at an unrelated origin. The field expansion is
valid only over a region containing both charges. If the dipole separation is $d$,
the next correction depends on how the gradient changes across that separation; the
condition $d|H|\ll|G+Hx_s|$ makes the leading point-dipole force reliable.

The support must prevent rotation if the calculation treats $\vec p$ as fixed.
A freely rotating permanent dipole changes its angle until the torque and mechanical
constraints balance, so the dot product in the force changes during motion. A
translation result based on the initial angle can then be valid only instantaneously.
A mechanical clamp, a fast measurement relative to rotation, or a coupled
translation-rotation calculation is required before assigning one scalar force to a
free object.

The same reasoning sets a limit on force-map interpretation. A map of arrows can
identify the direction in which an aligned dipole is pulled, but it does not provide
the numerical gradient unless arrows have a calibrated length scale or component data
are available. Use mapped values of the field, fit the local derivative, and state
the sample position and orientation. This turns a qualitative gradient sketch into a
force calculation with identifiable assumptions.


**Exact-versus-point-model audit.**

The point-dipole model should be checked against the exact charge-pair model before
its coefficient is used as a source property. Choose several observation points on a
common ray, compute the exact vector by adding the two point-charge fields, and
compare it with the dipole expression using the same moment. Record both the relative
magnitude error and the angular difference between vectors. Magnitude agreement alone
can conceal a direction error at off-axis points.

Set the stopping distance from the required accuracy. State that criterion alongside
the measurement or calculation. Increase the observation distance until changes
caused by the next multipole correction are below the uncertainty of the application.
The required distance is larger when the target quantity is a small difference of
field components, because cancellation amplifies relative error. It is smaller when
only a qualitative far-field direction is needed.

The same audit applies to the constrained gradient calculation. The dipole separation
must be small compared with both the observation distance used to define the source
field and the spatial scale of the applied gradient. State these two ratios
separately. Combining them into one vague statement that the dipole is “small” does
not identify which approximation controls the error.

## Determining a permanent moment with a torsion support

A torsion support converts electric torque into a measurable angular displacement.
Let the support exert restoring torque $-\kappa\varphi$, where $\kappa$ is the
torsion constant and $\varphi$ is the displacement from its zero-torque angle. If
the dipole rotates in a uniform field while its rotation axis is fixed, static
equilibrium gives

$$
\kappa\varphi=pE\sin\theta.
$$

The angle $\theta$ must be defined between the moment and applied field, not
between the rod and a laboratory reference that may have an offset. The relation can
be used to determine $p$ only after $\kappa$, the field magnitude, and the
mechanical zero have independent calibrations.

Use a field-reversal calibration sequence. Keep the support, sample position, and
readout procedure unchanged between the three readings. With the dipole initially
held near a stated angle, record the equilibrium displacement at $+E$, at zero
field, and at $-E$. Half the signed difference between the two energized positions
removes a fixed angular offset:

$$
\varphi_{\rm odd}=\frac{\varphi(+E)-\varphi(-E)}{2}.
$$

The offset-free response should reverse sign with field reversal. A nonreversing
shift can arise from gravity on an off-centre mount, thread creep, optical readout
drift, or a support bearing that changes friction with angle. Record the full
reversal sequence rather than fitting a moment to a single deflection.

The small-angle form clarifies the operating range. If the reference orientation is
parallel to the field, then $\sin\theta\simeq\theta$ and the electric torque has
the same form as an additional angular spring. The effective stiffness is

$$
\kappa_{\rm eff}=\kappa+pE
$$

for small oscillations about parallel alignment. Around an antiparallel orientation,
the electric contribution has the opposite sign and can reduce the mechanical
stiffness. A support that remains stable at zero field can become unstable when the
electric anti-alignment torque exceeds its restoring stiffness. The result follows
from mechanical stability, with the chosen sign convention already fixed by torque.

Static torque data should be collected well away from angle ranges where
$\sin\theta$ is too small to resolve. Near zero or $\pi$, a modest angle error
produces a large relative uncertainty in $p$ inferred from
$p=\kappa\varphi/(E\sin\theta)$. Midrange orientations give larger torque and
reduce the angular sensitivity. A graph of $\varphi_{\rm odd}$ against
$E\sin\theta$ should be linear for a fixed permanent moment and a calibrated
linear support. Curvature in that graph can indicate support nonlinearity, changing
dipole orientation, electrical leakage, or an applied field that varies across the
sample.

