An electric dipole consists of equal and opposite charges separated by a displacement. For charges $-q$ and $+q$, define the separation vector $\mathbf d$ to point from the negative charge to the positive charge.

The electric dipole moment is

$$\boxed{\mathbf p=q\mathbf d}.$$

Its direction is therefore from negative to positive charge. The SI unit of dipole moment is $\mathrm{C,m}$.

A dipole has zero net charge, but its separated positive and negative charges still create a nonzero electric field.

Field of a dipole

The dipole field is obtained by superposing the fields of its two charges:

$$\mathbf E(\mathbf r)=\mathbf E_{+}(\mathbf r)+\mathbf E_{-}(\mathbf r).$$

Near the charges, the detailed two-charge geometry matters. Far from a compact dipole, the fields of $+q$ and $-q$ nearly cancel, so the dipole field decreases faster with distance than the field of an isolated point charge.

The field-line pattern leaves the positive charge and curves toward the negative charge. Field lines visualize the direction of $\mathbf E$; they are not material connections between the charges.

Dipole in a uniform electric field

Place the dipole in a uniform external field $\mathbf E$. The positive charge feels

$$\mathbf F_+=q\mathbf E,$$

while the negative charge feels

$$\mathbf F_-=-q\mathbf E.$$

The forces cancel translationally:

$$\boxed{\mathbf F_{\rm net}=\mathbf0}.$$

However, if the dipole is not parallel to the field, the two forces act along different lines and form a couple that produces torque.

Let $\theta$ be the angle between $\mathbf p$ and $\mathbf E$. The torque magnitude is

$$\tau=qEd\sin\theta=pE\sin\theta,$$

so in vector form

$$\boxed{\boldsymbol\tau=\mathbf p\times\mathbf E}.$$

The torque tends to rotate the dipole so that $\mathbf p$ aligns with $\mathbf E$.

Stable and unstable orientations

If $\mathbf p$ is parallel to $\mathbf E$, then

$$\tau=0.$$

A small angular displacement produces a torque that tends to restore the parallel orientation, so it is stable.

If $\mathbf p$ is antiparallel to $\mathbf E$, the torque is also zero, but a small angular displacement makes the dipole rotate farther away from that orientation. The antiparallel orientation is unstable.

Worked example

A dipole has charges

$$q=2.0\times10^{-9},\mathrm C$$

separated by

$$d=3.0\times10^{-3},\mathrm m.$$

Its dipole moment magnitude is

$$p=qd=(2.0\times10^{-9})(3.0\times10^{-3}) =6.0\times10^{-12},\mathrm{C,m}.$$

Place it in a uniform field

$$E=5.0\times10^4,\mathrm{N/C}$$

at angle

$$\theta=30^\circ.$$

The torque magnitude is

$$\tau=pE\sin\theta =(6.0\times10^{-12})(5.0\times10^4)(0.5) =1.5\times10^{-7},\mathrm{N,m}.$$

Thus

$$\boxed{\tau=1.5\times10^{-7},\mathrm{N,m}}.$$

Nonuniform fields

In a nonuniform electric field, the fields at the two charges are not exactly equal. The two electric forces then need not cancel, so a dipole can experience both torque and a net force.

This distinction is important: a uniform field can rotate a dipole without translating its center of mass, while a spatial field gradient can also pull the dipole toward or away from regions of stronger field depending on its orientation.