KAHIBARO
Table of Contents
Dipoles as Two Opposite Charges
An electric dipole is a very simple but very important charge arrangement. It consists of two charges of equal magnitude and opposite sign, separated by a small distance. If the charges are $+q$ and $-q$, and the distance between them is $d$, then together they form a dipole.
A dipole is electrically neutral overall, because the total charge is
$$
(+q) + (-q) = 0
$$
Even though the total charge is zero, the charges are separated in space, so the dipole can still produce an electric field and can still interact strongly with other charges and fields.
Dipole Moment
The key quantity used to describe a dipole is the electric dipole moment. Its magnitude is defined as
$$
p = qd
$$
The dipole moment is a vector. Its direction points from the negative charge toward the positive charge.
The SI unit of dipole moment is coulomb meter, written as $\mathrm{C \cdot m}$.
For a dipole made of charges $\pm q$ separated by distance $d$,
$$
\vec p = q \, \vec d
$$
where $\vec d$ points from the negative charge to the positive charge.
The magnitude is
$$
p = qd
$$
Why Dipoles Matter
Dipoles appear in many physical situations. Some molecules have charge distributed unevenly, which gives them permanent dipole moments. Even objects that are neutral overall can behave like dipoles when positive and negative charges are slightly shifted apart.
This means that neutrality does not imply no electrical effect. A neutral object with separated charges can still feel forces and torques in electric fields.
Electric Field of a Dipole
The electric field of a dipole comes from adding the fields of the two charges. The exact expression can become complicated, but the main idea is simple. Near the positive charge, the field points away from it. Near the negative charge, the field points toward it. The total field is the combination of these two contributions.
Two especially important directions are the axial line and the equatorial line.
The axial line is the line through both charges. The equatorial line is the perpendicular bisector of the dipole.
Far from the dipole, the field becomes weaker very quickly. A single point charge produces a field that falls like $1/r^2$, but a dipole field falls faster, like $1/r^3$. This is one reason why separated opposite charges can cancel strongly at large distances.
A dipole has zero net charge, but its electric field is not zero.
Far from the dipole, the field decreases approximately as
$$
E \propto \frac{1}{r^3}
$$
which is faster than the field of a single point charge.
Field Direction on Important Lines
On the axial line, the electric field points in the same direction as the dipole moment on one side of the dipole. On the equatorial line, the electric field points opposite to the dipole moment.
For beginners, the most useful thing is to remember the pattern rather than memorize full formulas immediately.
| Location relative to dipole | Field direction |
|---|---|
| Axial line | Along the dipole axis |
| Equatorial line | Opposite to $\vec p$ |
For points very far away from the dipole, the magnitudes on these lines are commonly written as
$$
E_{\text{axial}} = \frac{1}{4\pi \varepsilon_0}\frac{2p}{r^3}
$$
and
$$
E_{\text{equatorial}} = \frac{1}{4\pi \varepsilon_0}\frac{p}{r^3}
$$
where $r$ is the distance from the center of the dipole to the observation point, assuming $r \gg d$.
For points far from the dipole, $r \gg d$,
$$
E_{\text{axial}} = \frac{1}{4\pi \varepsilon_0}\frac{2p}{r^3}
$$
$$
E_{\text{equatorial}} = \frac{1}{4\pi \varepsilon_0}\frac{p}{r^3}
$$
These are approximate formulas for an ideal or distant dipole.
Dipole in an External Electric Field
A dipole placed in an external electric field experiences a special effect. Because one charge is positive and the other is negative, the field pulls them in opposite directions. In a uniform field, these two forces are equal in magnitude and opposite in direction, so the net force is zero, but they produce a turning effect called torque.
This torque tends to rotate the dipole so that its dipole moment lines up with the electric field.
The magnitude of the torque is
$$
\tau = pE\sin\theta
$$
where $\theta$ is the angle between $\vec p$ and $\vec E$.
The torque is zero when the dipole is parallel or antiparallel to the field. It is maximum when the dipole is perpendicular to the field.
For a dipole in a uniform electric field,
$$
\tau = pE\sin\theta
$$
The torque tends to align $\vec p$ with $\vec E$.
Potential Energy of a Dipole
A dipole in an electric field also has electric potential energy that depends on its orientation. The potential energy is
$$
U = -\vec p \cdot \vec E
$$
or, in terms of the angle,
$$
U = -pE\cos\theta
$$
This tells us which orientation is stable. The lowest potential energy occurs when $\theta = 0$, meaning the dipole moment is aligned with the field. The highest potential energy occurs when $\theta = \pi$, meaning it points opposite to the field.
| Orientation | Angle $\theta$ | Potential energy |
|---|---|---|
| Aligned with field | $0$ | $U = -pE$ |
| Perpendicular to field | $\pi/2$ | $U = 0$ |
| Opposite to field | $\pi$ | $U = +pE$ |
Potential energy of a dipole in a uniform electric field:
$$
U = -\vec p \cdot \vec E = -pE\cos\theta
$$
Minimum energy occurs when the dipole is aligned with the field.
Uniform and Nonuniform Fields
In a uniform electric field, a dipole experiences torque but no net force. In a nonuniform electric field, the two charges may feel forces of different magnitudes, so the dipole can experience both torque and a net force.
This is important because a dipole can be pulled toward regions where the electric field is stronger, depending on its orientation.
Ideal Dipole Idea
Sometimes physicists treat a dipole as an ideal dipole. This means the separation $d$ is very small, while the dipole moment $p = qd$ remains finite. This model makes calculations easier and captures the essential behavior when the observation point is much farther away than the charge separation.
Physical Examples
A water molecule is a classic example of a permanent electric dipole. Its positive and negative charges are not centered at the same point, so it has a dipole moment. This helps explain why water responds strongly to electric effects and why polar molecules interact in special ways.
Another example is an atom or molecule placed in an external electric field. Even if it has no permanent dipole, the field can shift the electron cloud slightly relative to the nucleus, creating an induced dipole.
Summary Relations
The main ideas of electric dipoles can be collected in a compact form.
| Quantity | Expression | Meaning |
|---|---|---|
| Dipole moment magnitude | $p = qd$ | Strength of charge separation |
| Dipole moment direction | from $-q$ to $+q$ | Orientation of dipole |
| Torque in uniform field | $\tau = pE\sin\theta$ | Turning effect |
| Potential energy | $U = -pE\cos\theta$ | Energy of orientation |
| Far field dependence | $E \propto 1/r^3$ | Dipole field decreases rapidly |
Important dipole formulas:
$$
p = qd
$$
$$
\tau = pE\sin\theta
$$
$$
U = -pE\cos\theta
$$
A dipole is neutral overall, but it still creates an electric field and responds to external fields.