**Rotational transients and damping measurements.**

Release of a supported dipole produces rotational motion before it reaches static
equilibrium. For a moment of inertia $I_{\rm rot}$, viscous angular damping
$b\dot\varphi$, and a support stiffness $\kappa$, the small-angle equation near
parallel alignment is

$$
I_{\rm rot}\ddot\varphi+b\dot\varphi+
\left(\kappa+pE\right)\varphi=0.
$$

The undamped angular frequency is

$$
\omega_{\rm rot}=\sqrt{\frac{\kappa+pE}{I_{\rm rot}}}.
$$

Measuring the shift in $\omega_{\rm rot}^2$ between two field magnitudes provides
an independent estimate of $p$. It avoids an absolute angular-deflection
calibration but requires an accurate moment of inertia and a field that remains
constant over the oscillation. The frequency method and the static-deflection method
should agree within their separate uncertainties. A discrepancy points to a support
model error or an unaccounted electric response.

Separate damping from static torque. Damping controls the rate at which the dipole
approaches equilibrium and the width of a driven angular response. It does not alter
the ideal equilibrium condition when the support and applied field are unchanged.
Dry friction can violate the viscous model by producing a threshold torque and
different approach paths for increasing and decreasing field. Sweep field magnitude
upward and downward, pause for equilibration, and compare the two static curves to
detect hysteresis in the mechanical support.

The model also requires a time-scale separation between field switching and the
dipole response. An abrupt voltage step can excite support modes. A slowly swept
field can allow charge leakage or dielectric relaxation in the sample and its
mounting. State the switching waveform, settling time, sampling rate, and the
criterion used to label a reading as static. These details distinguish a torque
measurement from a transient trace fitted with a static formula.

**Thermal orientation of an ensemble.**

Molecules and small particles with permanent dipole moments are subject to thermal
agitation as well as electric torque. For an ensemble at temperature $T$ in a
uniform field, the orientational energy is

$$
U(\theta)=-pE\cos\theta.
$$

The probability density for orientation contains the Boltzmann factor

$$
P(\theta)\,d\theta\propto
\exp\!\left(\frac{pE\cos\theta}{k_{\rm B}T}\right)
\sin\theta\,d\theta.
$$

The $\sin\theta$ factor counts the solid-angle area at polar angle $\theta$.
It prevents a uniform distribution from appearing uniform in $\theta$ alone.
The dimensionless ratio

$$
\xi=\frac{pE}{k_{\rm B}T}
$$

sets the competition between electric alignment and thermal disorder. For an ideal
freely rotating ensemble,

$$
\left\langle\cos\theta\right\rangle=
\coth\xi-\frac{1}{\xi}.
$$

At $\xi\ll1$, the mean alignment is approximately $\xi/3$. At large $\xi$,
the mean direction approaches the applied field but remains below perfect alignment
at finite temperature. A permanent molecular dipole therefore produces an average
polarization without requiring every molecule to point in the same direction.

$$
% caption: Thermal orientation of permanent dipoles in an applied electric field.
% A warm ensemble has a broad range of orientations, while a larger ratio of dipole
% energy to thermal energy concentrates the orientation distribution toward the
% applied-field direction without making the alignment perfectly sharp.
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\draw[->,black] (0.5,0.5)--(6.0,0.5) node[right] {orientation};
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$$

The ensemble result differs from a fixed rigid dipole in two ways. The moment
direction is a statistical average, and the average itself changes with field and
temperature. A fixed-$\vec p$ energy formula applies to one constrained
orientation. An induced moment requires a separate material response model. A
molecular liquid can have permanent moments yet exhibit an induced and an orientational
polarization component at the same time. The separate mechanisms should not be
combined by assigning one constant dipole moment to every field strength.

**Exact force difference across a finite charge pair.**

The point-dipole gradient formula is the first term of an exact two-charge force
calculation. Put charges $-q$ and $+q$ at positions $x-d/2$ and $x+d/2$ in
an externally prescribed one-dimensional field $E(x)$ parallel to the pair. The
net external force is

$$
F_x=qE(x+d/2)-qE(x-d/2).
$$

Taylor expansion about the pair centre gives

$$
F_x=qd\frac{\d E}{\d x}
+\frac{qd^3}{24}\frac{\d^3 E}{\d x^3}
+\frac{qd^5}{1920}\frac{\d^5 E}{\d x^5}+\cdots.
$$

The leading term is $p\,\d E/\d x$. The first correction depends on the third
derivative, not the second derivative, because the symmetric charge positions cancel
the even-order terms. A field with constant gradient has zero higher corrections for
this parallel two-charge model. A quadratic field has a constant gradient derivative
at the centre and also cancels the third-derivative correction; a rapidly curved
field with nonzero third derivative requires the finite-pair expression.

The exact expression gives a direct experimental check. Map the external field at
both charge locations with the dipole absent, use those two measured values to
predict the finite-pair force, then compare with the local-gradient approximation.
Vary the separation $d$ while holding the centre coordinate fixed. The difference
between the two predictions should decrease rapidly with $d$ in a smooth field.
Holding the pair on a support prevents rotational changes from being mistaken for a
failure of the translational approximation.

