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Electric Field due to an Electric Dipole

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Estimated time: 12 minutes
CISCE: Class 12

Definition: Electric Dipole

A system consisting of two equal and opposite point charges (+q and −q) separated by a small distance (2a) is called an electric dipole.

CISCE: Class 12

Definition: Dipole Axis

The straight line passing through both charges, +q and −q, of a dipole is called the dipole axis.

CISCE: Class 12

Definition: Dipole Moment

The vector quantity equal to the product of the magnitude of either charge and the distance between the two charges, directed from −q to +q, is called the dipole moment.

CISCE: Class 12

Electric Field on the Axial Line

Seup

Consider point P on the axial line, at distance r from the dipole centre, on the side of charge +q.

Derivation (Step-by-Step)

Step 1: Field due to +q at P (directed away from +q):

E+q = \[\frac {1}{4πε_0}\cdot\frac {q}{(r−a)^2}\]​   (1)

Step 2: Field due to −q at P (directed toward −q):

E−q = \[\frac {1}{4πε_0}\cdot\frac {q}{(r+a)^2}\]​   (2)

Step 3: Both fields point in the same direction (along \[\hat p\]​), so the net field is:

\[E_{axial}=\frac{q}{4\pi\varepsilon_0}\left[\frac{1}{(r-a)^2}-\frac{1}{(r+a)^2}\right]\]   (3)

Step 4: Simplifying (using difference of squares):

\[E_{axial}=\frac{1}{4\pi\varepsilon_0}\cdot\frac{4qar}{(r^2-a^2)^2}\]   (4)

Step 5 (Short Dipole Approximation): For r ≫ a, (r2 − a2)2 ≈ r4:

\[{E_{axial}=\frac{1}{4\pi\varepsilon_0}\cdot\frac{2p}{r^3}}\]   (5)

Direction: Along \[\vec p\]​ (from −q to +q).

CISCE: Class 12

Electric Field on the Equatorial Line

Setup

Point P lies on the perpendicular bisector of the dipole axis, at distance r from the centre.

Derivation (Step-by-Step)

Step 1: Distance of P from each charge:

d = \[\sqrt{r^2+a^2}\]

Step 2: Magnitude of field due to each charge (equal for both):

E = ​\[\frac{1}{4\pi\varepsilon_0}\cdot\frac{q}{r^2+a^2}\]   (6)

Step 3: Components perpendicular to the dipole axis cancel out (equal and opposite); components parallel to the axis add up.

Step 4: Each contributes E cos ⁡θ, where cos ⁡θ = \[\frac{a}{\sqrt{r^2+a^2}}\]:

\[E_{eq}=\frac{1}{4\pi\varepsilon_0}\cdot\frac{p}{(r^2+a^2)^{3/2}}\]   ​(7)

Step 5 (Short Dipole Approximation): For r ≫ a:

\[{E_{eq}=\frac{1}{4\pi\varepsilon_0}\cdot\frac{p}{r^3}}\]   (8)

Direction: Opposite to \[\vec p\].

CISCE: Class 12

Axial vs Equatorial Field — Comparison Table

Parameter Axial Line Equatorial Line
Formula (short dipole) \[\frac {1}{4πε_0}\cdot\frac {2p}{r^3}\] \[\frac {1}{4πε_0}\cdot\frac {p}{r^3}\]
Direction relative to \[\vec p\]​ Same direction Opposite direction
Field magnitude (same r) Twice the equatorial field Half the axial field
Dependence on r 1/r3 1/r3
Physical location Along dipole axis Perpendicular bisector
CISCE: Class 12

General Field at Any Point (Angle θ)

For a short dipole, the resultant field at a point making angle θ with the dipole axis:

E = \[\frac{1}{4\pi\varepsilon_0}\cdot\frac{p}{r^3}\sqrt{1+3\cos^2\theta}\]

The angle α between the resultant field and the position vector satisfies:

tan⁡ α = \[\frac {tan⁡θ}{2}\]

This general formula reduces to the axial case at θ = 0° and the equatorial case at θ = 90°.

CISCE: Class 12

Real-Life Analogy

Analogy: Think of a dipole like a tug-of-war between two equal-strength teams (+q and −q) standing close together. Even though the "net pull" (total charge) is zero, anyone standing nearby still feels a directional "tug" — this is the dipole field. The pull feels strongest when you stand in line with the rope (axial line) and weaker, opposite-facing, when you stand sideways (equatorial line).

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