Dipole in a Uniform External Field

You already know that an electric dipole produces its own field. But what happens when you take a dipole and place it inside someone else’s field, a uniform external field $\mathbf{E}$? Does it feel a push? A twist? Both? The answer reveals something elegant: in a uniform field, the dipole feels no net push at all, but it does feel a turning effect that tries to swing it into alignment with the field. In a non-uniform field, however, the story changes, and this change explains a simple everyday observation you have probably seen many times.

Forces on a Dipole in a Uniform Field

Consider a permanent dipole (one whose dipole moment $\mathbf{p}$ exists on its own, not created by the applied field) sitting in a uniform external field $\mathbf{E}$. The dipole has charges $+q$ and $-q$ separated by distance $2a$, and the dipole moment $\mathbf{p}$ makes an angle $\theta$ with $\mathbf{E}$.

Fig 1.19: Dipole in a uniform electric field

Each charge responds to the field independently:

• The positive charge $+q$ feels a force $q\mathbf{E}$ in the direction of the field.
• The negative charge $-q$ feels a force $-q\mathbf{E}$, equal in size but pointing the opposite way.

Since the field is uniform (same everywhere), both forces have exactly the same magnitude $qE$. They point in opposite directions, so they cancel perfectly:

$\text{Net force} = q\mathbf{E} + (-q\mathbf{E}) = \mathbf{0}$

The dipole as a whole does not accelerate. It stays put.

Why a Torque Still Exists: The Turning Effect

Even though the net force is zero, these two forces do not act at the same point. One pushes the positive end forward and the other pulls the negative end backward. Because they act at different locations, they create a torque (a turning effect, also called a couple in mechanics).

Here is a useful fact from mechanics: when the net force on a system is zero, the torque about any point gives the same answer, no matter which point you pick as the origin. So the torque on the dipole does not depend on the choice of origin.

Deriving the Torque

The magnitude of the torque produced by a couple equals the magnitude of either force multiplied by the arm of the couple (the perpendicular distance between the two lines of action of the forces).

Each force has magnitude $qE$. The two charges are separated by distance $2a$, but what matters for the torque is the perpendicular distance between the force lines, not the full separation. When the dipole axis makes angle $\theta$ with the field, this perpendicular distance is:

$\text{Arm of couple} = 2a\sin\theta$

So the torque magnitude is:

$\tau = (\text{force}) \times (\text{arm}) = qE \times 2a\sin\theta$

Since the dipole moment has magnitude $p = q \times 2a$, this simplifies to:

$\tau = pE\sin\theta$

The direction of this torque is perpendicular to the plane containing $\mathbf{p}$ and $\mathbf{E}$, following the right-hand rule for cross products. You can verify that the cross product $\mathbf{p} \times \mathbf{E}$ gives exactly this magnitude and direction. So the vector form of the torque is:

$\boldsymbol{\tau} = \mathbf{p} \times \mathbf{E} \qquad \text{(1.22)}$

This compact formula captures everything: the magnitude ($pE\sin\theta$), the direction (perpendicular to both $\mathbf{p}$ and $\mathbf{E}$), and the sense of rotation (tending to align $\mathbf{p}$ with $\mathbf{E}$).

When Is the Torque Maximum, and When Does It Vanish?

• Maximum torque occurs when $\theta = 90°$ (dipole perpendicular to the field). The torque then equals $pE$.
• Zero torque occurs when $\theta = 0°$ (dipole aligned with the field) or $\theta = 180°$ (dipole antiparallel to the field). At $\theta = 0°$, the dipole is in stable equilibrium; at $\theta = 180°$, it is in unstable equilibrium.

The torque always acts to reduce $\theta$, rotating the dipole toward alignment with $\mathbf{E}$. Once $\mathbf{p}$ lines up with $\mathbf{E}$, the turning stops.

What Changes in a Non-Uniform Field

The analysis above assumed the field was the same at both charges. What if the field varies from place to place?

In a non-uniform field, the field strength at the location of $+q$ differs from the field strength at $-q$. The two forces no longer have the same magnitude, so their sum is not zero. The dipole now experiences a net force in addition to a torque.

Fig 1.20: Electric force on a dipole in a non-uniform field, (a) $\mathbf{p}$ parallel to $\mathbf{E}$ (b) $\mathbf{p}$ antiparallel to $\mathbf{E}$

The general case involves both torque and force and can get complicated. But two special orientations are straightforward to analyse: when $\mathbf{p}$ is parallel to $\mathbf{E}$, and when $\mathbf{p}$ is antiparallel to $\mathbf{E}$. In both cases, the torque is zero (since $\sin 0° = \sin 180° = 0$), but there is a net force.

Case 1: $\mathbf{p}$ Parallel to $\mathbf{E}$

The positive charge sits in the region where the field is stronger, and the negative charge sits where the field is weaker. Since $+q$ experiences a larger force (in the field direction) than $-q$ experiences (against the field direction), the net force points toward the region of increasing field.

Case 2: $\mathbf{p}$ Antiparallel to $\mathbf{E}$

Now the dipole is flipped. The positive charge is in the weaker-field region and the negative charge is in the stronger-field region. The force on $-q$ (directed opposite to $\mathbf{E}$) is larger in magnitude than the force on $+q$ (directed along $\mathbf{E}$). The net force points toward the region of decreasing field.

In general, the direction and magnitude of the net force depend on how the dipole is oriented relative to the field and on how rapidly the field changes from one point to another.

A Familiar Observation Explained: The Comb and Paper Trick

You have probably seen this: run a plastic comb through dry hair, then bring it near tiny bits of paper. The paper jumps toward the comb, even though the paper carries no charge. How does this work?

The answer uses both ideas from this topic:

1. Polarisation: The charged comb creates an electric field around it. This field pushes the electrons inside the paper slightly to one side and pulls the nuclei slightly the other way. The paper develops an induced dipole moment aligned with the field. (This is an induced dipole, not a permanent one, but the physics of force and torque apply just the same.)

2. Non-uniform field: The field produced by the comb is not uniform. It is stronger close to the comb and weaker farther away. The induced dipole moment is parallel to this non-uniform field.

Combining these two facts: a dipole whose moment is parallel to a non-uniform field feels a net force toward the region of stronger field, which is toward the comb. So the paper moves toward the comb, exactly as observed.

This simple observation, which you can try right now with any comb and some torn paper, ties together the concepts of polarisation, dipole moment, and the force on a dipole in a non-uniform field.