Derive an expression for the torque experienced by a dipole due to a uniform electric field.

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Q. Derive an expression for the torque experienced by a dipole due to a uniform electric field.

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Concept understanding — Torque On Dipole

Torque on a Dipole — From Intuition to the Formula

Imagine a bar magnet placed in a uniform magnetic field. You know that the north pole gets pulled one way and the south pole the opposite way. If the magnet is not aligned with the field, these two equal and opposite forces create a twist — a torque — that tries to rotate the magnet until it lines up with the field. That's the core idea.

The same thing happens with an electric dipole (two equal and opposite charges $+q$ and $-q$ separated by a small distance $d$ ) placed in a uniform electric field $\vec{E}$ . The two charges experience forces in opposite directions, and unless the dipole is already parallel to the field, those forces produce a torque.

Step 1: The Forces on the Two Charges

Let the dipole moment $\vec{p}$ point from the negative charge to the positive charge, with magnitude $p = qd$ .

In a uniform electric field $\vec{E}$ :

The positive charge $+q$ feels a force $\vec{F}_+ = +q\vec{E}$ (in the direction of $\vec{E}$ ).
The negative charge $-q$ feels a force $\vec{F}_- = -q\vec{E}$ (opposite to $\vec{E}$ ).

These two forces are equal in magnitude but opposite in direction. They form a couple — a pair of equal, opposite, parallel forces that do not share the same line of action. A couple always produces a pure torque, with no net force.

Step 2: Why a Torque Appears

If the dipole is at an angle $\theta$ to the field, the two forces are not along the same line. They are separated by the perpendicular distance between their lines of action. That perpendicular distance is $d \sin\theta$ , where $d$ is the separation between the charges.

The torque $\tau$ due to a couple is:

\tau = (\text{force magnitude}) \times (\text{perpendicular distance between forces})

Here:

Force magnitude on each charge: $F = qE$
Perpendicular distance: $d \sin\theta$

So:

\tau = (qE) \times (d \sin\theta) = qd \, E \sin\theta

But $qd = p$ , the magnitude of the dipole moment. Therefore:

\tau = pE \sin\theta

Step 3: The Vector Form

Torque is a vector — it has a direction. The direction of the torque is perpendicular to both $\vec{p}$ and $\vec{E}$ , following the right-hand rule. The complete vector equation is:

\vec{\tau} = \vec{p} \times \vec{E}

The magnitude is $|\vec{\tau}| = pE \sin\theta$ , where $\theta$ is the angle between $\vec{p}$ and $\vec{E}$ .

Step 4: What the Torque Does

When $\theta = 0^\circ$ (dipole aligned with the field): $\sin 0 = 0$ , so $\tau = 0$ . The dipole is in stable equilibrium — if you nudge it slightly, the torque brings it back.
When $\theta = 90^\circ$ (dipole perpendicular to the field): $\sin 90^\circ = 1$ , so torque is maximum: $\tau_{\text{max}} = pE$ . …

Why this formula?

Torque on a Dipole in a Uniform Electric Field

Let's build this from first principles — understanding why the torque formula is what it is, not just memorizing it.

What is a Dipole?

A dipole consists of two equal and opposite charges $+q$ and $-q$ , separated by a small distance $2a$ (or $d$ ). The dipole moment vector is:

\vec{p} = q \cdot \vec{d}

where $\vec{d}$ points from $-q$ to $+q$ , and $|\vec{d}| = 2a$ .

The Physical Situation

Place this dipole in a uniform external electric field $\vec{E}$ . Uniform means the field has the same magnitude and direction everywhere.

The $+q$ charge experiences a force: $\vec{F}_+ = +q\vec{E}$
The $-q$ charge experiences a force: $\vec{F}_- = -q\vec{E}$

These two forces are equal in magnitude but opposite in direction.

Why is there a Torque?

Since the forces are equal and opposite, the net force on the dipole is zero:

\vec{F}_{\text{net}} = q\vec{E} + (-q\vec{E}) = 0

So the dipole won't accelerate linearly. But — crucially — the two forces act at different points in space (the two charges are separated). This creates a couple (a pair of equal, opposite, parallel forces not acting along the same line). A couple always produces a torque (rotational effect).

Deriving the Torque Magnitude

Let the dipole be oriented at an angle $\theta$ with respect to the field $\vec{E}$ .

The line joining the charges makes angle $\theta$ with $\vec{E}$ .
The perpendicular distance between the lines of action of the two forces is the "lever arm."

Step 1: The force on each charge is $qE$ .

Step 2: The perpendicular distance between the two forces is:

\text{Lever arm} = 2a \sin\theta

Why $\sin\theta$ ? Because the separation vector $\vec{d}$ is at angle $\theta$ to $\vec{E}$ . The component of $\vec{d}$ perpendicular to $\vec{E}$ is $d\sin\theta = 2a\sin\theta$ .

Step 3: Torque = Force × Perpendicular distance (for one force about the midpoint):

\tau = (qE) \times (2a \sin\theta)

Step 4: But $q \times 2a = p$ , the dipole moment magnitude. So:

\boxed{\tau = pE \sin\theta}

Vector Form — The Full Picture

Torque is a vector. Its direction is given by the right-hand rule: it tends to rotate the dipole toward alignment with the field.

The vector form captures both magnitude and direction:

\boxed{\vec{\tau} = \vec{p} \times \vec{E}}

Magnitude: $|\vec{\tau}| = pE \sin\theta$ (as derived)
Direction: Perpendicular to both $\vec{p}$ and $\vec{E}$ , given by the cross product rule.

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Step 1. Forces on the two charges. In a uniform field $\vec{E}$ , the charge $+q$ feels $+q\vec{E}$ and $-q$ feels $-q\vec{E}$ -- equal in magnitude, opposite in direction, so the net translational force on the dipole is zero.

Step 2. A couple, not zero torque. Because these two equal-and-opposite forces act at two different points (the dipole's two ends, offset by $2a$ ), they form a couple, producing a net torque even though the net force is zero.

Step 3. Torque magnitude. Taking moments about the midpoint, with $\theta$ the angle between $\vec{p}$ and $\vec{E}$ , the perpendicular lever arm for each force works out to $a\sin\theta$ , giving total torque $\tau=(qE)(a\sin\theta)+(qE)(a\sin\theta)=q(2a)E\sin\theta=pE\sin\theta$ . …

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Q.Derive an expression for the torque experienced by a dipole due to a uniform electric field.

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