UY1: Electric Dipole

Define electric dipole moment, derive torque and potential energy in a uniform field, and solve angle-dependent dipole problems.

University Physics Year 1

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Learning objectives

Construct electric-field and potential models for discrete and continuous charge distributions.

This page gives the UY1 working model/result for Electric Dipole. You reuse it when you build fields/potentials by symmetry or superposition, and when you connect fields to forces, energy, and circuits.

Module path: Electromagnetism (UY1)
Practice: UY1 Electromagnetism Quiz
Full routing: UY1 Assessment Map
Math toolkit: Mathematics for Undergraduate Physics

1) At a glance

Prerequisites: point-charge fields (Field of a Point Charge), vector products (dot/cross) and angles (Vector Calculus)
Outcomes: define vector p, compute torque and potential energy in a uniform field, and identify stable vs unstable orientations
Key results: vector p = q vector d (from - to +), vector τ = vector p × vector E, U = - vector p · vector E
Common trap: reversing the direction of vector p (it points from negative to positive) or losing the minus sign in U = - vector p · vector E
A dipole is two equal and opposite charges separated by distance d.
Dipole moment magnitude:

p = qd

with direction from negative to positive charge.

In a uniform electric field:

vector τ = vector p × vector E, U = - vector p · vector E

Motivation / intuition

A dipole is the simplest “neutral but not trivial” charge arrangement: net charge is zero, so there is no big overall push in a uniform field, but the separated charges still feel opposite forces, creating a torque that tries to align the dipole with the field.

2) Setup

External field vector E is uniform.
Dipole axis makes angle θ with vector E.
Sign conventions:
τ = pE sin θ is the turning tendency.
U = -pE cos θ is minimum at θ = 0 (stable alignment).

3) Core derivation/explanation

Each charge experiences force magnitude qE in opposite directions, creating a couple. The perpendicular lever-arm component gives torque:

τ = qE(d/2 sin θ) + qE(d/2 sin θ) = qEd sin θ = pE sin θ

Vector form:

vector τ = vector p × vector E

For potential energy, use work-energy relation for rotational motion:

W_field = -Δ U

and integrate torque with angle to get:

U(θ) = -pE cos θ

Interpretation:

θ = 0: U = -pE (stable, lowest energy).
θ = π: U = +pE (unstable, highest energy).

Units: [p] = C m, so [pE] = (C m)(N/C) = N m = J (consistent for both τ and U).
Limits/signs: τ = pE sin θ is zero at θ = 0,π; U(θ) = -pE cos θ is minimum at θ = 0 (stable) and maximum at θ = π (unstable).

4) Worked example(s)

A dipole has q = 3.0 × 10⁻⁶ C and separation d = 4.0 × 10⁻³ m. It is in a uniform field E = 2.5 × 10⁴ N C⁻¹ at angle θ = 30°.

Dipole moment:

p = qd = (3.0 × 10⁻⁶)(4.0 × 10⁻³) = 1.2 × 10⁻⁸ C m

Torque magnitude:

τ = pE sin θ = (1.2 × 10⁻⁸)(2.5 × 10⁴)(0.5) = 1.5 × 10⁻⁴ N m

Potential energy:

U = -pE cos θ = -(1.2 × 10⁻⁸)(2.5 × 10⁴)(0.866) = -2.60 × 10⁻⁴ J

5) Practice set (with hints + answers)

A dipole with p = 5.0 × 10⁻⁹ C m is in E = 4.0 × 10⁵ N C⁻¹. Find maximum torque. Hint: max at sin θ = 1. Answer: τₘₐₓ = 2.0 × 10⁻³ N m.
For the same dipole/field, find U at θ = 0. Hint: U = -pE cos θ. Answer: U = -2.0 × 10⁻³ J.
Is θ = π stable or unstable? Hint: compare potential energy extrema. Answer: unstable (maximum potential energy).

6) Summary + next steps

Dipole moment sets orientation physics: torque rotates it, energy ranks orientations.
The minus sign in U = - vector p · vector E explains why dipoles align with the field.
These ideas lead directly to field patterns and dipole-field formulas.

Next: Electric Field Lines Previous: Electric Field Of A Point Charge Back To UY1: Electromagnetism