The value of electric potential at any point due to any electric dipole is $\left(\mathrm{k}=\frac{1}{4\pi {\epsilon }_0}\right)$

Given data: Electric potential at any point due to any electric dipole

Formula used: Electric potential $p:v=v_1+v_2$ ; dipole moment , $p=q\times 2l$

Detailed solution:

Let P,Q be the dipole will be a distance of 2L between then consider a point X at a distance ‘r’ from the midpoint of PQ.

$QX = RX= OX-OR = r – L \cos \theta$

$PX= SX= OX+OS = r + L \cos \theta$ (as dipoles are nearer )

Electric potential at X :

 $$\begin{gathered} v=v_1+v_2 \\ v=\frac{1}{4\pi \xi }\frac{-q}{PX}+\frac{1}{4\pi \xi }\frac{q}{QX} \\ v=\frac{kq2/\cos \left(\theta \right)}{r^2-l^2{\cos }^2\theta } \end{gathered}$$

We know dipole moment  

$$\begin{gathered} p=q\times 2l \\ v=\frac{kp\cos \left(\theta \right)}{r^2}=\frac{kpr\cos \left(\theta \right)}{r^3} \\ v=k\frac{\overrightarrow{p}·\overrightarrow{r}}{r^3} \end{gathered}$$