Electric Field due to an Electric Dipole | Physics for JAMB PDF Download

WHAT IS AN ELECTRIC DIPOLE
An electric dipole is defined as a pair of equal and opposite charges separated by a distance. However, a continuous charge distribution can also be approximated as an electric dipole from a large distance. These dipoles are characterized by their dipole moment, a vector quantity defined as the charge multiplied by their separation and the direction of this vector quantity is from the -Ve charge to the +ve charge. The total charge corresponding to a dipole is always zero. As the positive and negative charge centers are separated by a finite distance, the electric field at a test point does not cancel out completely leading to a finite electric field. Similarly, we also get finite electric potential due to a dipole.

ELECTRIC FIELD DUE TO A DIPOLE
The electric field due to a pair of equal and opposite charges at any test point can be calculated using the Coulomb’s law and the superposition principle. Let the test point P be at a distance r from the center of the dipole. The distance between +q and -q is d. We have shown the situation in the diagram below.
If    and  be the electric field at point P due to the positive and the negative charges separately then the total electric field  at Point P can be calculated by using the superposition principle.

Please note that the directions of  and  are along  and  respectively. This is the most general form of the electric field due to a dipole. However, we will express this vector in terms of radial and inclination vectors as shown in the diagram below.
In order to calculate the electric field in the polar coordinate, we will use the expression of the electric potential due to an electric dipole which we have calculated earlier.

Here p is the magnitude of the dipole moment and is given by qd

We can easily derive the electric field due to this dipole by calculating the negative gradient of this electric potential. In polar coordinate electric field will be independent of azimuthal (ϕ) coordinate.

The resultant electric field at point P is

As shown in the diagram, the resultant electric field makes an angle \alpha with the radial vector. Then


ELECTRIC FIELD AT AN AXIAL POINT
In this case, the test point P is on the axis of the dipole. Consequently θ= 0 or π. The electric field at point P is

SIMPLIFIED DERIVATION
The electric field at point P due to the positive charge is

Electric field at point P due to negative charge is

Total electric field due to the dipole at axial point P is

At a relatively large distance r>>d/2 and we can approximate the electric field as


ELECTRIC FIELD AT AN EQUATORIAL POINT
In this case, the test point P is on the perpendicular bisector of the dipole. Consequently θ= π/2. The electric field at point P is


SIMPLIFIED DERIVATION
The electric field at point P due to positive charge is

Electric field at point P due to negative charge is

Total electric field due to the dipole at equatorial point P is

At a relatively large distance r>>d/2 and we can approximate the electric field as