Electric dipole

What is an electric dipole?

In plain words: a dipole has no net charge, but its plus and minus are not in the same place. Close to it you feel both charges separately. Far away their fields almost cancel, and what is left falls off faster than the field of a single charge.

Dipole moment p = q × 2a

• The total charge of a dipole is zero, but its field is not zero: the two charges are apart, so their fields do not cancel exactly (NCERT page 23).
• Dipole moment: a vector of magnitude q × 2a, the charge times the separation, directed along the line from −q to q (NCERT Eq. (1.19), page 24). Its SI unit is C m (NCERT Example 1.9, page 26).
• Why this quantity? At large distances the dipole’s field depends on q and a only through the product qa, which suggests defining p (page 24).
• Point dipole: let the size 2a shrink to zero while q grows so that p = q × 2a stays finite. For such a dipole the far-field formulas below are exact for every r > 0 (page 25); the ideal point-dipole field is singular at r = 0.
• Chemistry draws bond dipoles the other way round, with a crossed arrow from the positive to the negative end: see polarity of bonds and dipole moment. In physics, p points from −q to q.

The field of an electric dipole

NCERT finds the field from Coulomb’s law and superposition at two kinds of points (Section 1.10.1, pages 23–24). Below, r is the distance from the centre of the dipole, k = 1/4πε₀, and p̂ is the unit vector from −q to q.

On the axis, at P outside the pair on the side of +q, with r > a and q > 0 (NCERT Eqs. (1.13) and (1.14)): the nearer charge q gives a field kq/(r − a)² along p̂, and −q gives kq/(r + a)² towards itself, against p̂. Over a common denominator, 1/(r − a)² − 1/(r + a)² = 4ar/(r² − a²)², so the total is

along p̂, the direction of the dipole moment.

On the equatorial plane, the plane through the centre perpendicular to the axis (NCERT Eqs. (1.16) and (1.17)): both charges are √(r² + a²) away, so their fields have equal size kq/(r² + a²). Each field makes an angle with the axis whose cosine is a/√(r² + a²). The parts across the axis cancel and the parts along it add:

which is

directed opposite to p̂ (page 24).

Field falls as 1/r^³ at large distances

When r ≫ a, the a² terms are negligible beside r², and with p = 2qa the two results become (NCERT Eqs. (1.20) and (1.21), page 24):

The axial field points along p; the equatorial field points opposite to p. At the same distance, the axial field is twice the equatorial field.

• The dipole field falls off as 1/r³, not 1/r² like the field of a single charge: far away, the fields of +q and −q nearly cancel (NCERT pages 23 and 25).
• Its size and direction depend on the distance r and also on the angle between the position vector r and p (page 25).

At any angle (for exams, beyond NCERT’s text): at distance r and angle θ from p (r ≫ a), split p into p cos θ along r and p sin θ across it. The first acts as an axial dipole and the second as an equatorial one, so

θ = 0 and θ = 90° give back the sizes in NCERT Eqs. (1.20) and (1.21).

Physical significance of dipoles

• In most molecules the centres of positive and negative charge lie at the same place, so the dipole moment is zero: carbon dioxide (CO₂) and methane (CH₄) are examples. Such molecules develop a dipole moment when an electric field is applied (NCERT Section 1.10.2, page 25).
• In some molecules the two centres do not coincide, so they have a permanent dipole moment even without a field. These are polar molecules; water (H₂O) is one (page 25).
• The centre of a set of positive point charges is defined like a centre of mass, with charges in place of masses: r_cm = Σq_ir_i/Σq_i (page 25, footnote).

Solved examples

Common mistakes to avoid

Quick revision summary

• Electric dipole: charges q and −q a distance 2a apart; total charge zero, field not zero.
• Dipole moment p = q × 2a, directed from −q to q; unit C m.
• For r ≫ a: on the axis E = 2p/(4πε_₀r^³) along p; on the equatorial plane E = p/(4πε_₀r^³) opposite to p.
• The dipole field falls as 1/r^³ and depends on the angle between r and p.
• CO_₂ and CH_₄ have no permanent dipole moment; polar molecules such as H_₂O do.

