Electric Dipole

PHYCoulomb's Law, Field & Dipole

An electric dipole is a pair of equal and opposite charges $\pm q$ separated by a small distance, characterised by the dipole moment $\vec{p}$ pointing from $-q$ to $+q$ . In a uniform field it feels a torque but no net force, tending to align with the field.

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ISC tests the axial and equatorial field formulas, the torque, and the work or energy of rotation.

Dipole moment

p = q\,(2a)

q

magnitude of either charge,

2a

the separation;

\vec{p}

points from

-q

+q

, units

\text{C m}

Axial field of a short dipole

E_{axial} = \dfrac{1}{4\pi\varepsilon_0}\dfrac{2p}{r^3}

r

is distance from centre along the axis; valid for

r\gg a

; direction along

\vec{p}

Derivation

On the axis at distance $r$ from the centre, $+q$ is at $r-a$ and $-q$ at $r+a$ . Both fields point along $\vec{p}$ , so their magnitudes subtract by distance:
$E_{axial} = \dfrac{q}{4\pi\varepsilon_0}\left[\dfrac{1}{(r-a)^2} - \dfrac{1}{(r+a)^2}\right]$
Combining over a common denominator:
$E_{axial} = \dfrac{q}{4\pi\varepsilon_0}\cdot\dfrac{4ar}{(r^2 - a^2)^2}$
For a short dipole $r \gg a$ , so $r^2 - a^2 \approx r^2$ , and with $q\,(2a) = p$ :
$E_{axial} = \dfrac{1}{4\pi\varepsilon_0}\dfrac{2p}{r^3}$

Equatorial field of a short dipole

E_{eq} = \dfrac{1}{4\pi\varepsilon_0}\dfrac{p}{r^3}

r

distance from centre perpendicular to the axis; direction opposite to

\vec{p}

; half of

E_{axial}

Derivation

On the equatorial line each charge lies at distance $\sqrt{r^2+a^2}$ . The components perpendicular to the axis cancel; those along $-\vec{p}$ add, each scaled by $\cos\theta = \dfrac{a}{\sqrt{r^2+a^2}}$ :
$E_{eq} = 2\cdot\dfrac{q}{4\pi\varepsilon_0(r^2+a^2)}\cdot\dfrac{a}{\sqrt{r^2+a^2}} = \dfrac{q\,(2a)}{4\pi\varepsilon_0(r^2+a^2)^{3/2}}$
With $p = q(2a)$ and the short-dipole limit $r \gg a$ , so $(r^2+a^2)^{3/2} \approx r^3$ :
$E_{eq} = \dfrac{1}{4\pi\varepsilon_0}\dfrac{p}{r^3}$
This is exactly half the axial field and points opposite to $\vec{p}$ .

Torque in a uniform field

\tau = pE\sin\theta

\theta

angle between

\vec{p}

and

\vec{E}

; torque is maximum at

90^{\circ}

, zero at

0^{\circ}

and

180^{\circ}

Derivation

In a uniform field the charges feel equal and opposite forces $F = qE$ — a couple. Their lines of action are separated by the perpendicular distance $(2a)\sin\theta$ :
$\tau = F\times(\text{perp. distance}) = qE\,(2a)\sin\theta$
Grouping $q\,(2a) = p$ :
$\tau = pE\sin\theta$
In vector form $\vec{\tau} = \vec{p}\times\vec{E}$ — maximum at $\theta = 90^\circ$ , zero when $\vec{p}$ is aligned with $\vec{E}$ .

Potential energy of a dipole

U = -pE\cos\theta

minimum (

-pE

) at

\theta=0^{\circ}

(stable), maximum (

+pE

) at

\theta=180^{\circ}

(unstable)

Derivation

Rotating the dipole by $d\theta$ against the torque needs work $dW = \tau\,d\theta = pE\sin\theta\,d\theta$ . Taking the zero of PE at $\theta = 90^\circ$ and integrating:
$U = \int_{90^\circ}^{\theta} pE\sin\theta'\,d\theta' = pE\big[-\cos\theta'\big]_{90^\circ}^{\theta}$
Since $\cos 90^\circ = 0$ :
$U = -pE\cos\theta$
Minimum ( $-pE$ , stable) at $\theta = 0^\circ$ ; maximum ( $+pE$ , unstable) at $\theta = 180^\circ$ .

