1. Concept Overview
An electric dipole creates a unique electric field pattern because it has two opposite charges placed close together. Unlike a single point charge whose field decreases as \(1/r^2\), a dipole’s field falls faster with distance. The field depends strongly on direction, so we usually calculate it separately along the dipole axis and along the perpendicular bisector.
2. Dipole Setup
A dipole consists of charges \(+q\) and \(-q\) separated by distance \(2a\). Its dipole moment is:
\( \vec{p} = q(2a) \hat{d} \)
The axis joining the charges is called the axial line, and the line through the centre perpendicular to the axis is the equatorial line.
3. Electric Field on the Axial Line
The axial line is the line along which the two charges lie. Points on this line experience a stronger field because the effects of the charges partially add up.
3.1. Formula
At a point far from the dipole (\( r \gg a \)) along the axis:
\( E_{\text{axial}} = \dfrac{1}{4\pi\varepsilon_0} \dfrac{2p}{r^3} \)
3.2. Direction
The electric field on the axial line points away from the positive charge and toward the negative charge. Overall direction is the same as the dipole moment \( \vec{p} \).
3.3. Interpretation
The field is stronger on the axial line because the components of the two charge fields reinforce each other.
4. Electric Field on the Equatorial Line
The equatorial line is perpendicular to the dipole axis and passes through the centre. Here, the fields from the two charges partially cancel, giving a different expression.
4.1. Formula
For points far from the dipole (\( r \gg a \)):
\( E_{\text{equatorial}} = \dfrac{1}{4\pi\varepsilon_0} \dfrac{p}{r^3} \)
4.2. Direction
The electric field on the equatorial line points opposite to the dipole moment \( \vec{p} \).
4.3. Interpretation
The two charge fields partially cancel, so the equatorial field is weaker than the axial field.
5. General Behaviour of Dipole Field
A dipole field decreases much faster than the field of a single charge:
- Point charge: \( E \propto 1/r^2 \)
- Dipole: \( E \propto 1/r^3 \)
This is why dipole effects dominate only when charges are close together or at moderate distances.
6. Field at Any Point (Qualitative)
At an arbitrary point, the dipole field is the vector sum of fields due to +q and -q. The resulting pattern is curved and symmetric, resembling the field of a small bar magnet but produced entirely by charges.
7. Worked Examples
7.1. Example 1: Field on the Axial Line
A dipole has dipole moment \( p = 4 \times 10^{-8} \text{ C·m} \). Find the field at a point \( 0.2 \text{ m} \) away on the axial line.
\( E = \dfrac{1}{4\pi\varepsilon_0} \dfrac{2p}{r^3} \)
\( E = 9 \times 10^9 \times \dfrac{2 \times 4 \times 10^{-8}}{(0.2)^3} = 9 \times 10^9 \times \dfrac{8 \times 10^{-8}}{0.008} \)
\( E = 9 \times 10^9 \times 1 \times 10^{-5} = 9 \times 10^{4} \text{ N/C} \).
7.2. Example 2: Field on the Equatorial Line
For the same dipole, find the field at the same distance on the equatorial line.
\( E = \dfrac{1}{4\pi\varepsilon_0} \dfrac{p}{r^3} \)
\( E = 9 \times 10^9 \times \dfrac{4 \times 10^{-8}}{0.008} = 9 \times 10^9 \times 5 \times 10^{-6} = 4.5 \times 10^{4} \text{ N/C} \).
This is half the axial field, matching the formulas.
8. Physical Interpretation
The electric field of a dipole reflects the balance between attraction toward the negative charge and repulsion from the positive charge. Far away, these two influences nearly cancel, leaving a weaker field that drops quickly with distance. Dipoles appear in many natural systems, especially molecules where charge separation occurs, making this field pattern very important in understanding matter at microscopic levels.