An unaccounted net charge adds a force $Q_{\rm net}E(x)$, which is often larger
than the dipole-gradient force in a nearly uniform field. Field reversal separates
some cases: a fixed permanent-dipole gradient force is odd in the applied field,
whereas a residual net-charge force is also odd and therefore cannot be eliminated
by reversal alone. Translate the pair through a region of reduced gradient, compare
with a neutral reference sample, and measure charge independently before assigning a
small force to a dipole moment.

**Reproducible electric-dipole analysis.**

An electric-dipole calculation begins by separating the source field from the dipole
response. List the external electrodes or charge distribution that creates
$\vec E(\vec r)$, then state whether the dipole is a rigid charge pair, a
permanent molecular moment, or an induced response of a material. The same symbol
$\vec p$ has different experimental meaning in those cases. A rigid permanent
moment can be held at a fixed laboratory angle. A free permanent moment rotates.
An induced moment changes with the applied field and with the material's local
environment.

State the coordinate origin and the moment direction before calculating torque or
field. For a charge pair, locate both charges or give their centre and separation
vector. For a distributed neutral object, state the charge-density model used to
compute $\vec p=\int\vec r'\,\d q$. The origin independence of that integral
depends on zero total charge. A measured net charge should therefore be reported
alongside a claimed dipole moment, especially when force data are used.

The applied-field model requires its own spatial domain. A uniform-field torque
calculation needs the field to vary little over the separation and over the object's
rotation. A gradient-force calculation needs a local derivative at the actual centre
coordinate. A field map with one vector at a nearby point cannot establish either
condition.
Measure or calculate the field at paired positions spanning the dipole, estimate the
variation over that span, and retain the exact two-charge force when the local
derivative does not dominate the finite-separation correction.

Use independent observables to identify a moment:

- **Static torque:** compare torsion-support displacement with
  $pE\sin\theta$ after the support constant and angular zero have been calibrated.
- **Angular frequency:** compare the field-dependent rotational stiffness with
  $pE$ after moment of inertia and damping have been measured.
- **Far electric field:** fit axial and equatorial vector components only at
  observation distances where the point-dipole approximation has been checked
  against the exact charge-pair model.
- **Gradient force:** compare an independently mapped field derivative with a
  constrained force measurement after net charge and support forces have been
  controlled.

Agreement among two of these measurements is more informative than a highly precise
fit to one data set. The methods depend on different apparatus quantities: support
stiffness for static torque, inertia for a frequency shift, sensor calibration for
field mapping, and position accuracy for a gradient force. A shared bias can still
remain, but disagreement directs attention toward a specific model boundary instead
of inviting a general scale adjustment.

Field reversal classifies responses by their parity in the applied field.
Permanent-dipole torque at fixed orientation reverses with $\vec E$.
An ideal induced-dipole force derived from $E^2$ remains unchanged. A net-charge
force also reverses. Thus a reversal separates field-odd from field-even responses;
it does not by itself distinguish a permanent dipole from a charged contaminant.
Combine reversal with translation through a region of known gradient, net-charge
measurement, and a neutral reference sample.

The far-field source model also has a clear reporting boundary. Give the source size,
observation distance, and both component directions. The statement that a field
decays as $r^{-3}$ applies to the leading dipole term after the monopole term has
been cancelled and higher terms have become small. A fit over a narrow near-source
range can return an apparent cubic exponent while still containing substantial
finite-separation corrections. Plot residuals against scaled distance $d/r$, not
only against raw distance, when assessing the point-dipole limit.

Uncertainty should preserve the distinction between vector direction and magnitude.
A field measurement must report the sensor axis, orientation uncertainty, active
area, and calibration scale. For a torque measurement, report angle in radians,
support stiffness, settling criterion, and repeated-reversal scatter. For a force
measurement, report the centre coordinate, field-gradient procedure, mechanical
zero, and net-charge test. A single relative uncertainty attached to $p$ is
incomplete if the source geometry, response constraint, and measured component have
not been identified.

These records allow an independent calculation. Another analysis can reconstruct the
source field, apply the stated mechanical constraint, evaluate the same field or
force component, and check the approximation limits without inferring missing
assumptions from a diagram.

A computational model should retain exact source positions and external-field data
separately from the dipole-response routine. First compute
$\vec E(\vec r)$ on a grid or at required sample coordinates. Then evaluate
the two charge forces or the stated point-dipole approximation. This separation
allows the same source map to be tested with several dipole separations and
orientations. It also exposes an error that can otherwise remain hidden: using the
dipole's own source field as though it were the prescribed external field in a
translation calculation.

Use SI units throughout the record. Dipole moment is in $\mathrm{C\,m}$, electric
field in $\mathrm{N/C}$, torque in $\mathrm{N\,m}$, and energy in joules. A
gradient $\d E/\d x$ has unit $\mathrm{N/(C\,m)}$, so $p\,\d E/\d x$ has unit newton.
Checking that unit identity catches a missing separation, a misplaced charge factor,
or a gradient evaluated against the wrong coordinate.
It also fixes the force sign after the chosen axis is recorded.