Work to rotate the dipole

W = pE(\cos\theta_1 - \cos\theta_2)

work done against the field to turn from

\theta_1

\theta_2

; for

0^{\circ}\to90^{\circ}

W=pE

Derivation

The work done against the field equals the change in the dipole's potential energy:
$W = U(\theta_2) - U(\theta_1) = (-pE\cos\theta_2) - (-pE\cos\theta_1)$
Simplifying:
$W = pE(\cos\theta_1 - \cos\theta_2)$
For $\theta_1 = 0^\circ \to \theta_2 = 90^\circ$ , this gives $W = pE(1-0) = pE$ .

Both axial and equatorial fields of a short dipole fall as $1/r^3$ , faster than a point charge's $1/r^2$ .
The axial field is exactly twice the equatorial field at the same distance, $E_{axial}=2E_{eq}$ .
On the axis $\vec{E}$ is parallel to $\vec{p}$ ; on the equatorial plane $\vec{E}$ is antiparallel to $\vec{p}$ .
In a uniform field the net force on a dipole is zero (the two charge forces cancel); only a torque acts.
Stable equilibrium is at $\theta=0^{\circ}$ ( $\vec{p}$ aligned with $\vec{E}$ ), unstable at $\theta=180^{\circ}$ .
Work to rotate from alignment $0^{\circ}$ to $90^{\circ}$ is $W=pE$ , and to fully reverse to $180^{\circ}$ is $2pE$ .
The 'short dipole' formulas assume $r\gg 2a$ ; convert the separation to metres and use it inside $p=q(2a)$ .

Where the marks go

Forgetting the factor of $2$ on the axial field, or swapping the axial and equatorial formulas.
Mishandling degrees in $\sin\theta$ / $\cos\theta$ — e.g. $\sin 30^{\circ}=0.5$ ; if a calculator is in radian mode it gives a wrong value.
Taking $U=+pE\cos\theta$ (wrong sign): the correct potential energy is $U=-pE\cos\theta$ , lowest when aligned.
Computing a net translational force on a dipole in a uniform field — there is none; only a torque exists.

How the board asks it

Numericalaxial and equatorial field formulas
An electric dipole consists of charges $\pm 4\,\mu C$ separated by $2\,mm$ . Calculate the electric field intensity at a point $20\,cm$ from its centre on (i) the axial line and (ii) the equatorial line of the dipole.
Derive / proveaxial field of a short dipole
Derive an expression for the electric field intensity at a point on the axial line of a short electric dipole, and hence show that it is twice the field at an equal distance on the equatorial line.
Numericaltorque and work to rotate
A dipole of moment $5 \times 10^{-8}\,C\,m$ is held at $30^{\circ}$ to a uniform electric field of $2 \times 10^{4}\,N\,C^{-1}$ . Calculate the torque acting on it and the work done in rotating it from $0^{\circ}$ to $90^{\circ}$ .
Derive / provepotential energy of a dipole
Obtain an expression for the potential energy of an electric dipole of moment $\vec{p}$ placed at an angle $\theta$ in a uniform electric field $\vec{E}$ , and state its orientations of stable and unstable equilibrium.
Give reasonszero net force in a uniform field
Account for the fact that an electric dipole placed in a uniform electric field experiences a torque but no net translational force.
Define / statedipole moment definition
Define electric dipole moment and state its SI unit and direction.

Practise this topic

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Written for Sublevo. Question text quoted anywhere in these notes is the Council’s and carries its year and paper; the board’s own diagrams are not reproduced.