Q.Establish the expression for electric potential at a point due to a small electric dipole.

Add the potentials of the +q and −q charges of a short dipole; for $r\gg 2a$ this gives $V=\dfrac{1}{4\pi\varepsilon_0}\dfrac{p\cos\theta}{r^2}$.

Consider an electric dipole made of charges $+q$ and $-q$ separated by a small distance $2a$, so its dipole moment is $p = q\cdot 2a$ directed from $-q$ to $+q$. Let P be a point at distance $r$ from the centre O of the dipole, with OP making angle $\theta$ with the dipole axis.

Let $r_1$ and $r_2$ be the distances of P from $+q$ and $-q$ respectively. The potential at P is the algebraic sum of the potentials due to the two charges:

$V = \frac{1}{4\pi\varepsilon_0}\left(\frac{q}{r_1} - \frac{q}{r_2}\right) = \frac{q}{4\pi\varepsilon_0}\cdot\frac{r_2 - r_1}{r_1 r_2}$

Approximation for a short dipole ($r \gg a$). Dropping perpendiculars from the two charges onto OP, geometry gives

$r_1 \approx r - a\cos\theta, \qquad r_2 \approx r + a\cos\theta$

Hence

$r_2 - r_1 \approx 2a\cos\theta, \qquad r_1 r_2 \approx r^2 - a^2\cos^2\theta \approx r^2 \ \ (\text{since } a \ll r)$

Substituting:

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