Field Due to an Electric Dipole

An electric dipole consists of two equal and opposite charges, +q and −q, separated by a small distance 2a. While the net charge of a dipole is zero, it creates a non-zero electric field in the surrounding space. This field is not uniform — its magnitude and direction depend strongly on where you are relative to the dipole's orientation. Understanding this field pattern is fundamental to explaining phenomena ranging from how polar molecules interact with each other to how antennas radiate electromagnetic waves. Physicists define two special lines of symmetry for a dipole. The axial line (also called the end-on position) runs along the axis connecting the two charges. The equatorial line (also called the broadside-on position) runs perpendicularly through the midpoint between the charges. These two directions give the simplest calculations and reveal the essential character of the dipole field: on the axial line the field points in the same direction as the dipole moment, while on the equatorial line it points opposite to it. The key quantity that captures the strength and orientation of a dipole is the dipole moment vector p⃗, defined as p⃗ = q·(2a) n̂, where n̂ points from the negative charge to the positive charge. The SI unit of dipole moment is the coulomb-metre (C·m). All field formulas for a dipole are elegantly expressed using p, and both axial and equatorial field magnitudes fall off as 1/r³ at large distances — much faster than the 1/r² fall-off of a single point charge. This rapid fall-off makes dipole fields short-ranged compared to monopole fields.

Key takeaways

Setting up coordinates
Field on the axial line
Field on the equatorial line
Comparing axial and equatorial fields
General field of a dipole (for context)

Setting up coordinates

Place the dipole at the origin with the negative charge at −a and the positive charge at +a along the x-axis. The dipole moment p = q(2a) points in the +x direction. Consider a point P on the axial line at distance r from the centre (r > a).

Field on the axial line

The field at P due to +q is E₊ = kq/(r−a)² directed away from +q (i.e., in +x). The field due to −q is E₋ = kq/(r+a)² directed toward −q (also in +x, since −q is to the left of P). Both contributions add. The net field is E_axial = kq[1/(r−a)² − 1/(r+a)²]. Simplifying: E_axial = kq · 4ar/(r²−a²)² = 2kpr/(r²−a²)². For r >> a this becomes E_axial ≈ 2kp/r³. The direction is along p⃗ (i.e., along the dipole axis).

Field on the equatorial line

Now take a point Q on the equatorial line at perpendicular distance r from the centre. Both charges are at distance √(r²+a²) from Q. The magnitudes of the fields due to each charge are equal: E = kq/(r²+a²). These fields point in symmetric directions — the components along the equatorial line (y-direction) cancel by symmetry, while the components along −x (opposite to p⃗) add. The resultant is E_eq = 2·kq/(r²+a²) · cos θ, where cos θ = a/√(r²+a²). So E_eq = kqa·2/(r²+a²)^(3/2) = kp/(r²+a²)^(3/2). For r >> a: E_eq ≈ kp/r³. The direction is antiparallel to p⃗.

Comparing axial and equatorial fields

At the same large distance r >> a, E_axial = 2kp/r³ and E_eq = kp/r³. Therefore E_axial = 2 · E_eq — the axial field is exactly twice the equatorial field at the same distance. This factor of 2 is a universal result for any dipole and is a direct consequence of both field components adding on the axis versus only the resolved component surviving on the equatorial line. Both fields decrease as 1/r³, confirming that the dipole field is a 'second-order' effect compared to a monopole.

General field of a dipole (for context)

For a point at distance r and angle θ measured from the dipole axis, the general expressions are: radial component E_r = (2kp cosθ)/r³ and tangential component E_θ = (kp sinθ)/r³. The axial case corresponds to θ = 0° (E_θ = 0, E_r = 2kp/r³) and the equatorial case to θ = 90° (E_r = 0, E_θ = kp/r³, pointing antiparallel to p⃗), consistent with our earlier results.

The Key Equation

E_{\text{axial}} = \frac{2kp\,r}{(r^2 - a^2)^2} \approx \frac{2kp}{r^3} \quad (r \gg a)

E_{\text{axial}} — Axial electric field (N C⁻¹): Magnitude of the electric field at a point on the dipole axis
k — Coulomb's constant (N m² C⁻²): Equal to 1/(4πε₀) ≈ 9 × 10⁹ N m² C⁻²
p — Dipole moment (C m): Product of charge magnitude q and separation 2a; p = q(2a)
r — Distance from centre of dipole (m): Distance from the midpoint of the dipole to the field point
a — Half-separation (m): Half the distance between the two charges of the dipole

Worked Example

Problem. A dipole consists of charges +2 nC and −2 nC separated by 4 mm. Calculate the electric field at (a) a point 20 cm from the centre on the axial line, and (b) a point 20 cm from the centre on the equatorial line. (Take k = 9 × 10⁹ N m² C⁻².)

E_{\text{axial}} = 18 \text{ N/C}, \quad E_{\text{eq}} = 9 \text{ N/C}, \quad \frac{E_{\text{axial}}}{E_{\text{eq}}} = 2

Answer. Axial field = 18 N/C (along dipole moment); Equatorial field = 9 N/C (opposite to dipole moment). Ratio = 2:1.

Figure

Diagram

Origins · 1 minCharles-Augustin de Coulomb and how this was discovered

Charles-Augustin de Coulomb

1736–1806 · French

Coulomb established the quantitative law of electrostatic force between point charges (Coulomb's Law) using a torsion balance he invented, laying the mathematical foundation upon which the theory of electric dipoles — and all of electrostatics — is built. The constant k = 1/(4πε₀) in every dipole field formula is named in his honour. Before turning to physics, Coulomb spent nine years as a military engineer in Martinique, where harsh tropical conditions permanently damaged his health — yet he went on to complete some of his most important scientific work after returning to France.

Electric Dipole Moment — The dipole moment p = q(2a) is the central quantity in all dipole field expressions; understanding it is prerequisite to calculating dipole fields.
Coulomb's Law — All dipole field derivations use Coulomb's Law as their starting point, applying the principle of superposition to the fields of +q and −q.
Electric Field Lines — Field lines of a dipole have a characteristic looped pattern that visually illustrates the axial and equatorial field results — they emerge from +q, curve around, and terminate at −q.
Torque on a Dipole in a Uniform Electric Field — Knowing the dipole field naturally leads to studying how an external field exerts a restoring torque τ = p × E on a dipole, a key concept in dielectric materials.