The Electric Dipole

Many physically important charge arrangements are not a single isolated charge but a pair of equal and opposite point charges held a small, fixed distance apart. This configuration is called an electric dipole. The most familiar natural example is the water molecule, H$_2$O, in which the shared electrons sit, on average, closer to the oxygen atom than to the two hydrogen atoms, leaving the oxygen end of the molecule slightly negative and the hydrogen end slightly positive -- a permanent dipole, and the reason water is such an effective solvent. Many molecules that have no permanent dipole of their own can still acquire a temporary, induced dipole moment when placed in an external electric field.

Definition and dipole moment. A dipole consists of a charge $+q$ and a charge $-q$ separated by a distance $2a$. The line joining the two charges is called the dipole axis, and the point exactly midway between them is the dipole's centre. The strength and orientation of a dipole are together captured by a single vector quantity, the electric dipole moment $\vec{p}$, defined as

where $\hat{p}$ is a unit vector directed from the negative charge toward the positive charge along the dipole axis. Its magnitude is simply

and its SI unit is coulomb-metre (C·m). The direction of $\vec{p}$ -- always from $-q$ to $+q$, never the other way round -- is a fixed convention that must be remembered carefully, since it is easy to instinctively reverse it.

A perpendicular line through the dipole's centre, at right angles to the dipole axis, is called the equatorial line (or perpendicular bisector) of the dipole; it plays an important role in Section 1.10. …