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Field of an Electric Dipole

Defines dipole moment and its field on axis and equatorial plane.

What is an Electric Dipole?

An electric dipole is a pair of equal and opposite point charges, $+q$ and $-q$ , separated by a fixed distance $2a$ .

Even though the total charge of a dipole is zero, its electric field is not zero. This is because the two charges are separated by a distance, so their individual electric fields don't perfectly cancel each other out in the surrounding space.

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Dipole Geometry

Visual of axial and equatorial point geometry for a dipole.

Clean scientific diagram showing an electric dipole. A horizontal line connects a negative charge -q on the left and a positive charge +q on the right, separated by distance 2a. A vector p points from -q to +q. On the extended horizontal axis to the right, a point P shows an electric field vector E pointing in the same direction as p. Above the center of the dipole on a vertical equatorial line, another point shows an electric field vector E pointing horizontally to the left, opposite to p.

Click to zoom

Electric field directions at axial and equatorial points relative to the dipole moment p.

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Dipole Field Formulas

Formulas for dipole field at large distances.

Axial Field

E_{axial} = \frac{1}{4 π ε _{0}} \frac{2 p}{r ^{3}}

Valid for large distances:

r ≫ a

Field is parallel to the dipole moment

p

.

Equatorial Field

E_{equatorial} = - \frac{1}{4 π ε _{0}} \frac{p}{r ^{3}}

Valid for large distances:

r ≫ a

Field is antiparallel to the dipole moment

p

.

Quick comparison: At the same distance

r

, the axial field has twice the magnitude of the equatorial field.

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Worked Example

Calculating Dipole Field

Calculates field at axial and equatorial points.

Problem Statement

Two charges $\pm 10\, \mu\text{C}$ are placed $5.0\, \text{mm}$ apart. Determine the electric field at: (a) a point P on the axis of the dipole $15\, \text{cm}$ away from its centre O on the side of the positive charge, and (b) a point Q, $15\, \text{cm}$ away from O on a line passing through O and normal to the axis of the dipole.

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Dipole Fall-off

MCQ testing the 1/r^3 rule.

An engineer measures the electric field magnitude of an ideal electric dipole to be $E_0$ at a distance $r$ from its center along its axis (where $r \gg a$ ). If the engineer measures the field magnitude again at a distance of $3r$ along the same axis, what will the new field magnitude be?

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Dipole in a Uniform External Field

Derives the torque acting on a dipole.

What happens when you place a permanent electric dipole (with dipole moment $\mathbf{p}$ ) inside a uniform external electric field $\mathbf{E}$ ?

Because the field is uniform, the electric field strength is exactly the same at the location of the positive charge $+q$ as it is at the negative charge $-q$ .

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Torque on a Dipole

Formula for cross product torque.

τ = p \times E

Torque on a Dipole

Magnitude:

τ = pE sin θ

Symbol Legend

$τ$ = Torque vector ( $N \cdot m$ )
$p$ = Dipole moment ( $C \cdot m$ )
$E$ = External Electric Field ( $N/C$ )
$θ$ = Angle between $p$ and $E$

Electric Dipole And External Fields

Field of an Electric Dipole

What is an Electric Dipole?

Dipole Geometry

Dipole Field Formulas

Calculating Dipole Field

Problem Statement

Dipole Fall-off

Dipole in a Uniform External Field

Torque on a Dipole

Symbol Legend