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Electric Dipoles

Level 1 - Physics topic page in Electric Charge and Fields.

Principle

An electric dipole is a separated pair of equal and opposite charges, summarized at large distances by its dipole moment.

Notation

\(+q,-q\)
charges forming the dipole
\(\mathrm{C}\)
\(\vec d\)
separation vector from negative charge to positive charge
\(\mathrm{m}\)
\(\vec p\)
electric dipole moment
\(\mathrm{C\,m}\)
\(\theta\)
angle between \(\vec p\) and \(\vec E\)
rad or deg
\(\vec\tau\)
torque on the dipole
\(\mathrm{N\,m}\)
\(U\)
dipole potential energy in a uniform field
\(\mathrm{J}\)

Method

Derivation 1: Build the dipole moment

The separation direction is defined from the negative charge to the positive charge. Multiplying by the charge magnitude gives a vector that tracks both size and orientation.

Separation direction
\[\vec d=\vec r_{+}-\vec r_{-}\]
Dipole moment
\[\vec p=q\vec d\]
Moment magnitude
\[p=qd\]

Derivation 2: Torque in a uniform field

The two charges feel equal and opposite forces in a uniform field. The net force is zero, but the forces form a couple that tends to align \(\\vec p\) with \(\\vec E\).

Forces on charges
\[\vec F_+=q\vec E,\qquad \vec F_-=-q\vec E\]
Net force
\[\vec F_{\mathrm{net}}=0\quad\text{in uniform }\vec E\]
Torque vector
\[\vec\tau=\vec p\times\vec E\]
Torque magnitude
\[\tau=pE\sin\theta\]

Derivation 3: Energy and far-field scale

A dipole has lowest energy when it is aligned with the field. Far from the dipole, the positive and negative point-charge fields nearly cancel, leaving a field that falls faster than \(1/r^2\).

Potential energy
\[U=-\vec p\cdot\vec E=-pE\cos\theta\]
Far axial field
\[E_{\mathrm{axis}}\approx\frac{2kp}{r^3}\]
Far equatorial field
\[E_{\mathrm{eq}}\approx\frac{kp}{r^3}\]

Rules

These are the compact dipole results.

Dipole moment
\[\vec p=q\vec d\]
Torque
\[\vec\tau=\vec p\times\vec E,\qquad \tau=pE\sin\theta\]
Dipole energy
\[U=-\vec p\cdot\vec E=-pE\cos\theta\]
Far-field scale
\[E_{\mathrm{dipole}}\propto\frac{p}{r^3}\]

Examples

Question
A dipole has
\[q=4.0\,\mathrm{nC}\]
and charge separation
\[3.0\,\mathrm{mm}\]
Find \(p\).
Answer
\[p=qd=(4.0\times10^{-9})(3.0\times10^{-3})=1.2\times10^{-11}\,\mathrm{C\,m}\]

Checks

  • \(\\vec p\) points from negative charge to positive charge.
  • A uniform field gives a dipole torque but no net force.
  • The aligned state \((\\theta=0)\) has minimum energy.
  • A dipole field falls as \(1/r^3\) far away, faster than a single point-charge field.

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Electric Flux