Electric Dipole

Many molecules in nature (H₂O, HCl, NH₃) carry separated positive and negative charge centres. This charge separation, even at the molecular scale, creates measurable electric fields, influences chemical bonding, and forms the basis of dielectrics, capacitors, and biological systems.

An electric dipole is a pair of equal and opposite point charges placed at a short distance apart.

OR

A system formed by two equal and opposite point charges placed at a small distance apart is called an electric dipole.

OR

A system of two equal and opposite point charges +q and −q separated by a small fixed distance 2a is called an electric dipole.

• The total charge of an electric dipole is zero
• Zero net charge does not mean zero electric field - the field exists because the charges are spatially separated​
• The midpoint of the line joining −q and +q is called the centre of the dipole

“The line joining the two charges, pointing from the negative charge to the positive charge. This is known as the ‘direction of dipole axis’.”

OR

The line passing through both charges +q and −q is called the dipole axis (also called the axial line or axis of the dipole).

The midpoint of the line joining the two charges is called the centre of the dipole.

The line passing through the centre of the dipole and perpendicular to the dipole axis is called the equatorial line.

OR

The plane passing through the centre of the dipole and perpendicular to the dipole axis is called the equatorial plane; the line along which the equatorial field is evaluated is the equatorial line (perpendicular bisector).

Electric dipole moment \[\vec p\] is a vector quantity defined as the product of the magnitude of either charge and the separation between them.

Mathematical definition: \[\vec p\] = q × 2a

Setup: Let P be a point on the axis of the dipole at a distance r from the centre O. The charge +q is at a distance (r − a) from P, and −q is at a distance (r + a) from P.​

Derivation (Steps):

1. Field due to +q at P (directed away from +q, i.e., along \[\hat p\]):
   \(E_+=\frac{1}{4\pi\varepsilon_0}\cdot\frac{q}{(r-a)^2}\)

2. Field due to −q at P (directed toward −q, i.e., along \[\hat p\]):
   \[E_-=\frac{1}{4\pi\varepsilon_0}\cdot\frac{q}{(r+a)^2}\]

3. Both fields are along the same line (axial direction). Net field:
   E_axial = E₊ −E₋ = \[\frac{q}{4\pi\varepsilon_0}\left[\frac{1}{(r-a)^2}-\frac{1}{(r+a)^2}\right]\]

4. Simplifying:
   E_axial = \[\frac{1}{4\pi\varepsilon_0}\cdot\frac{4qar}{(r^2-a^2)^2}\]

5. For a short dipole (r ≫ a), (r² − a²)² ≈ r⁴:
   \[E_{axial}=\frac{1}{4\pi\varepsilon_0}\cdot\frac{2p}{r^3}\]

Direction: Along the direction of \[\vec p\] (from −q to +q).

Setup: Let P be a point on the equatorial line at distance r from the centre O. Both charges are at equal distances from P: \[\sqrt{r^2+a^2}\].​​

Derivation (Steps):

1. Field due to each charge at P has equal magnitude:
   E = \[\frac{1}{4\pi\varepsilon_0}\cdot\frac{q}{(r^2+a^2)}\]

2. The components along the equatorial direction cancel (equal and opposite). Only the components parallel to the dipole axis survive.

3. Each contributes E cos ⁡θ, where cos ⁡θ = \[\frac{a}{\sqrt{r^2+a^2}}\]:
   E_eq = \[2\cdot\frac{q}{4\pi\varepsilon_0(r^2+a^2)}\cdot\frac{a}{\sqrt{r^2+a^2}}=\frac{1}{4\pi\varepsilon_0}\cdot\frac{p}{(r^2+a^2)^{3/2}}\]

4. For a short dipole (r ≫ a): E_eq = \[{\frac{1}{4\pi\varepsilon_0}\cdot\frac{p}{r^3}}\]

Direction: Opposite to \[\vec p\] (anti-parallel to dipole moment).

Force on the Dipole

When a dipole is placed in a uniform electric field \[\vec E\]:

• Force on +q: F = qE (along \[\vec E\])
• Force on −q: F = qE (opposite to \[\vec E\])
• Net translational force = 0 (forces are equal and opposite)​

The dipole does not translate in a uniform field — it only rotates.

Torque on the Dipole

When the dipole makes an angle θ with the field direction, each charge experiences a force qE. The perpendicular distance between the forces is 2a sin⁡ θ.

τ = qE ⋅ 2a sin⁡ θ = pE sin ⁡θ

Vector form:

\[\vec τ\] = \[\vec p\] × \[\vec E⃗\]

• Torque tends to align the dipole along the field direction
• Maximum torque: τ_max = pE when θ = 90°
• Torque = 0 when θ = 0° (aligned) or θ = 180° (anti-aligned)

The work done against the torque to rotate the dipole from θ₁ to θ₂ is stored as potential energy.​

W = \[-\int_{\theta_1}^{\theta_2}\tau d\theta=pE(\cos\theta_1-\cos\theta_2)\]

Taking θ₁ = 90° as reference (zero potential energy):

U = −pE cos⁡ θ = −\[\vec p\] ⋅ \[\vec E\]

Equilibrium Conditions

For a point P at distance r from the centre and at angle θ with the dipole axis:​

V = \[\frac{1}{4\pi\varepsilon_0}\cdot\frac{p\cos\theta}{r^2}\quad(r\gg a)\]

Potential falls as 1/r² for a dipole, compared to 1/r for a point charge.

Electric dipoles are not just theoretical — they appear in numerous real-world systems:​

• Water (H₂O): Bent molecular geometry causes unequal charge distribution → net dipole moment (~1.85 D)
• Hydrochloric acid (HCl): Highly polar covalent bond → large dipole moment
• Ammonia (NH₃): Pyramidal structure → net dipole
• Carbon dioxide (CO₂): Linear and symmetric → dipole moments cancel → zero net dipole

Polar molecules have a permanent dipole moment; non-polar molecules have zero dipole moment because bond dipoles cancel.