The electric field due to a short electric dipole at a large distance $(r)$ from center of dipole on the equatorial plane varies with distance as :

At a large distance $r$ from the center of a short electric dipole, the electric field on the equatorial plane can be approximated as: <br/><br/> $E = \frac{1}{4\pi\epsilon_0}\frac{2p}{r^3}$ <br/><br/> where $p$ is the dipole moment of the electric dipole, and $\epsilon_0$ is the permittivity of free space. <br/><br/> This formula is derived using the concept of electric dipole moment, which is defined as: <br/><br/> $\vec{p} = q\vec{d}$ <br/><br/> where $q$ is the magnitude of the electric charge, and $\vec{d}$ is the separation vector between the positive and negative charges of the dipole. The electric field at a point on the equatorial plane of the dipole is due to the electric field of the positive and negative charges at that point. <br/><br/>Since the charges are equal in magnitude and opposite in sign, their electric fields at a point on the equatorial plane cancel out along the axis of the dipole, leaving only the component perpendicular to the axis. <br/><br/>This perpendicular component of the electric field is proportional to the dipole moment $p$ and inversely proportional to the cube of the distance $r$ from the center of the dipole. <br/><br/> Therefore, the electric field due to a short electric dipole at a large distance $r$ from the center of the dipole on the equatorial plane varies with distance as: <br/><br/> $\boxed{E \propto \frac{1}{r^3}}$ <br/><br/> where the proportionality constant is $\frac{1}{4\pi\epsilon_0}$.