The electric dipole

The first non-point-like system we encounter is the electric dipole: two opposite charges $+q$ and $-q$ held at a fixed distance $d$. It appears in a great many physical contexts — water molecules, antennas, dielectric materials — and deserves separate treatment.

Principle — Dipole moment

For two opposite charges $\pm q$ at distance $d$ we define the dipole moment as the vector $\ev{\vec p = q\,\vec d}$ where $\vec d$ is the vector going from the negative charge to the positive one. Its magnitude is $\abs{p} = q\,d$, with units $\text{C}\cdot\text{m}$.

Key formula

$\vec p = q\vec d \qquad (q \to -q \text{ reverses } \vec p)$ At large distance the field decreases as $E \propto 1/r^3$. The energy in an external field is $U = -\vec p\cdot\vec E$.

Field at large distance

At distance $r \gg d$ from the centre of the dipole, the fields of the two charges almost cancel, and what remains decreases more rapidly than that of a single charge:

$E_\text{dipole}(r) \sim \frac{k\,p}{r^3} \qquad (r \gg d)$

that is, as $1/r^3$ rather than $1/r^2$. The direction of the residual field depends on the point of observation.

The two characteristic directions of the dipole field: along the axis (axial) and on the plane perpendicular to the moment (equatorial).

Along the dipole axis the field is $E_\text{ax} = 2kp/r^3$; on the equatorial plane instead $E_\text{eq} = -kp/r^3$, opposite to $\vec p$ and half the magnitude. The $1/r^3$ dependence is the unmistakable signature of the dipole.

Dipole in an external field

A dipole immersed in a uniform electric field $\vec E$ feels no net force — the two contributions $+q\vec E$ and $-q\vec E$ cancel — but it feels a torque that tends to align it with the field:

$\vec\tau = \vec p \wedge \vec E \qquad U = -\vec p\cdot\vec E = -p\,E\,\cos\theta$

with $\theta$ the angle between $\vec p$ and $\vec E$. The orientation of minimum energy is $\theta = 0$ (dipole aligned with the field), that of maximum energy is $\theta = \pi$ (antiparallel dipole).

Example — Water and microwaves

The H$_2$O molecule has a permanent dipole moment of magnitude $p \approx 6.2\cdot 10^{-30}\;\text{C}\cdot\text{m}$, because oxygen is more electronegative than hydrogen and “pulls” the electrons towards itself. In a microwave oven, an oscillating electric field at $f = 2.45$ GHz rotates all the water dipoles in the food back and forth: molecular friction converts this rotational motion into heat. This is why microwaves heat water (and hence moist foods) but not a dry ceramic plate (Bloomfield 2016).

Summary

Dipole moment: $\vec p = q\vec d$. Far field $\propto 1/r^3$. Torque in an external field: $\vec\tau = \vec p\wedge\vec E$. Energy: $U = -\vec p\cdot\vec E$.

Collegamenti

Argomenti: Campo elettrico e potenziale Concetti: Dipolo elettrico · Campo elettrico

Esercizi collegati: Dipole and test charge · Worked exercise — electric dipole on the axis · Dipole in a uniform field