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:
at the centre of the dipole (\( r = 0 \))
at a point \( r \gg a \), where \( 2a \) is the length of the dipole.
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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.
A dipole comprises 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 generated by a dipole at any point is calculated by summing the contributions from each charge.
Electric Field on the Equatorial Plane:
On the dipole's equatorial plane, the angle between the position vector and the dipole moment is \( 90^\circ \), and the distance from the dipole is \( r \).
1. The electric field intensity at a point on the equatorial plane, located at a distance \( r \) from the dipole's center, is given by:
\[
E = \frac{1}{4 \pi \epsilon_0} \times \frac{2p}{r^3}
\]
Here, \( p = q \times 2a \) represents the dipole moment, \( r \) is the distance from the dipole's center, and \( \epsilon_0 \) is the permittivity of free space.
2. Direction of the Electric Field:
The electric field on the equatorial plane is oriented perpendicular to the dipole's axis and lies within the plane defined by the dipole charges. It points away from the dipole axis.
(I) Electric Field at the Center of the Dipole (\( r = 0 \)):
At the dipole's center, the electric fields produced by each charge are equal in magnitude but opposite in direction. Consequently, the resultant electric field at the dipole's center 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 \( r \) significantly exceeds the charge separation \( a \), the dipole approximates the behavior of a point charge. In this scenario, the electric field follows the relationship:
\[
E = \frac{1}{4 \pi \epsilon_0} \times \frac{2p}{r^3}
\]
For large distances, the dipole field resembles the field of a point charge with the same total charge \( q \). However, for \( r \gg a \), the field expression diminishes significantly (inversely with \( r^3 \)) compared to that of a single charge.
Hence, the electric field at a considerable distance from the dipole is:
\[
E \propto \frac{1}{r^3}
\]
Therefore, at locations where \( r \gg a \), the dipole field attenuates rapidly with the cube of the distance.
