Q.Derive an expression for the electric potential at a point in axial position due to an electric dipole.

The electric potential of a dipole at a general point depends on $\vec p\cdot\hat r$, not just on $p$ and $r$.

For a short dipole of moment $\vec p$, the potential at a point whose position vector from the centre of the dipole is $\vec r$ (with $r \gg$ the dipole's charge separation) is

$V = \frac{1}{4\pi\epsilon_0}\,\frac{\vec p \cdot \hat r}{r^2}$ …

On the equatorial (broad-side-on) line the two charges are equidistant, so their potentials cancel → V = 0.

The potential due to a dipole at a point making angle θ with the axis is

$V = \frac{1}{4\pi\varepsilon_0}\,\frac{p\cos\theta}{r^2}$

…

On the axial line of an electric dipole, at a distance r from the centre (with r much larger than the charge separation), the potential falls off as p/r², unlike a single point charge's 1/r fall-off.

For a dipole of moment $p = q(2a)$, with charges $+q$ and $-q$ separated by $2a$, the potential at an axial point at distance $r$ from the centre O is obtained by superposing the potentials of the two point charges: …