Q.The magnitude of electric intensity at a distance 'r' from the centre of an electric dipole along its axial line is E. The distance of the point from the centre of the electric dipole along its equatorial line at which the electric intensity has the same value E is :

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: $\dfrac{2}{r^3} = \dfrac{1}{r'^3} \implies r'^3 = \dfrac{r^3}{2}$

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