Question

Obtain an expression for the electric field \( \vec{E} \) due to a dipole of dipole moment \( \vec{p} \) at a point on its equatorial plane and specify its direction.
Hence, find the value of electric field:

1. at the centre of the dipole (\( r = 0 \))
2. at a point \( r \gg a \), where \( 2a \) is the length of the dipole.

Hint:

The electric field due to a dipole is strongest near the dipole and decreases rapidly as \( \frac{1}{r^3} \) when you move farther from the dipole. The field on the equatorial plane is directed perpendicular to the dipole axis.

Answer 1

Consider a dipole consisting of two equal and opposite charges \( +q \) and \( -q \), separated by a distance \( 2a \). The dipole moment \( \vec{p} \) is defined as: \[ \vec{p} = q \times 2a \] The electric field due to a dipole at any point is derived by considering the contribution from both charges. Electric Field on the Equatorial Plane: On the equatorial plane of the dipole, the angle between the position vector and the dipole moment is \( 90^\circ \), and the distance from the dipole is \( r \). 1. The expression for the electric field at a point on the equatorial plane at a distance \( r \) from the center of the dipole is: \[ E = \frac{1}{4 \pi \epsilon_0} \times \frac{2p}{r^3} \] where \( p = q \times 2a \) is the dipole moment, \( r \) is the distance from the center of the dipole, and \( \epsilon_0 \) is the permittivity of free space. 2. Direction of the Electric Field: The electric field on the equatorial plane is directed perpendicular to the axis of the dipole and lies in the plane containing the dipole charges. Specifically, it points away from the dipole axis. (I) Electric Field at the Centre of the Dipole (\( r = 0 \)): At the center of the dipole, the electric field due to each charge is equal in magnitude but opposite in direction. Therefore, the net electric field at the center of the dipole is zero. Thus, the electric field at the center of the dipole is: \[ E = 0 \, \text{N/C} \] (II) Electric Field at a Point \( r \gg a \): When the distance \( r \) is much greater than the separation of the charges \( a \), the dipole behaves as though it were a point charge. In this case, the electric field behaves as: \[ E = \frac{1}{4 \pi \epsilon_0} \times \frac{2p}{r^3} \] For large distances, the dipole field behaves like the field due to a point charge with the same total charge \( q \). However, for \( r \gg a \), the field expression becomes much weaker (as \( r^3 \)) compared to that of a single charge. Hence, the electric field at a large distance from the dipole is: \[ E \propto \frac{1}{r^3} \] Thus, at points where \( r \gg a \), the dipole field decreases rapidly with the cube of the distance.