Q.Define 'electric dipole'. Give the expression for the magnitude of its electric dipole moment and its direction.

Equating the axial-line field at distance $r$ to the equatorial-line field gives the equatorial distance as $r/2^{1/3}$.

For a short electric dipole of dipole moment $p$, the electric field intensity at a point on the axial line at distance $r$ from the centre (for $r$ much greater than the dipole length) is $E_{axial} = \dfrac{1}{4\pi\epsilon_0}\dfrac{2p}{r^3} = \dfrac{2kp}{r^3}$, where $k = \dfrac{1}{4\pi\epsilon_0}$.

The field intensity at a point on the equatorial (perpendicular bisector) line at distance $r'$ from the centre is $E_{eq} = \dfrac{1}{4\pi\epsilon_0}\dfrac{p}{r'^3} = \dfrac{kp}{r'^3}$.

We are told these two fields are equal in magnitude: $\dfrac{2kp}{r^3} = \dfrac{kp}{r'^3}$

Cancelling $kp$ from both sides: $\dfrac{2}{r^3} = \dfrac{1}{r'^3} \implies r'^3 = \dfrac{r^3}{2}$

…