JEE Mains Physics · Electrostatics

Electric Dipoles: Field, Torque and Energy

A dipole is a pair of equal and opposite charges with moment p = q × separation, from − to +; its field falls as 1/r³, and in a uniform field it feels a torque p × E and has energy −p·E.

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Why this matters

Twenty-one PYQs, fifteen of them multiple choice, and four from 2026. Nine ask for a dipole moment, field or potential, including two dipoles at right angles or along one line. Twelve put a dipole in an external field: the torque on it, the work to turn it, its small oscillations, and the force on it when the field is not uniform.

Concept 1 of 2: Dipole moment, field and potential

Far from a dipole, the two charges almost cancel. What remains depends on the moment p, and falls one power of r faster than a single charge: the field as 1/r³, the potential as 1/r². Along the axis the two charges' fields work together; on the equatorial line they nearly cancel, leaving half as much field, pointing opposite to p.

Definition

$p = q \times d$ , pointing from −q to +q. For a group of charges, $\vec p = \sum q_i\vec r_i$ ; this depends on the origin unless the net charge is zero.
Short dipole, on the axis: $E = \dfrac{2kp}{r^{3}}$ , along $\vec p$ . On the equatorial line: $E = \dfrac{kp}{r^{3}}$ , opposite to $\vec p$ .
At angle θ from the axis: $E = \dfrac{kp}{r^{3}}\sqrt{1 + 3\cos^{2}\theta}$ and $V = \dfrac{kp\cos\theta}{r^{2}}$ . V is zero on the equatorial line.
Two dipoles: add their fields as vectors, or add their moments first when they sit at one point.

Short dipole

E_{\text{axial}} = \frac{2kp}{r^{3}}, \qquad E_{\text{equatorial}} = \frac{kp}{r^{3}}, \qquad V = \frac{kp\cos\theta}{r^{2}}

Worked example

A short dipole of moment

2 \times 10^{-9}

C m sits at the origin, pointing along +x. Find the field and the potential at (0.1, 0) m and at (0, 0.1) m.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2026 · 24 Jan 2026 Shift 1 · Q2Moderate

Example 1 · Electrostatics · Electric Dipoles: Field, Torque and Energy

Three charges

+2q

+3q

and

-4q

are situated at

(0, - 3a),(2a,0)

and

( - 2a,0)

respectively in the xy plane. The resultant dipole moment about origin is

\_\_\_\_

Axial is twice equatorial

At the same distance, the axial field is 2kp/r³ and the equatorial field kp/r³. The equatorial field also points opposite to p.

p points from − to +

The dipole moment runs from the negative charge to the positive one, the reverse of the field lines between them.

Net charge makes p depend on the origin

When the charges do not add to zero, Σqr changes with the origin. Use the origin the question names.

Drill 8 more on dipole moment, field and potential

Concept 2 of 2: Dipole in an external field: torque, energy and work

In a uniform field the two charges feel equal and opposite forces, so the net force is zero but the pair is twisted towards the field. The energy is lowest when p points along E and highest when it points against it. Turning the dipole needs work equal to the change in that energy. In a field that varies, the two forces no longer cancel and the dipole is also pulled.

Definition

Uniform field: net force zero; torque $\vec\tau = \vec p \times \vec E$ , of size $pE\sin\theta$ , largest (pE) at $90^{\circ}$ .
Energy $U = -\vec p \cdot \vec E = -pE\cos\theta$ : stable at θ = 0 (−pE), unstable at $180^{\circ}$ (+pE).
Work to turn from $\theta_1$ to $\theta_2$ : $W = pE(\cos\theta_1 - \cos\theta_2)$ . From aligned: pE to $90^{\circ}$ , 2pE to $180^{\circ}$ .
Small oscillations about the field: $\omega = \sqrt{\dfrac{pE}{I}}$ , so the frequency goes as $\sqrt E$ . A free dipole turns about its centre of mass.
Non-uniform field: a net force appears. A dipole aligned with a field that grows in one direction is pulled that way.

Torque, energy and work

\vec\tau = \vec p \times \vec E, \qquad U = -\vec p \cdot \vec E, \qquad W = pE(\cos\theta_1 - \cos\theta_2)

Worked example

Charges

-3\ \mu\text{C}

at (0, 0, 0) m and

+3\ \mu\text{C}

at (0, 2, 1) m sit in a uniform field

\vec E = 50\hat i

N/C. Find the torque and the potential energy.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2024 · 27 Jan 2024 · Q112Moderate

Example 2 · Electrostatics · Electric Dipoles: Field, Torque and Energy

Two charges of

- 4\mu C

and

+ 4\mu C

are placed at the points

A(1,0,4)m

and

B(2, - 1,5)m

located in an electric field

\overrightarrow{E}= 0.20\widehat{i}V/cm

. The magnitude of the torque acting on the dipole is

8\sqrt{\alpha}\times10^{- 5}Nm

, Where

\alpha =

. ________

Half a turn costs 2pE

From aligned to reversed, the energy goes from −pE to +pE, so the work is 2pE. Writing pE counts only the quarter turn.

The minimum energy is negative

U = −pE cos θ is −pE when aligned. A dipole set along the field is in stable equilibrium, at its lowest energy, not zero.

Unequal masses turn about the centre of mass

In a uniform field the net force is zero, so the centre of mass stays put. Take the moment of inertia about it, not about the midpoint.

Drill 11 more on dipole in an external field: torque, energy and work

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Dipole moment, field and potential
Short dipole
$E_{\text{axial}} = \frac{2kp}{r^{3}}, \qquad E_{\text{equatorial}} = \frac{kp}{r^{3}}, \qquad V = \frac{kp\cos\theta}{r^{2}}$
Dipole in an external field: torque, energy and work
Torque, energy and work
$\vec\tau = \vec p \times \vec E, \qquad U = -\vec p \cdot \vec E, \qquad W = pE(\cos\theta_1 - \cos\theta_2)$

Watch out for (6)

Axial is twice equatorial
→ Dipole moment, field and potential
p points from − to +
→ Dipole moment, field and potential
Net charge makes p depend on the origin
→ Dipole moment, field and potential
Half a turn costs 2pE
→ Dipole in an external field: torque, energy and work
The minimum energy is negative
→ Dipole in an external field: torque, energy and work
Unequal masses turn about the centre of mass
→ Dipole in an external field: torque, energy and work

Test yourself on Electrostatics

20 past JEE Mains questions from this chapter, timed at 48 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

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