Formula of Dipole Moment: p = q × 2a, Unit C m, Examples

The formula of dipole moment for an electric dipole is $p = q \times 2a$ , where +q and −q are the two equal and opposite charges and 2a is the distance between them. Many books write the same thing as $p = qd$ . The answer is in coulomb metre (C m).

Dipole moment measures how strongly a pair of separated charges behaves as a unit in a field. It decides the torque on the dipole, its stored energy and the field it produces far away. This page covers the dipole moment formula in physics, with a note on how chemistry draws its arrow. Related pages include the electric field formula and the electricity and magnetism formulas hub.

Quick facts

Formula
$\vec p = q \cdot 2\vec a$
SI unit
coulomb metre (C m)
Dimensional formula
[M⁰ L¹ T¹ I¹]
Type
Vector, from −q to +q in physics
Torque in a field
$\tau = pE\sin\theta$
Potential energy
$U = -pE\cos\theta$
Molecular unit
debye, 1 D ≈ $3.336 \times 10^{-30}$ C m

Each term in the formula of dipole moment

Symbols for an electric dipole
Symbol	Meaning	SI unit
$p$	Electric dipole moment	C m
$q$	Size of either charge (they are equal and opposite)	C
$2a$	Distance from −q to +q, also written d or l	m
$a$	Half the separation, from the centre to one charge	m

Use only one charge in the product, not the sum. The total charge of a dipole is zero, yet its dipole moment is not, because the charges sit at different places. If you are given a and not 2a, double it before you multiply. That single slip costs more marks on this topic than anything else.

Direction of the dipole moment vector

In physics the vector p points from the negative charge to the positive charge, along the axis of the dipole. This is the convention used in Class 12 physics and in AP and college physics courses.

Chemistry books have traditionally drawn the arrow the other way, from the positive end toward the more electronegative, negative end, with a small cross at the positive tail. Both describe the same molecule; only the arrow drawing differs. In a physics answer, always use negative to positive.

Water has a dipole moment of about 1.85 D, roughly 6.2 × 10⁻³⁰ C m, which is why it lines up in electric fields.

Dipole moment examples with full working

Example 1: two point charges

Charges of +2.5 μC and −2.5 μC are 3.0 mm apart. Find the dipole moment.

In SI units the charge is $2.5 \times 10^{-6}$ C and the gap is $3.0 \times 10^{-3}$ m.

$p = q \times 2a = (3.0 \times 10^{-3}\ \text{m})(2.5 \times 10^{-6}\ \text{C}) = 7.5 \times 10^{-9}\ \text{C m}$

Example 2: torque on a dipole in a field

A dipole has charges of ±5.0 nC separated by 4.0 cm. It sits at 30° to a steady field of strength $2.0 \times 10^5$ N/C. Find p and the torque on it.

Convert: q = $5.0 \times 10^{-9}$ C, 2a = 0.040 m.

$p = (5.0 \times 10^{-9})(0.040) = 2.0 \times 10^{-10}$ C m

$\tau = pE\sin\theta = (2.0 \times 10^{-10})(2.0 \times 10^5)(0.5) = 2.0 \times 10^{-5}\ \text{N m}$

Example 3: finding the separation

A dipole with p = $4.8 \times 10^{-29}$ C m is made of an electron and a proton. How far apart are they?

$2a = \frac{p}{q} = \frac{4.8 \times 10^{-29}}{1.602 \times 10^{-19}} \approx 3.0 \times 10^{-10}\ \text{m}$

That is about 0.30 nm, close to the size of a small molecule.

A dipole placed in a uniform field

In a uniform field E the forces qE and −qE are equal and opposite, so the net force is zero. They do not act along the same line, though, so they form a couple:

$\vec\tau = \vec p \times \vec E, \qquad \tau = pE\sin\theta$

The torque turns the dipole to line up with the field. Its potential energy is $U = -pE\cos\theta$ . It is least, −pE, when p points along E (stable), and greatest, +pE, when p points against E (unstable). The work needed to turn it from θ₁ to θ₂ is $W = pE(\cos\theta_1 - \cos\theta_2)$ . In a field that is not uniform there is also a net force, which is why a charged comb pulls on small bits of paper.

Units, dimensions and the field of a dipole

Charge times length gives C m. Charge has dimensions [I T], so p has [M⁰ L¹ T¹ I¹], with no mass dependence. A unit check on the torque formula: (C m)(N/C) = N m, which is correct for torque.

Far from a short dipole (r much larger than a), the field falls with the cube of distance, faster than the inverse square field of a single charge:

Field of a short dipole at distance r
Position	Field size	Direction
On the axis	$E = \frac{1}{4\pi\varepsilon_0}\frac{2p}{r^3}$	Along p
On the equatorial line	$E = \frac{1}{4\pi\varepsilon_0}\frac{p}{r^3}$	Opposite to p

The magnetic version, current loop times area, is covered on the magnetic moment formula page. For charge spread along a rod, see the linear charge density formula.

Slips to avoid

Using a instead of 2a. The separation is the full distance between the charges.
Adding the two charges. The product uses one charge only.
Pointing the arrow the chemistry way in a physics exam. Physics uses −q to +q.
Saying a uniform field moves a dipole. It only rotates it; the net force is zero.
Forgetting nC and mm conversions. 1 nC = 10⁻⁹ C and 1 mm = 10⁻³ m.

Frequently asked questions

What is the formula of dipole moment?

The electric dipole moment is p = q × 2a, the size of one charge times the distance between the two charges. It is often written p = qd.

What unit is dipole moment given in?

The coulomb metre, written C m. In chemistry, molecular dipole moments are often given in debye, where 1 D is about 3.336 × 10⁻³⁰ C m.

Is dipole moment a vector?

Yes. In physics it points along the dipole axis from the negative charge to the positive charge.

What is the dipole moment formula in physics compared with chemistry?

The size is the same, charge times separation. Physics draws the arrow from negative to positive, while chemistry has often drawn it from positive to negative.

What is the net force on a dipole in a uniform field?

Zero. The two forces are equal and opposite, but they form a couple that produces a torque τ = pE sin θ.