Q.An electric dipole is placed in a non-uniform electric field. The field lines all point in the same general direction (to the right, the $+x$ direction), but they are crowded together (closer spaced) on the left and spread farther apart on the right; hence the field is stronger on the left and weaker on the right. The dipole is oriented with its dipole moment $\vec{p}$ pointing to the right — that is, its negative charge $-q$ is on the left (in the stronger-field region) and its positive charge $+q$ is on the right (in the weaker-field region). Which statement is correct?

Electric Dipole in a Uniform Field — First Look

Imagine you have a tiny bar magnet. If you place it in a uniform magnetic field, it doesn't get pulled anywhere — but it does twist to align with the field. An electric dipole behaves in exactly the same way when placed in a uniform electric field.

An electric dipole is simply a pair of equal and opposite charges, $+q$ and $-q$, separated by a small distance $d$. Think of it as a tiny "stretched" charge. The dipole has a dipole moment $\vec{p}$, a vector that points from the negative charge to the positive charge, with magnitude $p = qd$.

Now place this dipole in a uniform electric field $\vec{E}$. Uniform means the field has the same strength and direction everywhere in that region.

What happens? Two forces, one twist

The positive charge feels a force $F_+ = +qE$ in the direction of the field. The negative charge feels a force $F_- = -qE$ opposite to the field. These two forces are equal in magnitude but opposite in direction — so they cancel out as far as net force is concerned. The dipole as a whole does not accelerate linearly.

But the forces are not along the same line. They are separated by the distance $d$, so they form a couple — a pair of equal, opposite, parallel forces that produce a torque. This torque tries to rotate the dipole so that its dipole moment $\vec{p}$ aligns with the field $\vec{E}$.

The torque formula

Let the dipole make an angle $\theta$ with the field direction (so $\theta = 0$ when $\vec{p}$ and $\vec{E}$ point the same way). The lever arm for each force about the centre is $(d/2)\sin\theta$. The torque from each force is $F \times \text{lever arm} = qE \cdot (d/2)\sin\theta$. Since both forces contribute in the same rotational sense, the total torque is:

But $qd = p$, the dipole moment. So:

The direction of the torque is such that it tries to reduce $\theta$ — to bring $\vec{p}$ into alignment with $\vec{E}$. In vector form:

What does this mean physically?

• When $\theta = 0$ (dipole aligned with field), $\sin 0 = 0$, so torque is zero. This is the stable equilibrium position.
• When $\theta = 90^\circ$ (dipole perpendicular to field), torque is maximum: $\tau_{\text{max}} = pE$. …