Questions students ask

Is electric dipole moment a vector, and what are its SI unit and dimensional formula?

Yes, the electric dipole moment is a vector: its size is q × 2a, the charge times the separation, and it points along the axis from −q to +q (NCERT page 24). Its SI unit is the coulomb metre, C m, and its dimensional formula is [L T A], charge [A T] times length [L] (NCERT page 40; see dimensional formulae). For molecules, chemistry usually quotes dipole moments in the debye, a non-SI unit: see polarity of bonds and dipole moment.

Why does the dipole moment point from −q to +q, when field lines run from +q to −q?

The electric dipole moment points from −q to +q by convention (NCERT page 23), a choice that makes p the sum of each charge times its position vector: with +q at r₂ and −q at r₁, qr₂ − qr₁ = q(r₂ − r₁), a vector from the negative to the positive charge (reasoned from NCERT Eq. (1.19), page 24). With it, the field on the axis outside the charges points along p (NCERT page 26), and in a uniform field the torque turns p towards E (page 27). Field lines run from +q to −q, but the field points exactly opposite to p only between the charges and on the equatorial plane.

What is the difference between an electric dipole and a magnetic dipole?

An electric dipole is a pair of equal and opposite charges, with moment p in C m, while a magnetic dipole is a current loop or a bar magnet, with magnetic moment m in A m^² (NCERT Chapter 4, page 134). Far away their fields match: NCERT’s dipole analogy turns the electric dipole’s 2p/4πε₀r^³ (axis) and p/4πε₀r^³ (equator) into a bar magnet’s by replacing p with m and 1/ε₀ with μ₀ (NCERT Chapter 5, Table 5.1, page 141). The sources differ: an electric dipole’s two charges can be separated, but isolated magnetic poles are not known to exist, so a bar magnet cut in two gives two smaller magnets (NCERT pages 129 and 151).

Why does a dipole’s field fall as 1/r^³ instead of 1/r^²?

A dipole’s field falls as 1/r^³ because its two charges are equal and opposite: far away their 1/r^² fields almost cancel, and what is left is only the small difference made by one charge being 2a nearer than the other (NCERT pages 23 and 25). Beyond the NCERT text, reasoned from Coulomb’s law: a 1/r^² field changes over a short distance 2a by about 4a/r times itself, so the leftover is a single charge’s field times a factor that shrinks as 1/r, one extra power of r. NCERT states the general rule: a charge configuration whose total charge is zero has a field that falls off faster than 1/r^² at large distances, and the dipole is the simplest example (NCERT page 41).

Is the electric field of a dipole zero at any point?

No, the electric field of an electric dipole is not zero at any point at a finite distance; it only approaches zero far away. Between the two charges both fields point the same way, from +q towards −q; on the axis outside them the fields are opposite, but the nearer charge’s field is always the stronger; and off the axis the two fields act along different lines, so they cannot cancel (reasoned from Coulomb’s law and superposition, NCERT page 23). Two equal charges of the same sign, by contrast, give zero field at their midpoint.

Practice questions

Try each question first, then open the answer to check your working.

1. Q1. Charges of ±3 nC are 2 mm apart. Find the dipole moment.

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   p = q × 2a = (3 × 10⁻⁹)(2 × 10⁻³) = 6 × 10⁻¹² C m, from −q to +q (NCERT Eq. (1.19), page 24).

2. Q2. Far from a dipole, by what factor does the field fall when the distance is tripled?

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   By 3³ = 27, since E ∝ 1/r^³ (NCERT page 25).

3. Q3. Name a molecule with a permanent dipole moment and one without.

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   Water (H₂O) has one; carbon dioxide (CO₂) or methane (CH₄) has none (NCERT page 25).