Potential due to an electric dipole

What is the potential of an electric dipole?

In plain words: far from a dipole, the positive and negative charges almost cancel each other’s potential. What is left depends on which side you are on: positive on the side of +q, negative on the side of −q, and exactly zero on the plane halfway between them.

Potential on the axis and on the equatorial plane

A dipole is a charge q and a charge −q a small distance 2a apart, with dipole moment of size p = q × 2a pointing from −q to q (NCERT page 49). With the origin at the centre, the potential at P, a distance r₁ from q and r₂ from −q, is the sum of the two point-charge potentials (NCERT Eq. (2.9), page 50):

• Geometry: r₁² = r² + a² − 2ar cos θ and r₂² = r² + a² + 2ar cos θ (NCERT Eq. (2.10)).
• Far away, r ≫ a: keep terms only to first order in a/r. The binomial theorem then gives 1/r₁ ≈ (1/r)(1 + (a/r) cos θ) and 1/r₂ ≈ (1/r)(1 − (a/r) cos θ) (NCERT Eqs. (2.11)–(2.13)).
• Subtracting, 1/r₁ − 1/r₂ ≈ 2a cos θ/r², and with p = 2qa this gives the result above. Since p cos θ = p·r̂, NCERT writes it as p·r̂/(4πε₀r²) (Eqs. (2.14)–(2.15), pages 50–51).

• On the axis, θ = 0 (the side of +q) gives +p/(4πε₀r²), and θ = π (the side of −q) gives the same with a minus sign (NCERT Eq. (2.16), page 51).
• On the equatorial plane, θ = π/2, the potential is zero (page 51).
• The formula is approximate, valid when r is large compared with the dipole; for a point dipole at the origin it is exact at r > 0 (page 51).

Potential falls as 1/r^²

NCERT contrasts the dipole with a single charge (page 51):

• The dipole potential depends not only on r but also on the angle between r and p. It is axially symmetric about p: turning r round p, keeping its length and θ fixed, carries P round a circle on a cone about p, and every point of that circle has the same potential as P.
• At large distances it falls off as 1/r², not as 1/r like the potential of a single charge. (The dipole’s field falls as 1/r³, one power faster again: NCERT Section 1.10, page 25.)
• NCERT’s text refers to “Fig. 2.5” for graphs of 1/r² and 1/r against r; those graphs are in Figure 2.4 (page 49), Figure 2.5 being the dipole diagram (page 50).

Solved examples

Common mistakes to avoid

Quick revision summary

• Potential obeys superposition, so a dipole’s potential is the sum of those of +q and −q.
• For r ≫ a: V = p cos θ/(4πε_₀r^²) = p·r̂/(4πε_₀r^²); exact for a point dipole.
• Far from a finite dipole, on the axis V = ±p/(4πε_₀r^²); on the equatorial plane V = 0.
• The dipole potential depends on the angle as well as r, is symmetric about p, and falls as 1/r^² (a single charge’s falls as 1/r).
• Doubling the distance divides the dipole potential by 4.

Questions students ask

Is the potential due to a dipole the same as the potential energy of a dipole?

No, the potential due to a dipole and the potential energy of a dipole are different quantities: V = p cos θ/4πε₀r^² is the far-field potential the dipole itself produces at a point a distance r away, in volts, with θ the angle between p and the line to that point (NCERT pages 50–51), while U = −pE cos θ is the energy of a dipole placed in a uniform external field E, in joules, with θ the angle between p and E (NCERT page 60; see potential energy of a dipole). The two meet when a charge q is placed near a dipole: its energy in the dipole’s field is qV.

How do you get a dipole’s electric field from its potential?

A dipole’s field follows from its potential because the field’s component in any direction is minus the rate at which V changes per metre in that direction (reasoned from NCERT page 55). Far along the positive axis V ≈ kp/r^², with k = 1/4πε₀, so −dV/dr = 2kp/r^³ along p, the axial field NCERT finds from Coulomb’s law (NCERT Chapter 1, page 24). Beyond the NCERT text, the same rule taken along the radius and across it at any angle θ, far from the dipole, gives back the field (kp/r^³)√(1 + 3 cos^² θ) that the dipole’s field page finds from Coulomb’s law.

What is a short dipole, and when can its potential formula be used?

A short dipole is one whose length 2a is small compared with the distance r to the point; V = p cos θ/4πε₀r^² is then a good approximation, and it is exact for a point dipole, the limit in which 2a shrinks to zero while p stays fixed (NCERT page 51; NCERT Chapter 1, page 25). On the positive axis beyond +q (r > a), the exact potential is kp/(r^² − a^²), with k = 1/4πε₀, so the short-dipole value kp/r^² is low by a^²/r^² of the exact value: 4% at r = 5a, 0.25% at r = 20a (reasoned from NCERT Eq. (2.9), page 50). Far away only the product p = q × 2a matters, so ±2 nC charges 1 cm apart and ±1 nC charges 2 cm apart give the same distant potential.

Practice questions

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

1. Q1. At what angle to the dipole moment is the potential of a small dipole zero?

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   At θ = 90°, on the equatorial plane (NCERT page 51).

2. Q2. On the axis of a small dipole, how does the potential on the −q side compare with the potential on the +q side at the same distance?

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   Same size, opposite sign: V = −p/(4πε_₀r^²) against +p/(4πε_₀r^²) (NCERT Eq. (2.16), page 51).

3. Q3. By what factor does the potential of a small dipole fall when the distance is tripled?

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   By 9, since V ∝ 1/r^² (NCERT page 51).